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New submissions

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New submissions for Tue, 20 Mar 18

[1]  arXiv:1803.06353 [pdf, other]
Title: Potentials for Moduli Spaces of A_m-local Systems on Surfaces
Authors: Efim Abrikosov
Comments: 24 figures
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG)

We study properties of potentials on quivers $Q_{\mathcal{T},m}$ arising from cluster coordinates on moduli spaces of $PGL_{m+1}$-local systems on a topological surface with punctures. To every quiver with potential one can associate a $3d$ Calabi-Yau $A_\infty$-category in such a way that a natural notion of equivalence for quivers with potentials (called "right-equivalence") translates to $A_\infty$-equivalence of associated categories.
For any quiver one can define a notion of a "primitive" potential. Our first result is the description of the space of equivalence classes of primitive potentials on quivers $Q_{\mathcal{T}, m}$. Then we provide a full description of the space of equivalence classes of all \emph{generic} potentials for the case $m = 2$ (corresponds to $PGL_3$-local systems). In particular, we show that it is finite-dimensional. This claim extends results of Gei\ss, Labardini-Fragoso and Schr\"oer who have proved analogous statement in $m=1$ case.
In many cases $3d$ Calabi-Yau $A_\infty$-categories constructed from quivers with potentials are expected to be realized geometrically as Fukaya categories of certain Calabi-Yau $3$-folds. Bridgeland and Smith gave an explicit construction of Fukaya categories for quivers $Q_{\mathcal{T},m=1}$. We propose a candidate for Calabi-Yau $3$-folds that would play analogous role in higher rank cases, $m > 1$. We study their (co)homology and describe a construction of collections of $3$-dimensional spheres that should play a role of generating collections of Lagrangian spheres in corresponding Fukaya categories.

[2]  arXiv:1803.06357 [pdf, ps, other]
Title: Maximal subalgebras of the exceptional Lie algebras in low characteristic
Authors: Thomas Purslow
Comments: 216 pages, PhD thesis
Subjects: Rings and Algebras (math.RA); Representation Theory (math.RT)

In this thesis we consider the maximal subalgebras of the exceptional Lie algebras in algebraically closed fields of positive characteristic. This begins with a quick recap of the article by Herpel and Stewart which considered the Cartan type maximal subalgebras in the exceptional Lie algebras for good characteristic, and then the article by Premet considering non-semisimple maximal subalgebras in good characteristic.
For $p=5$ we give an example of what appears to be a new maximal subalgebra in the exceptional Lie algebra of type $E_8$. We show that this maximal subalgebra is isomorphic to the $p$-closure of the non-restricted Witt algebra $W(1;2)$.
After this, we focus completely on characteristics $p=2$ and $p=3$ giving examples of new non-semisimple maximal subalgebras in the exceptional Lie algebras. We consider the Weisfeiler filtration associated to these maximal subalgebras and leave many open questions. There are one or two examples of simple maximal subalgebras in $F_4$ for $p=3$ and $E_8$ for $p=2$.

[3]  arXiv:1803.06360 [pdf, ps, other]
Title: Geometry of probability simplex via optimal transport
Authors: Wuchen Li
Subjects: Differential Geometry (math.DG)

We study the Riemannian structures of the probability simplex on a weighted graph introduced by $L^2$-Wasserstein metric. The main idea is to embed the probability simplex as a submanifold of the positive orthant. From this embedding, we establish the geometry formulas of the probability simplex in Euclidean coordinates. The geometry computations on discrete simplex guide us to introduce the ones in the Fr{\'e}chet manifold of densities supported on a finite dimensional base manifold. Following the steps of Nelson, Bakery-{\'E}mery, Lott-Villani-Strum and the geometry of density manifold, we demonstrate an identity that connects the Bakery-{\'E}mery $\Gamma_2$ operator (carr{\'e} du champ it{\'e}r{\'e}) and Yano's formula on the base manifold. Several examples of differential equations in probability simplex are demonstrated.

[4]  arXiv:1803.06361 [pdf, ps, other]
Title: Halving the bounds for the Markov, Chebyshev, and Chernoff Inequalities using smoothing
Authors: Mark Huber
Subjects: Probability (math.PR)

The Markov, Chebyshev, and Chernoff inequalities are some of the most widely used methods for bounding the tail probabilities of random variables. In all three cases, the bounds are tight in the sense that there exists easy examples where the inequalities become equality. Here we will show that through a simple smoothing using auxiliary randomness, that each of the three bounds can be cut in half. In many common cases, the halving can be achieved without the need for the auxiliary randomness.

[5]  arXiv:1803.06363 [pdf, ps, other]
Title: Geometric Adaptive Control for a Quadrotor UAV with Wind Disturbance Rejection
Subjects: Optimization and Control (math.OC)

This paper presents a geometric adaptive control scheme for a quadrotor unmanned aerial, where the effects of unknown, unstructured disturbances are mitigated by a multilayer neural network that is adjusted online. The stability of the proposed controller is analyzed with Lyapunov stability theory on the special Euclidean group, and it is shown that the tracking errors are uniformly ultimately bounded with an ultimate bound that can be abridged arbitrarily. A mathematical model of wind disturbance on the quadrotor dynamics is presented, and it is shown that the proposed adaptive controller is capable of mitigating the effects of wind disturbances successfully.

[6]  arXiv:1803.06366 [pdf, ps, other]
Title: Graphs, Ultrafilters and Colourability
Authors: Felix Dilke
Comments: 12 pages
Subjects: Category Theory (math.CT); Combinatorics (math.CO)

Let $\beta$ be the functor from Set to CHaus which maps each discrete set X to its Stone-Cech compactification, the set $\beta$ X of ultrafilters on X. Every graph G with vertex set V naturally gives rise to a graph $\beta G$ on the set $\beta V$ of ultrafilters on V . In what follows, we interrelate the properties of G and $\beta G$. Perhaps the most striking result is that G can be finitely coloured iff $\beta G$ has no loops.

[7]  arXiv:1803.06369 [pdf, other]
Title: A Family of Minimal and Renormalizable Rectangle Exchange Maps
Comments: 33 pages, 12 figures
Subjects: Dynamical Systems (math.DS)

A domain exchange map (DEM) is a dynamical system defined on a smooth Jordan domain which is a piecewise translation. We explain how to use cut-and-project sets to construct minimal DEMs. Specializing to the case in which the domain is a square and the cut-and-project set is associated to a Galois lattice, we construct an infinite family of DEMs in which each map is associated to a PV number. We develop a renormalization scheme for these DEMs. Certain DEMs in the family can be composed to create multistage, renormalizable DEMs.

[8]  arXiv:1803.06372 [pdf, other]
Title: Stochastic basins of attraction and generalized committor functions
Subjects: Dynamical Systems (math.DS); Chaotic Dynamics (nlin.CD)

We generalize the concept of basin of attraction of a stable state in order to facilitate the analysis of dynamical systems with noise and to assess stability properties of metastable states and long transients. To this end we examine the notions of mean sojourn times and absorption probabilities for Markov chains and study their convergence to the basin of attraction in the limiting cases. Since any dynamical system described by a transfer operator on a compact domain can be approximated by a Markov chain our approach is applicable to a large variety of problems.

[9]  arXiv:1803.06380 [pdf, ps, other]
Title: Distributed Optimization for Second-Order Multi-Agent Systems with Dynamic Event-Triggered Communication
Subjects: Optimization and Control (math.OC)

In this paper, we propose a fully distributed continuous-time algorithm to solve the distributed optimization problem for second-order multi-agent systems. The optimization objective function is a sum of private cost functions associated to the individual agents and the interaction between agents is described by a weighted undirected graph. We show the exponential convergence of the proposed algorithm if the underlying graph is connected and the private cost functions are strongly convex and have locally Lipschitz gradients. Moreover, to reduce the overall need of communication, we then propose a dynamic event-triggered communication scheme that is free of Zeno behavior. It is shown that the exponential convergence is achieved if the private cost functions are globally Lipschitz. Numerical simulations are provided to illustrate the effectiveness of the theoretical results.

[10]  arXiv:1803.06382 [pdf, ps, other]
Title: Harmonic spinors on the Davis hyperbolic 4-manifold
Comments: 33 pages and 2 figures
Subjects: Geometric Topology (math.GT); Differential Geometry (math.DG)

In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the G-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold.

[11]  arXiv:1803.06385 [pdf, ps, other]
Title: The $α$-normal labeling method for computing the $p$-spectral radii of uniform hypergraphs
Authors: Lele Liu, Linyuan Lu
Comments: 24 pages
Subjects: Combinatorics (math.CO)

Let $G$ be an $r$-uniform hypergraph of order $n$. For each $p\geq 1$, the $p$-spectral radius $\lambda^{(p)}(G)$ is defined as \[ \lambda^{(p)}(G):=\max_{|x_1|^p+\cdots+|x_n|^p=1} r\sum_{\{i_1,\ldots,i_r\}\in E(G)}x_{i_1}\cdots x_{i_r}. \] The $p$-spectral radius was introduced by Keevash-Lenz-Mubayi, and subsequently studied by Nikiforov in 2014. The most extensively studied case is when $p=r$, and $\lambda^{(r)}(G)$ is called the spectral radius of $G$. The $\alpha$-normal labeling method, which was introduced by Lu and Man in 2014, is effective method for computing the spectral radii of uniform hypergraphs. It labels each corner of an edge by a positive number so that the sum of the corner labels at any vertex is $1$ while the product of all corner labels at any edge is $\alpha$. Since then, this method has been used by many researchers in studying $\lambda^{(r)}(G)$. In this paper, we extend Lu and Man's $\alpha$-normal labeling method to the $p$-spectral radii of uniform hypergraphs for $p\ne r$; and find some applications.

[12]  arXiv:1803.06394 [pdf, ps, other]
Title: Combinatorial proofs of two Euler type identities due to Andrews
Comments: 14 pages
Subjects: Combinatorics (math.CO); Number Theory (math.NT)

We prove combinatorially some identities related to Euler's partition identity (the number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts). They were conjectured by Beck and proved by Andrews via generating functions.
Let $a(n)$ be the number of partitions of $n$ such that the set of even parts has exactly one element, $b(n)$ be the difference between the number of parts in all odd partitions of $n$ and the number of parts in all distinct partitions of $n$, and $c(n)$ be the number of partitions of $n$ in which exactly one part is repeated. Then, $a(n)=b(n)=c(n)$. The identity $a(n)=c(n)$ was proved combinatorially (in greater generality) by Fu and Tang. We prove combinatorially that $a(n)=b(n)$ and $b(n)=c(n)$. Our proof relies on bijections between a set and a multiset, where the partitions in the multiset are decorated with bit strings. Let $c_1(n)$ be the number of partitions of $n$ such that there is exactly one part occurring three times while all other parts occur only once and let $b_1(n)$ to be the difference between the total number of parts in the partitions of $n$ into distinct parts and the total number of different parts in the partitions of $n$ into odd parts. We prove combinatorially that $c_1(n)=b_1(n)$. In addition to these results by Andrews, we prove combinatorially that $b_1(n)=a_1(n)$, where $a_1(n)$ counts partitions of $n$ such that the set of even parts has exactly one element and satisfying some additional conditions. Moreover, we offer an analog of these results for the number of partitions of $n$ with exactly one part occurring two times while all other parts occur only once.

[13]  arXiv:1803.06395 [pdf, ps, other]
Title: Singular genuine rigidity
Comments: 17 pages
Subjects: Differential Geometry (math.DG)

We extend the concept of genuine rigidity of submanifolds by allowing mild singularities, mainly to obtain new global rigidity results and unify the known ones. As one of the consequences, we simultaneously extend and unify Sacksteder and Dajczer-Gromoll theorems by showing that any compact $n$-dimensional submanifold of ${\mathbb R}^{n+p}$ is singularly genuinely rigid in ${\mathbb R}^{n+q}$, for any $q < \min\{5,n\} - p$. Unexpectedly, the singular theory becomes much simpler and natural than the regular one, even though all technical codimension assumptions, needed in the regular case, are removed.

[14]  arXiv:1803.06398 [pdf, ps, other]
Title: Additive invariants of logarithmic schemes
Comments: 44 pages, comments welcome!
Subjects: Algebraic Geometry (math.AG)

We lift the decomposition theorems in logarithmic algebraic K-theory obtained by Hagihara and Nizio{\l} to a semi-orthogonal decomposition of the category of perfect complexes over the Kummer flat topos. As a byproduct, we extend Hagihara and Nizio{\l}'s results to a much wider class of log schemes and stacks. Further, we obtain uniform structure theorems which apply, in addition to algebraic K-theory, to all additive invariants of log schemes.

[15]  arXiv:1803.06402 [pdf, ps, other]
Title: Shadowing for nonautonomous dynamics
Subjects: Dynamical Systems (math.DS)

We prove that whenever a sequence of invertible and bounded operators $(A_m)_{m\in \mathbb{Z}}$ acting on a Banach space $X$ admits an exponential dichotomy and a sequence of differentiable maps $f_m \colon X\to X$, $m\in \mathbb{Z}$, has bounded and H\"{o}lder derivatives, the nonautonomous dynamics given by $x_{m+1}=A_mx_m+f_m(x_m)$, $m\in \mathbb{Z}$ has various shadowing properties. Hence, we extend recent results of Bernardes Jr. et al. in several directions. As a nontrivial application of our results, we give a new proof of the nonautonomous Grobman-Hartman theorem.

[16]  arXiv:1803.06408 [pdf, ps, other]
Title: Three Études on a sequence transformation pipeline
Authors: Paul Barry
Comments: 41 pages
Subjects: Combinatorics (math.CO)

We study a sequence transformation pipeline that maps certain sequences with rational generating functions to permutation-based sequence families of combinatorial significance. Many of the number triangles we encounter can be related to simplicial objects such as the associahedron and the permutahedron. The linkages between these objects is facilitated by the use of the previously introduced $\mathcal{T}$ transform.

[17]  arXiv:1803.06409 [pdf, ps, other]
Title: Integral comparisons of nonnegative positive definite functions on locally compact abelian groups
Comments: 30 pages
Subjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA)

In this paper, we discuss the following general questions. Let $\mu, \nu$ be two regular Borel measures of finite total variation. When do we have a constant $C$ satisfying that $$\int f d\nu \le C \int f d\mu$$ whenever $f$ is a continuous nonnegative positive definite function? How the admissible constants $C$ can be characterized and what is the best value?
First we discuss the problem in locally compact Abelian groups and then apply the results to the case when $\mu, \nu$ are the traces of the usual Lebesgue measure over centered and arbitrary intervals, respectively. This special case was earlier investigated by Shapiro, Montgomery, Hal\'asz and Logan. It is a close companion of the more familiar problem of Wiener, as well.

[18]  arXiv:1803.06411 [pdf, ps, other]
Title: The 21 reducible polars of Klein's quartic
Comments: 28 pages, contains Singular's script
Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Combinatorics (math.CO)

We describe the singularities and related properties of the arrangement of 21 reducible polars of Klein's quartic, containing Klein's well-known arrangement of $21$ lines.

[19]  arXiv:1803.06412 [pdf, ps, other]
Title: Symplectic invariance of rational surfaces on Kähler manifolds
Comments: 21 pages
Subjects: Algebraic Geometry (math.AG)

By work of Koll\'{a}r and Ruan, uniruledness of K\"{a}hler manifolds is an invariant of the underlying symplectic manifold. Zhiyu Tian proved that rational connectedness is a symplectic invariant if the dimension is $\leq 3$. We prove existence of a covering family of rational surfaces assuming positivity of certain gravitational descendants.

[20]  arXiv:1803.06415 [pdf, ps, other]
Title: Connection Blocking In Quotients of $Sol$
Comments: 10 pages. arXiv admin note: text overlap with arXiv:1706.07996; text overlap with arXiv:1211.7291 by other authors
Subjects: Differential Geometry (math.DG)

Let $G$ be a connected Lie group and $\Gamma \subset G$ a lattice. Connection curves of the homogeneous space $M=G/\Gamma$ are the orbits of one parameter subgroups of $G$. To $block$ a pair of points $m_1,m_2 \in M$ is to find a finite set $B \subset M\setminus \{m_1, m_2 \}$ such that every connecting curve joining $m_1$ and $m_2$ intersects $B$. The homogeneous space $M$ is $blockable$ if every pair of points in $M$ can be blocked, otherwise we call it $non-blockable$. $Sol$ is an important Lie group and one of the eight homogeneous Thurston 3-geometries. It is a unimodular solvable Lie group diffeomorphic to $R^3$, and together with the left invariant metric $ds^2=e^{-2z}dx^2+e^{2z}dy^2+dz^2$ includes copies of the hyperbolic plane, which makes studying its geometrical properties more interesting. In this paper we prove that all quotients of $Sol$ are non-blockable. In particular, we show that for any lattice $\Gamma \subset Sol$, the set of non-blockable pairs is a dense subset of $Sol/\Gamma \times Sol/\Gamma$.

[21]  arXiv:1803.06417 [pdf, other]
Title: Coding for Channels with SNR Variation: Spatial Coupling and Efficient Interleaving
Comments: 8 pages, Submitted to IEEE Transactions on Magnetics (TMAG)
Subjects: Information Theory (cs.IT)

In magnetic-recording systems, consecutive sections experience different signal to noise ratios (SNRs). To perform error correction over these systems, one approach is to use an individual block code for each section. However, a section affected by a lower SNR shows a weaker performance compared to a section affected by a higher SNR. A commonly used technique is to perform interleaving across blocks to alleviate negative effects of varying SNR. However, this scheme is typically costly to implement and does not result in the best performance. Spatially-coupled (SC) codes are a family of graph-based codes with capacity approaching performance and low latency decoding. An SC code is constructed by partitioning an underlying block code to several component matrices, and coupling copies of the component matrices together. The contribution of this paper is threefold. First, we present a new partitioning technique to efficiently construct SC codes with column weights 4 and 6. Second, we present an SC code construction for channels with SNR variation. Our SC code construction provides local error correction for each section by means of the underlying codes that cover one section each, and simultaneously, an added level of error correction by means of coupling among the underlying codes. Consequently, and because of the structure of SC codes, more reliable sections can help unreliable ones to achieve an improved performance. Third, we introduce a low-complexity interleaving scheme specific to SC codes that further improves their performance over channels with SNR variation. Our simulation results show that our SC codes outperform individual block codes by more than 1 and 2 orders of magnitudes in the error floor region compared to the block codes with and without regular interleaving, respectively. This improvement is more pronounced by increasing the memory and column weight.

[22]  arXiv:1803.06432 [pdf, ps, other]
Title: Pseudo-differential operators with nonlinear quantizing functions
Comments: 26 pages
Subjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP); Operator Algebras (math.OA)

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$ Au(x)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}e^{i(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)dyd\xi, $$ where $\tau:\mathbb{R}^n\to\mathbb{R}^n$ is a general function. In particular, for the linear choices $\tau(x)=0$, $\tau(x)=x$, and $\tau(x)=\frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $\tau$ and here we investigate the corresponding calculus in the model case of $\mathbb{R}^n$. We also give examples of nonlinear $\tau$ appearing on the polarised and non-polarised Heisenberg groups, inspired by the recent joint work with Marius Mantoiu.

[23]  arXiv:1803.06439 [pdf, other]
Title: Global surfaces of section for dynamically convex Reeb flows on lens spaces
Comments: 29 pages, 4 figures
Subjects: Symplectic Geometry (math.SG); Dynamical Systems (math.DS)

We show that a dynamically convex Reeb flow on a lens space $L(p, 1), p>1,$ admits a $p$-unknotted closed Reeb orbit $P$ which is the binding of a rational open book decomposition with disk-like pages. Each page is a rational global surface of section for the Reeb flow and the Conley-Zehnder index of the $p$-th iterate of $P$ is $3$. This result applies to the H\'enon-Heiles Hamiltonian whose energy level presents $\mathbb{Z}_3$-symmetric and for all energies $<1/6$ the flow restricted to the sphere-like component descends to a dynamically convex Reeb flow on $L(3,1)$. Due to a $\mathbb{Z}_4$-symmetry the result also applies to Hill's lunar problem.

[24]  arXiv:1803.06446 [pdf, ps, other]
Title: On Polyhedral Estimation of Signals via Indirect Observations
Subjects: Statistics Theory (math.ST)

We consider the problem of recovering linear image of unknown signal belonging to a given convex compact signal set from noisy observation of another linear image of the signal. We develop a simple generic efficiently computable nonlinear in observations "polyhedral" estimate along with computation-friendly techniques for its design and risk analysis. We demonstrate that under favorable circumstances the resulting estimate is provably near-optimal in the minimax sense, the "favorable circumstances" being less restrictive than the weakest known so far assumptions ensuring near-optimality of estimates which are linear in observations.

[25]  arXiv:1803.06448 [pdf, ps, other]
Title: Frequency-Domain Decoupling for MIMO-GFDM Spatial Multiplexing
Comments: 4 pages, 4 figures
Subjects: Information Theory (cs.IT)

Generalized frequency division multiplexing (GFDM) is considered a non-orthogonal waveform and known to encounter difficulties when using in the spatial multiplexing mode of multiple-input-multiple-output (MIMO) scenario. In this paper, a class of GFDM prototype filters, under which the GFDM system is free from inter-subcarrier interference, is investigated, enabling frequency-domain decoupling in the processing at the GFDM receiver. An efficient MIMO-GFDM detection method based on depth-first sphere decoding is then proposed with such class of filters. Numerical results confirm a significant reduction in complexity, especially when the number of subcarriers is large, compared with existing methods presented in recent years.

[26]  arXiv:1803.06451 [pdf, other]
Title: Instability of the solitary waves for the generalized derivative nonlinear Schrödinger equation in the degenerate case
Comments: 33 pages, 2 figures, all comments are welcome
Subjects: Analysis of PDEs (math.AP)

In this paper, we develop the modulation analysis, the perturbation argument and the Virial identity similar as those in \cite{MartelM:Instab:gKdV} to show the orbital instability of the solitary waves $\Q\sts{x-ct}\e^{\i\omega t}$ of the generalized derivative nonlinear Schr\"odinger equation (gDNLS) in the degenerate case $c=2z_0\sqrt{\omega}$, where $z_0=z_0\sts{\sigma} $ is the unique zero point of $F\sts{z;~\sigma}$ in $\sts{-1, ~ 1}$. The new ingredients in the proof are the refined modulation decomposition of the solution near $\Q$ according to the spectrum property of the linearized operator $\Scal_{\omega, c}"\sts{\Q}$ and the refined construction of the Virial identity in the degenerate case. Our argument is qualitative, and we improve the result in \cite{Fukaya2017}.

[27]  arXiv:1803.06454 [pdf, ps, other]
Title: Some Questions in $l-$adic Cohomology
Subjects: Algebraic Geometry (math.AG)

The comparison theorem for a smooth projective variety $X$ over $\mathbb{C}$ tells us that the Betti numbers are independent of $l$. We aim to understand the $l$ independence of Betti numbers for smooth projective varieties $X$ over $k$, where $k$ is an algebraic extension of $\mathbb{F}_p$.

[28]  arXiv:1803.06455 [pdf, ps, other]
Title: A-infinity algebras, strand algebras, and contact categories
Comments: 83 pages, 23 figures, 6 tables
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Symplectic Geometry (math.SG)

In previous work we showed that the contact category algebra of a quadrangulated surface is isomorphic to the homology of a strand algebra from bordered Floer theory. Being isomorphic to the homology of a differential graded algebra, this contact category algebra has an A-infinity structure. In this paper we investigate such A-infinity structures in detail. We give explicit constructions of such A-infinity structures, and establish some of their properties, including conditions for the nonvanishing of A-infinity operations. Along the way we develop several related notions, including a detailed consideration of tensor products of strand diagrams.

[29]  arXiv:1803.06457 [pdf, ps, other]
Title: Generalization of a Real-Analysis Result to a Class of Topological Vector Spaces
Authors: Leonard T. Huang
Comments: 6 pages, no figures
Subjects: Functional Analysis (math.FA); Probability (math.PR)

In this paper, we generalize an elementary real-analysis result to a class of topological vector spaces. We also give an example of a topological vector space to which the result cannot be generalized.

[30]  arXiv:1803.06461 [pdf, ps, other]
Title: Constraints on the cohomological correspondence associated to a self map
Authors: K. V. Shuddhodan
Comments: Comments are welcome!
Subjects: Algebraic Geometry (math.AG)

In this article we establish some constraints on the eigenvalues for the action of a self map of a proper variety on its $\ell$-adic cohomology. The essential ingredients are a trace formula due to Fujiwara, and the theory of weights.

[31]  arXiv:1803.06463 [pdf, ps, other]
Title: Multiplication formulas and semisimplicity for q-Schur superalgebras
Comments: 22 pages. Nagoya J. Math. (to appear)
Subjects: Quantum Algebra (math.QA); Group Theory (math.GR); Rings and Algebras (math.RA); Representation Theory (math.RT)

We investigate products of certain double cosets for the symmetric group and use the findings to derive some multiplication formulas for q-Schur superalgebras. This gives a combinatorialisation of the relative norm approach developed by the first two authors. We then give several applications of the multiplication formulas, including the matrix representation of the regular representation and a semisimplicity criterion for q-Schur superalgebras. We also construct infinitesimal and little q-Schur superalgebras directly from the multiplication formulas and develop their semisimplicity criteria.

[32]  arXiv:1803.06465 [pdf, ps, other]
Title: Local Continuity and Asymptotic Behaviour of Degenerate Parabolic Systems
Comments: 32p
Subjects: Analysis of PDEs (math.AP)

We study the local continuity and asymptotic behavior of solutions, $\bold{u}=(u^1,\cdots, u^k)$, of degenerate system \begin{equation*} u^i_t=\nabla\cdot\left(U^{m-1}\nabla u^i\right) \qquad \text{for $m>1$ and $i=1,\cdots,k$} \end{equation*} describing the degenerate diffusion of the populations density vector, $\bold{u}$, of $k$-species whose diffusion is determined by their total population density $U=u^1+\cdots+u^k$. We adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and Dibenedetto for the local continuity of solutions, $u^i$. Under some regularity condition, we also prove that the population density function, $\bold{u}$, of $i$-th species with the population $M_i$ converges to $\frac{M_i}{M}\mathcal{B}_M(x,t)$ in the space of differentiable functions of all order where $\mathcal{B}_M$ is the Barenblatt profile of the Porous Medium Equation with $L^1$ mass $M=M_1+\cdots+M_k$ while $U$ converges to $\mathcal{B}_M$. As a consequence, each $u^i$ becomes a concave function after a finite time.

[33]  arXiv:1803.06467 [pdf, other]
Title: Optimizing Information Freshness in Wireless Networks under General Interference Constraints
Subjects: Information Theory (cs.IT); Networking and Internet Architecture (cs.NI)

Age of information (AoI) is a recently proposed metric for measuring information freshness. AoI measures the time that elapsed since the last received update was generated. We consider the problem of minimizing average and peak AoI in wireless networks under general interference constraints. When fresh information is always available for transmission, we show that a stationary scheduling policy is peak age optimal. We also prove that this policy achieves average age that is within a factor of two of the optimal average age. In the case where fresh information is not always available, and packet/information generation rate has to be controlled along with scheduling links for transmission, we prove an important separation principle: the optimal scheduling policy can be designed assuming fresh information, and independently, the packet generation rate control can be done by ignoring interference. Peak and average AoI for discrete time G/Ber/1 queue is analyzed for the first time, which may be of independent interest.

[34]  arXiv:1803.06468 [pdf, ps, other]
Title: Rings additively generated by idempotents and nilpotents
Subjects: Rings and Algebras (math.RA)

A ring R is a strongly 2-nil-clean if every element in R is the sum of two idempotents and a nilpotent that commute. A ring R is feebly clean if every element in R is the sum of two orthogonal idempotents and a unit. In this paper, strongly 2-nil-clean rings are studied with an emphasis on their relations with feebly clean rings. This work shows new interesting connections between strongly 2-nil-clean rings and weakly exchange rings

[35]  arXiv:1803.06469 [pdf, other]
Title: Distributed Scheduling Algorithms for Optimizing Information Freshness in Wireless Networks
Subjects: Information Theory (cs.IT); Networking and Internet Architecture (cs.NI)

Age of Information (AoI), measures the time elapsed since the last received information packet was generated at the source. We consider the problem of AoI minimization for single-hop flows in a wireless network, under pairwise interference constraints and time varying channel. We consider simple, yet broad, class of distributed scheduling policies, in which a transmission is attempted over each link with a certain attempt probability. We obtain an interesting relation between the optimal attempt probability and the optimal AoI of the link, and its neighboring links. We then show that the optimal attempt probabilities can be computed by solving a convex optimization problem, which can be done distributively.

[36]  arXiv:1803.06470 [pdf, ps, other]
Title: Twisted Alexander Polynomials of $(-2,3,2n+1)$-Pretzel Knots
Authors: Airi Aso
Comments: 8 pages, 2 figures
Subjects: Geometric Topology (math.GT)

We calculate the twisted Alexander polynomials of $(-2,3,2n+1)$-pretzel knots associated to their holonomy representations. As a corollary, we obtain new supporting evidences of Dunfield, Friedl and Jackson's conjecture, that is, the twisted Alexander polynomials of hyperbolic knots associated to their holonomy representations determine the genus and fiberedness of the knots.

[37]  arXiv:1803.06471 [pdf, other]
Title: Optimizing Age of Information in Wireless Networks with Perfect Channel State Information
Subjects: Information Theory (cs.IT); Networking and Internet Architecture (cs.NI)

Age of information (AoI), defined as the time elapsed since the last received update was generated, is a newly proposed metric to measure the timeliness of information updates in a network. We consider AoI minimization problem for a network with general interference constraints, and time varying channels. We propose two policies, namely, virtual-queue based policy and age-based policy when the channel state is available to the network scheduler at each time step. We prove that the virtual-queue based policy is nearly optimal, up to a constant additive factor, and the age-based policy is at-most factor 4 away from optimality. Comparing with our previous work, which derived age optimal policies when channel state information is not available to the scheduler, we demonstrate a 4 fold improvement in age due to the availability of channel state information.

[38]  arXiv:1803.06472 [pdf, ps, other]
Title: Transversely holomorphic branched Cartan geometry
Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)

Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension $d$ admits, away from a closed analytic subset of positive codimension, a nonsingular holomorphic foliation of complex codimension $d$ endowed with a transversely flat branched complex projective geometry (equivalently, a ${\mathbb C}P^d$-geometry). We also prove that transversely branched holomorphic Cartan geometries on compact complex projective rationally connected varieties and on compact simply connected Calabi-Yau manifolds are always flat (consequently, they are defined by holomorphic maps into homogeneous spaces).

