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Classical Analysis and ODEs

New submissions

[ total of 13 entries: 1-13 ]
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New submissions for Tue, 20 Mar 18

[1]  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).

[2]  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.

[3]  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.

[4]  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 $n-3\le r<n-2.$

[5]  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.

[6]  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 $1<p\le 2$.

Cross-lists for Tue, 20 Mar 18

[7]  arXiv:1803.06409 (cross-list from math.FA) [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.

[8]  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.

Replacements for Tue, 20 Mar 18

[9]  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)
[10]  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)
[11]  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)
[12]  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)
[13]  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)
[ total of 13 entries: 1-13 ]
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