Proceedings of the American Mathematical Society

Journal Information
ISSN / EISSN: 00029939 / 10886826
Total articles ≅ 36,957

Latest articles in this journal

Published: 18 August 2022
Proceedings of the American Mathematical Society; https://doi.org/10.1090/proc/15729

Abstract:
The ‘popcorn function’ is a well-known and important example in real analysis with many interesting features. We prove that the box dimension of the graph of the popcorn function is 4/3, as well as computing the Assouad dimension and Assouad spectrum. The main ingredients include Duffin-Schaeffer type estimates from Diophantine approximation and the Chung-Erdős inequality from probability theory.
Zhiliang Deng, Xiaomei Yang
Published: 29 July 2022
Proceedings of the American Mathematical Society; https://doi.org/10.1090/proc/14517

Abstract:
In this paper, q-Gaussian distribution, q-analogy of Gaussian distribution, is introduced to characterize the prior information of unknown parameters for inverse problems. Based on q-Hermite polynomials, we propose a spectral likelihood approximation (SLA) algorithm of Bayesian inversion. Convergence results of the approximated posterior distribution in the sense of Kullback–Leibler divergence are obtained when the likelihood function is replaced with the SLA and the prior density function is truncated to its partial sum. In the end, two numerical examples are displayed, which verify our results.
Ryan Causey
Published: 15 July 2022
Proceedings of the American Mathematical Society; https://doi.org/10.1090/proc/14353

Abstract:
Given a Banach space $X$, a $w^*$-compact subset of $X^*$, and $1>p>\infty$, we provide an optimal relationship between the Szlenk index of $K$ and the Szlenk index of an associated subset of $L_p(X)^*$. As an application, given a Banach space $X$, we prove an optimal estimate of the Szlenk index of $L_p(X)$ in terms of the Szlenk index of $X$. This extends a result of Hájek and Schlumprecht to uncountable ordinals. More generally, given an operator $A:X\to Y$, we provide an estimate of the Szlenk index of the “pointwise $A$” operator $A_p:L_p(X)\to L_p(Y)$ in terms of the Szlenk index of $A$.
Michael Brandenbursky, Jarek Kędra
Published: 22 June 2022
Proceedings of the American Mathematical Society; https://doi.org/10.1090/proc/14683

Abstract:
We prove that manifolds with complicated enough fundamental group admit measure-preserving homeomorphisms which have positive stable fragmentation norm with respect to balls of bounded measure.
Alfonso Artigue, Bernardo Carvalho, Welington Cordeiro, José Vieitez
Published: 13 May 2022
Proceedings of the American Mathematical Society, Volume 150, pp 3369-3378; https://doi.org/10.1090/proc/15326

Abstract:
We discuss the dynamics beyond topological hyperbolicity considering homeomorphisms satisfying the shadowing property and generalizations of expansivity. It is proved that transitive countably expansive homeomorphisms satisfying the shadowing property are expansive in the set of transitive points. This is in contrast with pseudo-Anosov diffeomorphisms of the two-dimensional sphere that are transitive, cw-expansive, satisfy the shadowing property but the dynamical ball in each transitive point contains a Cantor subset. We exhibit examples of countably expansive homeomorphisms that are not finite expansive, satisfy the shadowing property and admits an infinite number of chain-recurrent classes. We further explore the relation between countable and entropy expansivity and prove that for surface homeomorphisms $f\colon S\to S$ satisfying the shadowing property and $\Omega (f)=S$, both countably expansive and entropy cw-expansive are equivalent to being topologically conjugate to an Anosov diffeomorphism.
Xuedong Chai, Yufeng Zhang, Yong Chen, Shiyin Zhao
Published: 14 April 2022
Proceedings of the American Mathematical Society, Volume 150, pp 2879-2887; https://doi.org/10.1090/proc/15716

Abstract:
The (2+1)-dimensional Jimbo-Miwa equation is analyzed by means of the $\bar {{\partial }}$-dressing method. By means of the characteristic function and Green’s function of the Lax representation, the problem has been transformed into a new $\bar {{\partial }}$ problem. A solution is constructed based on solving the $\bar {{\partial }}$ problem with the help of Cauchy-Green formula and choosing the proper spectral transformation. Furthermore, we can obtain the solution formally of the Jimbo-Miwa equation when the time evolution of the spectral data is determined.
Yu Yang, Lan Zou, Cheng-Hsiung Hsu
Published: 7 April 2022
Proceedings of the American Mathematical Society, Volume 150, pp 2901-2911; https://doi.org/10.1090/proc/15730

Abstract:
This paper is concerned with the global attractivity of a nonlocal reaction-diffusion viral infection model. By constructing suitable Lyapunov functionals, we show that the solutions of the model converge to a unique endemic equilibrium when the basic reproduction number is greater than one. The global attractivity for certain models with specific net growth rate and cell-to-cell transmissions are investigated as examples for illustration. Our results improve and generalize some known results.
Aparajita Dasgupta, Vishvesh Kumar
Published: 29 March 2022
Proceedings of the American Mathematical Society, Volume 150, pp 2849-2860; https://doi.org/10.1090/proc/15661

Abstract:
The minimal operator and the maximal operator of an elliptic pseudo-differential operator with symbols on $\mathbb {Z}^n\times \mathbb {T}^n$ are proved to coincide and the domain is given in terms of a Sobolev space. Ellipticity and Fredholmness are proved to be equivalent for pseudo-differential operators on $\mathbb {Z}^n$. The index of an elliptic pseudo-differential operator on $\mathbb {Z}^n$ is also computed.
Frank Stephan, Guohua Wu, Bowen Yuan
Published: 29 March 2022
Proceedings of the American Mathematical Society, Volume 150, pp 3125-3131; https://doi.org/10.1090/proc/15325

Abstract:
A $\Pi ^{0}_{1}$ class $P$ is thin if every $\Pi ^{0}_{1}$ subclass $Q$ of $P$ is the intersection of $P$ with some clopen set. In 1993, Cenzer, Downey, Jockusch and Shore initiated the study of Turing degrees of members of thin $\Pi ^{0}_{1}$ classes, and proved that degrees containing no members of thin $\Pi ^{0}_{1}$ classes can be recursively enumerable, and can be minimal degree below $\mathbf {0}’$. In this paper, we work on this topic in terms of genericity, and prove that all 2-generic degrees contain no members of thin $\Pi ^{0}_{1}$ classes. In contrast to this, we show that all 1-generic degrees below $\mathbf {0}’$ contain members of thin $\Pi ^{0}_{1}$ classes.
D. Bongiorno, E. D’Aniello, U. Darji, L. Di Piazza
Published: 28 March 2022
Proceedings of the American Mathematical Society, Volume 150, pp 2823-2837; https://doi.org/10.1090/proc/15354

Abstract:
Weighted shifts are an important concrete class of operators in linear dynamics. In particular, they are an essential tool in distinguishing a variety of dynamical properties. Recently, a systematic study of dynamical properties of composition operators on $L^p$ spaces has been initiated. This class of operators includes weighted shifts and also allows flexibility in construction of other concrete examples. In this article, we study one such concrete class of operators, namely composition operators induced by measures on odometers. In particular, we study measures on odometers which induce mixing and transitive linear operators on $L^p$ spaces.