[39]  arXiv:1803.06477 [pdf, ps, other]
Title: On the homotopy type of $\mathrm{Sp}(n)$ gauge groups
Comments: 9 pages
Subjects: Algebraic Topology (math.AT)

Let $\mathcal{G}_{k,n}$ be the gauge group of the principal $\mathrm{Sp}(n)$-bundle over $S^4$ corresponding to $k\in\mathbb{Z}\cong\pi_3(\mathrm{Sp}(n))$. We refine the result of Sutherland on the homotopy types of $\mathcal{G}_{k,n}$ and relate it with the order of a certain Samelson product in $\mathrm{Sp}(n)$. Then we classify the $p$-local homotopy types of $\mathcal{G}_{k,n}$ for $(p-1)^2+1\ge 2n$.

[40]  arXiv:1803.06479 [pdf, ps, other]
Title: On the definition of a solution to a rough differential equation
Authors: I. Bailleul
Comments: 7 pages
Subjects: Classical Analysis and ODEs (math.CA)

We give an elementary proof that Davie's definition of a solution to a rough differential equation and the notion of solution given by Bailleul in (Flows driven by rough paths) coincide. This provides an alternative point on view on the deep algebraic insights of Cass and Weidner in their work (Tree algebras over topological vector spaces in rough path theory).

[41]  arXiv:1803.06481 [pdf, ps, other]
Title: The density of visible points in the Ammann-Beenker point set
Subjects: Number Theory (math.NT)

The relative density of visible points of the integer lattice $\mathbb{Z}^d$ is known to be $1/\zeta(d)$ for $d\geq 2$, where $\zeta$ is Riemann's zeta function. In this paper we prove that the relative density of visible points in the Ammann-Beenker point set is given by $2(\sqrt{2}-1)/\zeta_K(2)$, where $\zeta_K$ is Dedekind's zeta function over $K=\mathbb{Q}(\sqrt{2})$.

[42]  arXiv:1803.06482 [pdf, other]
Title: Asynchronous Distributed Method of Multipliers for Constrained Nonconvex Optimization
Subjects: Optimization and Control (math.OC)

This paper addresses a class of constrained optimization problems over networks in which local cost functions and constraints can be nonconvex. We propose an asynchronous distributed optimization algorithm, relying on the centralized Method of Multipliers, in which each node wakes up in an uncoordinated fashion and performs either a descent step on a local Augmented Lagrangian or an ascent step on the local multiplier vector. These two phases are regulated by a distributed logic-AND, which allows nodes to understand when the descent on the (whole) Augmented Lagrangian is sufficiently small. We show that this distributed algorithm is equivalent to a block coordinate descent algorithm for the minimization of the Augmented Lagrangian followed by an update of the whole multiplier vector. Thus, the proposed algorithm inherits the convergence properties of the Method of Multipliers.

[43]  arXiv:1803.06483 [pdf, ps, other]
Title: Products of $H$-separable spaces in the Laver model
Journal-ref: Topology Appl. 239 (2018), 115-119
Subjects: General Topology (math.GN); Logic (math.LO)

We prove that in the Laver model for the consistency of the Borel's conjecture, the product of any two $H$-separable spaces is $M$-separable.

[44]  arXiv:1803.06491 [pdf, ps, other]
Title: Solutions of the $U_q(\widehat{\mathfrak{sl}}_N)$ reflection equations
Comments: 36 pages
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Exactly Solvable and Integrable Systems (nlin.SI)

We find the complete set of invertible solutions of the untwisted and twisted reflection equations for the Bazhanov-Jimbo R-matrix of type ${\mathrm A}^{(1)}_{N-1}$. We also show that all invertible solutions can be obtained by an appropriate affinization procedure from solutions of the constant untwisted and twisted reflection equations.

[45]  arXiv:1803.06496 [pdf, ps, other]
Title: A simple algorithm for Max Cut
Comments: Submitted to Conference on March 17, 2018
Subjects: Optimization and Control (math.OC); Combinatorics (math.CO); Numerical Analysis (math.NA)

Based on an explicit equivalent continuous optimization problem, we propose a simple continuous iterative algorithm for Max Cut, which converges to a local optimum in finite steps. The inner subproblem is solved analytically and thus no optimization solver is called. Preliminary results on G-set demonstrate the performance. In particular, the ratio between the best cut values achieved by the simple algorithm without any local breakout techniques and the best known ones is of at least $0.986$.

[46]  arXiv:1803.06497 [pdf, other]
Title: Variational Bayesian Inference of Line Spectral Estimation with Multiple Measurement Vectors
Subjects: Information Theory (cs.IT)

In this paper, the line spectral estimation (LSE) problem with multiple measurement vectors (MMVs) is studied utilizing the Bayesian methods. Motivated by the recently proposed variational line spectral estimation (VALSE) method, we extend it to deal with the MMVs setting, which is especially important in array signal processing. The VALSE method can automatically estimate the model order and nuisance parameters such as noise variance and weight variance. In addition, by approximating the probability density function (PDF) of the frequencies with the mixture of von Mises PDFs, closed-form update equation and the uncertainty degree of the estimates can be obtained. Interestingly, we find that the VALSE with MMVs can be viewed as applying the VALSE with single measurement vector (SMV) to each snapshot, and combining the intermediate data appropriately. Furthermore, the proposed prior distribution provides a good interpretation of tradeoff between grid and off-grid based methods. Finally, numerical results demonstrate the effectiveness of the VALSE method, compared to the state-of-the-art methods in the MMVs setting.

[47]  arXiv:1803.06499 [pdf, ps, other]
Title: Kähler submanifolds of the symmetrized polydisc
Comments: To appear in Comptes Rendus Mathematique
Subjects: Complex Variables (math.CV); Differential Geometry (math.DG)

This paper proves the non-existence of common K\"ahler submanifolds of the complex Euclidean space and the symmetrized polydisc endowed with their canonical metrics.

[48]  arXiv:1803.06507 [pdf, ps, other]
Title: Covering Arrays for Equivalence Classes of Words
Comments: 17 pages
Subjects: Combinatorics (math.CO)

Covering arrays for words of length $t$ over a $d$ letter alphabet are $k \times n$ arrays with entries from the alphabet so that for each choice of $t$ columns, each of the $d^t$ $t$-letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case words are equivalent if they induce the same partition of a $t$ element set. In the second case, words of the same weight are equivalent. In both cases we produce logarithmic upper bounds on the minimum size $k=k(n)$ of a covering array. Definitive results for $t=2,3,4$, as well as general results, are provided.

[49]  arXiv:1803.06517 [pdf, other]
Title: Optimal Designs for the Generalized Partial Credit Model
Subjects: Statistics Theory (math.ST); Methodology (stat.ME)

Analyzing ordinal data becomes increasingly important in psychology, especially in the context of item response theory. The generalized partial credit model (GPCM) is probably the most widely used ordinal model and finds application in many large scale educational assessment studies such as PISA. In the present paper, optimal test designs are investigated for estimating persons' abilities with the GPCM for calibrated tests when item parameters are known from previous studies. We will derive that local optimality may be achieved by assigning non-zero probability only to the first and last category independently of a person's ability. That is, when using such a design, the GPCM reduces to the dichotomous 2PL model. Since locally optimal designs require the true ability to be known, we consider alternative Bayesian design criteria using weight distributions over the ability parameter space. For symmetric weight distributions, we derive necessary conditions for the optimal one-point design of two response categories to be Bayes optimal. Furthermore, we discuss examples of common symmetric weight distributions and investigate, in which cases the necessary conditions are also sufficient. Since the 2PL model is a special case of the GPCM, all of these results hold for the 2PL model as well.

[50]  arXiv:1803.06519 [pdf, other]
Title: Signal detection via Phi-divergences for general mixtures
Authors: Marc Ditzhaus
Subjects: Statistics Theory (math.ST)

In this paper we are interested in testing whether there are any signals hidden in high dimensional noise data. Therefore we study the family of goodness-of-fit tests based on $\Phi$-divergences including the test of Berk and Jones as well as Tukey's higher criticism test. The optimality of this family is already known for the heterogeneous normal mixture model. We now present a technique to transfer this optimality to more general models. For illustration we apply our results to dense signal and sparse signal models including the exponential-$\chi^2$ mixture model and general exponential families as the normal, exponential and Gumbel distribution. Beside the optimality of the whole family we discuss the power behavior on the detection boundary and show that the whole family has no power there, whereas the likelihood ratio test does.

[51]  arXiv:1803.06523 [pdf, ps, other]
Title: Stochastic model-based minimization of weakly convex functions
Comments: 9 pages
Subjects: Optimization and Control (math.OC); Learning (cs.LG)

We consider an algorithm that successively samples and minimizes stochastic models of the objective function. We show that under weak-convexity and Lipschitz conditions, the algorithm drives the expected norm of the gradient of the Moreau envelope to zero at the rate $O(k^{-1/4})$. Our result yields new complexity guarantees for the stochastic proximal point algorithm on weakly convex problems and for the stochastic prox-linear algorithm for minimizing compositions of convex functions with smooth maps. Moreover, our result also recovers the recently obtained complexity estimate for the stochastic proximal subgradient method on weakly convex problems.

[52]  arXiv:1803.06533 [pdf, ps, other]
Title: Moduli of quiver representations for exceptional collections on surfaces
Comments: 69 pages, comments are welcome
Subjects: Algebraic Geometry (math.AG)

Suppose $S$ is a smooth projective surface over an algebraically closed field $k$, $\mathcal{L}=\{L_1,\ldots,L_n\}$ is a full strong exceptional collection of line bundles on $S$. Let $Q$ be the quiver associated to this collection. One might hope that $S$ is the moduli space of representation of $Q$ with dimension vector $(1,\ldots,1)$ for a suitably chosen stability condition $\theta$: $S\cong M_\theta$. In this paper, we show that this is the case for many surfaces with such collections.

[53]  arXiv:1803.06537 [pdf, other]
Title: Renormalization of the Hutchinson Operator
Authors: Yann Demichel
Subjects: Dynamical Systems (math.DS)

One of the easiest and common ways of generating fractal sets in $\mathbb{R}^D$ is as attractors of affine iterated function systems (IFS). The classic theory of IFS's requires that they are made with contractive functions. In this paper, we relax this hypothesis considering a new operator $H_\rho$ obtained by renormalizing the usual Hutchinson operator $H$. Namely, the $H_\rho$-orbit of a given compact set $K_0$ is built from the original sequence $(H^n(K_0))_n$ by rescaling each set by its distance from $0$. We state several results for the convergence of these orbits and give a geometrical description of the corresponding limit sets. In particular, it provides a way to construct some eigensets for $H$. Our strategy to tackle the problem is to link these new sequences to some classic ones but it will depend on whether the IFS is strictly linear or not. We illustrate the different results with various detailed examples. Finally, we discuss some possible generalizations.

[54]  arXiv:1803.06543 [pdf, ps, other]
Title: The parametrix method for parabolic SPDEs
Comments: Submitted to Stochastics and Partial Differential Equations: Analysis and Computations
Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in H\"older classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the It\^o-Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.

[55]  arXiv:1803.06549 [pdf, ps, other]
Title: Low-Order Control Design using a Reduced-Order Model with a Stability Constraint on the Full-Order Model
Subjects: Optimization and Control (math.OC)

We consider low-order controller design for large-scale linear time-invariant dynamical systems with inputs and outputs. Model order reduction is a popular technique, but controllers designed for reduced-order models may result in unstable closed-loop plants when applied to the full-order system. We introduce a new method to design a fixed-order controller by minimizing the $L_\infty$ norm of a reduced-order closed-loop transfer matrix function subject to stability constraints on the closed-loop systems for both the reduced-order and the full-order models. Since the optimization objective and the constraints are all nonsmooth and nonconvex we use a sequential quadratic programming method based on quasi-Newton updating that is intended for this problem class, available in the open-source software package GRANSO. Using a publicly available test set, the controllers obtained by the new method are compared with those computed by the HIFOO (H-Infinity Fixed-Order Optimization) toolbox applied to a reduced-order model alone, which frequently fail to stabilize the closed-loop system for the associated full-order model.

[56]  arXiv:1803.06552 [pdf, ps, other]
Title: Generators of semigroups on Banach spaces inducing holomorphic semiflows
Comments: 14 pages, Accepted, Israel Journal of Mathematics 2018
Subjects: Functional Analysis (math.FA)

Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of analytic functions on the open unit disc. If $T$ consists of composition operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication.

[57]  arXiv:1803.06556 [pdf, ps, other]
Title: Linearization of third-order ordinary differential equations u'''=f(x,u,u',u'') via point transformations
Subjects: Classical Analysis and ODEs (math.CA); Differential Geometry (math.DG)

The linearization problem by use of the Cartan equivalence method for scalar third-order ODEs via point transformations was solved partially in [1,2]. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third-order ordinary differential equation u'''=f(x,u,u',u'') which admits a four-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is also presented. The method provides auxiliary functions which can be utilized to efficiently determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.

[58]  arXiv:1803.06566 [pdf, other]
Title: Computing the Best Approximation Over the Intersection of a Polyhedral Set and the Doubly Nonnegative Cone
Subjects: Optimization and Control (math.OC)

This paper introduces an efficient algorithm for computing the best approximation of a given matrix onto the intersection of linear equalities, inequalities and the doubly nonnegative cone (the cone of all positive semidefinite matrices whose elements are nonnegative). In contrast to directly applying the block coordinate descent type methods, we propose an inexact accelerated (two-)block coordinate descent algorithm to tackle the four-block unconstrained nonsmooth dual program. The proposed algorithm hinges on the efficient semismooth Newton method to solve the subproblems, which have no closed form solutions since the original four blocks are merged into two larger blocks. The $O(1/k^2)$ iteration complexity of the proposed algorithm is established. Extensive numerical results over various large scale semidefinite programming instances from relaxations of combinatorial problems demonstrate the effectiveness of the proposed algorithm.

[59]  arXiv:1803.06568 [pdf, ps, other]
Title: Splittable and unsplittable graphs and configurations
Comments: 19 pages, 10 figures
Subjects: Combinatorics (math.CO)

We prove that there exist infinitely many splittable and also infinitely many unsplittable cyclic $(n_3)$ configurations. We also present a complete study of trivalent cyclic Haar graphs on at most 60 vertices with respect to splittability. Finally, we show that all cyclic flag-transitive configurations with the exception of the Fano plane and the M\"obius-Kantor configuration are splittable.

[60]  arXiv:1803.06572 [pdf, ps, other]
Title: Non-hyperbolic ergodic measures and horseshoes in partially homoclinic classes
Comments: 25 pages and 1 figure
Subjects: Dynamical Systems (math.DS)

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$*$-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures.

[61]  arXiv:1803.06573 [pdf, ps, other]
Title: On the Fenchel Duality between Strong Convexity and Lipschitz Continuous Gradient
Authors: Xingyu Zhou
Subjects: Optimization and Control (math.OC)

We provide a simple proof for the Fenchel duality between strong convexity and Lipschitz continuous gradient. To this end, we first establish equivalent conditions of convexity for a general function that may not be differentiable. By utilizing these equivalent conditions, we can directly obtain equivalent conditions for strong convexity and Lipschitz continuous gradient. Based on these results, we can easily prove Fenchel duality. Beside this main result, we also identify several conditions that are implied by strong convexity or Lipschitz continuous gradient, but are not necessarily equivalent to them. This means that these conditions are more general than strong convexity or Lipschitz continuous gradient themselves.

[62]  arXiv:1803.06576 [pdf, other]
Title: Projection-Based Finite Elements for Nonlinear Function Spaces
Subjects: Numerical Analysis (math.NA)

We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise onto the manifold. We show optimal interpolation error bounds with respect to Lebesgue and Sobolev norms. Additionally, we show similar bounds for the test functions, i.e., variations of discrete functions. Combining these results with a nonlinear C\'ea lemma, we prove optimal $L^2$ and $H^1$ discretization error bounds for harmonic maps from a planar domain into a smooth manifold. All these error bounds are also verified numerically.

[63]  arXiv:1803.06577 [pdf, ps, other]
Title: Multi-user Multi-task Offloading and Resource Allocation in Mobile Cloud Systems
Comments: arXiv admin note: text overlap with arXiv:1712.00030
Subjects: Information Theory (cs.IT)

We consider a general multi-user Mobile Cloud Computing (MCC) system where each mobile user has multiple independent tasks. These mobile users share the computation and communication resources while offloading tasks to the cloud. We study both the conventional MCC where tasks are offloaded to the cloud through a wireless access point, and MCC with a computing access point (CAP), where the CAP serves both as the network access gateway and a computation service provider to the mobile users. We aim to jointly optimize the offloading decisions of all users as well as the allocation of computation and communication resources, to minimize the overall cost of energy, computation, and delay for all users. The optimization problem is formulated as a non-convex quadratically constrained quadratic program, which is NP-hard in general. For the case without a CAP, an efficient approximate solution named MUMTO is proposed by using separable semidefinite relaxation (SDR), followed by recovery of the binary offloading decision and optimal allocation of the communication resource. To solve the more complicated problem with a CAP, we further propose an efficient three-step algorithm named MUMTO-C comprising of generalized MUMTO SDR with CAP, alternating optimization, and sequential tuning, which always computes a locally optimal solution. For performance benchmarking, we further present numerical lower bounds of the minimum system cost with and without the CAP. By comparison with this lower bound, our simulation results show that the proposed solutions for both scenarios give nearly optimal performance under various parameter settings, and the resultant efficient utilization of a CAP can bring substantial cost benefit.

[64]  arXiv:1803.06582 [pdf, other]
Title: Contrasting Various Notions of Convergence in Geometric Analysis
Comments: 7 figures by Penelope Chang of Hunter College High School
Subjects: Metric Geometry (math.MG); Differential Geometry (math.DG)

We explore the distinctions between $L^p$ convergence of metric tensors on a fixed Riemannian manifold versus Gromov-Hausdorff, uniform, and intrinsic flat convergence of the corresponding sequence of metric spaces. We provide a number of examples which demonstrate these notions of convergence do not agree even for two dimensional warped product manifolds with warping functions converging in the $L^p$ sense. We then prove a theorem which requires $L^p$ bounds from above and $C^0$ bounds from below on the warping functions to obtain enough control for all these limits to agree.

[65]  arXiv:1803.06583 [pdf, ps, other]
Title: Circular orders, ultrahomogeneity and topological groups
Comments: 17 pages
Subjects: Dynamical Systems (math.DS)

We study topological groups $G$ for which the universal minimal $G$-system $M(G)$, or the universal irreducible affine $G$-system $IA(G)$ are tame. We call such groups intrinsically tame and convexly intrinsically tame. These notions are generalized versions of extreme amenability and amenability, respectively. When $M(G)$, as a $G$-system, admits a circular order we say that $G$ is intrinsically circularly ordered. This implies that $G$ is intrinsically tame.
We show that for every circularly ultrahomogeneous action $G \curvearrowright X$ on a circularly ordered set $X$ the topological group $G$, in its pointwise convergence topology, is intrinsically circularly ordered. This result is a "circular" analog of Pestov's result about the extremal amenability of ultrahomogeneous actions on linearly ordered sets by linear order preserving transformations.
In the case where $X$ is countable, the corresponding Polish group of circular automorphisms $G$ admits a concrete description. Using the Kechris-Pestov-Todorcevic construction we show that $M(G)$ is a circularly ordered compact space obtained by splitting the rational points on the circle. We show also that $G$ is Roelcke precompact, satisfies Kazhdan's property $T$ (using results of Evans-Tsankov) and has the automatic continuity property (using results of Rosendal-Solecki).

[66]  arXiv:1803.06590 [pdf, ps, other]
Title: Cell Decompositions for Rank Two Quiver Grassmannians
Comments: 35 pages
Subjects: Representation Theory (math.RT); Algebraic Geometry (math.AG); Rings and Algebras (math.RA)

We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these "truncated preprojectives" are also established. We also provide two natural combinatorial labelings for these cells. On the one hand, they are labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The natural bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.

[67]  arXiv:1803.06592 [pdf, ps, other]
Title: Layer structure of irreducible Lie algebra modules
Authors: Jorgen Rasmussen
Comments: 23 pages
Subjects: Representation Theory (math.RT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Combinatorics (math.CO)

Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of every finite-dimensional irreducible $\mathfrak{g}$-module admits a decomposition in terms of layer sums, with only non-negative integer coefficients. Ensuing results include a new approach to the computation of Weyl characters and weight multiplicities, and a closed-form expression for the number of distinct weights in a finite-dimensional irreducible $\mathfrak{g}$-module. The latter is given by a polynomial in the Dynkin labels, of degree equal to the rank of $\mathfrak{g}$.

[68]  arXiv:1803.06600 [pdf, ps, other]
Title: Optimizing the Efficiency of First-order Methods for Decreasing the Gradient of Smooth Convex Functions
Subjects: Optimization and Control (math.OC)

This paper optimizes the step coefficients of first-order methods for smooth convex minimization in terms of the worst-case convergence bound (i.e., efficiency) of the decrease of the gradient norm. This work is based on the performance estimation problem approach. The corresponding worst-case gradient bound of the optimized method is optimal up to a constant for large-dimensional smooth convex minimization problems. This paper then illustrates that the resulting method, named OGM-G, has a computationally efficient form that is similar to the optimized gradient method (OGM).

[69]  arXiv:1803.06601 [pdf, ps, other]
Title: Convergence of Heisenberg Modules over Quantum 2-tori for the Modular Gromov-Hausdorff Propinquity
Comments: Third part of 1608.04881v1; second part of arXiv:1703.07073v1; split due to length. 34 pages
Subjects: Operator Algebras (math.OA)

The modular Gromov-Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a continuous family for the modular propinquity.

[70]  arXiv:1803.06602 [pdf, ps, other]
Title: Two new classes of quantum MDS codes
Subjects: Information Theory (cs.IT)

Let $p$ be a prime and let $q$ be a power of $p$. In this paper, by using generalized Reed-Solomon (GRS for short) codes and extended GRS codes, we construct two new classes of quantum maximum-distance- separable (MDS) codes with parameters \[ [[tq, tq-2d+2, d]]_{q} \] for any $1 \leq t \leq q, 2 \leq d \leq \lfloor \frac{tq+q-1}{q+1}\rfloor+1$, and \[ [[t(q+1)+2, t(q+1)-2d+4, d]]_{q} \] for any $1 \leq t \leq q-1, 2 \leq d \leq t+2$ with $(p,t,d) \neq (2, q-1, q)$. Our quantum codes have flexible parameters, and have minimum distances larger than $\frac{q}{2}+1$ when $t > \frac{q}{2}$. Furthermore, it turns out that our constructions generalize and improve some previous results.

[71]  arXiv:1803.06603 [pdf, other]
Title: Energy-aware networked control systems under temporal logic specifications
Subjects: Optimization and Control (math.OC)

In recent years, event and self-triggered control have been proposed as energy-aware control strategies to expand the life-time of battery powered devices in Networked Control Systems (NCSs). In contrast to the previous works in which their control objective is to achieve stability, this paper presents a novel energy-aware control scheme for achieving high level specifications, or more specifically, temporal logic specifications. Inspired by the standard hierarchical strategy that has been proposed in the field of formal control synthesis paradigm, we propose a new abstraction procedure for jointly synthesizing control and communication strategies, such that the communication reduction in NCSs and the satisfaction of the temporal logic specifications are guaranteed. The benefits of the proposal are illustrated through a numerical example.

[72]  arXiv:1803.06609 [pdf, ps, other]
Title: Fundamental group of non-singular locus of Lauricella's $F_C$
Comments: 11 pages, 4 figures
Subjects: Algebraic Geometry (math.AG)

In this paper, we give a set of generators and relations of the fundamental group of the non-singular locus of Lauricella's hypergeometric functions

[73]  arXiv:1803.06610 [pdf, ps, other]
Title: Can You Pave the Plane Nicely with Identical Tiles
Authors: Chuanming Zong
Comments: 10 pages, 16 figures
Subjects: Metric Geometry (math.MG); History and Overview (math.HO)

Every body knows that identical regular triangles or squares can tile the whole plane. Many people know that identical regular hexagons can tile the plane properly as well. In fact, even the bees know and use this fact! Is there any other convex domain which can tile the Euclidean plane? Yes, there is a long list of them! To find the list and to show the completeness of the list is a unique drama in mathematics, which has lasted for more than one century and the completeness of the list has been mistakenly announced not only once! Up to now, the list consists of triangles, quadrilaterals, three types of hexagons, and fifteen types of pentagons. In 2017, Michael Rao announced a computer proof for the completeness of the list. Meanwhile, Qi Yang and Chuanming Zong made a series of unexpected discoveries in multiple tilings in the Euclidean plane. For examples, besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form any two-, three- or four-fold translative tiling in the plane; there are only two types of octagons and one type of decagons which can form five-fold translative tilings.

[74]  arXiv:1803.06620 [pdf, ps, other]
Title: Characterizations of the Logistic and Related Distributions
Comments: 17 pages, Journal of Mathematical Analysis and Applications (2018)
Subjects: Statistics Theory (math.ST); Probability (math.PR)

It is known that few characterization results of the logistic distribution were available before, although it is similar in shape to the normal one whose characteristic properties have been well investigated. Fortunately, in the last decade, several authors have made great progress in this topic. Some interesting characterization results of the logistic distribution have been developed recently. In this paper, we further provide some new results by the distributional equalities in terms of order statistics of the underlying distribution and the random exponential shifts. The characterization of the closely related Pareto type II distribution is also investigated.

[75]  arXiv:1803.06623 [pdf, ps, other]
Title: The invariant subspaces of the shift plus integer multiple of Volterra operator on Hardy spaces
Authors: Qingze Lin
Comments: 8 pages
Subjects: Complex Variables (math.CV)

\v{C}u\v{c}kovi\'{c} and Paudyal recently characterized the lattice of invariant subspaces of the shift plus a complex Volterra operator on the Hilbert space $H^2$ on the unit disk. Motivated by the idea of Ong, in this paper, we give a complete characterization of the lattice of invariant subspaces of the shift operator plus a positive integer multiple of the Volterra operator on Hardy spaces $H^p$, which essentially extends their works to the more general cases when $1\leq p<\infty$.

[76]  arXiv:1803.06627 [pdf, other]
Title: How to sheafify an elliptic quantum group
Comments: These lecture notes are based on Yang's talk at the MATRIX program Geometric R-Matrices: from Geometry to Probability, at the University of Melbourne, Dec.18-22, 2017, and Zhao's talk at Perimeter Institute for Theoretical Physics in January 2018. 12 pages, expository paper, submitted to the MATRIX Annals
Subjects: Representation Theory (math.RT)

We give an introductory survey of the results in arXiv: 1708.01418. We discuss a sheafified elliptic quantum group associated to any symmetric Kac-Moody Lie algebra. The sheafification is obtained by applying the equivariant elliptic cohomological theory to the moduli space of representations of a preprojective algebra. By construction, the elliptic quantum group naturally acts on the equivariant elliptic cohomology of Nakajima quiver varieties. As an application, we obtain a relation between the sheafified elliptic quantum group and the global affine Grassmannian over an elliptic curve.

[77]  arXiv:1803.06631 [pdf, other]
Title: A Deal with the Devil: From Divergent Perturbation Theory to an Exponentially-Convergent Self-Consistent Expansion
Comments: 37 pages, 11 figures
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)

For many nonlinear physical systems, approximate solutions are pursued by conventional perturbation theory in powers of the non-linear terms. Unfortunately, this often produces divergent asymptotic series, collectively dismissed by Abel as "an invention of the devil." An alternative method, the self-consistent expansion, has been introduced by Schwartz and Edwards. Its basic idea is a rescaling of the zeroth-order system around which the solution is expanded, to achieve optimal results. While low-order self-consistent calculations have been remarkably successful in describing the dynamics of non-equilibrium many-body systems (e.g., the Kardar-Parisi-Zhang equation), its convergence properties have not been elucidated before. To address this issue we apply this technique to the canonical partition function of the classical harmonic oscillator with a quartic $gx^{4}$ anharmonicity, for which perturbation theory's divergence is well-known. We explicitly obtain the $N^{\text{th}}$ order self-consistent expansion for the partition function, which is rigorously found to converge exponentially fast in $N$, and uniformly in $g$, for any coupling $g>0$. Comparing the self-consistent expansion with other methods that improve upon perturbation theory (Borel resummation, hyperasymptotics, Pad\'e approximants, and the Lanczos $\tau$-method), it compares favorably with all of them for small $g$ and dominates over them for large $g$. Remarkably, the self-consistent expansion is shown to successfully capture the correct partition function for the double-well potential case, where no perturbative expansion exists. Our treatment is generalized to the case of many oscillators, as well as to a more general nonlinearity of the form $g|x|^{q}$ with $q\ge0$ and complex $g$. These results allow us to treat the Airy function, and to see the fingerprints of Stokes lines in the self-consistent expansion.

[78]  arXiv:1803.06635 [pdf, other]
Title: A stabilized cut discontinuous Galerkin framework: I. Elliptic boundary value and interface problems
Comments: 35 pages, 12 figures, 2 tables
Subjects: Numerical Analysis (math.NA)

We develop a stabilized cut discontinuous Galerkin framework for the numerical solution of el- liptic boundary value and interface problems on complicated domains. The domain of interest is embedded in a structured, unfitted background mesh in R d , so that the boundary or interface can cut through it in an arbitrary fashion. The method is based on an unfitted variant of the classical symmetric interior penalty method using piecewise discontinuous polynomials defined on the back- ground mesh. Instead of the cell agglomeration technique commonly used in previously introduced unfitted discontinuous Galerkin methods, we employ and extend ghost penalty techniques from recently developed continuous cut finite element methods, which allows for a minimal extension of existing fitted discontinuous Galerkin software to handle unfitted geometries. Identifying four abstract assumptions on the ghost penalty, we derive geometrically robust a priori error and con- dition number estimates for the Poisson boundary value problem which hold irrespective of the particular cut configuration. Possible realizations of suitable ghost penalties are discussed. We also demonstrate how the framework can be elegantly applied to discretize high contrast interface problems. The theoretical results are illustrated by a number of numerical experiments for various approximation orders and for two and three-dimensional test problems.

