On the power of standard information for tractability for L∞ approximation of periodic functions in the worst case setting

We study multivariate approximation of periodic functions in the worst case setting with the error measured in the $L_{\infty }$ norm. We consider algorithms that use standard information ${\mathrm{\Lambda }}^{\mathrm{std}}$ consisting of function values or general linear information ${\mathrm{\Lambda }}^{\mathrm{all}}$ consisting of arbitrary continuous linear functionals. We investigate equivalences of various notions of algebraic and exponential tractability for ${\mathrm{\Lambda }}^{\mathrm{std}}$ and ${\mathrm{\Lambda }}^{\mathrm{all}}$ under the absolute or normalized error criterion, and show that the power of ${\mathrm{\Lambda }}^{\mathrm{std}}$ is the same as the one of ${\mathrm{\Lambda }}^{\mathrm{all}}$ for various notions of algebraic and exponential tractability. Our results can be applied to weighted Korobov spaces and Korobov spaces with exponential weights. This gives a special solution to Open Problem 145 as posed by Novak and Woźniakowski (2012) [40].     

Energy conservation for the weak solutions to the ideal inhomogeneous magnetohydrodynamic equations in a bounded domain

In this paper, we prove the energy conservation for the weak solutions of the three-dimensional ideal inhomogeneous magnetohydrodynamic (MHD) equations in a bounded domain. Two types of sufficient conditions on the regularity of the weak solutions are provided to ensure the energy conservation. Due to the presence of the boundary, we need to impose the boundedness in $L^p$ and the continuity in $L^{\frac{p}{p-2}}$ for the velocity and magnetic fields near the boundary.     

Dilations of semigroups on von Neumann algebras and noncommutative Lp-spaces

We prove that any weak* continuous semigroup ${(T_t)}_{t?0}$ of factorizable Markov maps acting on a von Neumann algebra M equipped with a normal faithful state can be dilated by a group of Markov ?-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative ${\mathrm{L}}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results imply the boundedness of the McIntosh's ${\mathrm{H}}^{\infty }$ functional calculus of the generators of these semigroups on the associated noncommutative ${\mathrm{L}}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of ${\mathbb{R}}^n$.     

On the local geometry of graphs in terms of their spectra

In this paper we consider the relation between the spectrum and the number of short cycles in large graphs. Suppose $G_1,G_2,G_3,\dots$ is a sequence of finite and connected graphs that share a common universal cover $T$ and such that the proportion of eigenvalues of $G_n$ that lie within the support of the spectrum of $T$ tends to 1 in the large $n$ limit. This is a weak notion of being Ramanujan. We prove such a sequence of graphs is asymptotically locally tree-like. This is deduced by way of an analogous theorem proved for certain infinite sofic graphs and unimodular networks, which extends results for regular graphs and certain infinite Cayley graphs.     

Non-uniqueness for a critical heat equation in two dimensions with singular data

Nonlinear heat equations in two dimensions with singular initial data are studied. In recent works nonlinearities with exponential growth of Trudinger-Moser type have been shown to manifest critical behavior: well-posedness in the subcritical case and non-existence for certain supercritical data. In this article we propose a specific model nonlinearity with Trudinger-Moser growth for which we obtain surprisingly complete results: a) for initial data strictly below a certain singular threshold function $\stackrel{?}{u}$ the problem is well-posed, b) for initial data above this threshold function $\stackrel{?}{u}$, there exists no solution, c) for the singular initial datum $\stackrel{?}{u}$ there is non-uniqueness. The function $\stackrel{?}{u}$ is a weak stationary singular solution of the problem, and we show that there exists also a regularizing classical solution with the same initial datum $\stackrel{?}{u}$.     

Global stability of an SAIRS epidemic model with vaccinations,transient immunity and treatment

We consider an SAIRS epidemic model with vaccinations and treatment, where asymptomatic and symptomatic infectious individuals are considered in the transmission of the disease. We found the basic reproduction number, ${ℛ}_0$ and using ${ℛ}_0$, we conducted global stability analysis. We proved when ${ℛ}_0<1$, the disease-free equilibrium is globally stable. If ${ℛ}_0>1$, the disease-free equilibrium in unstable and a unique endemic equilibrium exists. We explored the global stability of the endemic equilibrium and noticed it is globally stable under certain conditions. Moreover, we then considered a special case of the SAIRS model, the SAIR model. We proved the disease-free equilibrium is globally stability when ${ℛ}_0<1$ and the endemic equilibrium is globally stable when ${ℛ}_0>1$. Next, we numerically simulated our analytical results and plotted these for various cases. Finally, we performed sensitivity analysis to tell us how each parameter in the system affects disease transmission.     

A local condition for k-contractible edges

An edge of a $k$-connected graph is said to be $k$-contractible if the contraction of the edge results in a $k$-connected graph. For a graph $G$ and a vertex $x$ of $G$, let $G\left[N_G\left(x\right)\right]$ be the subgraph induced by the neighborhood of $x$. We prove that if $G\left[N_G\left(x\right)\right]$ has less than $?\frac{k}{2}?$ edges for any vertex $x$ of a $k$-connected graph $G$, then $G$ has a $k$-contractible edge. We also show that the bound $?\frac{k}{2}?$ is sharp.     

The cotype zeta function of Zd

We give an asymptotic formula for the number of sublattices $\Lambda \subseteq {\mathbb{Z}}^d$ of index at most $X$ for which ${\mathbb{Z}}^d/\Lambda$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to work of Stanley and Wang on Smith normal forms of random integral matrices and discuss connections to the Cohen–Lenstra heuristics. Our arguments are based on Petrogradsky’s formulas for the cotype zeta function of ${\mathbb{Z}}^d$, a multivariable generalization of the subgroup growth zeta function of ${\mathbb{Z}}^d$.     

A model-theoretic characterization of constant-depth arithmetic circuits

We study the class ${\mathrm{\#}\text{AC}}^0$ of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fan-in addition and multiplication gates. No model-theoretic characterization for arithmetic circuit classes is known so far. Inspired by Immerman's characterization of the Boolean circuit class ${\text{AC}}^0$, we remedy this situation and develop such a characterization of ${\mathrm{\#}\text{AC}}^0$. Our characterization can be interpreted as follows: Functions in ${\mathrm{\#}\text{AC}}^0$ are exactly those functions counting winning strategies in first-order model checking games. A consequence of our results is a new model-theoretic characterization of ${\text{TC}}^0$, the class of languages accepted by constant-depth polynomial-size majority circuits.     

Additive G-codes over Fq and their dualities

We define additive G-codes over finite fields. We prove that if C is an additive G-code over ${\mathbb{F}}_q$ with duality M then its dual with respect to this duality $C^M$ is an additive G-code. We prove that if M and $M^{\prime }$ are two dualities, then $C^M$ and $C^{M^{\prime }}$ are equivalent codes. Finally, we study the existence of self-dual codes for a variety of dualities and relate them to formally self-dual and linear self-dual codes.