[79]  arXiv:1803.06636 [pdf, ps, other]
Title: Complexity problems in enumerative combinatorics
Authors: Igor Pak
Comments: 30 pages; an expanded version of the ICM 2018 paper (Section 4 added, refs expanded)
Subjects: Combinatorics (math.CO); Computational Complexity (cs.CC); Discrete Mathematics (cs.DM); History and Overview (math.HO); Probability (math.PR)

We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.

[80]  arXiv:1803.06637 [pdf, ps, other]
Title: The nodal set of solutions to some elliptic problems: singular nonlinearities
Subjects: Analysis of PDEs (math.AP)

This paper deals with solutions to the equation \begin{equation*} -\Delta u = \lambda_+ \left(u^+\right)^{q-1} - \lambda_- \left(u^-\right)^{q-1} \quad \text{in $B_1$} \end{equation*} where $\lambda_+,\lambda_- > 0$, $q \in (0,1)$, $B_1=B_1(0)$ is the unit ball in $\mathbb{R}^N$, $N \ge 2$, and $u^+:= \max\{u,0\}$, $u^-:= \max\{-u,0\}$ are the positive and the negative part of $u$, respectively. We extend to this class of \emph{singular} equations the results recently obtained in \cite{SoTe2018} for \emph{sublinear and discontinuous} equations, $1\leq q<2$, namely: (a) the finiteness of the vanishing order at every point and the complete characterization of the order spectrum; (b) a weak non-degeneracy property; (c) regularity of the nodal set of any solution: the nodal set is a locally finite collection of regular codimension one manifolds up to a residual singular set having Hausdorff dimension at most $N-2$ (locally finite when $N=2$). As an intermediate step, we establish the regularity of a class of \emph{not necessarily minimal} solutions.
The proofs are based on a priori bounds, monotonicity formul\ae \ for a $2$-parameter family of Weiss-type functionals, blow-up arguments, and the classification of homogenous solutions.

[81]  arXiv:1803.06639 [pdf, ps, other]
Title: Stability of Energy Stable Flux Reconstruction for the Diffusion Problem using the Interior Penalty and Bassi and Rebay II Numerical Fluxes
Subjects: Numerical Analysis (math.NA)

Recovering some prominent high-order approaches such as the discontinuous Galerkin (DG) or the spectral difference (SD) methods, the flux reconstruction (FR) approach has been adopted by many individuals in the research community and is now commonly used to solve problems on unstructured grids over complex geometries. This approach relies on the use of correction functions to obtain a differential form for the discrete problem. A class of correction functions, named energy stable flux reconstruction (ESFR) functions, has been proven stable for the linear advection problem. This proof has then been extended for the diffusion equation using the local discontinuous Galerkin (LDG) scheme to compute the numerical fluxes. Although the LDG scheme is commonly used, many prefer the interior penalty (IP), as well as the Bassi and Rebay II (BR2) schemes. Similarly to the LDG proof, this article provides a stability analysis for the IP and the BR2 numerical fluxes. In fact, we obtain a theoretical condition on the penalty term to ensure stability. This result is then verified through numerical simulations. To complete this study, a von Neumann analysis is conducted to provide a combination of parameters producing the maximal time step while converging at the correct order. All things considered, this article has for purpose to provide the community with a stability condition while using the IP and the BR2 schemes.

[82]  arXiv:1803.06646 [pdf, ps, other]
Title: Toric geometry of $G_2$-manifolds
Comments: 33 pages
Subjects: Differential Geometry (math.DG); High Energy Physics - Theory (hep-th)

We consider $G_2$-manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of $T^3$-actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three- and four-form, we derive a Gibbons-Hawking type ansatz giving the geometry on an open dense set in terms a symmetric $3\times 3$-matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to $G_2$. We prove that the multi-moment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples.

[83]  arXiv:1803.06651 [pdf, ps, other]
Title: Limits in dagger categories
Subjects: Category Theory (math.CT)

We develop a notion of limit for dagger categories, that we show is suitable in the following ways: it subsumes special cases known from the literature; dagger limits are unique up to unitary isomorphism; a wide class of dagger limits can be built from a small selection of them; dagger limits of a fixed shape can be phrased as dagger adjoints to a diagonal functor; dagger limits can be built from ordinary limits in the presence of polar decomposition; dagger limits commute with dagger colimits in many cases.

[84]  arXiv:1803.06659 [pdf, ps, other]
Title: New characterizations of operator monotone functions
Comments: Linear Algebra and Its Applications, 2018
Subjects: Functional Analysis (math.FA); Operator Algebras (math.OA)

If $\sigma$ is a symmetric mean and $f$ is an operator monotone function on $[0, \infty)$, then $$f(2(A^{-1}+B^{-1})^{-1})\le f(A\sigma B)\le f((A+B)/2).$$ Conversely, Ando and Hiai showed that if $f$ is a function that satisfies either one of these inequalities for all positive operators $A$ and $B$ and a symmetric mean different than the arithmetic and the harmonic mean, then the function is operator monotone.
In this paper, we show that the arithmetic and the harmonic means can be replaced by the geometric mean to obtain similar characterizations. Moreover, we give characterizations of operator monotone functions using self-adjoint means and general means subject to a constraint due to Kubo and Ando.

[85]  arXiv:1803.06661 [pdf, ps, other]
Title: New Approach To Fixed Point Theorems
Subjects: Functional Analysis (math.FA)

In this article, we discuss a new version of metric fixed point theory especially of Banach Contraction Principle, Ran-Reurings Theorem and others.

[86]  arXiv:1803.06662 [pdf, ps, other]
Title: Fixed Point Theorems In Ordered Partial b-Metric Spaces With New Setting
Subjects: Functional Analysis (math.FA)

The main purpose of this paper is to find the fixed point in such cases where existing literature remain silent. In this paper we introduce partial completeness, a new type of contraction and many other definitions. Using this approach the existence of fixed point can be proved in incomplete metric spaces with non-contraction map on it. We have reported an example in support our result.

[87]  arXiv:1803.06664 [pdf, ps, other]
Title: An Introduction to the Moebius Function
Authors: Chris Godsil
Comments: This is a slightly revised version of an old set of notes that have been available on my webpage for some time. The changes are not substantial. I've eliminated some old typos and introduced some ne ones (doubtless), and added one new section
Subjects: Combinatorics (math.CO)

This is an introduction to the M\"obius function of a poset. The chief novelty is in the exposition. We show how order-preserving maps from one poset to another can be used to relate their M\"obius functions. We derive the basic results on the M\"obius function, applying them in particular to geometric lattices.

[88]  arXiv:1803.06665 [pdf, ps, other]
Title: Les exposants de Liapounoff du flot de Teichmüller
Comments: in French
Journal-ref: Ast\'erisque, Soci\'et\'e Math\'ematique de France, 2014, 361 (1060), pp.43-75
Subjects: Dynamical Systems (math.DS)

The purpose of the paper under review is to explain the main ideas and the main ingredients of the involved and delicate work of A. Eskin, M. Kontsevich and A. Zorich concerning the sum of the positive Lyapunov exponents of the so-called Kontsevich- Zorich cocycle acting on the first cohomology spaces of translation surfaces.

[89]  arXiv:1803.06668 [pdf, ps, other]
Title: Local derivations on Solvable Lie algebras
Subjects: Rings and Algebras (math.RA)

We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation.

[90]  arXiv:1803.06670 [pdf, ps, other]
Title: Stone-type representations and dualities for varieties of bisemilattices
Authors: Antonio Ledda
Subjects: Logic (math.LO)

In this article we will focus our attention on the variety of distributive bisemilattices and some linguistic expansions thereof: bounded, De Morgan, and involutive bisemilattices. After extending Balbes' representation theorem to bounded, De Morgan, and involutive bisemilattices, we make use of Hartonas-Dunn duality and introduce the categories of 2spaces and 2spaces$^{\star}$. The categories of 2spaces and 2spaces$^{\star}$ will play with respect to the categories of distributive bisemilattices and De Morgan bisemilattices, respectively, a role analogous to the category of Stone spaces with respect to the category of Boolean algebras. Actually, the aim of this work is to show that these categories are, in fact, dually equivalent.

[91]  arXiv:1803.06671 [pdf, ps, other]
Title: On some properties of PBZ*-lattices
Journal-ref: International Journal of Theoretical Physics, December 2017, Volume 56, Issue 12, pp 3895 3911
Subjects: Logic (math.LO)

We continue the algebraic investigation of PBZ*-lattices, a notion introduced in [12] in order to obtain insights into the structure of certain algebras of effects of a Hilbert space, lattice-ordered under the spectral ordering.

[92]  arXiv:1803.06676 [pdf, other]
Title: Scenario-Based Uncertainty Set for Two-Stage Robust Energy and Reserve Scheduling: A Data-Driven Approach
Subjects: Optimization and Control (math.OC)

Two-stage robust unit commitment (RUC) models have been widely used for day-ahead energy and reserve scheduling under high renewable integration. The current state of the art relies on budget-constrained polyhedral uncertainty sets to control the conservativeness of the solutions. The associated lack of interpretability and parameter specification procedures, as well as the high computational burden exhibited by available exact solution techniques call for new approaches. In this work, we use an alternative scenario-based framework whereby uncertain renewable generation is characterized by a polyhedral uncertainty set relying on the direct specification of its vertexes. Moreover, we present a simple, yet efficient, adaptive data-driven procedure to dynamically update the uncertainty set vertexes with observed daily renewable-output profiles. Within this setting, the proposed data-driven RUC ensures protection against the convex hull of realistic scenarios empirically capturing the complex and time-varying intra-day spatial and temporal interdependencies among units. The resulting counterpart features advantageous properties from a computational perspective and can be effectively solved by the column-and-constraint generation algorithm until $\epsilon$-global optimality. Out-of-sample experiments reveal that the proposed approach is capable of producing efficient solutions in terms of cost and robustness while keeping the model tractable and scalable.

[93]  arXiv:1803.06677 [pdf, ps, other]
Title: A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications
Authors: Dmitry Ostrovsky
Comments: 64 pages
Subjects: Probability (math.PR)

Selberg and Morris integral probability distributions are long conjectured to be distributions of the total mass of the Bacry-Muzy Gaussian Multiplicative Chaos measures with non-random logarithmic potentials on the unit interval and circle, respectively. The construction and properties of these distributions are reviewed from three perspectives: analytic based on several representations of the Mellin transform, asymptotic based on low intermittency expansions, and probabilistic based on the theory of Barnes beta probability distributions. In particular, positive and negative integer moments, infinite factorizations and involution invariance of the Mellin transform, analytic and probabilistic proofs of infinite divisibility of the logarithm, factorizations into products of Barnes beta distributions, and Stieltjes moment problems of these distributions are presented in detail. Applications are given in the form of conjectured mod-Gaussian limit theorems, laws of derivative martingales, distribution of extrema of $1/f$ noises, and calculations of inverse participation ratios in the Fyodorov-Bouchaud model.

[94]  arXiv:1803.06679 [pdf, ps, other]
Title: Kirchberg--Wassermann exactness vs exactness: reduction to the unimodular totally disconnected case
Comments: 7 pages
Subjects: Group Theory (math.GR); Operator Algebras (math.OA)

We show that in order to prove that every second countable locally compact groups with exact reduced group C*-algebra is exact in the dynamical sense (i.e. KW-exact) it suffices to show this for totally disconnected groups.

[95]  arXiv:1803.06681 [pdf, ps, other]
Title: Local-in-time well-posedness for Compressible MHD boundary layer
Comments: 29pp
Subjects: Analysis of PDEs (math.AP)

In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane, with no-slip condition for velocity field, perfect conducting condition for magnetic field and Dirichlet boundary condition for temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer that is described by a Prandtl-type system. By applying a coordinate transformation in terms of stream function as motivated by the recent work \cite{liu2016mhdboundarylayer} on the incompressible MHD system, under the non-degeneracy condition on the tangential magnetic field, we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.

[96]  arXiv:1803.06683 [pdf, ps, other]
Title: Conformal slant submersions in contact geometry
Comments: 17 pages
Subjects: Differential Geometry (math.DG)

Akyol M.A. [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistic, 46(2), (2017), 177-192.] defined and studied conformal anti-invariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $\xi$ is a vertical vector field) from almost contact metric manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions and conformal anti-invariant submersions. More precisely, we mention lots of examples and obtain the geometries of the leaves of $\ker\pi_{*}$ and $(\ker\pi_{*})^\perp,$ including the integrability of the distributions, the geometry of foliations, some conditions related to totally geodesicness and harmonicty of the submersions. Finally, we consider a decomposition theorem on total space of the new submersion.

[97]  arXiv:1803.06684 [pdf, ps, other]
Title: The decomposition formula for Verlinde Sums
Comments: 31 pages
Subjects: Symplectic Geometry (math.SG)

We prove a decomposition formula for Verlinde sums (rational trigonometric sums), as a discrete counterpart to the Boysal-Vergne decomposition formula for Bernoulli series. Motivated by applications to fixed point formulas in Hamiltonian geometry, we develop differential form valued version of Bernoulli series and Verlinde sums, and extend the decomposition formula to this wider context.

[98]  arXiv:1803.06685 [pdf, other]
Title: Shifted Poisson structures on differentiable stacks
Comments: 49 pages
Subjects: Differential Geometry (math.DG)

The purpose of this paper is to investigate shifted $(+1)$ Poisson structures in context of differential geometry. The relevant notion is shifted $(+1)$ Poisson structures on differentiable stacks. More precisely, we develop the notion of Morita equivalence of quasi-Poisson groupoids. Thus isomorphism classes of $(+1)$ Poisson stack correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following programs of independent interests:
(1) We introduce a $\mathbb Z$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under Morita equivalence of Lie groupoids, thus can be considered as polyvector fields on the corresponding differentiable stack ${\mathfrak X}$. It turns out that shifted $(+1)$ Poisson structures on ${\mathfrak X}$ correspond exactly to elements of the Maurer-Cartan moduli set of the corresponding dgla.
(2) We introduce the notion of tangent complex $T_{\mathfrak X}$ and cotangent complex $L_{\mathfrak X}$ of a differentiable stack ${\mathfrak X}$ in terms of any Lie groupoid $\Gamma{\rightrightarrows} M$ representing ${\mathfrak X}$. They correspond to homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\rightarrow TM$ and $T^\vee M\rightarrow A^\vee[-1]$, respectively. We prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak X}$, defines a morphism ${L_{{\mathfrak X}}}[1]\to {T_{{\mathfrak X}}}$. We rely on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids.

[99]  arXiv:1803.06688 [pdf, ps, other]
Title: Height estimates for mean curvature graphs in $\mathrm{Nil}_3$ and $\widetilde{PSL}_2(\mathbb{R})$
Authors: Antonio Bueno
Subjects: Differential Geometry (math.DG)

In this paper we obtain height estimates for compact, constant mean curvature vertical graphs in the homogeneous spaces $\mathrm{Nil}_3$ and $\widetilde{PSL}_2(\mathbb{R})$. As a straightforward consequence, we announce a structure-type result for proper graphs defined on relatively compact domains.

[100]  arXiv:1803.06691 [pdf, ps, other]
Title: On the Absence of a Normal Nonabelian Sylow Subgroup
Comments: 6 pages, 1 figure
Subjects: Group Theory (math.GR)

Let $G$ be a finite solvable group. We show that $G$ does not have a normal nonabelian Sylow $p$-subgroup when its prime character degree graph $\Delta(G)$ satisfies a technical hypothesis.

[101]  arXiv:1803.06692 [pdf, ps, other]
Title: Radial Schur multipliers on some generalisations of trees
Authors: Ignacio Vergara
Comments: 43 pages, 2 figures (created using GeoGebra)
Subjects: Operator Algebras (math.OA)

We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs and provide a sufficient condition for a function to define a radial Schur multiplier on a finite dimensional CAT(0) cube complex.

[102]  arXiv:1803.06694 [pdf, ps, other]
Title: Asymptotic properties of integrals of quotients, when the numerator oscillates and denominator degenerates
Authors: Sergei Kuksin
Subjects: Mathematical Physics (math-ph)

We study asymptotical expansion as $\nu\to0$ for integrals over ${ \mathbb{R} }^{2d}=\{(x,y)\}$ of quotients of the form $F(x,y) \cos(\lambda x\cdot y) \big/ \big( (x\cdot y)^2+\nu^2\big)$, where $\lambda\ge 0$ and $F$ decays at infinity sufficiently fast. Integrals of this kind appear in the theory of wave turbulence.

[103]  arXiv:1803.06697 [pdf, ps, other]
Title: Higher-order estimates for collapsing Calabi-Yau metrics
Comments: 52 pages
Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Complex Variables (math.CV)

We prove a uniform C^alpha estimate for collapsing Calabi-Yau metrics on the total space of a proper holomorphic submersion over the unit ball in C^m. The usual methods of Calabi, Evans-Krylov, and Caffarelli do not apply to this setting because the background geometry degenerates. We instead rely on blowup arguments and on linear and nonlinear Liouville theorems on cylinders. In particular, as an intermediate step, we use such arguments to prove sharp new Schauder estimates for the Laplacian on cylinders. If the fibers of the submersion are pairwise biholomorphic, our method yields a uniform C^infinity estimate. We then apply these local results to the case of collapsing Calabi-Yau metrics on compact Calabi-Yau manifolds. In this global setting, the C^0 estimate required as a hypothesis in our new local C^alpha and C^infinity estimates is known to hold thanks to earlier work of the second-named author.

[104]  arXiv:1803.06700 [pdf, ps, other]
Title: Categoricity of Shimura Varieties
Comments: 20 pages. Comments welcome
Subjects: Logic (math.LO); Algebraic Geometry (math.AG); Number Theory (math.NT)

We propose a model-theoretic structure for Shimura varieties and give necessary and sufficient conditions to obtain categoricity.

[105]  arXiv:1803.06701 [pdf, ps, other]
Title: Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling
Comments: Submitted version in Discrete Contin. Dyn. Syst. Ser. B
Subjects: Analysis of PDEs (math.AP)

Hysteresis is an important issue in modeling piezoelectric materials, for example, in applications to energy harvesting, where hysteresis losses may influence the efficiency of the process. The main problem in numerical simulations is the inversion of the underlying hysteresis operator. Moreover, hysteresis dissipation is accompanied with heat production, which in turn increases the temperature of the device and may change its physical characteristics. More accurate models therefore have to take the temperature dependence into account for a correct energy balance. We prove here that the classical Preisach operator with a fairly general parameter-dependence admits a Lipschitz continuous inverse in the space of right-continuous regulated functions, propose a time-discrete and memory-discrete inversion algorithm, and show that higher regularity of the inputs leads to a higher regularity of the output of the inverse.

[106]  arXiv:1803.06702 [pdf, ps, other]
Title: On Infinite Divisibility of the Distribution of Some Inverse Subordinators
Comments: 11 pages; submitted for publication
Subjects: Probability (math.PR)

We consider the infinite divisibility of the distributions of some well known inverse subordinators. Using a tail probability bound, we establish that the distributions of many of the inverse subordinators used in the literature are not infinitely divisible. We further show that the distribution of a renewal process time-changed by an inverse stable subordinator is not infinitely divisible, which in particular implies that the distribution of fractional Poisson process is not infinitely divisible.

[107]  arXiv:1803.06705 [pdf, other]
Title: Hierarchical Predictive Control Algorithms for Optimal Design and Operation of Microgrids
Comments: To appear in "Power Systems Computation Conference", Dublin, Ireland
Subjects: Optimization and Control (math.OC); Systems and Control (cs.SY)

In recent years, microgrids, i.e., disconnected distribution systems, have received increasing interest from power system utilities to support the economic and resiliency posture of their systems. The economics of long distance transmission lines prevent many remote communities from connecting to bulk transmission systems and these communities rely on off-grid microgrid technology. Furthermore, communities that are connected to the bulk transmission system are investigating microgrid technologies that will support their ability to disconnect and operate independently during extreme events. In each of these cases, it is important to develop methodologies that support the capability to design and operate microgrids in the absence of transmission over long periods of time. Unfortunately, such planning problems tend to be computationally difficult to solve and those that are straightforward to solve often lack the modeling fidelity that inspires confidence in the results. To address these issues, we first develop a high fidelity model for design and operations of a microgrid that include component efficiencies, component operating limits, battery modeling, unit commitment, capacity expansion, and power flow physics; the resulting model is a mixed-integer quadratically-constrained quadratic program (MIQCQP). We then develop an iterative algorithm, referred to as the Model Predictive Control (MPC) algorithm, that allows us to solve the resulting MIQCQP. We show, through extensive computational experiments, that the MPC-based method can scale to problems that have a very long planning horizon and provide high quality solutions that lie within 5\% of optimal.

[108]  arXiv:1803.06706 [pdf, ps, other]
Title: Descent distribution on Catalan words avoiding a pattern of length at most three
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)

Catalan words are particular growth-restricted words over the set of non-negative integers, and they represent still another combinatorial class counted by the Catalan numbers. We study the distribution of descents on the sets of Catalan words avoiding a pattern of length at most three: for each such a pattern $p$ we provide a bivariate generating function where the coefficient of $x^ny^k$ in its series expansion is the number of length $n$ Catalan words with $k$ descents and avoiding $p$. As a byproduct, we enumerate the set of Catalan words avoiding $p$, and we provide the popularity of descents on this set. Some of the obtained enumerating sequences are not yet recorded in the On-line Encyclopedia of Integer Sequences.

[109]  arXiv:1803.06710 [pdf, other]
Title: Almost all string graphs are intersection graphs of plane convex sets
Comments: This is the full version of a paper appearing in the proceedings of SoCG 2018
Subjects: Combinatorics (math.CO); Computational Geometry (cs.CG)

A {\em string graph} is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of {\em almost all} string graphs on $n$ vertices can be partitioned into {\em five} cliques such that some pair of them is not connected by any edge ($n\rightarrow\infty$). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that {\em almost all} string graphs on $n$ vertices are intersection graphs of plane convex sets.

[110]  arXiv:1803.06712 [pdf, ps, other]
Title: Haar-$\mathcal I$ sets: looking at small sets in Polish groups through compact glasses
Comments: 44 pages
Subjects: General Topology (math.GN); Group Theory (math.GR)

Generalizing Christensen's notion of a Haar-null set and Darji's notion of a Haar-meager set, we introduce and study the notion of a Haar-$\mathcal I$ set in a Polish group. Here $\mathcal I$ is an ideal of subsets of some compact metrizable space $K$. A Borel subset $B\subset X$ of a Polish group $X$ is called Haar-$\mathcal I$ if there exists a continuous map $f:K\to X$ such that $f^{-1}(B+x)\in\mathcal I$ for all $x\in X$. Moreover, $B$ is generically Haar-$\mathcal I$ if the set of witness functions $\{f\in C(K,X):\forall x\in X\;\;f^{-1}(B+x)\in\mathcal I\}$ is comeager in the function space $C(K,X)$. We study (generically) Haar-$\mathcal I$ sets in Polish groups for many concrete and abstract ideals $\mathcal I$, and construct the corresponding distinguishing examples. Also we establish various Steinhaus properties of the families of (generically) Haar-$\mathcal I$ sets in Polish groups for various ideals $\mathcal I$.

[111]  arXiv:1803.06713 [pdf, other]
Title: Four-manifolds with shadow-complexity one
Comments: 75 pages, 77 figures
Subjects: Geometric Topology (math.GT)

We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev shadows. This complexity roughly measures how complicated the 2-skeleton of the 4-manifold is. We characterise all the closed oriented 4-manifolds that have complexity $\leq 1$. These are precisely the 4-manifolds that are generated by a certain set of 20 blocks, that is some basic 4-manifolds with boundary consisting of copies of $S^2 \times S^1$, plus connected sums with some copies of $\mathbb {CP}^2$ with either orientation.

[112]  arXiv:1803.06715 [pdf, ps, other]
Title: Restricting homology to hypersurfaces
Comments: 17 pages. To appear in "Geometric and topological aspects of the representation theory of finite groups", to be published by Springer in the series titled "Proceedings in Mathematics"
Subjects: Commutative Algebra (math.AC); Group Theory (math.GR)

This paper concerns the homological properties of a module $M$ over a commutative noetherian ring $R$ relative to a presentation $R\cong P/I$, where $P$ is local ring. It is proved that the Betti sequence of $M$ with respect to $P/(f)$ for a regular element $f$ in $I$ depends only on the class of $f$ in $I/\mathfrak{n} I$, where $\mathfrak{n}$ is the maximal ideal of $P$. Applications to the theory of supports sets in local algebra and in the modular representation theory of elementary abelian groups are presented.

[113]  arXiv:1803.06716 [pdf, ps, other]
Title: High Dimensional Linear Regression using Lattice Basis Reduction
Subjects: Statistics Theory (math.ST); Probability (math.PR); Machine Learning (stat.ML)

We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector $\beta^*$ from $n$ noisy linear observations $Y=X\beta^*+W \in \mathbb{R}^n$, for known $X \in \mathbb{R}^{n \times p}$ and unknown $W \in \mathbb{R}^n$. Unlike most of the literature on this model we make no sparsity assumption on $\beta^*$. Instead we adopt a regularization based on assuming that the underlying vectors $\beta^*$ have rational entries with the same denominator $Q \in \mathbb{Z}_{>0}$. We call this $Q$-rationality assumption.
We propose a new polynomial-time algorithm for this task which is based on the seminal Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction algorithm. We establish that under the $Q$-rationality assumption, our algorithm recovers exactly the vector $\beta^*$ for a large class of distributions for the iid entries of $X$ and non-zero noise $W$. We prove that it is successful under small noise, even when the learner has access to only one observation ($n=1$). Furthermore, we prove that in the case of the Gaussian white noise for $W$, $n=o\left(p/\log p\right)$ and $Q$ sufficiently large, our algorithm tolerates a nearly optimal information-theoretic level of the noise.

[114]  arXiv:1803.06717 [pdf, other]
Title: High frequency limits for invariant Ruelle densities
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Dynamical Systems (math.DS); Spectral Theory (math.SP)

We establish an equidistribution result for Ruelle resonant states on compact locally symmetric spaces of rank one. More precisely, we prove that among the first band Ruelle resonances there is a density one subsequence such that the respective products of resonant and co-resonant states converge weakly to the Liouville measure. We prove this result by establishing an explicit quantum-classical correspondence between eigenspaces of the scalar Laplacian and the resonant states of the first band of Ruelle resonances which also leads to a new description of Patterson-Sullivan distributions.

[115]  arXiv:1803.06719 [pdf, ps, other]
Title: Summability in a monomial for some classes of singularly perturbed partial differential equation
Subjects: Classical Analysis and ODEs (math.CA)

The aim of this paper is to complete the study of asymptotic expansions and summability in a monomial in any number of variables. In particular we characterize these expansions in terms of bounded derivatives and we develop tauberian theorems for the summability processes involved. Furthermore, we develop and apply the Borel-Laplace analysis in this framework to prove the monomial summability of solutions of a specific class of singularly perturbed PDEs.

[116]  arXiv:1803.06728 [pdf, ps, other]
Title: A non-intersecting random walk on the Manhattan lattice and SLE_6
Authors: Tom Kennedy
Comments: 14 figures
Subjects: Probability (math.PR); Mathematical Physics (math-ph)

We consider a random walk on the Manhattan lattice. The walker must follow the orientations of the bonds in this lattice, and the walker is not allowed to visit a site more than once. When both possible steps are allowed, the walker chooses between them with equal probability. The walks generated by this model are known to be related to interfaces in a certain percolation model. So it is natural to conjecture that the scaling limit is SLE$_6$. We test this conjecture with Monte Carlo simulations of the random walk model and find strong support for the conjecture.

[117]  arXiv:1803.06733 [pdf, ps, other]
Title: Combinatorial bases of principal subspaces of modules for twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$
Subjects: Quantum Algebra (math.QA); Representation Theory (math.RT)

We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $k\Lambda_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.

[118]  arXiv:1803.06739 [pdf, ps, other]
Title: A Construction of the Stable Web
Subjects: Probability (math.PR)

We provide a process on the space of coalescing cadlag stable paths and show convergence in the appropriate topology for coalescing stable random walks on the integer lattice.

[119]  arXiv:1803.06742 [pdf, ps, other]
Title: Inventory Control with Modulated Demand and a Partially Observed Modulation Process
Subjects: Optimization and Control (math.OC)

We consider a periodic review inventory control problem having an underlying modulation process that affects demand and that is partially observed by the uncensored demand process and a novel additional observation data (AOD) process. Letting $K$ be the reorder cost, we present a condition, A1, which is a generalization of the Veinott attainability assumption, that guarantees the existence of an optimal myopic base stock policy if $K = 0$ and the existence of an optimal $(s, S)$ policy if $K > 0$, where both policies depend on the belief function of the modulation process. Assuming A1 holds, we show that (i) when $K = 0$, the value of the optimal base stock level is constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities and (ii) when $K > 0$, the values of $s$ and $S$ and upper and lower bounds on these values are constant within regions of the belief space and that these regions can be described by a finite set of linear inequalities. Computational procedures for $K \geq 0$ are outlined, and results for the $K = 0$ case are presented when A1 does not hold. Special cases of this inventory control problem include problems considered in the Markov-modulated demand and Bayesian updating literatures.

[120]  arXiv:1803.06746 [pdf, other]
Title: Experimental Verification of Rate Flexibility and Probabilistic Shaping by 4D Signaling
Comments: Presented at OFC'18, San Diego, CA, USA
Subjects: Information Theory (cs.IT)

The rate flexibility and probabilistic shaping gain of $4$-dimensional signaling is experimentally tested for short-reach, unrepeated transmission. A rate granularity of 0.5 bits/QAM symbol is achieved with a distribution matcher based on a simple look-up table.

[121]  arXiv:1803.06750 [pdf, ps, other]
Title: Determining both the source of a wave and its speed in a medium from boundary measurements
Subjects: Analysis of PDEs (math.AP)

We study the inverse problem of determining both the source of a wave and its speed inside a medium from measurements of the solution of the wave equation on the boundary. This problem arises in photoacoustic and thermoacoustic tomography, and has important applications in medical imaging. We prove that if $c^{-2}$ is harmonic in $\omega \subset R^3$ and identically 1 on $\omega^c$, where $\omega$ is a simply connected region, then a non-trapping wave speed $c$ can be uniquely determined from the solution of the wave equation on boundary of $\Omega \supset \supset \omega$ without the knowledge of the source. We also show that if the wave speed $c$ is known and only assumed to be bounded then, under mild assumptions on the set of discontinuous points of $c$, the source of the wave can be uniquely determined from boundary measurements.

[122]  arXiv:1803.06754 [pdf, ps, other]
Title: Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan-Lusztig algorithm
Comments: 63 pages
Subjects: Representation Theory (math.RT); Quantum Algebra (math.QA); Rings and Algebras (math.RA)

We establish ring isomorphisms between quantum Grothendieck rings of certain remarkable monoidal subcategories of finite-dimensional representations of quantum affine algebras of types $A_{2n-1}^{(1)}$ and $B_n^{(1)}$. Our proof relies in part on the corresponding quantum cluster algebra structures. Moreover, we prove that our isomorphisms are specialized at $t = 1$ to the isomorphisms of (classical) Grothendieck rings obtained recently by Kashiwara, Kim and Oh by other methods. As a consequence, we prove a conjecture formulated by the first author in 2002 : the multiplicities of simple modules in standard modules in the categories above for type $B_n^{(1)}$ are given by the specialization of certain analogues of Kazhdan-Lusztig polynomials and the coefficients of these polynomials are positive.

[123]  arXiv:1803.06755 [pdf, ps, other]
Title: Witt groups of abelian categories and perverse sheaves
Authors: Jörg Schürmann (Universität Münster), Jon Woolf (University of Liverpool)
Comments: 35 pages
Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)

In this paper we study the Witt groups of symmetric and anti-symmetric forms on perverse sheaves on a finite-dimensional topologically stratified space with even dimensional strata. We show that the Witt group has a canonical decomposition as a direct sum of the Witt groups of shifted local systems on strata. We compare this with another `splitting decomposition' for Witt classes of perverse sheaves obtained inductively from our main new tool, a `splitting relation' which is a generalisation of isotropic reduction.
The Witt groups we study are identified with the (non-trivial) Balmer-Witt groups of the constructible derived category of sheaves on the stratified space, and also with the corresponding cobordism groups defined by Youssin.
Our methods are primarily algebraic and apply more widely. The general context in which we work is that of a triangulated category with duality, equipped with a self-dual t-structure with noetherian heart, glued from self-dual t-structures on a thick subcategory and its quotient.

[124]  arXiv:1803.06757 [pdf, ps, other]
Title: On stabilization of solutions of higher order evolution inequalities
Subjects: Analysis of PDEs (math.AP)

We obtain sharp conditions guaranteeing that every non-negative weak solution of the inequality $$ \sum_{|\alpha| = m}
\partial^\alpha
a_\alpha (x, t, u)
-
u_t
\ge
f (x, t) g (u)
\quad
\mbox{in} {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty),
\quad
m,n \ge 1, $$ stabilizes to zero as $t \to \infty$. These conditions generalize the well-known Keller-Osserman condition on the grows of the function $g$ at infinity.

[125]  arXiv:1803.06760 [pdf, other]
Title: A Machine Learning Approach for Power Allocation in HetNets Considering QoS
Comments: 7 pages, 7 figures, IEEE ICC'18
Subjects: Information Theory (cs.IT)

There is an increase in usage of smaller cells or femtocells to improve performance and coverage of next-generation heterogeneous wireless networks (HetNets). However, the interference caused by femtocells to neighboring cells is a limiting performance factor in dense HetNets. This interference is being managed via distributed resource allocation methods. However, as the density of the network increases so does the complexity of such resource allocation methods. Yet, unplanned deployment of femtocells requires an adaptable and self-organizing algorithm to make HetNets viable. As such, we propose to use a machine learning approach based on Q-learning to solve the resource allocation problem in such complex networks. By defining each base station as an agent, a cellular network is modelled as a multi-agent network. Subsequently, cooperative Q-learning can be applied as an efficient approach to manage the resources of a multi-agent network. Furthermore, the proposed approach considers the quality of service (QoS) for each user and fairness in the network. In comparison with prior work, the proposed approach can bring more than a four-fold increase in the number of supported femtocells while using cooperative Q-learning to reduce resource allocation overhead.

[126]  arXiv:1803.06765 [pdf, ps, other]
Title: Sparse Regularization via Convex Analysis
Authors: Ivan Selesnick
Journal-ref: IEEE Transactions on Signal Processing, vol. 65, no. 17, pp. 4481-4494, 2017
Subjects: Optimization and Control (math.OC)

Sparse approximate solutions to linear equations are classically obtained via L1 norm regularized least squares, but this method often underestimates the true solution. As an alternative to the L1 norm, this paper proposes a class of non-convex penalty functions that maintain the convexity of the least squares cost function to be minimized, and avoids the systematic underestimation characteristic of L1 norm regularization. The proposed penalty function is a multivariate generalization of the minimax-concave (MC) penalty. It is defined in terms of a new multivariate generalization of the Huber function, which in turn is defined via infimal convolution. The proposed sparse-regularized least squares cost function can be minimized by proximal algorithms comprising simple computations.

[127]  arXiv:1803.06768 [pdf, ps, other]
Title: Gallai's path decomposition conjecture for triangle-free planar graphs
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)

A path decomposition of a graph $G$ is a collection of edge-disjoint paths of $G$ that covers the edge set of $G$. Gallai (1968) conjectured that every connected graph on $n$ vertices admits a path decomposition of cardinality at most $\lfloor (n+1)/2\rfloor$. Gallai's Conjecture has been verified for many classes of graphs. In particular, Lov\'asz (1968) verified this conjecture for graphs with at most one vertex with even degree, and Pyber (1996) verified it for graphs in which every cycle contains a vertex with odd degree. Recently, Bonamy and Perrett (2016) verified Gallai's Conjecture for graphs with maximum degree at most $5$, and Botler et al. (2017) verified it for graphs with treewidth at most $3$. In this paper, we verify Gallai's Conjecture for triangle-free planar graphs.

[128]  arXiv:1803.06774 [pdf, ps, other]
Title: Toda type equations over multi-dimensional lattices
Comments: 13 pages
Subjects: Mathematical Physics (math-ph); Commutative Algebra (math.AC); Exactly Solvable and Integrable Systems (nlin.SI)

We introduce a class of recursions defined over the $d$-dimensional integer lattice. The discrete equations we study are interpreted as higher dimensional extensions to the discrete Toda lattice equation. We shall prove that the equations satisfy the coprimeness property, which is one of integrability detectors analogous to the singularity confinement test. While the degree of their iterates grows exponentially, they exhibit pseudo-integrable nature in terms of the coprimeness property. We also prove that the equations can be expressed as mutations of a seed in the sense of the Laurent phenomenon algebra.

[129]  arXiv:1803.06778 [pdf, ps, other]
Title: Weighted composition operator on quaternionic Fock space
Comments: 39 pages
Subjects: Functional Analysis (math.FA)

In this paper, we study the weighted composition operator on the Fock space $\mf$ of slice regular functions. First, we characterize the boundedness and compactness of the weighted composition operator. Subsequently, we describe all the isometric composition operators. Finally, we introduce a kind of (right)-anti-complex-linear weighted composition operator on $\mf$ and obtain some concrete forms such that this (right)-anti-linear weighted composition operator is a (right)-conjugation. Specially, we present equivalent conditions ensuring weighted composition operators which are conjugate $\mathcal{C}_{a,b,c}-$commuting or complex $\mathcal{C}_{a,b,c}-$ symmetric on $\mf$, which generalized the classical results on $\mathcal{F}^2(\mathbb{C}).$ At last part of the paper, we exhibit the closed expression for the kernel function of $\mf.$

[130]  arXiv:1803.06786 [pdf, ps, other]
Title: The Optimal Compression Rate of Variable-to-Fixed Length Source Coding with a Non-Vanishing Excess-Distortion Probability
Comments: 10 pages
Subjects: Information Theory (cs.IT)

We consider the variable-to-fixed length lossy source coding (VFSC) problem. The optimal compression rate of the average length of variable-to-fixed source coding, allowing a non-vanishing probability of excess-distortion $\varepsilon$, is shown to be equal to $(1-\varepsilon)R(D)$, where $R(D)$ is the rate-distortion function of the source. In comparison to the related results of Koga and Yamamoto as well as Kostina, Polyanskiy, and Verd\'{u} for fixed-to-variable length source coding, our results demonstrate an interesting feature that variable-to-fixed length source coding has the same first-order compression rate as fixed-to-variable length source coding.

[131]  arXiv:1803.06788 [pdf, other]
Title: The Cohomology for Wu Characteristics
Authors: Oliver Knill
Comments: 40 pages, 12 figures
Subjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Algebraic Topology (math.AT)

While Euler characteristic X(G)=sum_x w(x) super counts simplices, Wu characteristics w_k(G) = sum_(x_1,x_2,...,x_k) w(x_1)...w(x_k) super counts simultaneously pairwise interacting k-tuples of simplices in a finite abstract simplicial complex G. More general is the k-intersection number w_k(G_1,...G_k), where x_i in G_i. We define interaction cohomology H^p(G_1,...,G_k) compatible with w_k and invariant under Barycentric subdivison. It allows to distinguish spaces which simplicial cohomology can not: it can identify algebraically the Moebius strip and the cylinder for example. The cohomology satisfies the Kuenneth formula: the Poincare polynomials p_k(t) are ring homomorphisms from the strong ring to the ring of polynomials in t. The Dirac operator D=d+d^* defines the block diagonal Hodge Laplacian L=D^2 which leads to the generalized Hodge correspondence b_p(G)=dim(H^p_k(G)) = dim(ker(L_p)) and Euler-Poincare w_k(G)=sum_p (-1)^p dim(H^p_k(G)) for Wu characteristic. Also, like for traditional simplicial cohomology, isospectral Lax deformation D' = [B(D),D], with B(t)=d(t)-d^*(t)-ib(t), D(t)=d(t)+d(t)^* + b(t) can deform the exterior derivative d. The Brouwer-Lefschetz fixed point theorem generalizes to all Wu characteristics: given an endomorphism T of G, the super trace of its induced map on k'th cohomology defines a Lefschetz number L_k(T). The Brouwer index i_T,k(x_1,...,x_k) = product_j=1^k w(x_j) sign(T|x_j) attached to simplex tuple which is invariant under T leads to the formula L_k(T) = sum_T(x)=x i_T,k(x). For T=Id, the Lefschetz number L_k(Id) is equal to the k'th Wu characteristic w_k(G) of the graph G and the Lefschetz formula reduces to the Euler-Poincare formula for Wu characteristic.

[132]  arXiv:1803.06790 [pdf, other]
Title: Towards "simultaneous selective inference": post-hoc bounds on the false discovery proportion
Subjects: Statistics Theory (math.ST)

Some pitfalls of the false discovery rate (FDR) as an error criterion for multiple testing of $n$ hypotheses include (a) committing to an error level $q$ in advance limits its use in exploratory data analysis, and (b) controlling the false discovery proportion (FDP) on average provides no guarantee on its variability. We take a step towards overcoming these barriers using a new perspective we call "simultaneous selective inference." Many FDR procedures (such as Benjamini-Hochberg) can be viewed as carving out a $\textit{path}$ of potential rejection sets $\varnothing = \mathcal R_0 \subseteq \mathcal R_1 \subseteq \cdots \subseteq \mathcal R_n \subseteq [n]$, assigning some algorithm-dependent estimate $\widehat{\text{FDP}}(\mathcal R_k)$ to each one. Then, they choose $k^* = \max\{k: \widehat{\text{FDP}}(\mathcal R_k) \leq q\}$. We prove that for all these algorithms, given independent null p-values and a confidence level $\alpha$, either the same $\widehat{FDP}$ or a minor variant thereof bounds the unknown FDP to within a small explicit (algorithm-dependent) constant factor $c_{\text{alg}}(\alpha)$, uniformly across the entire path, with probability $1-\alpha$. Our bounds open up a middle ground between fully simultaneous inference (guarantees for all $2^n$ possible rejection sets), and fully selective inference (guarantees only for $\mathcal R_{k^*}$). They allow the scientist to $\textit{spot}$ one or more suitable rejection sets (Select Post-hoc On the algorithm's Trajectory), by picking data-dependent sizes or error-levels, after examining the entire path of $\widehat{\text{FDP}}(\mathcal R_k)$ and the uniform upper band on $\text{FDP}$. The price for the additional flexibility of spotting is small, for example the multiplier for BH corresponding to 95% confidence is approximately 2.

[133]  arXiv:1803.06794 [pdf, ps, other]
Title: Short Proof of a Conjecture Concerning Split-By-Nilpotent Extensions
Authors: Stephen Zito
Subjects: Representation Theory (math.RT)

Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of mod B inherited by mod C. We show if B is a tilted algebra, then C is a tilted algebra.

[134]  arXiv:1803.06796 [pdf, ps, other]
Title: Different Statistical Future of Dynamical Orbits over Expanding or Hyperbolic Systems (II): Nonempty Syndetic Center
Comments: The present paper is a continuation of arXiv:1701.01910 which consider dynamical orbits without syndetic center. Here sixteen different statistical behavior of dynamical orbits with nonempty syndetic center may appear but up to now only two cases are well-studied
Subjects: Dynamical Systems (math.DS)

In [30] different statistical behavior of dynamical orbits without syndetic center are considered. In present paper we continue this project and consider different statistical behavior of dynamical orbits with nonempty syndetic center: Two of sixteen cases appear (for which other fourteen cases are still unknown) in transive topologically expanding or hyperbolic systems and are discovered to have full topological entropy for which it is also true if combined with non-recurrence and multifractal analysis such as quasi-regular set, irregular set and level sets. In this process a strong entropy-dense property, called minimal-entropy-dense, is established. In particular, we show that points that are minimal (or called almost periodic), a classical and important concept in the study of dynamical systems, form a set with full topological entropy if the dynamical system satisfies shadowing or almost specification property.

[135]  arXiv:1803.06801 [pdf, other]
Title: On the existence problem of Einstein-Maxwell Kähler metrics
Comments: 19 pages, 6 figures
Subjects: Differential Geometry (math.DG)

In this expository paper we review on the existence problem of Einstein-Maxwell K\"ahler metrics, and make several remarks. Firstly, we consider a slightly more general set-up than Einstein-Maxwell K\"ahler metrics, and give extensions of volume minimization principle, the notion of toric K-stability and other related results to the general set-up. Secondly, we consider the toric case when the manifold is the one point blow-up of the complex project plane and the K\"ahler class $\Omega$ is chosen so that the area of the exceptional curve is sufficiently close to the area of the rational curve of self-intersection number 1. We observe by numerical analysis that there should be a Killing vector field $K$ which gives a toric K-stable pair $(\Omega, K)$ in the sense of Apostolov-Maschler.

[136]  arXiv:1803.06803 [pdf, ps, other]
Title: Hadamard powers of some positive matrices
Authors: Tanvi Jain
Journal-ref: Linear Algebra and its Applications, 528, (2017) 147-158
Subjects: Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)

Positivity properties of the Hadamard powers of the matrix $\begin{bmatrix}1+x_ix_j\end{bmatrix}$ for distinct positive real numbers $x_1,\ldots,x_n$ and the matrix $\begin{bmatrix}|\cos((i-j)\pi/n)|\end{bmatrix}$ are studied. In particular, it is shown that $\begin{bmatrix}(1+x_ix_j)^r\end{bmatrix}$ is not positive semidefinite for any positive real number $r<n-2$ that is not an integer, and $\begin{bmatrix}|\cos((i-j)\pi/n)|^r\end{bmatrix}$ is positive semidefinite for every odd integer $n\ge 3$ and $n-3\le r<n-2.$

[137]  arXiv:1803.06804 [pdf, ps, other]
Title: Stochastic maximum principle, dynamic programming principle, and their relationship for fully coupled forward-backward stochastic control systems
Subjects: Optimization and Control (math.OC)

In this paper, we consider stochastic optimal control problems for fully coupled forward-backward stochastic control systems with a nonconvex control domain. Within the framework of viscosity solution, the relationship between the maximum principle and dynamic programming principle is investigated, and the set inclusions among the value function and the adjoint processes are obtained. Three special cases are studied. In the first case, the value function W is supposed to be smooth. In the second case, the diffusion term {\sigma} of the forward stochastic differential equation does not include the term z. Finally, we study the local case in which the control domain is convex.

[138]  arXiv:1803.06806 [pdf, ps, other]
Title: On certain unimodal sequences and strict partitions
Comments: 11 pages, 2 figures, 1 table
Subjects: Combinatorics (math.CO)

Building on a bijection of Vandervelde, we enumerate certain unimodal sequences whose alternating sum equals zero. This enables us to refine the enumeration of strict partitions with respect to the number of parts and the BG-rank.

[139]  arXiv:1803.06807 [pdf, other]
Title: Centralized Caching with Unequal Cache Sizes
Subjects: Information Theory (cs.IT)

We address centralized caching problem with unequal cache sizes. We consider a system with a server of files connected through a shared error-free link to a group of cache-enabled users where one subgroup has a larger cache size than the rest. We investigate caching schemes with uncoded cache placement which minimize the load of worst-case demands over the shared link. We propose a caching scheme which improves upon existing schemes by either having a lower worst-case load, or decreasing the complexity of the scheme while performing within 1.1 multiplicative factor suggested by our numerical simulations.

[140]  arXiv:1803.06808 [pdf, other]
Title: Local martingales associated with SLE with internal symmetry
Authors: Shinji Koshida
Comments: 45 pages
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Probability (math.PR); Quantum Algebra (math.QA); Representation Theory (math.RT)

We consider Schramm-Loewner evolutions with internal degrees of freedom that are associated with representations of affine Lie algebras, following the group theoretical formulation of SLE. We observe that SLEs considered by Bettelheim et al. [PRL 95, 251601 (2005)] and Alekseev et al. [Lett. Math. Phys. 97, 243-261 (2011)] in correlation function formulation are reconstrunced. We also explicitly write down stochastic differential equations on internal degrees of freedom for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. Our formulation enables to write down several local martingales associated with the solution of SLE from computation on a representation of an affine Lie algebra. Indeed, we write down local martingales associated with solution of SLE for Heisenberg algebras and the affine $\mathfrak{sl}_{2}$. We also find affine $\mathfrak{sl}_{2}$ symmetry of a space of SLE local martingales for the affine $\mathfrak{sl}_{2}$, which can be extended to other affine Lie algebras.

[141]  arXiv:1803.06817 [pdf, ps, other]
Title: Annular Representations of Free Product Categories
Comments: 21 pages, 1 figure
Subjects: Quantum Algebra (math.QA); Category Theory (math.CT); Operator Algebras (math.OA)

We provide a description of the annular representation category of the free product of two rigid C*-tensor categories.

[142]  arXiv:1803.06820 [pdf, ps, other]
Title: Rescaled weighted determinantal random balls
Authors: Adrien Clarenne (IRMAR)
Subjects: Probability (math.PR)

We consider a collection of weighted Euclidian random balls in R^d distributed according a determinantal point process. We perform a zoom-out procedure by shrinking the radii while increasing the number of balls. We observe that the repulsion between the balls is erased and three different regimes are obtained, the same as in the weighted Poissonian case.

[143]  arXiv:1803.06825 [pdf, ps, other]
Title: A counterexample to Las Vergnas' strong map conjecture on realizable oriented matroids
Authors: Pei Wu
Comments: 5 pages
Subjects: Combinatorics (math.CO)

The Las Vergnas' strong map conjecture, states that any strong map of oriented matroids $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$ can be factored into extensions and contractions. The conjecture is known to be false due to a construction by Richter-Gebert, he find a non-factorizable strong map $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, however in his example $\mathcal{M}_1$ is not realizable. The problem that whether there exists a non-factorizable strong map between realizable oriented matroids still remains open. In this paper we provide a counterexample to the strong map conjecture on realizable oriented matroids, which is a strong map $f:\mathcal{M}_1\rightarrow\mathcal{M}_2$, $\mathcal{M}_1$ is an alternating oriented matroid of rank $4$ and $f$ has corank $2$. We prove it is not factorizable by showing that there is no uniform oriented matroid $\mathcal{M}^{\prime}$ of rank $3$ such that $\mathcal{M}_1\rightarrow\mathcal{M}^{\prime}\rightarrow\mathcal{M}_2$.

[144]  arXiv:1803.06832 [pdf, ps, other]
Title: Boosting the Maxwell double layer potential using a right spin factor
Authors: Andreas Rosén
Subjects: Analysis of PDEs (math.AP); Numerical Analysis (math.NA)

We construct new spin singular integral equations for solving scattering problems for Maxwell's equations, both against perfect conductors and in media with piecewise constant permittivity, permeability and conductivity, improving and extending earlier formulations by the author. These differ in a fundamental way from classical integral equations, which use double layer potential operators, and have the advantage of having a better condition number, in particular in Fredholm sense and on Lipschitz regular interfaces, and do not suffer from spurious resonances. The construction of the integral equations builds on the observation that the double layer potential factorises into a boundary value problem and an ansatz. We modify the ansatz, inspired by a non-selfadjoint local elliptic boundary condition for Dirac equations.

[145]  arXiv:1803.06833 [pdf, other]
Title: An adaptive minimum spanning tree multi-element method for uncertainty quantification of smooth and discontinuous responses
Comments: 20 pages, 18 figures
Subjects: Numerical Analysis (math.NA)

A novel approach for non-intrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalised polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest depends discontinuously on the input parameters. As a remedy we propose an adaptive sampling algorithm based on minimum spanning trees combined with a domain decomposition method based on support vector machines. The minimum spanning tree determines new sample locations based on both the probability density of the input parameters and the gradient in the quantity of interest. The support vector machine efficiently decomposes the random space in multiple elements, avoiding the appearance of Gibbs phenomena near discontinuities. On each element, local approximations are constructed by means of least orthogonal interpolation, in order to produce stable interpolation on the unstructured sample set. The resulting minimum spanning tree multi-element method does not require initial knowledge of the behaviour of the quantity of interest and automatically detects whether discontinuities are present. We present several numerical examples that demonstrate accuracy, efficiency and generality of the method.

[146]  arXiv:1803.06837 [pdf, ps, other]
Title: Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition
Journal-ref: Topol. Methods Nonlinear Anal. 51:1 (2018), 55-77
Subjects: Analysis of PDEs (math.AP)

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: $-\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \text{ in }\Omega,$ and $u=0,\text{ on }\partial \Omega.$ We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. Our results extend previous work by Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comp. Math. Appl. 69 (2015), 1-12, and we establish the existence of solutions under weaker hypotheses on the nonlinear term.

[147]  arXiv:1803.06840 [pdf, ps, other]
Title: On n-Hom-Leibniz algebras and cohomology
Subjects: Rings and Algebras (math.RA); Mathematical Physics (math-ph); Quantum Algebra (math.QA)

The purpose of this paper is to provide a cohomology of $n$-Hom-Leibniz algebras. Moreover, we study some higher operations on cohomology spaces and deformations.

[148]  arXiv:1803.06846 [pdf, other]
Title: A primal discontinuous Galerkin method with static condensation on very general meshes
Authors: Alexei Lozinski
Subjects: Numerical Analysis (math.NA)

We propose an efficient variant of a primal Discontinuous Galerkin method with interior penalty for the second order elliptic equations on very general meshes (polytopes with eventually curved boundaries). Efficiency, especially when higher order polynomials are used, is achieved by static condensation, i.e. a local elimination of certain degrees of freedom element by element. This alters the original method in a way that preserves the optimal error estimates. Numerical experiments confirm that the solutions produced by the new method are indeed very close to that produced by the classical one.

[149]  arXiv:1803.06847 [pdf, ps, other]
Title: The square negative correlation on l_p^n balls
Subjects: Metric Geometry (math.MG); Probability (math.PR)

In this paper we prove that for any $p\in[2,\infty)$ the $\ell_p^n$ unit ball, $B_p^n$, satisfies the square negative correlation property with respect to every orthonormal basis, while we show it is not always the case for $1\le p\le 2$. In order to do that we regard $B_p^n$ as the orthogonal projection of $B_p^{n+1}$ onto the hyperplane $e_{n+1}^\perp$. We will also study the orthogonal projection of $B_p^n$ onto the hyperplane orthogonal to the diagonal vector $(1,\dots,1)$. In this case, the property holds for all $p\ge 1$ and $n$ large enough.

[150]  arXiv:1803.06849 [pdf, ps, other]
Title: The arithmetic derivative and Leibniz-additive functions
Subjects: Number Theory (math.NT)

An arithmetic function $f$ is Leibniz-additive if there is a completely multiplicative function $h_f$, i.e., $h_f(1)=1$ and $h_f(mn)=h_f(m)h_f(n)$ for all positive integers $m$ and $n$, satisfying $$ f(mn)=f(m)h_f(n)+f(n)h_f(m) $$ for all positive integers $m$ and $n$. A motivation for the present study is the fact that Leibniz-additive functions are generalizations of the arithmetic derivative $D$; namely, $D$ is Leibniz-additive with $h_D(n)=n$. In this paper, we study the basic properties of Leibniz-additive functions and, among other things, show that a Leibniz-additive function $f$ is totally determined by the values of $f$ and $h_f$ at primes. We also consider properties of Leibniz-additive functions with respect to the usual product, composition and Dirichlet convolution of arithmetic functions.

[151]  arXiv:1803.06859 [pdf, ps, other]
Title: Approximation of non-archimedean Lyapunov exponents and applications over global fields
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)

Let $K$ be an algebraically closed field of characteristic 0 that is complete with respect to a non-archimedean absolute value. We establish a locally uniform approximation formula of the Lyapunov exponent of a rational map $f$ of $\mathbb{P}^1$ of degree $d>1$ over $K$, in terms of the multipliers of $n$-periodic points of $f$, with an explicit control in terms of $n$, $f$ and $K$. As an immediate consequence, we obtain an estimate for the blow-up of the Lyapunov exponent near a pole in one-dimensional families of rational maps over $K$. Combined with our former archimedean version, this non-archimedean quantitative approximation allows us to show:
- a quantified version of Silverman's and Ingram's recent comparison between the critical height and any ample height on the moduli space $\mathcal{M}_d(\bar{\mathbb{Q}})$,
- two improvements of McMullen's finiteness of the mutiplier maps: reduction to multipliers of cycles of exact given period and an effective bound from below on the period,
- a characterization of non-affine isotrivial rational maps defined over the function field $\mathbb{C}(X)$ of a normal projective variety $X$ in terms of the growth of the degree of the multipliers.

[152]  arXiv:1803.06863 [pdf, ps, other]
Title: Approximation property on entropies for surface diffeomorphisms
Subjects: Dynamical Systems (math.DS)

In this paper, we prove that for any $C^1$ surface diffeomorphism $f$ with positive topological entropy, there exists a diffeomorphism $g$ arbitrarily close (in the $C^1$ topology) to $f$ exhibiting a horseshoe $\Lambda$, such that the topological entropy of $g$ restricted on $\Lambda$ can arbitrarily approximate the topological entropy of $f$. This extends the Theorem \cite[Theorem 1.1]{Gan} of Gan.

[153]  arXiv:1803.06864 [pdf, other]
Title: On the hierarchical structure of Pareto critical sets
Subjects: Optimization and Control (math.OC)

In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems considering subsets of the objective functions. If the Pareto critical set is completely described by its boundary (e.g. if we have more objective functions than dimensions in the parameter space), this can be used to solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set. This technique is especially useful for efficiently solving many-objective optimization problems by breaking them down into MOPs with a reduced number of objective functions.

[154]  arXiv:1803.06865 [pdf, other]
Title: Spatio-temporal Poisson processes for visits to small sets
Subjects: Dynamical Systems (math.DS)

For many measure preserving dynamical systems $(\Omega,T,m)$ the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new process obtained from recording not only the successive times $n$ of visits to a set $A$, but also the position $T^n(x)$ in $A$ of the orbit, in the limit where $m(A)\to0$.
We obtain a convergence of this process, suitably normalized, to a Poisson point process in time and space under some decorrelation condition. We present several new applications to hyperbolic maps and SRB measures, including the case of a neighborhood of a periodic point, and some billiards such as Sinai billiards, Bunimovich stadium and diamond billiard.

[155]  arXiv:1803.06866 [pdf, ps, other]
Title: The planar 3-body problem II:reduction to pure shape and spherical geometry (2nd version)
Comments: 34 pages
Subjects: Mathematical Physics (math-ph); Differential Geometry (math.DG)

Geometric reduction of the Newtonian planar three-body problem is investigated in the framework of equivariant Riemannian geometry, which reduces the study of trajectories of three-body motions to the study of their moduli curves, that is, curves which record the change of size and shape, in the moduli space of oriented mass-triangles. The latter space is a Riemannian cone over the shape 2-sphere, and the shape curve is the image curve on this sphere. It is shown that the time parametrized moduli curve is in general determined by the relative geometry of the shape curve and the shape potential function. This also entails the reconstruction of time, namely the geometric shape curve determines the time parametrization of the moduli curve, hence also the three-body motion itself, modulo a fixed rotation of the plane. The first version of this work is an (unpublished) paper from 2012, and the present version is an editorial revision of this.

[156]  arXiv:1803.06870 [pdf, ps, other]
Title: An invitation to higher Teichmüller theory
Authors: Anna Wienhard
Comments: written for the Proceedings of the ICM
Subjects: Geometric Topology (math.GT)

The goal of this article is to invite the reader to get to know and to get involved into higher Teichm\"uller theory by describing some of its many facets.

[157]  arXiv:1803.06871 [pdf, ps, other]
Title: Symbol-Level Precoding Design for Max-Min SINR in Multiuser MISO Broadcast Channels
Comments: Submitted to SPAWC 2018, 7 pages, 2 figures
Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)

In this paper, we address the symbol level precoding (SLP) design problem under max-min SINR criterion in the downlink of multiuser multiple-input single-output (MISO) channels. First, we show that the distance preserving constructive interference regions (DPCIR) are always polyhedral angles (shifted pointed cones) for any given constellation point with unbounded decision region. Then we prove that any signal in a given unbounded DPCIR has a norm larger than the norm of the corresponding vertex if and only if the convex hull of the constellation contains the origin. Using these properties, we show that the power of the noiseless received signal lying on an unbounded DPCIR is an strictly increasing function of two parameters. This allows us to reformulate the originally non-convex SLP max-min SINR as a convex optimization problem. We discuss the loss due to our proposed convex reformulation and provide some simulation results.

[158]  arXiv:1803.06872 [pdf, ps, other]
Title: The group generated by Riordan involutions
Comments: 18 pages
Subjects: Group Theory (math.GR)

We prove that any element in the group generated by the Riordan involutions is the product of at most four of them. We also give a description of this subgroup as a semidirect product of a special subgroup of the commutator subgroup and the Klein four-group.

[159]  arXiv:1803.06876 [pdf, ps, other]
Title: Generalised Net Convergence Structures in Posets
Subjects: General Topology (math.GN)

In this paper, we introduce the notion of $\mathcal{M}$-convergence and $\mathcal{MN}$-convergence structures in posets, which, in some sense, generalise the well-known Scott-convergence and order-convergence structures. As results, we give a necessary and sufficient conditions for each generalised convergence structures being topological. These results then imply the following two well-established results: (1) The Scott-convergence structure in a poset $P$ is topological if and only if $P$ is continuous, and (2) The order-convergence structure in a poset $P$ is topological if and only if $P$ is $\mathcal{R}^*$-doubly continuous.

[160]  arXiv:1803.06879 [pdf, other]
Title: The tree of numerical semigroups with low multiplicity
Subjects: Combinatorics (math.CO); Commutative Algebra (math.AC)

We show that the number of numerical semigroups with multiplicity three, four or five and fixed genus is increasing as a function in the genus. To this end we use the Kunz polytope for these multiplicities. Counting numerical semigroups with fixed multiplicity and genus is then an integer partition problem with some extra conditions (those of membership to the Kunz polytope). For the particular case of multiplicity four, we are able to prove that the number of numerical semigroups with multiplicity four and genus $g$ is the number of partitions $x+y+z=g+6$ with $0<x\le y\le z$, $x\neq 1$, $y\neq 2$ and $z\neq 3$.

[161]  arXiv:1803.06882 [pdf, ps, other]
Title: The correct formulation of Gleason's theorem in quaternionic Hilbert spaces
Authors: Valter Moretti, Marco Oppio (Trento U. and TIFPA-INFN)
Comments: 33 pages, no figures
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Operator Algebras (math.OA); Quantum Physics (quant-ph)

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

[162]  arXiv:1803.06886 [pdf, ps, other]
Title: Exchanging the phase space and symmetry group of integrable Hamiltonian systems related to Lie bialgebra of bi-symplectic type
Comments: 10 pages
Subjects: Mathematical Physics (math-ph)

We construct integrable Hamiltonian systems with Lie bialgebra of bi-symplectic type for which the Poisson-Lie group $G$ plays the role of phase space and its dual Lie group $\tilde{G}$ plays the role of symmetry group of the system. We give the new transformation to exchange the role of phase space and symmetry group. We obtain relation between integrals of motion of these two integrable systems. Finally we give some examples about real four dimensional Lie bialgebras of bi-symplectic type.

[163]  arXiv:1803.06887 [pdf, other]
Title: Lossless Analog Compression
Subjects: Functional Analysis (math.FA); Information Theory (cs.IT)

We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary m-dimensional real random vectors x from the noiseless linear measurements y=Ax with n x m measurement matrix A. Our theory is inspired by the groundbreaking work of Wu and Verdu (2010) on almost lossless analog compression, but applies to the nonasymptotic, i.e., fixed-m case, and considers zero error probability. Specifically, our achievability result states that, for almost all A, the random vector x can be recovered with zero error probability provided that n > K(x), where the description complexity K(x) is given by the infimum of the lower modified Minkowski dimensions over all support sets U of x. We then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. (2016); these are random vectors of absolutely continuous distribution---with respect to the s-dimensional Hausdorff measure---supported on countable unions of s-dimensional differentiable manifolds. Countable unions of differentiable manifolds include essentially all signal models used in compressed sensing theory, in spectrum-blind sampling, and in the matrix completion problem. Specifically, we prove that, for almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n>s linear measurements. This threshold is, however, found not to be tight as exemplified by the construction of an s-rectifiable random vector that can be recovered with zero error probability from n<s linear measurements. This leads us to the introduction of the new class of s-analytic random vectors, which admit a strong converse in the sense of n greater than or equal to s being necessary for recovery with probability of error smaller than one. The central conceptual tool in the development of our theory is geometric measure theory.

[164]  arXiv:1803.06893 [pdf, other]
Title: On reference solutions and the sensitivity of the 2d Kelvin-Helmholtz instability problem
Comments: 20 pages, 11 figures, 1 table
Subjects: Numerical Analysis (math.NA); Computational Physics (physics.comp-ph); Fluid Dynamics (physics.flu-dyn)

Two-dimensional Kelvin-Helmholtz instability problems are popular examples for assessing discretizations or turbulence models for incompressible flows. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin-Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2d turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed.

[165]  arXiv:1803.06895 [pdf, ps, other]
Title: Global multiplicity bounds and Spectral Statistics Random Operators
Comments: 23 pages
Subjects: Spectral Theory (math.SP)

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on $\mathbb{R}$. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular, we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory.
In particular higher rank Anderson models with pure-point spectrum, with the randomness having support equal to $\mathbb{R}$, there is a uniform lower bound on spectral multiplicity and in case this is larger than one the local statistics is not Poisson.

[166]  arXiv:1803.06899 [pdf, ps, other]
Title: Limit Theorems for Cylindrical Martingale Problems associated with Lévy Generators
Authors: David Criens
Subjects: Probability (math.PR)

We derive limit theorems for cylindrical martingale problems associated to L\'evy generators. Furthermore, we give sufficient and necessary conditions for the Feller property of well-posed problems with continuous coefficients and limit theorems for solution measures to stochastic (partial) differential equations.

[167]  arXiv:1803.06901 [pdf, ps, other]
Title: Cyclic Sieving and Cluster Duality for Grassmannian
Comments: 40 pages
Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Combinatorics (math.CO)

We introduce a decorated configuration space $\mathscr{C}onf_n^\times(a)$ with a potential function $\mathcal{W}$. We prove the cluster duality conjecture of Fock-Goncharov for Grassmannians, that is, the tropicalization of $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ canonically parametrizes a linear basis of the homogenous coordinate ring of the Grassmannian ${\rm Gr}_a(n)$. We prove that $(\mathscr{C}onf_n^\times(a), \mathcal{W})$ is equivalent to the mirror Landau-Ginzburg model of Grassmannian considered by Marsh-Rietsch and Rietsch-Williams. As an application, we show a cyclic sieving phenomenon involving plane partitions under a sequence of piecewise-linear toggles.

[168]  arXiv:1803.06902 [pdf, other]
Title: Filters for anisotropic wavelet decompositions
Subjects: Numerical Analysis (math.NA)

Like the continous shearlet transform and their relatives, discrete transformations based on the interplay between several filterbanks with anisotropic dilations provide a high potential to recover directed features in two and more dimensions. Due to simplicity, most of the directional systems constructed so far were using prediction--correction methods based on interpolatory subdivision schemes. In this paper, we give a simple but effective construction for QMF (quadrature mirror filter) filterbanks which are the discrete object between orthogonal wavelet analysis. We also characterize when the filterbank gives rise to the existence of refinable functions and hence wavelets and give a generalized shearlet construction for arbitrary dimensions and arbitrary scalings for which the filterbank construction ensures the existence of an orthogonal wavelet analysis.

[169]  arXiv:1803.06903 [pdf, ps, other]
Title: On class groups of random number fields
Comments: 28 pages; comments welcome!
Subjects: Number Theory (math.NT)

The aim of the present paper is to add, in several ways, to our understanding of the Cohen-Lenstra-Martinet heuristics on class groups of random number fields. Firstly, we point out several difficulties with the original formulation, and offer possible corrections. Secondly, we recast the heuristics in terms of Arakelov class groups of number fields. Thirdly, we propose a rigorously formulated Cohen-Lenstra-Martinet conjecture.

[170]  arXiv:1803.06907 [pdf, ps, other]
Title: Auxiliary information : the raking-ratio empirical process
Comments: 45 pages
Subjects: Statistics Theory (math.ST)

We study the empirical measure associated to a sample of size $n$ and modified by $N$ iterations of the raking-ratio method. The empirical measure is adjusted to match the true probability of sets in a finite partition which changes each step. We establish asymptotic properties of the raking-ratio empirical process indexed by functions as $n\rightarrow +\infty$, for $N$ fixed. A closed-form expression of the limiting covariance matrices is derived as $N\rightarrow +\infty$. The nonasymptotic Gaussian approximation we use also yields uniform Berry-Esseen type bounds in $n, N$ and sharp estimates of the uniform quadratic risk reduction. In the two-way contingency table formulas characterizing the limiting process are very simple.

[171]  arXiv:1803.06909 [pdf, other]
Title: Row-finite systems of ordinary differential equations in a scale of Banach spaces
Subjects: Functional Analysis (math.FA)

We study an infinite system of first-order differential equations in a Euclidean space, parameterized by elements $x$ of a fixed countable set. We suppose that the system is row-finite, that is, the right-hand side of the $x$-equation depends on a finite but in general unbounded number $n_x$ of variables. Under certain dissipativity-type conditions on the right-hand side and a bound on the growth of $n_x$, we show the existence of the solutions with infinite lifetime and prove that they live in a scale of increasing Banach spaces. For this, we approximate our system by finite systems and obtain uniform estimates of the corresponding solutions using the version of Ovsyannikov's method for linear systems in a scale of Banach spaces. As a by-product, we develop an infinite-time generalization of the Ovsyannikov method.

[172]  arXiv:1803.06914 [pdf, ps, other]
Title: Mixing Time of Markov chain of the Knapsack Problem
Comments: 9 pages, 2 figures
Subjects: Combinatorics (math.CO); Data Structures and Algorithms (cs.DS); Probability (math.PR)

To find the number of assignments of zeros and ones satisfying a specific Knapsack Problem is $\#P$ hard, so only approximations are envisageable. A Markov chain allowing uniform sampling of all possible solutions is given by Luby, Randall and Sinclair. In 2005, Morris and Sinclair, by using a flow argument, have shown that the mixing time of this Markov chain is $\mathcal{O}(n^{9/2+\epsilon})$, for any $\epsilon > 0$. By using a canonical path argument on the distributive lattice structure of the set of solutions, we obtain an improved bound, the mixing time is given as $\tau_{_{x}}(\epsilon) \leq n^{3} \ln (16 \epsilon^{-1})$.

[173]  arXiv:1803.06915 [pdf, other]
Title: Exploiting symmetry in network analysis
Comments: Main Text (7 pages) plus Supplementary Information (24 pages)
Subjects: Combinatorics (math.CO); Social and Information Networks (cs.SI); Data Analysis, Statistics and Probability (physics.data-an); Physics and Society (physics.soc-ph)

Virtually all network analyses involve structural measures or metrics between pairs of vertices, or of the vertices themselves. The large amount of redundancy present in real-world networks is inherited by such measures, and this has practical consequences which have not yet been explored in full generality, nor systematically exploited by network practitioners. Here we develop a complete framework to study and quantify the effect of redundancy on arbitrary network measures, and explain how to exploit redundancy in practice, achieving, for instance, remarkable lossless compression and computational reduction ratios in several real-world networks against some popular measures.

[174]  arXiv:1803.06918 [pdf, other]
Title: Correcting Observation Model Error in Data Assimilation
Subjects: Dynamical Systems (math.DS); Data Analysis, Statistics and Probability (physics.data-an)

Standard methods of data assimilation assume prior knowledge of a model that describes the system dynamics and an observation function that maps the model state to a predicted output. An accurate mapping from model state to observation space is crucial in filtering schemes when adjusting the estimate of the system state during the filter's analysis step. However, in many applications the true observation function may be unknown and the available observation model may have significant errors, resulting in a suboptimal state estimate. We propose a method for observation model error correction within the filtering framework. The procedure involves an alternating minimization algorithm used to iteratively update a given observation function to increase consistency with the model and prior observations, using ideas from attractor reconstruction. The method is demonstrated on the Lorenz 1963 and Lorenz 1996 models, and on a single-column radiative transfer model with multicloud parameterization.

[175]  arXiv:1803.06925 [pdf, other]
Title: (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods
Subjects: Numerical Analysis (math.NA)

We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method.

[176]  arXiv:1803.06928 [pdf]
Title: Optimization Based Solutions for Control and State Estimation in Non-holonomic Mobile Robots: Stability, Distributed Control, and Relative Localization
Comments: This a preprint of Mohamed Said's PhD thesis
Subjects: Optimization and Control (math.OC); Robotics (cs.RO)

Interest in designing, manufacturing, and using autonomous robots has been rapidly growing during the most recent decade. The main motivation for this interest is the wide range of potential applications these autonomous systems can serve in. The applications include, but are not limited to, area coverage, patrolling missions, perimeter surveillance, search and rescue missions, and situational awareness. In this thesis, the area of control and state estimation in non-holonomic mobile robots is tackled. Herein, optimization based solutions for control and state estimation are designed, analyzed, and implemented to such systems. One of the main motivations for considering such solutions is their ability of handling constrained and nonlinear systems such as non-holonomic mobile robots. Moreover, the recent developments in dynamic optimization algorithms as well as in computer processing facilitated the real-time implementation of such optimization based methods in embedded computer systems.

[177]  arXiv:1803.06929 [pdf, other]
Title: Stochastic filtering and optimal control of pure jump Markov processes with noise-free partial observation
Comments: arXiv admin note: text overlap with arXiv:1707.07202
Subjects: Optimization and Control (math.OC)

We consider an infinite horizon optimal control problem for a pure jump Markov process $X$, taking values in a complete and separable metric space $I$, with noise-free partial observation. The observation process is defined as $Y_t = h(X_t)$, $t \geq 0$, where $h$ is a given map defined on $I$. The observation is noise-free in the sense that the only source of randomness is the process $X$ itself. The aim is to minimize a discounted cost functional. In the first part of the paper we write down an explicit filtering equation and characterize the filtering process as a Piecewise Deterministic Process. In the second part, after transforming the original control problem with partial observation into one with complete observation (the separated problem) using filtering equations, we prove the equivalence of the original and separated problems through an explicit formula linking their respective value functions. The value function of the separated problem is also characterized as the unique fixed point of a suitably defined contraction mapping.

[178]  arXiv:1803.06930 [pdf, ps, other]
Title: Explicit formula for the density of local times of Markov Jump Processes
Subjects: Probability (math.PR)

In this note we show a simple formula for the joint density of local times, last exit tree and cycling numbers of continuous-time Markov Chains on finite graphs, which involves the modified Bessel function of the first type.

[179]  arXiv:1803.06931 [pdf, ps, other]
Title: Note on Calderón's inverse problem for measurable conductivities
Comments: 11 pages
Subjects: Analysis of PDEs (math.AP)

The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation $\mathrm{div} (\sigma \nabla u) = 0$ is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calder\'on problem in three and higher dimensions in the $L^\infty$ case.

[180]  arXiv:1803.06932 [pdf, other]
Title: Orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$
Comments: Banach Center Publ. (to appear). arXiv admin note: text overlap with arXiv:1611.07840, arXiv:1611.08181
Subjects: Number Theory (math.NT)

This paper continues the authors previous investigations concerning orders of Tate-Shafarevich groups in quadratic twists of a given elliptic curve, and for the family of the Neumann-Setzer type elliptic curves. Here we present the results of our search for the (analytic) orders of Tate-Shafarevich groups for the cubic twists of $X_0(27)$. Our calculations extend those given by Zagier and Kramarz \cite{ZK} and by Watkins \cite{Wat}. Our main observations concern the asymptotic formula for the frequency of orders of Tate-Shafarevich groups. In the last section we propose a similar asymptotic formula for the class numbers of real quadratic fields.

[181]  arXiv:1803.06933 [pdf, ps, other]
Title: The canonical projection associated to certain possibly infinite generalized iterated function system as a fixed point
Subjects: Classical Analysis and ODEs (math.CA)

In this paper, influenced by the ideas from A. Mihail, The canonical projection between the shift space of an IIFS and its attractor as a fixed point, Fixed Point Theory Appl., 2015, Paper No. 75, 15 p., we associate to every generalized iterated function system F (of order m) an operator H defined on C^m and taking values on C, where C stands for the space of continuous functions from the shift space on the metric space corresponding to the system. We provide sufficient conditions (on the constitutive functions of F) for the operator H to be continuous, contraction, phi-contraction, Meir-Keeler or contractive. We also give sufficient condition under which H has a unique fixed point. Moreover, we prove that, under these circumstances, the closer of the imagine of the fixed point is the attractor of F and that the fixed point is the canonical projection associated to F. In this way we give a partial answer to the open problem raised on the last paragraph of the above mentioned Mihail's paper.

[182]  arXiv:1803.06947 [pdf, ps, other]
Title: Differentiability of SDEs with drifts of super-linear growth
Comments: 41 Pages
Subjects: Probability (math.PR)

We close an unexpected gap in the literature of stochastic differential equations (SDEs) with drifts of super linear growth (and random coefficients), namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Ray Absolute Continuity and Stochastic G\^ateaux Differentiability. This method enables one to take limits in probability rather than mean square which bypasses the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology from Nualart's 2006 work, Lemma 1.2.3 for this setting. Several examples illustrating the range and scope of our results are presented.
We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.

[183]  arXiv:1803.06949 [pdf, ps, other]
Title: Graded Identities and Isomorphisms on Algebras of Upper Block-Triangular Matrices
Subjects: Rings and Algebras (math.RA)

Let $G$ be an abelian group and $\mathbb{K}$ an algebraically closed field of characteristic zero. A. Valenti and M. Zaicev described the $G$-gradings on upper block-triangular matrix algebras provided that $G$ is finite. We prove that their result holds for any abelian group $G$: any grading is isomorphic to the tensor product $A\otimes B$ of an elementary grading $A$ on an upper block-triangular matrix algebra and a division grading $B$ on a matrix algebra. We then consider the question of whether graded identities $A\otimes B$, where $B$ is an algebra with a division grading, determine $A\otimes B$ up to graded isomorphism. In our main result, Theorem 3, we reduce this question to the case of elementary gradings on upper block-triangular matrix algebras which was previously studied by O. M. Di Vincenzo and E. Spinelli.

[184]  arXiv:1803.06953 [pdf, ps, other]
Title: Entropy solutions for stochastic porous media equations
Comments: 25 pages
Subjects: Probability (math.PR); Analysis of PDEs (math.AP)

We provide an entropy formulation for porous medium-type equations with a stochastic, non-linear, spatially inhomogeneous forcing. Well - posedness and $L_1$-contraction is obtained in the class of entropy solutions. Our scope allows for porous medium operators $\Delta (|u|^{m-1}u)$ for all $m\in(1,\infty)$, and H\"older continuous diffusion nonlinearity with exponent $1/2$.

[185]  arXiv:1803.06956 [pdf, ps, other]
Title: On a question of Swan
Authors: Dorin Popescu
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)

We show that a regular local ring is a filtered inductive limit of regular local rings, essentially of finite type over $\bf Z$.

[186]  arXiv:1803.06964 [pdf, other]
Title: A modern maximum-likelihood theory for high-dimensional logistic regression
Comments: 25 pages, 12 figures, 4 tables
Subjects: Statistics Theory (math.ST); Methodology (stat.ME)

Every student in statistics or data science learns early on that when the sample size largely exceeds the number of variables, fitting a logistic model produces estimates that are approximately unbiased. Every student also learns that there are formulas to predict the variability of these estimates which are used for the purpose of statistical inference; for instance, to produce p-values for testing the significance of regression coefficients. Although these formulas come from large sample asymptotics, we are often told that we are on reasonably safe grounds when $n$ is large in such a way that $n \ge 5p$ or $n \ge 10p$. This paper shows that this is absolutely not the case. Consequently, inferences routinely produced by common software packages are unreliable.
Consider a logistic model with independent features in which $n$ and $p$ become increasingly large in a fixed ratio. Then we show that (1) the MLE is biased, (2) the variability of the MLE is far greater than classically predicted, and (3) the commonly used likelihood-ratio test (LRT) is not distributed as a chi-square. The bias of the MLE is extremely problematic as it yields completely wrong predictions for the probability of a case based on observed values of the covariates. We develop a new theory, which asymptotically predicts (1) the bias of the MLE, (2) the variability of the MLE, and (3) the distribution of the LRT. We empirically also demonstrate that these predictions are extremely accurate in finite samples. Further, an appealing feature is that these novel predictions depend on the unknown sequence of regression coefficients only through a single scalar, the overall strength of the signal. This suggests very concrete procedures to adjust inference; we describe one such procedure learning a single parameter from data and producing accurate inference.

[187]  arXiv:1803.06965 [pdf, ps, other]
Title: The normal hull and commutator group for nonconnected group schemes
Authors: Giulia Battiston
Comments: 4 pages, comments are welcome
Subjects: Algebraic Geometry (math.AG); Group Theory (math.GR)

In this short note, we prove that there is a well behaved notion of normal hull for smooth algebraic group schemes over a field and that the commutator group $(G,H)$ is well defined for $H\subset G$ smooth, even when both of them are not connected.

[188]  arXiv:1803.06968 [pdf, ps, other]
Title: Bounded error uniformity of the linear flow on the torus
Authors: Bence Borda
Comments: 18 pages
Subjects: Number Theory (math.NT)

A linear flow on the torus $\mathbb{R}^d / \mathbb{Z}^d$ is uniformly distributed in the Weyl sense if the direction of the flow has linearly independent coordinates over $\mathbb{Q}$. In this paper we combine Fourier analysis and the subspace theorem of Schmidt to prove bounded error uniformity of linear flows with respect to certain polytopes if, in addition, the coordinates of the direction are all algebraic. In particular, we show that there is no van Aardenne--Ehrenfest type theorem for the mod $1$ discrepancy of continuous curves in any dimension, demonstrating a fundamental difference between continuous and discrete uniform distribution theory.

[189]  arXiv:1803.06970 [pdf, ps, other]
Title: Higher symmetries of symplectic Dirac operator
Comments: Symplectic Dirac operator, Higher symmetry algebra, Projective differential geometry, Minimal nilpotent orbit, $\mathfrak{sl}(3,\mR)$
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Functional Analysis (math.FA); Representation Theory (math.RT); Symplectic Geometry (math.SG)

We construct in projective differential geometry of the real dimension $2$ higher symmetry algebra of the symplectic Dirac operator ${D}\kern-0.5em\raise0.22ex\hbox{/}_s$ acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $\mathfrak{sl}(3,{\mathbb{R}})$.

[190]  arXiv:1803.06974 [pdf, ps, other]
Title: Quasi boundary triples, self-adjoint extensions, and Robin Laplacians on the half-space
Subjects: Spectral Theory (math.SP)

In this note self-adjoint extensions of symmetric operators are investigated by using the abstract technique of quasi boundary triples and their Weyl functions. The main result is an extension of Theorem 2.6 in [5] which provides sufficient conditions on the parameter in the boundary space to induce self-adjoint realizations. As an example self-adjoint Robin Laplacians on the half-space with boundary conditions involving an unbounded coefficient are considered.

[191]  arXiv:1803.06980 [pdf, other]
Title: Second order ensemble simulation for MHD flow in Elsässer variable with noisy input data
Subjects: Numerical Analysis (math.NA)

We propose, analyze and test a fully discrete, efficient second-order algorithm for computing flow ensembles average of viscous, incompressible, and time-dependent magnetohydrodynamic (MHD) flows under uncertainties in initial conditions. The scheme is decoupled and based on Els\"asser variable formulation. The algorithm uses the breakthrough idea of Jiang and Layton, 2014 to approximate the ensemble average of $J$ realizations. That is, at each time step, each of the $J$ realization shares the same coefficient matrix for different right-hand side matrices. Thus, storage requirements and computational time are reduced by building preconditioners once per time step and reuse them. We prove stability and optimal convergence with respect to the time step restriction. On some manufactured solutions, numerical experiments are given to verify the predicted convergence rates of our analysis. Finally, we test the scheme on a benchmark channel flow over a step and it performs well.

[192]  arXiv:1803.06981 [pdf, ps, other]
Title: Quantitative weighted estimates for the Littlewood-Paley square function and Marcinkiewicz multipliers
Authors: Andrei K. Lerner
Subjects: Classical Analysis and ODEs (math.CA)

Quantitative weighted estimates are obtained for the Littlewood-Paley square function $S$ associated with a lacunary decomposition of ${\mathbb R}$ and for the Marcinkiewicz multiplier operator. In particular, we find the sharp dependence on $[w]_{A_p}$ for the $L^p(w)$ operator norm of $S$ for $1<p\le 2$.

[193]  arXiv:1803.06984 [pdf, other]
Title: Robust Optimization and Control for Electricity Generation and Transmission
Subjects: Optimization and Control (math.OC)

In this paper we present the theoretical results of solving a robust optimization problem for the power system under uncertainty. Solving the deterministic alternating current optimal power flow (ACOPF) problem has been considered a hard problem since 1960s because the optimization problem is nonlinear and highly nonconvex. Linear approximation of the AC power flow system (DC approximation) has been deployed in the industry but does not guarantee a physically feasible system configuration. In recently years, different convex relaxation schemes of the ACOPF problem have been researched, and under some assumptions, a physically feasible solution can be recovered. Based on these convex relaxation schemes, we construct a robust convex optimization problem with recourse to solve for optimal controllable injections (fossil fuel, nuclear etc.) in electricity power systems under uncertainty (renewable energy generation, demand fluctuation, etc.). We propose a cutting-plane method to solve this robust optimization problem and prove its convergence property. Extensive experiment results indicate that the robust convex relaxation of the ACOPF problem will provide a tight lower bound, and for the test cases where the nominal relaxation is tight, the convex relaxation solution can also be used for the non-convex robust ACOPF problem.

[194]  arXiv:1803.06985 [pdf, other]
Title: The Hessian discretisation method for fourth order linear elliptic equations
Subjects: Numerical Analysis (math.NA)

In this paper, we propose a unified framework, the Hessian discretisation method (HDM), which is based on four discrete elements (called altogether a Hessian discretisation) and a few intrinsic indicators of accuracy, independent of the considered model. An error estimate is obtained, using only these intrinsic indicators, when the HDM framework is applied to linear fourth order problems. It is shown that HDM encompasses a large number of numerical methods for fourth order elliptic problems: finite element method (conforming and non-conforming) as well as finite volume method. Finally, we use the HDM to design a novel method, based on conforming $\mathbb{P}_1$ finite element space and gradient recovery operators. Results of numerical experiments are presented for this novel scheme.

[195]  arXiv:1803.06987 [pdf, ps, other]
Title: Synthesis of Logical Clifford Operators via Symplectic Geometry
Comments: Single column, main text: 20 pages, full length with appendices: 32 pages. Includes pseudo-codes for all algorithms. Part of this work has been submitted to the 2018 IEEE International Symposium on Information Theory
Subjects: Information Theory (cs.IT); Quantum Physics (quant-ph)

Quantum error-correcting codes can be used to protect qubits involved in quantum computation. This requires that logical operators acting on protected qubits be translated to physical operators (circuits) acting on physical quantum states. We propose a mathematical framework for synthesizing physical circuits that implement logical Clifford operators for stabilizer codes. Circuit synthesis is enabled by representing the desired physical Clifford operator in $\mathbb{C}^{N \times N}$ as a partial $2m \times 2m$ binary symplectic matrix, where $N = 2^m$. We state and prove two theorems that use symplectic transvections to efficiently enumerate all symplectic matrices that satisfy a system of linear equations. As an important corollary of these results, we prove that for an $[\![ m,m-k ]\!]$ stabilizer code every logical Clifford operator has $2^{k(k+1)/2}$ symplectic solutions. The desired physical circuits are then obtained by decomposing each solution as a product of elementary symplectic matrices. Our assembly of the possible physical realizations enables optimization over them with respect to a suitable metric. Furthermore, we show that any circuit that normalizes the stabilizer of the code can be transformed into a circuit that centralizes the stabilizer, while realizing the same logical operation. Our method of circuit synthesis can be applied to any stabilizer code, and this paper provides a proof of concept synthesis of universal Clifford gates for the $[\![ 6,4,2 ]\!]$ CSS code. We conclude with a classical coding-theoretic perspective for constructing logical Pauli operators for CSS codes. Since our circuit synthesis algorithm builds on the logical Pauli operators for the code, this paper provides a complete framework for constructing all logical Clifford operators for CSS codes. Programs implementing our algorithms can be found at https://github.com/nrenga/symplectic-arxiv18a.

[196]  arXiv:1803.06988 [pdf, ps, other]
Title: Maximal Symmetry and Unimodular Solvmanifolds
Comments: 9 pages
Subjects: Differential Geometry (math.DG)

Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily non-unimodular. In this work we consider unimodular solvable Lie groups and prove that there is always some metric with maximal symmetry. Further, if the group at hand admits a Ricci soliton, then it is the isometry group of the Ricci soliton which is maximal.

[197]  arXiv:1803.06989 [pdf, other]
Title: Numerical Integration on Graphs: where to sample and how to weigh
Subjects: Statistics Theory (math.ST); Learning (cs.LG); Numerical Analysis (math.NA); Machine Learning (stat.ML)

Let $G=(V,E,w)$ be a finite, connected graph with weighted edges. We are interested in the problem of finding a subset $W \subset V$ of vertices and weights $a_w$ such that $$ \frac{1}{|V|}\sum_{v \in V}^{}{f(v)} \sim \sum_{w \in W}{a_w f(w)}$$ for functions $f:V \rightarrow \mathbb{R}$ that are `smooth' with respect to the geometry of the graph. The main application are problems where $f$ is known to somehow depend on the underlying graph but is expensive to evaluate on even a single vertex. We prove an inequality showing that the integration problem can be rewritten as a geometric problem (`the optimal packing of heat balls'). We discuss how one would construct approximate solutions of the heat ball packing problem; numerical examples demonstrate the efficiency of the method.

[198]  arXiv:1803.06996 [pdf, ps, other]
Title: Existence and Ergodic properties of equilibrium measures for maps associated with inducing schemes of hyperbolic type
Subjects: Dynamical Systems (math.DS)

We consider maps $f:X\to X$ admitting inducing schemes of hyperbolic type introduced in \cite{ind} as well as the induced maps $\tilde{f}:\tilde{X}\to \tilde{X}$ and the associated tower maps $\hat{f}:\hat{X} \to \hat {X}$. For a certain class of potential functions $\varphi$ on $X$, that includes all H\"older continuous functions, we establish thermodynamic formalism for the above three systems. We study relations among the corresponding equilibrium measures and their ergodic properties. We establish decay of correlations, the Central Limit Theorem (CLT), the Bernoulli property for the three systems with respect to their corresponding equilibrium measures. Finally, we prove analyticity of the pressure function for the three systems.

[199]  arXiv:1803.07001 [pdf, other]
Title: A short survey on Newton polytopes, tropical geometry and ring of conditions of algebraic torus
Comments: 15 pages, 4 figures
Subjects: Algebraic Geometry (math.AG)

The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in several variables over complex numbers. The exposition is aimed for a general audience in mathematics and we hope to be accessible to undergraduate as well as advance high school students. The topics discussed belong to relatively new, and closely related branches of algebraic geometry which are usually referred to as tropical geometry and toric geometry. These areas make connections between the study of algebra and geometry of polynomials and the combinatorial and convex geometric study of piecewise linear functions. The main results discussed in this note are descriptions of the so-called "ring of conditions" of algebraic torus.

[200]  arXiv:1803.07002 [pdf, ps, other]
Title: Auslander-Reiten $(d+2)$-angles in subcategories and a $(d+2)$-angulated generalisation of a theorem by Brüning
Authors: Francesca Fedele
Subjects: Representation Theory (math.RT)

Let $\Phi$ be a finite dimensional algebra over an algebraically closed field $k$ and assume gldim$\,\Phi\leq d$, for some fixed positive integer $d$. For $d=1$, Br\"uning proved that there is a bijection between the wide subcategories of the abelian category mod$\,\Phi$ and those of the triangulated category $\mathcal{D}^b(\text{mod}\Phi)$. Moreover, for a suitable triangulated category $\mathcal{M}$, J{\o}rgensen gave a description of Auslander-Reiten triangles in the extension closed subcategories of $\mathcal{M}$.
In this paper, we generalise these results for $d$-abelian and $(d+2)$-angulated categories, where kernels and cokernels are replaced by complexes of $d+1$ objects and triangles are replaced by complexes of $d+2$ objects. The categories are obtained as follows: if $\mathcal{F}\subseteq \text{mod} \Phi$ is a $d$-cluster tilting subcategory, consider $\overline{\mathcal{F}}:=\text{add} \{\Sigma^{id}\mathcal{F}\mid i\in\mathbb{Z} \}\subseteq \mathcal{D}^b(\text{mod}\Phi)$. Then $\mathcal{F}$ is $d$-abelian and plays the role of a higher mod$\,\Phi$ having for higher derived category the $(d+2)$-angulated category $\overline{\mathcal{F}}$.

[201]  arXiv:1803.07005 [pdf, ps, other]
Title: Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions
Authors: Jonas M. Tölle
Comments: 25 pages, 53 references
Subjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA); Probability (math.PR)

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the $d$-dimensional torus with singular $p$-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with $C^{1}$-vector field coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative It\^o-formula.

[202]  arXiv:1803.07011 [pdf]
Title: Radio bearing of sources with directional antennas in urban environment
Authors: Cezary Ziolkowski, Jan M. Kelner (Military University of Technology, Faculty of Electronics, Institute of Telecommunications, Warsaw, Poland)
Comments: 24 pages, 16 figures, 3 tables; Accepted in International Journal of Microwave and Wireless Technologies (from Cambridge University Press)
Subjects: Information Theory (cs.IT)

This paper focuses on assessing the limitations in the direction-finding process of radio sources with directional antennas in an urbanized environment, demonstrating how signal source antenna parameters, such as beamwidth and maximum radiation direction affect bearing accuracy in non-line-of-sight (NLOS) conditions. These evaluations are based on simulation studies, which use measurement-tested signal processing procedures. These procedures are based on a multi-elliptical propagation model, the geometry of which is related to the environment by the power delay profile or spectrum. The probability density function of the angle of arrival for different parameters of the transmitting antenna is the simulation result. This characteristic allows assessing the effect of the signal source antenna parameters on bearing error. The obtained results are the basis for practical correction bearing error and these show the possibility of improving the efficiency of the radio source location in the urbanized environment.

[203]  arXiv:1803.07017 [pdf, ps, other]
Title: A Positive Proportion of Hasse Principle Failures in a Family of Châtelet Surfaces
Authors: Nick Rome
Comments: 13 pages, comments welcome
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)

We investigate the family of surfaces defined by the affine equation $$Y^2 + Z^2 = (aT^2 + b)(cT^2 +d)$$ where $\vert ad-bc \vert=1$ and develop asymptotic formulae for the frequency of Hasse principle failures. We show that a positive proportion (roughly 23.7%) of such surfaces fail the Hasse principle, by building on previous work of la Bret\`{e}che and Browning.

[204]  arXiv:1803.07020 [pdf, ps, other]
Title: An efficient algorithm for packing cuts and (2,3)-metrics in a planar graph with three holes
Comments: 25 pages, 10 figures
Subjects: Combinatorics (math.CO)

We consider a planar graph $G$ in which the edges have nonnegative integer lengths such that the length of every cycle of $G$ is even, and three faces are distinguished, called holes in $G$. It is known that there exists a packing of cuts and (2,3)-metrics with nonnegative integer weights in $G$ which realizes the distances within each hole. We develop a strongly polynomial purely combinatorial algorithm to find such a packing.

[205]  arXiv:1803.07024 [pdf, ps, other]
Title: A note on vague convergence of measures
Comments: 15 pages
Subjects: Probability (math.PR)

We propose a notion of convergence of measures with intention of generalizing and unifying several frequently used types of vague convergence. We explain that by general theory of boundedness due to Hu (1966), in Polish spaces, this notion of convergence can be always formulated as follows: $\mu_n \stackrel{v}{\longrightarrow} \mu$ if $\int f d\mu_n \to \int f d\mu$ for all continuous bounded functions $f$ with support bounded in some suitably chosen metric. This brings all the related types of vague convergence into the framework of Daley and Vere-Jones (2003) and Kallenberg (2017). In the rest of the note we discuss the vague topology and the corresponding notion of convergence in distribution, complementing the theory developed in those two references.

[206]  arXiv:1803.07027 [pdf, other]
Title: Brownian Motions on Star Graphs with Non-Local Boundary Conditions
Authors: Florian Werner
Subjects: Probability (math.PR)

Brownian motions on star graphs in the sense of It\^o-McKean, that is, Walsh processes admitting a generalized boundary behavior including stickiness and jumps and having an angular distribution with finite support, are examined. Their generators are identified as Laplace operators on the graph subject to non-local Feller-Wentzell boundary conditions. A pathwise description is achieved for every admissible boundary condition: For finite jump measures, a construction of Kostrykin, Potthoff and Schrader in the continuous setting is expanded via a technique of successive killings and revivals; for infinite jump measures, the pathwise solution of It\^o-McKean for the half line is analyzed and extended to the star graph. These processes can then be used as main building blocks for Brownian motions on general metric graphs with non-local boundary conditions.

[207]  arXiv:1803.07033 [pdf, other]
Title: Natural gradient via optimal transport I
Subjects: Optimization and Control (math.OC); Information Theory (cs.IT)

We study a natural Wasserstein gradient flow on manifolds of probability distributions with discrete sample spaces. We derive the Riemannian structure for the probability simplex from the dynamical formulation of the Wasserstein distance on a weighted graph. We pull back the geometric structure to the parameter space of any given probability model, which allows us to define a natural gradient flow there. In contrast to the natural Fisher-Rao gradient, the natural Wasserstein gradient incorporates a ground metric on sample space. We discuss implementations following the forward and backward Euler methods. We illustrate the analysis on elementary exponential family examples.

[208]  arXiv:1803.07042 [pdf, ps, other]
Title: On the $k$-independence number of graphs
Subjects: Combinatorics (math.CO)

This paper improves and unifies the existing spectral bounds on the $k$-independence number of a graph, which is the maximum size of a set of vertices at pairwise distance greater than $k$. The previous bounds known in the literature follow as a corollary of the main results in this work.

[209]  arXiv:1803.07043 [pdf, other]
Title: Projective Splitting with Forward Steps: Asynchronous and Block-Iterative Operator Splitting
Subjects: Optimization and Control (math.OC); Learning (cs.LG)

This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the same kind of coordination procedure and can be implemented in the same block-iterative and potentially distributed and asynchronous manner, but may perform backward steps on some operators and forward steps on others. Prior algorithms in the projective splitting family have used only backward steps. Forward steps can be used for any Lipschitz-continuous operators provided the stepsize is bounded by the inverse of the Lipschitz constant. If the Lipschitz constant is unknown, a simple backtracking linesearch procedure may be used. For affine operators, the stepsize can be chosen adaptively without knowledge of the Lipschitz constant and without any additional forward steps. We close the paper by empirically studying the performance of several kinds of splitting algorithms on the lasso problem.

[210]  arXiv:1803.07044 [pdf, other]
Title: Positive neighborhoods of curves
Authors: M. Falla Luza, P. Sad
Subjects: Complex Variables (math.CV)

In this work we study neighborhoods of curves in surfaces with positive self-intersection that can be embeeded as a germ of neighborhood of a curve on the projective plane.

[211]  arXiv:1803.07046 [pdf, other]
Title: Time-Domain Multi-Beam Selection and Its Performance Improvement for mmWave Systems
Comments: Accepted by IEEE International Conference on Communications (ICC), Kansas City, MO, USA, May 2018
Subjects: Information Theory (cs.IT)

Multi-beam selection is one of the crucial technologies in hybrid beamforming systems for frequency-selective fading channels. Addressing the problem in the frequency domain facilitates the procedure of acquiring observations for analog beam selection. However, it is difficult to improve the quality of the contaminated observations at low SNR. To this end, this paper uses an idea that the significant observations are sparse in the time domain to further enhance the quality of signals as well as the beam selection performance. By exploiting properties of channel impulse responses and circular convolutions in the time domain, we can reduce the size of a Toeplitz matrix in deconvolution to generate periodic true values of coupling coefficients plus random noise signals. An arithmetic mean of these signals yields refined observations with minor noise effects and provides more accurate sparse multipath delay information. As a result, only the refined observations associated with the estimated multipath delay indices have to be taken into account for the analog beam selection problem.

[212]  arXiv:1803.07049 [pdf, ps, other]
Title: Kinematics and Dynamics of Quantum Walks in terms of Systems of Imprimitivity
Comments: 16 pages, 3 figures
Subjects: Mathematical Physics (math-ph)

We build systems of imprimitivity (SI) in the context of quantum walks and provide geometric constructions for their configuration space. We consider three systems, an evolution of unitaries from the group SO3 on a low dimensional de Sitter space where the walk happens on the dual of SO3, standard quantum walk whose SI live on the orbits of stabilizer subgroups (little groups) of semidirect products describing the symmetries of 1+1 spacetime, and automorphisms (walks are specific automorphisms) on distant-transitive graphs as application of the constructions.

[213]  arXiv:1803.07053 [pdf, ps, other]
Title: Attack-Resilient H2, H-infinity, and L1 State Estimator
Subjects: Optimization and Control (math.OC)

This paper considers the secure state estimation problem for noisy systems in the presence of sparse sensor integrity attacks. We show a fundamental limitation: that is, 2r-detectability is necessary for achieving bounded estimation errors, where r is the number of attacks. This condition is weaker than the 2r-observability condition typically assumed in the literature. Conversely, we propose a real-time state estimator that achieves the fundamental limitation. The proposed state estimator is inspired by robust control and FDI: that is, it consists of local Luenberger estimators, local residual detectors, and a global fusion process. We show its performance guarantees for H2, H-infinity, and L1 systems. Finally, numerical examples show that it has relatively low estimation errors among existing algorithms and average computation time for systems with a sufficiently small number of compromised sensors.

[214]  arXiv:1803.07054 [pdf, ps, other]
Title: Optimal link prediction with matrix logistic regression
Subjects: Statistics Theory (math.ST); Machine Learning (stat.ML)

We consider the problem of link prediction, based on partial observation of a large network, and on side information associated to its vertices. The generative model is formulated as a matrix logistic regression. The performance of the model is analysed in a high-dimensional regime under a structural assumption. The minimax rate for the Frobenius-norm risk is established and a combinatorial estimator based on the penalised maximum likelihood approach is shown to achieve it. Furthermore, it is shown that this rate cannot be attained by any (randomised) algorithm computable in polynomial time under a computational complexity assumption.

[215]  arXiv:1803.07062 [pdf, ps, other]
Title: Asymptotic behaviour of neuron population models structured by elapsed-time
Subjects: Analysis of PDEs (math.AP)

We study two population models describing the dynamics of interacting neurons, initially proposed by Pakdaman, Perthame, and Salort (2010, 2014). In the first model, the structuring variable $s$ represents the time elapsed since its last discharge, while in the second one neurons exhibit a fatigue property and the structuring variable is a generic "state". We prove existence of solutions and steady states in the space of finite, nonnegative measures. Furthermore, we show that solutions converge to the equilibrium exponentially in time in the case of weak nonlinearity (i.e., weak connectivity). The main innovation is the use of Doeblin's theorem from probability in order to show the existence of a spectral gap property in the linear (no-connectivity) setting. Relaxation to the steady state for the nonlinear models is then proved by a constructive perturbation argument.

Cross-lists for Tue, 20 Mar 18

[216]  arXiv:1803.05999 (cross-list from cs.LG) [pdf, other]
Title: Escaping Saddles with Stochastic Gradients
Subjects: Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)

We analyze the variance of stochastic gradients along negative curvature directions in certain non-convex machine learning models and show that stochastic gradients exhibit a strong component along these directions. Furthermore, we show that - contrary to the case of isotropic noise - this variance is proportional to the magnitude of the corresponding eigenvalues and not decreasing in the dimensionality. Based upon this observation we propose a new assumption under which we show that the injection of explicit, isotropic noise usually applied to make gradient descent escape saddle points can successfully be replaced by a simple SGD step. Additionally - and under the same condition - we derive the first convergence rate for plain SGD to a second-order stationary point in a number of iterations that is independent of the problem dimension.

[217]  arXiv:1803.06377 (cross-list from cs.SI) [pdf, other]
Title: Spread of Information with Confirmation Bias in Cyber-Social Networks
Subjects: Social and Information Networks (cs.SI); Multiagent Systems (cs.MA); Systems and Control (cs.SY); Optimization and Control (math.OC)

This paper provides a model to investigate information spreading over cyber-social network of agents communicating with each other. The cyber-social network considered here comprises individuals and news agencies. Each individual holds a belief represented by a scalar. Individuals receive information from news agencies that are closer to their belief, confirmation bias is explicitly incorporated into the model. The proposed dynamics of cyber-social networks is adopted from DeGroot-Friedkin model, where the individual's opinion update mechanism is a convex combination of his innate opinion, his neighbors' opinions at the previous time step (obtained from the social network), and the opinions passed along by news agencies from cyber layer which he follows. The characteristics of the interdependent social and cyber networks are radically different here: the social network relies on trust and hence static while the news agencies are highly dynamic since they are weighted as a function of the distance between an individual state and the state of news agency to account for confirmation bias. The conditions for convergence of the aforementioned dynamics to a unique equilibrium are characterized. The estimation and exact computation of the steady-state values under non-linear and linear state-dependent weight functions are provided. Finally, the impact of polarization in the opinions of news agencies on the public opinion evolution is numerically analyzed in the context of the well-known Krackhardt's advice network.

[218]  arXiv:1803.06460 (cross-list from q-fin.PM) [pdf, other]
Title: Mean Reverting Portfolios via Penalized OU-Likelihood Estimation
Comments: 7 pages, 6 figures
Subjects: Portfolio Management (q-fin.PM); Optimization and Control (math.OC); Machine Learning (stat.ML)

We study an optimization-based approach to con- struct a mean-reverting portfolio of assets. Our objectives are threefold: (1) design a portfolio that is well-represented by an Ornstein-Uhlenbeck process with parameters estimated by maximum likelihood, (2) select portfolios with desirable characteristics of high mean reversion and low variance, and (3) select a parsimonious portfolio, i.e. find a small subset of a larger universe of assets that can be used for long and short positions. We present the full problem formulation, a specialized algorithm that exploits partial minimization, and numerical examples using both simulated and empirical price data.

[219]  arXiv:1803.06476 (cross-list from quant-ph) [pdf, other]
Title: PT-symmetric phase transition, hysteresis and bound states in the continuum
Comments: 8 pages, 2 figures
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)

We explicate the merging of levels and spectral bifurcation near an exceptional point as also the appearance of resonances and bound states in the PT-broken phase for the PT-symmetric complex Scarf II potential. The bound states in the PT-broken phase are manifested as spectral singularities and are found to be bound states in continuum and thus seen as zero width resonances, starting to first appear at the exceptional points. The intimate connection of PT-symmetry breaking and breaking of supersymmetry is pointed out as also the phenomenon of hysteresis near exceptional point. Intriguingly, the PT-symmetric Hamiltonians related by SUSY are also found to be isospectrally deformed counterparts for a specific parametric condition with the deformation satisfying the Korteweg-deVries equation.

[220]  arXiv:1803.06505 (cross-list from stat.CO) [pdf, ps, other]
Title: A simulated annealing procedure based on the ABC Shadow algorithm for statistical inference of point processes
Subjects: Computation (stat.CO); Statistics Theory (math.ST)

Recently a new algorithm for sampling posteriors of unnormalised probability densities, called ABC Shadow, was proposed in [8]. This talk introduces a global optimisation procedure based on the ABC Shadow simulation dynamics. First the general method is explained, and then results on simulated and real data are presented. The method is rather general, in the sense that it applies for probability densities that are continuously differentiable with respect to their parameters

[221]  arXiv:1803.06510 (cross-list from stat.ML) [pdf, other]
Title: Hidden Integrality of SDP Relaxation for Sub-Gaussian Mixture Models
Subjects: Machine Learning (stat.ML); Information Theory (cs.IT); Learning (cs.LG); Optimization and Control (math.OC); Statistics Theory (math.ST)

We consider the problem of estimating the discrete clustering structures under Sub-Gaussian Mixture Models. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solutions to the SDP are not integer-valued in general, their estimation errors can be upper bounded in terms of the error of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. To the best of our knowledge, this is the first exponentially decaying error bound for convex relaxations of mixture models, and our results reveal the "global-to-local" mechanism that drives the performance of the SDP relaxation.
A corollary of our results shows that in certain regimes the SDP solutions are in fact integral and exact, improving on existing exact recovery results for convex relaxations. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of $(1-\delta)$ fraction of the points for any $\delta\in(0,1)$. As a special case, we show that under the $d$-dimensional Stochastic Ball Model, SDP achieves non-trivial (sometimes exact) recovery when the center separation is as small as $\sqrt{1/d}$, which complements previous exact recovery results that require constant separation.

[222]  arXiv:1803.06511 (cross-list from nlin.SI) [pdf, ps, other]
Title: On discretization of the Euler top
Authors: A.V. Tsiganov
Comments: 12 pages, 2 figures, LaTeX with AMS fonts
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph); Dynamical Systems (math.DS)

Application of the intersection theory to construction of n-point finite-difference equations associated with classical integrable systems is discussed. As an example, we present a few new discretizations of motion of the Euler top sharing the integrals of motion with the continuous time system and the Poisson bracket up to the integer scaling factor.

[223]  arXiv:1803.06515 (cross-list from quant-ph) [pdf, ps, other]
Title: From nonholonomic quantum constraint to canonical variables of photons I: true intrinsic degree of freedom
Comments: 18 pages, 1 figure
Subjects: Quantum Physics (quant-ph); Representation Theory (math.RT); Optics (physics.optics)

We report that the true intrinsic degree of freedom of the photon is neither the polarization nor the spin. It describes a local property in momentum space and is represented in the local representation by the Pauli matrices. This result is achieved by treating the transversality condition on the vector wavefunction as a nonholonomic quantum constraint. We find that the quantum constraint makes it possible to generalize the Stokes parameters to characterize the polarization of a general state. Unexpectedly, the generalized Stokes parameters are specified in a momentum-space local reference system that is fixed by another degree of freedom, called Stratton vector. Only constant Stokes parameters in one particular local reference system can convey the intrinsic degree of freedom of the photon. We show that the optical rotation is one of such processes that change the Stratton vector with the intrinsic quantum number remaining fixed. Changing the Stratton vector of the eigenstate of the helicity will give rise to a Berry's phase.

[224]  arXiv:1803.06531 (cross-list from cs.SY) [pdf, ps, other]
Title: Topology Estimation using Graphical Models in Multi-Phase Power Distribution Grids
Comments: 12 pages 9 figures
Subjects: Systems and Control (cs.SY); Optimization and Control (math.OC); Machine Learning (stat.ML)

Distribution grid is the medium and low voltage part of a large power system. Structurally, the majority of distribution networks operate radially, such that energized lines form a collection of trees, i.e. forest, with a substation being at the root of any tree. The operational topology/forest may change from time to time, however tracking these changes, even though important for the distribution grid operation and control, is hindered by limited real-time monitoring. This paper develops a learning framework to reconstruct radial operational structure of the distribution grid from synchronized voltage measurements in the grid subject to the exogenous fluctuations in nodal power consumption. To detect operational lines our learning algorithm uses conditional independence tests for continuous random variables that is applicable to a wide class of probability distributions of the nodal consumption and Gaussian injections in particular. Moreover, our algorithm applies to the practical case of unbalanced three-phase power flow. Algorithm performance is validated on AC power flow simulations over IEEE distribution grid test cases.

[225]  arXiv:1803.06539 (cross-list from cs.DM) [pdf, other]
Title: The Graph Structure of Chebyshev Polynomials over Finite Fields and Applications
Subjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO)

We completely describe the functional graph associated to iterations of Chebyshev polynomials over finite fields. Then, we use our structural results to obtain estimates for the average rho length, average number of connected components and the expected value for the period and preperiod of iterating Chebyshev polynomials.

[226]  arXiv:1803.06584 (cross-list from nlin.SI) [pdf, ps, other]
Title: Linear Instability of the Peregrine Breather: Numerical and Analytical Investigations
Comments: 12 pages, 4 figures
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)

We study the linear stability of the Peregrine breather both numerically and with analytical arguments based on its derivation as the singular limit of a single-mode spatially periodic breather as the spatial period becomes infinite. By constructing solutions of the linearization of the nonlinear Schr\"odinger equation in terms of quadratic products of components of the eigenfunctions of the Zakharov-Shabat system, we show that the Peregrine breather is linearly unstable. A numerical study employing a highly accurate Chebychev pseudo-spectral integrator confirms exponential growth of random initial perturbations of the Peregrine breather.

[227]  arXiv:1803.06675 (cross-list from stat.ME) [pdf, other]
Title: Rare Feature Selection in High Dimensions
Comments: 32 pages, 9 figures
Subjects: Methodology (stat.ME); Statistics Theory (math.ST); Computation (stat.CO); Machine Learning (stat.ML)

It is common in modern prediction problems for many predictor variables to be counts of rarely occurring events. This leads to design matrices in which many columns are highly sparse. The challenge posed by such "rare features" has received little attention despite its prevalence in diverse areas, ranging from natural language processing (e.g., rare words) to biology (e.g., rare species). We show, both theoretically and empirically, that not explicitly accounting for the rareness of features can greatly reduce the effectiveness of an analysis. We next propose a framework for aggregating rare features into denser features in a flexible manner that creates better predictors of the response. Our strategy leverages side information in the form of a tree that encodes feature similarity.
We apply our method to data from TripAdvisor, in which we predict the numerical rating of a hotel based on the text of the associated review. Our method achieves high accuracy by making effective use of rare words; by contrast, the lasso is unable to identify highly predictive words if they are too rare. A companion R package, called rare, implements our new estimator, using the alternating direction method of multipliers.

[228]  arXiv:1803.06687 (cross-list from quant-ph) [pdf, other]
Title: Sub-Riemannian Geodesics on SU(n)/S(U(n-1)xU(1)) and Optimal Control of Three Level Quantum Systems
Subjects: Quantum Physics (quant-ph); Optimization and Control (math.OC)

We study the time optimal control problem for the evolution operator of an n-level quantum system from the identity to any desired final condition. For the considered class of quantum systems the control couples all the energy levels to a given one and is assumed to be bounded in Euclidean norm. From a mathematical perspective, such a problem is a sub-Riemannian K-P problem, whose underlying symmetric space is SU(n)/S(U(n-1) x U(1)). Following the method of symmetry reduction, we consider the action of S(U(n-1) xU(1)) on SU(n) as a conjugation X ---> AXA^{-1}. This allows us to do a symmetry reduction and consider the problem on a quotient space. We give an explicit description of such a quotient space which has the structure of a stratified space. We prove several properties of sub-Riemannian problems with the given structure. We derive the explicit optimal control for the case of three level quantum systems where the desired operation is on the lowest two energy levels (Lambda-systems). We solve this latter problem by reducing it to an integer quadratic optimization problem with linear constraints.

[229]  arXiv:1803.06689 (cross-list from quant-ph) [pdf, ps, other]
Title: Controllability of Symmetric Spin Networks
Subjects: Quantum Physics (quant-ph); Optimization and Control (math.OC)

We consider a network of n spin 1/2 systems which are pairwise interacting via Ising interaction and are controlled by the same electro-magnetic control field. Such a system presents symmetries since the Hamiltonian is unchanged if we permute two spins. This prevents full (operator) controllability in that not every unitary evolution can be obtained. We prove however that controllability is verified if we restrict ourselves to unitary evolutions which preserve the above permutation invariance. For low dimensional cases, n=2 and n=3, we provide an analysis of the Lie group of available evolutions and give explicit control laws to transfer between any two permutation invariant states. This class of states includes highly entangled states such as GHZ states and W states, which are of interest in quantum information.

[230]  arXiv:1803.06714 (cross-list from physics.comp-ph) [pdf, other]
Title: Mathematics for cryo-electron microscopy
Authors: Amit Singer
Comments: Proceedings of the International Congress of Mathematicians 2018
Subjects: Computational Physics (physics.comp-ph); History and Overview (math.HO)

Single-particle cryo-electron microscopy (cryo-EM) has recently joined X-ray crystallography and NMR spectroscopy as a high-resolution structural method for biological macromolecules. Cryo-EM was selected by Nature Methods as Method of the Year 2015, large scale investments in cryo-EM facilities are being made all over the world, and the Nobel Prize in Chemistry 2017 was awarded to Jacques Dubochet, Joachim Frank and Richard Henderson "for developing cryo-electron microscopy for the high-resolution structure determination of biomolecules in solution". This paper focuses on the mathematical principles underlying existing algorithms for structure determination using single particle cryo-EM.

[231]  arXiv:1803.06727 (cross-list from cs.LG) [pdf, other]
Title: Aggregating Strategies for Long-term Forecasting
Comments: 20 pages, 4 figures
Subjects: Learning (cs.LG); Probability (math.PR); Statistics Theory (math.ST); Machine Learning (stat.ML)

The article is devoted to investigating the application of aggregating algorithms to the problem of the long-term forecasting. We examine the classic aggregating algorithms based on the exponential reweighing. For the general Vovk's aggregating algorithm we provide its generalization for the long-term forecasting. For the special basic case of Vovk's algorithm we provide its two modifications for the long-term forecasting. The first one is theoretically close to an optimal algorithm and is based on replication of independent copies. It provides the time-independent regret bound with respect to the best expert in the pool. The second one is not optimal but is more practical and has $O(\sqrt{T})$ regret bound, where $T$ is the length of the game.

[232]  arXiv:1803.06764 (cross-list from hep-th) [pdf, other]
Title: Exact holographic RG flows and the $A_{1}\times A_{1}$ Toda chain
Comments: latex, 55 pages, 34 figures
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)

We construct analytic solutions of Einstein gravity coupled to a dilaton field with a potential given by a sum of two exponentials, by rewriting the equations of motion in terms of an integrable Toda chain. Such solutions can be interpreted as domain walls interpolating between different asymptotics, and as such they can have interesting applications in holography. In some cases we can find flows that start from an AdS fixed point. We also find analytic black brane solutions at finite temperature. We discuss the properties of the solutions and the interpretation in terms of RG flow.

[233]  arXiv:1803.06800 (cross-list from cs.CC) [pdf, other]
Title: Computational topology and the Unique Games Conjecture
Comments: Full version of a conference paper in 34th International Symposium on Computational Geometry (SoCG 2018)
Subjects: Computational Complexity (cs.CC); Computational Geometry (cs.CG); Discrete Mathematics (cs.DM); Algebraic Topology (math.AT)

Covering spaces of graphs have long been useful for studying expanders (as "graph lifts") and unique games (as the "label-extended graph"). In this paper we advocate for the thesis that there is a much deeper relationship between computational topology and the Unique Games Conjecture. Our starting point is Linial's 2005 observation that the only known problems whose inapproximability is equivalent to the Unique Games Conjecture - Unique Games and Max-2Lin - are instances of Maximum Section of a Covering Space on graphs. We then observe that the reduction between these two problems (Khot-Kindler-Mossel-O'Donnell, FOCS 2004; SICOMP, 2007) gives a well-defined map of covering spaces. We further prove that inapproximability for Maximum Section of a Covering Space on (cell decompositions of) closed 2-manifolds is also equivalent to the Unique Games Conjecture. This gives the first new "Unique Games-complete" problem in over a decade.
Our results partially settle an open question of Chen and Freedman (SODA 2010; Disc. Comput. Geom., 2011) from computational topology, by showing that their question is almost equivalent to the Unique Games Conjecture. (The main difference is that they ask for inapproximability over $\mathbb{Z}/2\mathbb{Z}$, and we show Unique Games-completeness over $\mathbb{Z}/k\mathbb{Z}$ for large $k$.) This equivalence comes from the fact that when the structure group $G$ of the covering space is Abelian - or more generally for principal $G$-bundles - Maximum Section of a $G$-Covering Space is the same as the well-studied problem of 1-Homology Localization.
Although our most technically demanding result is an application of Unique Games to computational topology, we hope that our observations on the topological nature of the Unique Games Conjecture will lead to applications of algebraic topology to the Unique Games Conjecture in the future.

[234]  arXiv:1803.06829 (cross-list from cond-mat.stat-mech) [pdf, ps, other]
Title: Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process
Comments: 6 pages, 3 figures
Subjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Probability (math.PR)

We consider current statistics for a two species exclusion process of particles hopping in opposite directions on a one-dimensional lattice. We derive an exact formula for the Green's function as well as for a joint current distribution of the model, and study its long time behavior. For a step type initial condition, we show that the limiting distribution is a product of the Gaussian and the GUE Tracy-Widom distribution. This is the first analytic confirmation for a multi-component system of a prediction from the recently proposed non-linear fluctuating hydrodynamics for one dimensional systems.

[235]  arXiv:1803.06857 (cross-list from cond-mat.stat-mech) [pdf, other]
Title: Anomalous heat equation in a system connected to thermal reservoirs
Comments: Main text: 5 pages. Supplementary: 9 page. 5 Figures
Subjects: Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Chaotic Dynamics (nlin.CD)

We study anomalous transport in a one-dimensional system with two conserved quantities in presence of thermal baths. In this system we derive exact expressions of the temperature profile and the two point correlations in steady state as well as in the non-stationary state where the later describes the relaxation to the steady state. In contrast to the Fourier heat equation in the diffusive case, here we show that the evolution of the temperature profile is governed by a non-local anomalous heat equation. We provide numerical verifications of our results.

[236]  arXiv:1803.06858 (cross-list from cs.DS) [pdf, ps, other]
Title: An improved isomorphism test for bounded-tree-width graphs
Comments: 34 pages, 1 figures
Subjects: Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Combinatorics (math.CO)

We give a new fpt algorithm testing isomorphism of $n$-vertex graphs of tree width $k$ in time $2^{k\operatorname{polylog} (k)}\operatorname{poly} (n)$, improving the fpt algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time $2^{\mathcal{O}(k^5\log k)}\operatorname{poly} (n)$. Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree width $k$. Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai's algorithm as a black box in one place.
We also give a second algorithm which, at the price of a slightly worse running time $2^{\mathcal{O}(k^2 \log k)}\operatorname{poly} (n)$, avoids the use of Babai's algorithm and, more importantly, has the additional benefit that it can also used as a canonization algorithm.

[237]  arXiv:1803.06920 (cross-list from nlin.CD) [pdf, other]
Title: On the Keldysh Problem of Flutter Suppression
Subjects: Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS)

This work is devoted to the Keldysh model of flutter suppression and rigorous approaches to its analysis. To solve the stabilization problem in the Keldysh model we use an analog of direct Lyapunov method for differential inclusions. The results obtained here are compared with the results of Keldysh obtained by the method of harmonic balance (describing function method), which is an approximate method for analyzing the existence of periodic solutions. The limitations of the use of describing function method for the study of systems with dry friction and stationary segment are demonstrated.

[238]  arXiv:1803.06921 (cross-list from cs.SY) [pdf, ps, other]
Title: Approximating Flexibility in Distributed Energy Resources: A Geometric Approach
Comments: accepted for presentation at the Power Systems Computations Conference 2018
Subjects: Systems and Control (cs.SY); Optimization and Control (math.OC)

With increasing availability of communication and control infrastructure at the distribution systems, it is expected that the distributed energy resources (DERs) will take an active part in future power systems operations. One of the main challenges associated with integration of DERs in grid planning and control is in estimating the available flexibility in a collection of (heterogeneous) DERs, each of which may have local constraints that vary over time. In this work, we present a geometric approach for approximating the flexibility of a DER in modulating its active and reactive power consumption. The proposed method is agnostic about the type and model of the DERs, thereby facilitating a plug-and-play approach, and allows scalable aggregation of the flexibility of a collection of (heterogeneous) DERs at the distributed system level. Simulation results are presented to demonstrate the performance of the proposed method.

[239]  arXiv:1803.06922 (cross-list from q-fin.RM) [pdf, ps, other]
Title: Approximation of Some Multivariate Risk Measures for Gaussian Risks
Authors: E. Hashorva
Subjects: Risk Management (q-fin.RM); Probability (math.PR)

Gaussian random vectors exhibit the loss of dimension phenomena, which relate to their joint survival tail behaviour. Besides, the fact that the components of such vectors are light-tailed complicates the approximations of various multivariate risk measures significantly. In this contribution we derive precise approximations of marginal mean excess, marginal expected shortfall and multivariate conditional tail expectation of Gaussian random vectors and highlight links with conditional limit theorems. Our study indicates that similar results hold for elliptical and Gaussian like multivariate risks.

[240]  arXiv:1803.06934 (cross-list from cs.MS) [pdf, ps, other]
Title: PyGOM - A Python Package for Simplifying Modelling with Systems of Ordinary Differential Equations
Comments: 23 pages, 6 figures
Subjects: Mathematical Software (cs.MS); Classical Analysis and ODEs (math.CA)

Ordinary Differential Equations (ODE) are used throughout science where the capture of rates of change in states is sought. While both pieces of commercial and open software exist to study such systems, their efficient and accurate usage frequently requires deep understanding of mathematics and programming. The package we present here, PyGOM, seeks to remove these obstacles for models based on ODE systems. We provide a simple interface for the construction of such systems backed by a comprehensive and easy to use tool--box. This tool--box implements functions to easily perform common operations for ODE systems such as solving, parameter estimation, and stochastic simulation. The package source is freely available and organized in a way that permits easy extension. With both the algebraic and numeric calculations performed automatically (but still accessible), the end user is freed to focus on model development.

[241]  arXiv:1803.06938 (cross-list from hep-th) [pdf, other]
Title: Conformal amplitude hierarchy and the Poincare disk
Authors: Hirohiko Shimada
Comments: 13 pages, 2 figures. see this version; corrections made; references added
Journal-ref: J. Phys.: Conf. Ser. 965 (2018) 012036
Subjects: High Energy Physics - Theory (hep-th); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

The amplitude for the singlet channels in the 4-point function of the fundamental field in the conformal field theory of the 2d $O(n)$ model is studied as a function of $n$. For a generic value of $n$, the 4-point function has infinitely many amplitudes, whose landscape can be very spiky as the higher amplitude changes its sign many times at the simple poles, which generalize the unique pole of the energy operator amplitude at $n=0$. In the stadard parameterization of $n$ by angle in unit of $\pi$, we find that the zeros and poles happen at the rational angles, forming a hierarchical tree structure inherent in the Poincar\'{e} disk. Some relation between the amplitude and the Farey path, a piecewise geodesic that visits these zeros and poles, is suggested. In this hierarchy, the symmetry of the congruence subgroup $\Gamma(2)$ of $SL(2,\mathbb{Z})$ naturally arises from the two clearly distinct even/odd classes of the rational angles, in which one respectively gets the truncated operator algebras and the logarithmic 4-point functions.

[242]  arXiv:1803.06971 (cross-list from stat.ML) [pdf, other]
Title: What Doubling Tricks Can and Can't Do for Multi-Armed Bandits
Authors: Lilian Besson (IETR), Emilie Kaufmann (SEQUEL, CNRS)
Subjects: Machine Learning (stat.ML); Learning (cs.LG); Statistics Theory (math.ST)

An online reinforcement learning algorithm is anytime if it does not need to know in advance the horizon T of the experiment. A well-known technique to obtain an anytime algorithm from any non-anytime algorithm is the "Doubling Trick". In the context of adversarial or stochastic multi-armed bandits, the performance of an algorithm is measured by its regret, and we study two families of sequences of growing horizons (geometric and exponential) to generalize previously known results that certain doubling tricks can be used to conserve certain regret bounds. In a broad setting, we prove that a geometric doubling trick can be used to conserve (minimax) bounds in $R\_T = O(\sqrt{T})$ but cannot conserve (distribution-dependent) bounds in $R\_T = O(\log T)$. We give insights as to why exponential doubling tricks may be better, as they conserve bounds in $R\_T = O(\log T)$, and are close to conserving bounds in $R\_T = O(\sqrt{T})$.

[243]  arXiv:1803.06982 (cross-list from quant-ph) [pdf, other]
Title: Quantifying coherence with quantum addition
Comments: 6 pages + references, comments/suggestions most welcome
Subjects: Quantum Physics (quant-ph); Information Theory (cs.IT)

Quantum addition channels have been recently introduced in the context of deriving entropic power inequalities for finite dimensional quantum systems. We prove a reverse entropy power equality which can be used to analytically prove an inequality conjectured recently for arbitrary dimension and arbitrary addition weight. We show that the relative entropic difference between the output of such a quantum additon channel and the corresponding classical mixture quantitatively captures the amount of coherence present in a quantum system. This new coherence measure admits an upper bound in terms of the relative entropy of coherence and is utilized to formulate a state-dependent uncertainty relation for two observables. Our results may provide deep insights to the origin of quantum coherence for mixed states that truly come from the discrepancy between quantum addition and the classical mixture.

[244]  arXiv:1803.07055 (cross-list from cs.LG) [pdf, other]
Title: Simple random search provides a competitive approach to reinforcement learning
Comments: 22 pages, 5 figures, 9 tables
Subjects: Learning (cs.LG); Artificial Intelligence (cs.AI); Optimization and Control (math.OC); Machine Learning (stat.ML)

A common belief in model-free reinforcement learning is that methods based on random search in the parameter space of policies exhibit significantly worse sample complexity than those that explore the space of actions. We dispel such beliefs by introducing a random search method for training static, linear policies for continuous control problems, matching state-of-the-art sample efficiency on the benchmark MuJoCo locomotion tasks. Our method also finds a nearly optimal controller for a challenging instance of the Linear Quadratic Regulator, a classical problem in control theory, when the dynamics are not known. Computationally, our random search algorithm is at least 15 times more efficient than the fastest competing model-free methods on these benchmarks. We take advantage of this computational efficiency to evaluate the performance of our method over hundreds of random seeds and many different hyperparameter configurations for each benchmark task. Our simulations highlight a high variability in performance in these benchmark tasks, suggesting that commonly used estimations of sample efficiency do not adequately evaluate the performance of RL algorithms.

[245]  arXiv:1803.07065 (cross-list from physics.app-ph) [pdf]
Title: Sensorless Resonance Tracking of Resonant Electromagnetic Actuator through Back-EMF Estimation for Mobile Devices
Authors: Youngjun Cho
Subjects: Applied Physics (physics.app-ph); Systems and Control (cs.SY); Signal Processing (eess.SP); Dynamical Systems (math.DS)

Resonant electromagnetic actuators have been broadly used as vibration motors for mobile devices given their ability of generating relatively fast, strong, and controllable vibration force at a given resonant frequency. Mechanism of the actuators that is based on mechanical resonance, however, limits their use to a situation where their resonant frequencies are known and unshifted. In reality, there are many factors that alter the resonant frequency: for example, manufacturing tolerances, worn mechanical components such as a spring, nonlinearity in association with different input voltage levels. Here, we describe a sensorless resonance tracking method that actuates the motor and automatically detects its unknown damped natural frequency through the estimation of back electromotive force (EMF) and inner mass movements. We demonstrate the tracking performance of the proposed method through a series of experiments. This approach has the potential to control residual vibrations and then improve vibrotactile feedback, which can potentially be used for human-computer interaction, cognitive and affective neuroscience research.

Replacements for Tue, 20 Mar 18

[246]  arXiv:1408.1989 (replaced) [pdf, ps, other]
Title: On the stability of L^p-norms of Curvature Tensor at Rank one symmetrics spaces
Authors: Soma Maity
Comments: 15 Pages and 1 figure
Subjects: Differential Geometry (math.DG)
[247]  arXiv:1409.6943 (replaced) [pdf, ps, other]
Title: Koszul duality between $E_n$-algebras and coalgebras in a filtered category
Authors: Takuo Matsuoka
Comments: Was previously part of arXiv:1312.2562. 35 pages. Technical improvements, more descriptive terminology
Subjects: Algebraic Topology (math.AT); K-Theory and Homology (math.KT)
[248]  arXiv:1410.2615 (replaced) [pdf, ps, other]
Title: O-asymptotic classes of finite structures
Authors: Darío García
Comments: 28 pages
Subjects: Logic (math.LO)
[249]  arXiv:1501.01368 (replaced) [pdf, ps, other]
Title: On parameter loci of the Hénon family
Comments: 70 pages, 38 figures, 3 tables
Subjects: Dynamical Systems (math.DS)
[250]  arXiv:1503.06556 (replaced) [pdf, ps, other]
Title: 3-connected Reduction for Regular Graph Covers
Comments: The journal version of the first part of arXiv:1402.3774
Subjects: Combinatorics (math.CO)
[251]  arXiv:1505.01692 (replaced) [pdf, ps, other]
Title: Rough flows
Comments: v4, 55 pages; final version. The exposition has been polished to make the work easier to read
Subjects: Probability (math.PR); Classical Analysis and ODEs (math.CA)
[252]  arXiv:1505.03898 (replaced) [pdf, ps, other]
Title: Pinball Loss Minimization for One-bit Compressive Sensing: Convex Models and Algorithms
Comments: 11 pages
Subjects: Information Theory (cs.IT); Numerical Analysis (math.NA); Optimization and Control (math.OC); Machine Learning (stat.ML)
[253]  arXiv:1506.00563 (replaced) [pdf, other]
Title: Rational degeneration of M-curves, totally positive Grassmannians and KP2-solitons
Comments: 49 pages, 10 figures. Minor revisions
Subjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)
[254]  arXiv:1507.04915 (replaced) [pdf, other]
Title: The boundary model for the continuous cohomology of Isom$^+(\mathbb{H}^n)$
Authors: Hester Pieters
Comments: 25 pages, 1 figure, changed title, revised and extended version
Subjects: Group Theory (math.GR); Algebraic Topology (math.AT)
[255]  arXiv:1508.06269 (replaced) [pdf, other]
Title: A systematic process for evaluating structured perfect Bayesian equilibria in dynamic games with asymmetric information
Comments: 36 pages, 3 figures, IEEE Transactions on Automatic Control, vol. PP, no. 99, pp. 1-1, 2018
Subjects: Optimization and Control (math.OC); Computer Science and Game Theory (cs.GT); Systems and Control (cs.SY)
[256]  arXiv:1512.04234 (replaced) [pdf, ps, other]
Title: Two applications of the spectrum of numbers
Subjects: Number Theory (math.NT); Formal Languages and Automata Theory (cs.FL)
[257]  arXiv:1512.08390 (replaced) [pdf, ps, other]
Title: Dwork families and $\mathcal{D}$-modules
Comments: 24 pages, final version
Subjects: Algebraic Geometry (math.AG)
[258]  arXiv:1601.00050 (replaced) [pdf, ps, other]
Title: The proof-theoretic strength of Ramsey's theorem for pairs and two colors
Comments: 32 pages
Subjects: Logic (math.LO)
[259]  arXiv:1601.05661 (replaced) [pdf, other]
Title: Distortion Bounds for Source Broadcast Problems
Comments: Revision after the first round of review
Subjects: Information Theory (cs.IT)
[260]  arXiv:1601.06544 (replaced) [pdf, ps, other]
Title: Goal-oriented a posteriori error control for nonstationary convection-dominated transport problems
Subjects: Numerical Analysis (math.NA)
[261]  arXiv:1602.07500 (replaced) [pdf, ps, other]
Title: Topological comparison theorems for Bredon motivic cohomology
Comments: Corrected indices in main theorem and a few minor changes. To appear, Transactions AMS
Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG)
[262]  arXiv:1602.08072 (replaced) [pdf, ps, other]
Title: Model theory of $\mathrm{C}^*$-algebras
Comments: Various bug fixes and performance improvements
Subjects: Logic (math.LO); Operator Algebras (math.OA)
[263]  arXiv:1603.02301 (replaced) [pdf, ps, other]
Title: Constructing Reducible Brill--Noether Curves
Authors: Eric Larson
Subjects: Algebraic Geometry (math.AG)
[264]  arXiv:1603.08360 (replaced) [pdf, other]
Title: Lyapunov spectrum of Markov and Euclid trees
Comments: Slightly improved version with more details added to the proof of Theorem 3
Journal-ref: Nonlinearity, Volume 30, Number 12 (2017), 4428-53
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
[265]  arXiv:1604.01938 (replaced) [pdf, ps, other]
Title: On the Noether number of $p$-groups
Comments: Several corrections and simplifications made
Subjects: Group Theory (math.GR); Commutative Algebra (math.AC); Number Theory (math.NT); Representation Theory (math.RT)
[266]  arXiv:1605.07543 (replaced) [pdf, other]
Title: On tea, donuts and non-commutative geometry
Authors: Igor Nikolaev
Comments: to appear Surveys in Mathematics and its Applications
Subjects: Algebraic Geometry (math.AG); History and Overview (math.HO); Operator Algebras (math.OA)
[267]  arXiv:1605.07879 (replaced) [pdf, ps, other]
Title: Cubulating mapping tori of polynomial growth free group automorphisms
Comments: The proof has been significantly simplified
Subjects: Group Theory (math.GR)
[268]  arXiv:1606.00300 (replaced) [pdf, ps, other]
Title: Del Pezzo surfaces over finite fields and their Frobenius traces
Comments: 25 pages. Fixed various typos and improved exposition
Subjects: Number Theory (math.NT)
[269]  arXiv:1606.01200 (replaced) [pdf, ps, other]
Title: Simple and Honest Confidence Intervals in Nonparametric Regression
Subjects: Applications (stat.AP); Statistics Theory (math.ST)
[270]  arXiv:1606.09494 (replaced) [pdf, other]
Title: Matched Metrics to the Binary Asymmetric Channels
Subjects: Information Theory (cs.IT)
[271]  arXiv:1607.02970 (replaced) [pdf, other]
Title: The sequential functionals of type $(ι\rightarrow ι)^n \rightarrow ι$ form a dcpo for all $n \in \Bbb N$
Authors: Dag Normann
Comments: 10 pages
Subjects: Logic in Computer Science (cs.LO); Logic (math.LO)
[272]  arXiv:1607.03972 (replaced) [pdf, ps, other]
Title: $F$-singularities under generic linkage
Comments: 11 pages, final version
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
[273]  arXiv:1607.04026 (replaced) [pdf, ps, other]
Title: A new characterization of convexity with respect to Chebyshev systems
Journal-ref: J. Math. Inequal. 12 (2018)
Subjects: Classical Analysis and ODEs (math.CA)
[274]  arXiv:1607.08369 (replaced) [pdf, ps, other]
Title: Probabilistic logic of quantum observations
Subjects: Logic (math.LO); Quantum Physics (quant-ph)
[275]  arXiv:1608.02199 (replaced) [pdf, other]
Title: An EM algorithm for absolutely continuous Marshall-Olkin bivariate Pareto distribution with location and scale
Comments: 17 pages, 4 figures
Subjects: Computation (stat.CO); Statistics Theory (math.ST)
[276]  arXiv:1608.08163 (replaced) [pdf, ps, other]
Title: Singular Knots and Involutive Quandles
Comments: 13 pages; v4 adds axioms required for symmetry at unoriented singular crossings
Subjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
[277]  arXiv:1609.02360 (replaced) [pdf, ps, other]
Title: Canonical syzygies of smooth curves on toric surfaces
Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)
[278]  arXiv:1609.04201 (replaced) [pdf, ps, other]
Title: Quotients of orders in algebras obtained from skew polynomials with applications to coding theory
Authors: Susanne Pumpluen
Comments: The title changed from "Quotients of orders in algebras obtained from skew polynomials and possible applications" to "Quotients of orders in algebras obtained from skew polynomials with applications to coding theory". This version contains some minor corrections of the previous one, mostly typos
Subjects: Rings and Algebras (math.RA)
[279]  arXiv:1609.07232 (replaced) [pdf, ps, other]
Title: From visco to perfect plasticity in thermoviscoelastic materials
Authors: Riccarda Rossi
Comments: To appear in Zeitschrift fur Angewandte Mathematik und Mechanik
Subjects: Analysis of PDEs (math.AP)
[280]  arXiv:1609.08593 (replaced) [pdf, other]
Title: Multi-graded nilpotent tuples
Authors: Magdalena Boos
Comments: arXiv admin note: text overlap with arXiv:1509.04584
Subjects: Representation Theory (math.RT)
[281]  arXiv:1610.00351 (replaced) [pdf, ps, other]
Title: Quantitative stratification of stationary connections
Authors: Yu Wang
Comments: 27 pages
Subjects: Differential Geometry (math.DG)
[282]  arXiv:1611.03103 (replaced) [pdf, ps, other]
Title: An introduction to matrix convex sets and free spectrahedra
Authors: Tom-Lukas Kriel
Comments: 70 pages, an old version of this article was named "Free spectahedra, determinants of monic linear pencils and decomposition of pencils"
Subjects: Algebraic Geometry (math.AG)
[283]  arXiv:1611.04874 (replaced) [pdf, ps, other]
Title: The damped stochastic wave equation on p.c.f. fractals
Subjects: Probability (math.PR)
[284]  arXiv:1611.06470 (replaced) [pdf, ps, other]
Title: Bounded orbits of Diagonalizable Flows on finite volume quotients of products of $SL_2(\mathbb{R})$
Comments: Added Tue Ly as author. Other minor changes
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
[285]  arXiv:1611.09680 (replaced) [pdf, ps, other]
Title: Topological invariants and corner states for Hamiltonians on a three-dimensional lattice
Authors: Shin Hayashi
Comments: v3: section 4 added, references and typos corrected. 15 pages, 1 figure
Subjects: Mathematical Physics (math-ph); K-Theory and Homology (math.KT)
[286]  arXiv:1611.10268 (replaced) [pdf, other]
Title: On Equivalence of Binary Asymmetric Channels regarding the Maximum Likelihood Decoding
Comments: The presentation was improved with respect to the previous version. New examples, applications and references were included
Subjects: Information Theory (cs.IT)
[287]  arXiv:1612.06256 (replaced) [pdf, ps, other]
Title: Triviality of Equivariant Maps in Crossed Products and Matrix Algebras
Authors: Benjamin Passer
Comments: 12 pages. Version 2 implemented the following changes: stated definitions and theorems in greater generality, cleaned up proofs, and highlighted key examples (as opposed to referring to them casually in the text)
Subjects: Operator Algebras (math.OA)
[288]  arXiv:1612.08485 (replaced) [pdf, ps, other]
Title: Limit theorems for random cubical homology
Comments: 19 pages, 14 figures
Subjects: Probability (math.PR); Algebraic Topology (math.AT)
[289]  arXiv:1701.04102 (replaced) [pdf, ps, other]
Title: Two-stage Linear Decision Rules for Multi-stage Stochastic Programming
Subjects: Optimization and Control (math.OC)
[290]  arXiv:1701.07735 (replaced) [pdf, ps, other]
Title: Notes on finitely generated flat modules
Comments: 9 pages
Subjects: Commutative Algebra (math.AC)
[291]  arXiv:1702.00131 (replaced) [pdf, ps, other]
Title: Optimal Caching in Content-Centric Mobile Hybrid Networks: A Variable Decoupling Approach
Comments: 19 pages, 7 figures, Part of this paper was presented at the IEEE International Symposium on Information Theory, Barcelona, Spain, July 2016
Subjects: Information Theory (cs.IT); Networking and Internet Architecture (cs.NI)
[292]  arXiv:1702.03062 (replaced) [pdf, other]
Title: Sparsity/Undersampling Tradeoffs in Anisotropic Undersampling, with Applications in MR Imaging/Spectroscopy
Subjects: Information Theory (cs.IT)
[293]  arXiv:1702.08501 (replaced) [pdf, other]
Title: Formal Synthesis of Control Strategies for Positive Monotone Systems
Comments: To appear in IEEE Transactions on Automatic Control (TAC) (2018), 16 pages, double column
Subjects: Systems and Control (cs.SY); Optimization and Control (math.OC)
[294]  arXiv:1703.02382 (replaced) [pdf, other]
Title: Assessing the Privacy Cost in Centralized Event-Based Demand Response for Microgrids
Subjects: Systems and Control (cs.SY); Cryptography and Security (cs.CR); Optimization and Control (math.OC)
[295]  arXiv:1703.03436 (replaced) [pdf, ps, other]
Title: Forward-Backward-Half Forward Algorithm with non Self-Adjoint Linear Operators for Solving Monotone Inclusions
Comments: 34 Pages
Subjects: Optimization and Control (math.OC)
[296]  arXiv:1703.07073 (replaced) [pdf, ps, other]
Title: Heisenberg Modules over Quantum 2-tori are metrized quantum vector bundles
Comments: 38 Pages. Second part of arXiv:1608.04881v1; first part of arXiv:1703.07073v1; split due to length of paper
Subjects: Operator Algebras (math.OA)
[297]  arXiv:1703.07301 (replaced) [pdf, other]
Title: Linearly many rainbow trees in properly edge-coloured complete graphs
Subjects: Combinatorics (math.CO)
[298]  arXiv:1703.08281 (replaced) [pdf, ps, other]
Title: Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic
Comments: Final version
Subjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
[299]  arXiv:1703.09481 (replaced) [pdf, ps, other]
Title: Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions
Subjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech)
[300]  arXiv:1704.00889 (replaced) [pdf, ps, other]
Title: Crystal analysis of type $C$ Stanley symmetric functions
Comments: 39 pages
Journal-ref: Electronic Journal of Combinatorics 24(3) (2017) #P3.51
Subjects: Combinatorics (math.CO)
[301]  arXiv:1704.01597 (replaced) [pdf, ps, other]
Title: Fourier Series of Gegenbauer-Sobolev Polynomials
Journal-ref: SIGMA 14 (2018), 024, 11 pages
Subjects: Functional Analysis (math.FA)
[302]  arXiv:1704.05130 (replaced) [pdf, ps, other]
Title: Rotation number of interval contracted rotations
Subjects: Dynamical Systems (math.DS); Number Theory (math.NT)
[303]  arXiv:1704.06762 (replaced) [pdf, ps, other]
Title: Estimation for multiplicative models under multinomial sampling
Authors: Antonio Forcina
Subjects: Statistics Theory (math.ST)
[304]  arXiv:1704.07174 (replaced) [pdf, ps, other]
Title: On a priori estimates and existence of periodic solutions to the modified Benjamin-Ono equation below $H^{1/2}(\mathbb{T})$
Authors: Robert Schippa
Comments: 33 pages, existence of solutions and a priori estimates for large initial data added
Subjects: Analysis of PDEs (math.AP)
[305]  arXiv:1704.07531 (replaced) [pdf, other]
Title: Sufficient Markov Decision Processes with Alternating Deep Neural Networks
Comments: 31 pages, 3 figures, extended abstract in the proceedings of RLDM2017. (v2 revisions: Fixed a minor bug in the code w.r.t. setting seed, as a result numbers in the simulation experiments had some slight changes, but conclusions stayed the same. Corrected typos. Improved notations.)
Subjects: Methodology (stat.ME); Statistics Theory (math.ST); Machine Learning (stat.ML)
[306]  arXiv:1704.08485 (replaced) [pdf, ps, other]
Title: 2-Verma modules and the Khovanov-Rozansky link homologies
Comments: v1, 32 pages, colored figures. v2, 42 pages, Proof of the main result expanded into a new subsection, minor corrections
Subjects: Quantum Algebra (math.QA); Geometric Topology (math.GT); Representation Theory (math.RT)
[307]  arXiv:1705.00429 (replaced) [src]
Title: Polynomial-Time Algorithms for Sliding Tokens on Cactus Graphs and Block Graphs
Comments: The algorithm for block graphs in this manuscript contains some flaws. More precisely, Proposition 20 is not correct. Therefore, we withdraw this manuscript
Subjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO)
[308]  arXiv:1705.01023 (replaced) [pdf, ps, other]
Title: Expectations, Concave Transforms, Chow weights, and Roth's theorem for varieties
Authors: Nathan Grieve
Subjects: Algebraic Geometry (math.AG)
[309]  arXiv:1705.02166 (replaced) [pdf, ps, other]
Title: Lines in Euclidean Ramsey theory
Comments: 7 pages
Subjects: Combinatorics (math.CO)
[310]  arXiv:1705.02457 (replaced) [pdf, ps, other]
Title: On nonlinear cross-diffusion systems: an optimal transport approach
Comments: improved version; some well-known results shortened
Subjects: Analysis of PDEs (math.AP)
[311]  arXiv:1705.04735 (replaced) [pdf, ps, other]
Title: On the weak Roman domination number of lexicographic product graphs
Subjects: Combinatorics (math.CO)
[312]  arXiv:1705.08159 (replaced) [pdf, ps, other]
Title: Diagonal forms of higher degree over function fields of $p$-adic curves
Authors: Susanne Pumpluen
Comments: Some small corrections/changes have been done with respect to the first version
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
[313]  arXiv:1706.01015 (replaced) [pdf, other]
Title: The split-and-drift random graph, a null model for speciation
Comments: added Proposition 2.4 and formal proofs of Proposition 2.3 and 2.6
Subjects: Probability (math.PR); Combinatorics (math.CO); Populations and Evolution (q-bio.PE)
[314]  arXiv:1706.01354 (replaced) [pdf, ps, other]
Title: Non Projected Calabi-Yau Supermanifolds over $\mathbb{P}^2$
Comments: 17 pages. Exposition of the main theorem improved. Typos fixed
Subjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[315]  arXiv:1706.01822 (replaced) [pdf, other]
Title: Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation
Comments: Published version, 7 pages, 2 figures
Subjects: Quantum Gases (cond-mat.quant-gas); Mathematical Physics (math-ph)
[316]  arXiv:1706.03844 (replaced) [pdf, ps, other]
Title: Interpolation and Fatou-Zygmund property for completely Sidon subsets of discrete groups (New title: Completely Sidon sets in discrete groups)
Authors: Gilles Pisier
Comments: This new version contains a significant addition, namely the operator space version of a result of Varopoulos, showing that if the closed span of a subset of G in C*(G) is completely isomorphic to $\ell_1$ (by an arbitrary isomorphism) or if the dual operator space is exact then the set is completely Sidon. v3: more polished, longer and more detailed version with a new title
Subjects: Operator Algebras (math.OA); Functional Analysis (math.FA)
[317]  arXiv:1706.04629 (replaced) [pdf, other]
Title: Verification Studies for the Noh Problem using Non-ideal Equations of State and Finite Strength Shocks
Comments: 14 pages, 7 figures, 19 images
Subjects: Fluid Dynamics (physics.flu-dyn); Computational Engineering, Finance, and Science (cs.CE); Mathematical Physics (math-ph)
[318]  arXiv:1706.05589 (replaced) [pdf, ps, other]
Title: Higher congruences between newforms and Eisenstein series of squarefree level
Authors: Catherine Hsu
Comments: 17 pages. Significant revisions. Comments welcome
Subjects: Number Theory (math.NT)
[319]  arXiv:1706.09130 (replaced) [pdf, other]
Title: Potts models with a defect line
Comments: Revised version, with additional explanations and details. Should hopefully be easier to read
Subjects: Mathematical Physics (math-ph); Probability (math.PR)
[320]  arXiv:1707.00739 (replaced) [pdf, ps, other]
Title: The odd primary order of the commutator on low rank Lie groups
Authors: Tse Leung So
Comments: 18 pages; Accepted by Topology and its Applications
Subjects: Algebraic Topology (math.AT)
[321]  arXiv:1707.00977 (replaced) [pdf, ps, other]
Title: Electric-Magnetic Aspects On Yang-Mills Fields
Authors: Tosiaki Kori
Comments: arXiv admin note: text overlap with arXiv:1312.4121
Subjects: Symplectic Geometry (math.SG); Mathematical Physics (math-ph); Differential Geometry (math.DG)
[322]  arXiv:1707.01954 (replaced) [pdf, ps, other]
Title: Analysis of level-dependent subdivision schemes near extraordinary vertices and faces
Subjects: Numerical Analysis (math.NA)
[323]  arXiv:1707.04830 (replaced) [pdf, ps, other]
Title: On Banach-Mazur distance between planar convex bodies
Subjects: Metric Geometry (math.MG)
[324]  arXiv:1707.04963 (replaced) [pdf, other]
Title: A large class of solvable multistate Landau-Zener models and quantum integrability
Comments: The 2nd version contains considerable changes, including new sections, that are based on recent advances in arXiv/1711.09945 (new Ref.[1])
Subjects: Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Mathematical Physics (math-ph)
[325]  arXiv:1707.05351 (replaced) [pdf, ps, other]
Title: The reduced phase space of Palatini-Cartan-Holst theory
Comments: Corrected a minor mistake and reformulated some of the results. Main theorem restructured and overall improvements. 26 pages
Subjects: Mathematical Physics (math-ph); General Relativity and Quantum Cosmology (gr-qc); Symplectic Geometry (math.SG)
[326]  arXiv:1707.05530 (replaced) [pdf, ps, other]
Title: Chain varieties of monoids
Comments: 76 pages, 3 figures, 3 tables. In comparison with the previous version, we made a number of linguistic corrections only
Subjects: Group Theory (math.GR)
[327]  arXiv:1707.05797 (replaced) [pdf]
Title: Low-complexity implementation of convex optimization-based phase retrieval
Subjects: Information Theory (cs.IT); Optimization and Control (math.OC)
[328]  arXiv:1708.01084 (replaced) [pdf, ps, other]
Title: Square function estimates for the Bochner-Riesz means
Authors: Sanghyuk Lee
Comments: 43 pages, revised from the earlier version, additional references, fixing typos and improving presentations, to appear in APDE
Subjects: Classical Analysis and ODEs (math.CA)
[329]  arXiv:1708.02487 (replaced) [pdf, ps, other]
Title: Spectral density of mixtures of random density matrices for qubits
Comments: 21 pages, LaTex, 6 figures
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
[330]  arXiv:1708.04853 (replaced) [pdf, other]
Title: A polynomial time knot polynomial
Comments: Typos fixed, length reduced for publication in PAMS
Subjects: Geometric Topology (math.GT)
[331]  arXiv:1708.05531 (replaced) [pdf, ps, other]
Title: Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI
Subjects: Numerical Analysis (math.NA)
[332]  arXiv:1708.06172 (replaced) [pdf, ps, other]
Title: Global small solutions of 3D incompressible Oldroyd-B model without damping mechanism
Authors: Yi Zhu
Comments: 20 pages
Subjects: Analysis of PDEs (math.AP)
[333]  arXiv:1708.07573 (replaced) [pdf, other]
Title: Reconstruction of a compact Riemannian manifold from the scattering data of internal sources
Subjects: Differential Geometry (math.DG)
[334]  arXiv:1709.00483 (replaced) [pdf, other]
Title: Iteratively Linearized Reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems
Subjects: Numerical Analysis (cs.NA); Computer Vision and Pattern Recognition (cs.CV); Numerical Analysis (math.NA); Optimization and Control (math.OC); Machine Learning (stat.ML)
[335]  arXiv:1709.01685 (replaced) [pdf, ps, other]
Title: Regular characters of classical groups over complete discrete valuation rings
Authors: Shai Shechter
Comments: 47 pages. Substantial changes to Sections 2 and 3, following referee report. Some structural changes to Sections 1 and 4 as well
Subjects: Representation Theory (math.RT)
[336]  arXiv:1709.01774 (replaced) [pdf, ps, other]
Title: Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian
Comments: 31 pages, 4 figures
Subjects: Spectral Theory (math.SP)
[337]  arXiv:1709.01781 (replaced) [pdf, other]
Title: Parameterizations for Ensemble Kalman Inversion
Subjects: Numerical Analysis (math.NA); Optimization and Control (math.OC); Methodology (stat.ME)
[338]  arXiv:1709.02006 (replaced) [pdf, ps, other]
Title: Quotients of del Pezzo surfaces of degree 2
Authors: Andrey Trepalin
Comments: Final version, accepted to MMJ. 39 pages, 5 tables
Subjects: Algebraic Geometry (math.AG)
[339]  arXiv:1709.03364 (replaced) [pdf, other]
Title: Detecting localized eigenstates of linear operators
Subjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph); Spectral Theory (math.SP)
[340]  arXiv:1709.04489 (replaced) [pdf, ps, other]
Title: A remark on the Tate conjecture
Authors: Ben Moonen
Comments: 3 pages; updated version that includes a result in characteristic p. To appear in the J. of Alg. Geom
Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
[341]  arXiv:1709.05780 (replaced) [pdf, ps, other]
Title: On the indivisibility of derived Kato's Euler systems and the main conjecture for modular forms
Comments: Some errors and typos are corrected. Two non-ordinary examples are added at the end
Subjects: Number Theory (math.NT)
[342]  arXiv:1709.09226 (replaced) [pdf, ps, other]
Title: Second quantized quantum field theory based on invariance properties of locally conformally flat space-times
Authors: John Mashford
Comments: 50 pages, version for submission to journal. Minor modifications including adding a reference
Subjects: Mathematical Physics (math-ph)
[343]  arXiv:1709.09278 (replaced) [pdf, other]
Title: Classifying Character Degree Graphs With 6 Vertices
Comments: 18 pages. arXiv admin note: text overlap with arXiv:1707.03020
Subjects: Group Theory (math.GR); Representation Theory (math.RT)
[344]  arXiv:1709.10362 (replaced) [pdf, ps, other]
Title: Some analytic aspects of automorphic forms on GL(2) of minimal type
Comments: 28 pages. To appear in Comm. Math. Helv
Subjects: Number Theory (math.NT)
[345]  arXiv:1710.03830 (replaced) [pdf, other]
Title: Inference on Auctions with Weak Assumptions on Information
Subjects: Econometrics (econ.EM); Computer Science and Game Theory (cs.GT); Learning (cs.LG); Statistics Theory (math.ST)
[346]  arXiv:1710.04965 (replaced) [pdf, ps, other]
Title: Lorentz signature and twisted spectral triples
Comments: minor corrections. To be published in JHEP
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
[347]  arXiv:1710.05318 (replaced) [pdf, ps, other]
Title: On Finsler spacetimes with a timelike Killing vector field
Comments: 28 pages, AMSLaTex. v3: First part of the introduction and conclusions section expanded, some new references added; v3 matches the published version
Journal-ref: Class. Quantum Grav. 35 (2018) 085007 (28pp)
Subjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc)
[348]  arXiv:1710.05506 (replaced) [pdf, ps, other]
Title: Multiple Lattice Tilings in Euclidean Spaces
Comments: 6 pages, 2 figures
Subjects: Metric Geometry (math.MG)
[349]  arXiv:1710.05537 (replaced) [pdf, ps, other]
Title: Hamiltonian stability for weighted measure and generalized Lagrangian mean curvature flow
Comments: 40 pages; (ver.3) minor corrections. (ver.2) Tex file format is changed, Corollary 2.5 and a reference are added, minor corrections
Subjects: Differential Geometry (math.DG)
[350]  arXiv:1710.06135 (replaced) [pdf, ps, other]
Title: The depth structure of motivic multiple zeta values
Authors: Jiangtao Li
Comments: 24 pages
Subjects: Number Theory (math.NT)
[351]  arXiv:1710.07402 (replaced) [pdf, ps, other]
Title: The Collatz-Wielandt quotient for pairs of nonnegative operators
Authors: Shmuel Friedland
Comments: 26 pages
Subjects: Optimization and Control (math.OC)
[352]  arXiv:1710.07858 (replaced) [pdf, ps, other]
Title: Gradient flows, second order gradient systems and convexity
Subjects: Optimization and Control (math.OC)
[353]  arXiv:1710.09542 (replaced) [pdf, ps, other]
Title: Differential operators on the algebra of densities and factorization of the generalized Sturm-Liouville operator
Comments: 15 p. LaTeX
Subjects: Mathematical Physics (math-ph)
[354]  arXiv:1710.10132 (replaced) [pdf, ps, other]
Title: An unfitted Hybrid High-Order method for elliptic interface problems
Subjects: Numerical Analysis (math.NA)
[355]  arXiv:1710.10481 (replaced) [pdf, ps, other]
Title: Quantum Newton duality
Subjects: Mathematical Physics (math-ph)
[356]  arXiv:1710.10698 (replaced) [pdf, ps, other]
Title: Extended nilHecke algebra and symmetric functions in type B
Authors: Michael Reeks
Comments: 14 pages. Revised version for publication in Journal of Pure and Applied Algebra. Includes a new section on differentials and DG structure
Subjects: Representation Theory (math.RT)
[357]  arXiv:1710.10823 (replaced) [pdf, ps, other]
Title: Some remarks on the notions of boundary systems and boundary triple(t)s
Comments: 8 pages
Subjects: Functional Analysis (math.FA)
[358]  arXiv:1711.00778 (replaced) [pdf, other]
Title: Asymptotic behaviour of a network of oscillators coupled to thermostats of finite energy
Authors: Andrey V. Dymov
Comments: 22 pages. In comparison with the previous version where a chain of oscillators was considered, the result is generalized to the case when the oscillators form arbitrary network
Subjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS)
[359]  arXiv:1711.02514 (replaced) [pdf, ps, other]
Title: Multiple Translative Tilings in Euclidean Spaces
Comments: 12 pages, 6 figures. arXiv admin note: substantial text overlap with arXiv:1712.01122, arXiv:1710.05506
Subjects: Metric Geometry (math.MG)
[360]  arXiv:1711.02752 (replaced) [pdf, ps, other]
Title: Constructing Reducible Brill--Noether Curves II
Authors: Eric Larson
Subjects: Algebraic Geometry (math.AG)
[361]  arXiv:1711.02907 (replaced) [pdf, ps, other]
Title: Strong convergence rate of Runge--Kutta methods and simplified step-$N$ Euler schemes for SDEs driven by fractional Brownian motions
Subjects: Numerical Analysis (math.NA)
[362]  arXiv:1711.03493 (replaced) [pdf, ps, other]
Title: On Kedlaya type inequalities for weighted means
Comments: J. Inequal. Appl. (2018)
Subjects: Classical Analysis and ODEs (math.CA)
[363]  arXiv:1711.04449 (replaced) [pdf, other]
Title: A General Framework for Covariance Matrix Optimization in MIMO Systems
Comments: 16 Pages, 4 Figures, IEEE Communications
Subjects: Information Theory (cs.IT)
[364]  arXiv:1711.05450 (replaced) [pdf, ps, other]
Title: Scattering Theory and $\mathcal{P}\mathcal{T}$-Symmetry
Authors: Ali Mostafazadeh
Comments: Slightly expanded revised version, 38 pages
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optics (physics.optics)
[365]  arXiv:1711.05850 (replaced) [pdf, other]
Title: Local eigenvalue statistics of one-dimensional random non-selfadjoint pseudo-differential operators
Comments: 81 pages, 7 figures
Subjects: Spectral Theory (math.SP); Analysis of PDEs (math.AP); Probability (math.PR)
[366]  arXiv:1711.07587 (replaced) [pdf, other]
Title: Domination structure for number three
Authors: Misa Nakanishi
Subjects: Combinatorics (math.CO)
[367]  arXiv:1711.07725 (replaced) [pdf, ps, other]
Title: Effective cycles on the symmetric product of a curve, II: the Abel-Jacobi faces
Comments: 26 pages, 1 figure; v2: strenghtened Theorem A by adding some new Abel-Jacobi facets; improved and clarified some results; fixed a few typos
Subjects: Algebraic Geometry (math.AG)
[368]  arXiv:1711.08024 (replaced) [pdf, ps, other]
Title: New complex analytic methods in the theory of minimal surfaces: a survey
Comments: To appear in J. Aust. Math. Soc
Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)
[369]  arXiv:1711.11416 (replaced) [pdf, ps, other]
Title: A pseudo zeta function and Riemann hypothesis
Authors: A. Durmagambetov
Comments: 11 pages
Subjects: General Mathematics (math.GM)
[370]  arXiv:1712.00434 (replaced) [pdf, other]
Title: On the treewidth of triangulated 3-manifolds
Comments: 23 pages, 3 figures, 1 table. An extended abstract of this paper is to appear in the Proceedings of the 34th International Symposium on Computational Geometry (SoCG 2018), Budapest, June 11-14 2018
Subjects: Geometric Topology (math.GT); Computational Geometry (cs.CG); Combinatorics (math.CO)
[371]  arXiv:1712.01122 (replaced) [pdf, ps, other]
Title: Characterization of the Two-Dimensional Five-Fold Lattice Tiles
Authors: Chuanming Zong
Comments: 20 pages, 14 figures. arXiv admin note: text overlap with arXiv:1711.02514, arXiv:1710.05506
Subjects: Metric Geometry (math.MG)
[372]  arXiv:1712.01975 (replaced) [pdf, other]
Title: Sparsity Regularization and feature selection in large dimensional data
Subjects: Learning (cs.LG); Numerical Analysis (cs.NA); Optimization and Control (math.OC)
[373]  arXiv:1712.06356 (replaced) [pdf, other]
Title: On convergence of infinite matrix products with alternating factors from two sets of matrices
Authors: Victor Kozyakin
Comments: 7 pages, 13 bibliography references, expanded Introduction and Section 4 "Remarks and Open Questions", accepted for publication in Discrete Dynamics in Nature and Society
Subjects: Optimization and Control (math.OC); Rings and Algebras (math.RA)
[374]  arXiv:1712.06841 (replaced) [pdf, other]
Title: Graphons, permutons and the Thoma simplex: three mod-Gaussian moduli spaces
Comments: New version: the paper has been slightly shortened, and a few references were added. 52 pages, 13 figures
Subjects: Probability (math.PR)
[375]  arXiv:1712.07734 (replaced) [pdf, ps, other]
Title: Sheaf-Theoretic Stratification Learning
Authors: Adam Brown, Bei Wang
Subjects: Computational Geometry (cs.CG); Algebraic Topology (math.AT)
[376]  arXiv:1712.07768 (replaced) [pdf, other]
Title: Electric field concentration in the presence of an inclusion with eccentric core-shell geometry
Comments: 32 pages
Subjects: Analysis of PDEs (math.AP)
[377]  arXiv:1712.09203 (replaced) [pdf, ps, other]
Title: Algorithmic Regularization in Over-parameterized Matrix Sensing and Neural Networks with Quadratic Activations
Comments: clarified that the sensing matrices can be symmetric wlog and revised the quadratic neural nets section
Subjects: Learning (cs.LG); Data Structures and Algorithms (cs.DS); Optimization and Control (math.OC); Machine Learning (stat.ML)
[378]  arXiv:1712.09732 (replaced) [pdf, ps, other]
Title: Characterization of the Two-Dimensional Five-Fold Translative Tiles
Comments: 18 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:1711.02514, arXiv:1712.01122
Subjects: Metric Geometry (math.MG)
[379]  arXiv:1801.00335 (replaced) [pdf, ps, other]
Title: Plato's cave and differential forms
Authors: Fedor Manin
Comments: 33 pages, 1 figure; comments welcome! This version corrects an error pointed out by A. Berdnikov
Subjects: Geometric Topology (math.GT); Algebraic Topology (math.AT); Differential Geometry (math.DG); Metric Geometry (math.MG)
[380]  arXiv:1801.01165 (replaced) [pdf, other]
Title: Automorphism groups and Ramsey properties of sparse graphs
Comments: 40 pages, 3 figures, new proof of Theorem 3.19 and other minor revisions
Subjects: Group Theory (math.GR); Discrete Mathematics (cs.DM); Combinatorics (math.CO); Dynamical Systems (math.DS); Logic (math.LO)
[381]  arXiv:1801.02932 (replaced) [pdf, ps, other]
Title: Polynomials describing the multiplication in finitely generated torsion free nilpotent groups
Subjects: Group Theory (math.GR)
[382]  arXiv:1801.03183 (replaced) [pdf, other]
Title: Discrete Stratified Morse Theory: A User's Guide
Subjects: Computational Geometry (cs.CG); Algebraic Topology (math.AT)
[383]  arXiv:1801.03972 (replaced) [pdf, ps, other]
Title: Extremal $G$-free induced subgraphs of Kneser graphs
Comments: Minor changes
Subjects: Combinatorics (math.CO)
[384]  arXiv:1801.04176 (replaced) [pdf, ps, other]
Title: Non-meagre subgroups of reals disjoint with meagre sets
Authors: Ziemowit Kostana
Subjects: General Topology (math.GN)
[385]  arXiv:1801.04738 (replaced) [pdf, ps, other]
Title: Tilting modules over Auslander-Gorenstein Algebras
Subjects: Representation Theory (math.RT)
[386]  arXiv:1801.05682 (replaced) [pdf, ps, other]
Title: Automorphisms of Hilbert schemes of points on a generic projective K3 surface
Authors: Alberto Cattaneo
Comments: 19 pages. v2: Sections 2 and 6 added, with several new major results
Subjects: Algebraic Geometry (math.AG)
[387]  arXiv:1801.09269 (replaced) [pdf, ps, other]
Title: Wasserstein Riemannian Geometry of Positive Definite Matrices
Comments: Second version. Submitted
Subjects: Statistics Theory (math.ST)
[388]  arXiv:1801.09665 (replaced) [pdf, ps, other]
Title: Cooperative repair: Constructions of optimal MDS codes for all admissible parameters
Comments: This version forms a substantive extension of Version 1. We provide a construction of MDS codes that support optimal cooperative repair for all admissible parameters. This resolves the general problem of constructing codes for cooperative repair with minimum repair bandwidth
Subjects: Information Theory (cs.IT)
[389]  arXiv:1802.00085 (replaced) [pdf, other]
Title: Explicit bounds for primes in arithmetic progressions
Comments: 67 pages. A typo detected in the initial version of Lemma 2.18 was repaired; its effects propagated through the paper, resulting in a change to many of the constants in our theorems. Results of computations, and the code used for those computations, can be found at: this http URL
Subjects: Number Theory (math.NT)
[390]  arXiv:1802.00215 (replaced) [pdf, other]
Title: Discontinuous traveling waves as weak solutions to the Fornberg-Whitham equation
Comments: 1 figure
Subjects: Analysis of PDEs (math.AP)
[391]  arXiv:1802.00982 (replaced) [pdf, ps, other]
Title: Stochastic integral with respect to the mixed fractional Brownian motion and drift estimation of the mixed fraction Ornstein-Ulenbeck process
Subjects: Statistics Theory (math.ST)
[392]  arXiv:1802.03186 (replaced) [pdf, ps, other]
Title: A simplification problem in manifold theory
Comments: 35 pages. Small improvements and new references
Subjects: Geometric Topology (math.GT)
[393]  arXiv:1802.05502 (replaced) [pdf, ps, other]
Title: Sharp Lower Bounds for the First Eigenvalues of the Bi-Laplace Operator
Comments: 15 pages
Subjects: Analysis of PDEs (math.AP)
[394]  arXiv:1802.05856 (replaced) [pdf, other]
Title: Algorithmic Complexity and Reprogrammability of Chemical Structure Networks
Comments: 19 pages + Appendix
Subjects: Molecular Networks (q-bio.MN); Computational Engineering, Finance, and Science (cs.CE); Information Theory (cs.IT)
[395]  arXiv:1802.06381 (replaced) [pdf, ps, other]
Title: Inverse images of generic maps and homology groups of the Reeb spaces
Authors: Naoki Kitazawa
Comments: 6 pages, 1 figure, the author corrected some mistakes in the previous version
Subjects: Algebraic Topology (math.AT)
[396]  arXiv:1802.06848 (replaced) [pdf, ps, other]
Title: Stable constant mean curvature surfaces with free boundary in slabs
Authors: Rabah Souam
Comments: minor modifications, one reference added
Subjects: Differential Geometry (math.DG)
[397]  arXiv:1802.07049 (replaced) [pdf, ps, other]
Title: The Bieri-Neumann-Strebel invariants via Newton polytopes
Authors: Dawid Kielak
Comments: 39 pages, no figures. v2. improved exposition
Subjects: Group Theory (math.GR); Geometric Topology (math.GT); Rings and Algebras (math.RA)
[398]  arXiv:1802.07219 (replaced) [pdf, ps, other]
Title: Leibniz algebras as non-associative algebras
Authors: Jorg Feldvoss
Comments: 33 pages
Subjects: Rings and Algebras (math.RA)
[399]  arXiv:1802.07529 (replaced) [pdf, ps, other]
Title: Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review
Comments: We dedicate this paper to Adi Ben-Israel, our scientific father and grandfather, respectively. Revised version March 19, 2018
Subjects: Optimization and Control (math.OC)
[400]  arXiv:1802.07654 (replaced) [pdf, ps, other]
Title: Intersecting limit sets of Kleinian subgroups and Susskind's conjecture
Subjects: Dynamical Systems (math.DS)
[401]  arXiv:1802.08185 (replaced) [pdf, ps, other]
Title: On quaternion algebra over the composite of quadratic number fields and over some dihedral fields
Comments: This is a preliminary form of the article. Categories: number theory; rings and algebras
Subjects: Number Theory (math.NT)
[402]  arXiv:1802.08667 (replaced) [pdf, ps, other]
Title: Double/De-Biased Machine Learning Using Regularized Riesz Representers
Comments: 15 pages; fixed several typos + updated references
Subjects: Machine Learning (stat.ML); Econometrics (econ.EM); Statistics Theory (math.ST)
[403]  arXiv:1802.08922 (replaced) [pdf, ps, other]
Title: Positively curved Killing foliations via deformations
Comments: 24 pages, corrected typos, example 8.1 removed
Subjects: Differential Geometry (math.DG)
[404]  arXiv:1802.08934 (replaced) [pdf, other]
Title: The Archimedean limit of random sorting networks
Authors: Duncan Dauvergne
Comments: 58 pages, 5 figures. Changes from v1: the proof of Lemma 7.2 has been lengthened for clarity; otherwise, only minor edits have been made
Subjects: Probability (math.PR); Combinatorics (math.CO)
[405]  arXiv:1802.09051 (replaced) [pdf, ps, other]
Title: Graphs with equal domination and covering numbers
Comments: 17 pages, 6 figures
Subjects: Combinatorics (math.CO)
[406]  arXiv:1802.09312 (replaced) [pdf, other]
Title: DP-3-coloring of some planar graphs
Comments: 15 pages, five figures
Subjects: Combinatorics (math.CO)
[407]  arXiv:1802.10012 (replaced) [pdf, ps, other]
Title: Integral points on generalised affine Châtelet surfaces
Comments: 21 pages, submitted
Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
[408]  arXiv:1802.10248 (replaced) [pdf, ps, other]
Title: M-eigenvalues of The Riemann Curvature Tensor
Subjects: Differential Geometry (math.DG); Spectral Theory (math.SP)
[409]  arXiv:1803.00332 (replaced) [pdf, other]
Title: Geometry Transition in Covariant Loop Quantum Gravity
Comments: PhD thesis submitted for the degree of Doctor in Theoretical and Mathematical Physics. Defended at the Center for Theoretical Physics/CNRS/Aix-Marseille University, the 23rd of October 2017. The manuscript is written in English and begins with a short summary in French
Subjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
[410]  arXiv:1803.00402 (replaced) [pdf, ps, other]
Title: On Periodic Solutions to Lagrangian System With Constraints
Authors: Oleg Zubelevich
Subjects: Dynamical Systems (math.DS)
[411]  arXiv:1803.00856 (replaced) [pdf, ps, other]
Title: Embedded loops in the hyperbolic plane with prescribed, almost constant curvature
Comments: Theorems numbering has been changed; few misprints were fixed
Subjects: Differential Geometry (math.DG)
[412]  arXiv:1803.01173 (replaced) [pdf, ps, other]
Title: Lattices of coarse structures
Comments: coarse structure, ballean, lattice of coarse structures
Subjects: General Topology (math.GN)
[413]  arXiv:1803.01238 (replaced) [pdf, ps, other]
Title: Dynamic risk measure for BSVIE with jumps and semimartingale issues
Authors: Nacira Agram
Comments: 11 pages
Subjects: Optimization and Control (math.OC)
[414]  arXiv:1803.01964 (replaced) [pdf, ps, other]
Title: Harmonic Analysis on the Adèle Ring of $\Q$
Comments: 27 pages, 1 figure
Subjects: Classical Analysis and ODEs (math.CA)
[415]  arXiv:1803.02040 (replaced) [pdf, ps, other]
Title: Principal eigenvalues of a class of nonlinear integro-differential operators
Authors: Anup Biswas
Comments: Theorem 2.6 updated
Subjects: Analysis of PDEs (math.AP)
[416]  arXiv:1803.02570 (replaced) [pdf, ps, other]
Title: Why Black Swan events must occur
Subjects: Risk Management (q-fin.RM); Logic (math.LO)
[417]  arXiv:1803.02792 (replaced) [pdf, ps, other]
Title: Lozenge tilings of hexagons with central holes and dents
Authors: Tri Lai
Comments: Version 2: 93 pages and many figures; list of references and main proofs are updated. Comments are welcome!
Subjects: Combinatorics (math.CO)
[418]  arXiv:1803.03231 (replaced) [pdf, other]
Title: Principal eigenvalue and maximum principle for cooperative periodic-parabolic systems
Comments: 38 pages, 2 figures, Remarks on the generation of the paper at the end of the Introduction
Subjects: Analysis of PDEs (math.AP)
[419]  arXiv:1803.03606 (replaced) [pdf, ps, other]
Title: A Simple proof of Johnson-Lindenstrauss extension
Authors: Manor Mendel
Comments: 2 pages. Elimination of typos
Subjects: Metric Geometry (math.MG); Functional Analysis (math.FA)
[420]  arXiv:1803.03944 (replaced) [pdf, ps, other]
Title: Extendible cardinals and the mantle
Authors: Toshimichi Usuba
Subjects: Logic (math.LO)
[421]  arXiv:1803.03974 (replaced) [pdf, ps, other]
Title: Formality conjecture for K3 surfaces
Comments: 26 pages, comments are welcome
Subjects: Algebraic Geometry (math.AG)
[422]  arXiv:1803.04016 (replaced) [pdf, ps, other]
Title: Homological invariants of powers of fiber products
Comments: 18 pages, plus an appendix and a bibliography. Minor typos corrected and references updated
Subjects: Commutative Algebra (math.AC)
[423]  arXiv:1803.04315 (replaced) [pdf, other]
Title: Power-Efficient Deployment of UAVs as Relays
Authors: Erdem Koyuncu
Subjects: Information Theory (cs.IT); Signal Processing (eess.SP)
[424]  arXiv:1803.04516 (replaced) [pdf, ps, other]
Title: Explicit inverse of tridiagonal matrix with applications in autoregressive modeling
Authors: Linda S. L. Tan
Subjects: Numerical Analysis (math.NA)
[425]  arXiv:1803.04750 (replaced) [pdf, other]
Title: Electric Vehicle Charge Scheduling Mechanism to Maximize Cost Efficiency and User Convenience
Comments: 11 pages. accepted by IEEE Transactions on Smart Grid
Subjects: Optimization and Control (math.OC)
[426]  arXiv:1803.04828 (replaced) [pdf, ps, other]
Title: Amenable actions of discrete quantum groups on von Neumann algebras
Subjects: Operator Algebras (math.OA)
[427]  arXiv:1803.04913 (replaced) [pdf, other]
Title: On non-commutativity in quantum theory (I): from classical to quantum probability
Authors: Luca Curcuraci
Comments: 19 pages, 1 figure
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Probability (math.PR)
[428]  arXiv:1803.04916 (replaced) [pdf, other]
Title: On non-commutativity in quantum theory (II): toy models for non-commutative kinematics
Authors: Luca Curcuraci
Comments: 31 pages, 7 figures
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Probability (math.PR)
[429]  arXiv:1803.04921 (replaced) [pdf, other]
Title: On non-commutativity in quantum theory (III): determinantal point processes and non-relativistic quantum mechanics
Authors: Luca Curcuraci
Comments: 23 pages, 1 figure
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
[430]  arXiv:1803.05116 (replaced) [pdf, ps, other]
Title: Extending automorphisms of the genus-2 surface over the 3-sphere
Comments: 21 pages, 15 fugures; typos corrected
Subjects: Geometric Topology (math.GT)
[431]  arXiv:1803.05182 (replaced) [pdf, ps, other]
Title: Approximative Theorem of Incomplete Riemann-Stieltjes Sum of Stochastic Integral
Authors: Jingwei Liu
Comments: 4 figures
Subjects: Probability (math.PR)
[432]  arXiv:1803.05236 (replaced) [pdf, other]
Title: Some negative results related to Poissonian pair correlation problems
Comments: 18 pages, 1 figure
Subjects: Number Theory (math.NT)
[433]  arXiv:1803.05410 (replaced) [pdf, ps, other]
Title: Effective non-linear spinor dynamics in a spin-1 Bose-Einstein condensate
Subjects: Mathematical Physics (math-ph)
[434]  arXiv:1803.05742 (replaced) [pdf, ps, other]
Title: On the piecewise pseudo almost periodic solution of nondensely impulsive integro-differential systems with infinite delay
Subjects: Functional Analysis (math.FA)
[435]  arXiv:1803.05775 (replaced) [pdf, ps, other]
Title: $\mathfrak{q}$-crystal structure on primed tableaux and on signed unimodal factorizations of reduced words of type $B$
Authors: Toya Hiroshima
Comments: 25 pages. Reference [1] added; Remark 3.2 deleted; arXiv admin note: text overlap with arXiv:1704.00889 by other authors
Subjects: Combinatorics (math.CO)
[436]  arXiv:1803.05844 (replaced) [pdf, other]
Title: Joint Turbo Receiver for LDPC-Coded MIMO Systems Based on Semi-definite Relaxation
Authors: Kun Wang, Zhi Ding
Comments: 5 pages, 4 figures, conference
Subjects: Information Theory (cs.IT)
[437]  arXiv:1803.06299 (replaced) [pdf, other]
Title: On the existence of a scalar pressure field in the Bredinger problem
Authors: Aymeric Baradat
Subjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC); Probability (math.PR)
[438]  arXiv:1803.06322 (replaced) [pdf, other]
Title: Computing performability measures in Markov chains by means of matrix functions
Subjects: Numerical Analysis (math.NA)
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