Main Theorem Research Articles

A Simple Linear Controller Design to Ensure Global Exponential Stability for a Class of Uncertain Fifth-Order Nonlinear Systems

This paper intends to explore the controller design problem for a class of uncertain fifth-order nonlinear control systems. Combining the theory of differential and integral inequalities, a linear controller will be proposed to promote a class of nonlinear control systems with multiple uncertainties to achieve the goal of global exponential stability. In addition, the guaranteed exponential convergence rate of such uncertain nonlinear systems will also be precisely calculated. Finally, some numerical simulation results will also be presented to verify and illustrate the correctness of this main theorem and the design process of the controller

Complete convergence for the maximal partial sums without maximal inequalities

This work provides the necessary and sufficient conditions for complete convergence for the maximal partial sums of dependent random variables. The results are proved without using maximal inequalities. The main theorems can be applied to sequences of (i) m-pairwise negatively dependent random variables and (ii) m-extended negatively dependent random variables. While the result for case (i) unifies and improves many existing ones, the result for case (ii) complements the main theorem of Chen et al. [J. Appl. Probab., 2010]. Affirmative answers to open questions raised by Chen et al. [J. Math. Anal. Appl., 2014], and Wu and Rosalsky [Glas. Mat. Ser. III, 2015] are also given. Two examples illustrating the sharpness of the main result are presented.

A Functional Inequality and a New Class of Probabilities in the N-Person Red-and-Black Game

In this paper, we explore a model of an N-player, non-cooperative stochastic game, drawing inspiration from the discrete formulation of the red-and-black gambling problem, as initially introduced by Dubins and Savage in 1965. We extend upon the work of Pontiggia from 2007, presenting a main theorem that broadens the conditions under which bold strategies by all players can achieve a Nash equilibrium. This is obtained through the introduction of a novel functional inequality, which serves as a key analytical tool in our study. This inequality enables us to circumvent the restrictive conditions of super-multiplicativity and super-additivity prevalent in the works of Pontiggia and others. We conclude this paper with a series of illustrative examples that demonstrate the efficacy of our approach, notably highlighting its ability to accommodate a broader spectrum of probability functions than previously recognized in the existing literature.

On Suzuki-Proinov Type Contractions in Modular $b$-Metric Spaces with an Application

In this paper, by taking ${{\mathcal C}_\mathcal{A}}-$simulation function and Proinov type function into account, we set up a new contraction mapping called Suzuki$-$Proinov $\mathpzc{Z^*}_{\aE^*}^{\aR}(\alpha)-$contraction, including both rational expressions that possess quadratic terms and $\aE-$type contractions. Furthermore, we demonstrate a common fixed point theorem through the mappings endowed with triangular $\alpha-$admissibility in the setting of modular $b-$metric spaces. Besides that, we achieve some new outcomes that contribute to the current ones in the literature through the main theorem, and, as an application, we examine the existence of solutions to a class of functional equations emerging in dynamic programming.

On the Geometry and Topology of Discrete Groups: An Overview

In this paper, we provide a brief introduction to the main notions of geometric group theory and of asymptotic topology of finitely generated groups. We will start by presenting the basis of discrete groups and of the topology at infinity, then we will state some of the main theorems in these fields. Our aim is to give a sample of how the presence of a group action may affect the geometry of the underlying space and how in many cases topological methods may help the determine solutions of algebraic problems which may appear unrelated.

Fixed points of $G$-monotone mappings in metric and modular spaces

Let $C$ be a bounded, closed and convex subset of a reflexive metric space with a digraph $G$ such that $G$-intervals along walks are closed and convex. In the main theorem we show that if $T\colon C\rightarrow C$ is a monotone $G$-nonexpansive mapping and there exists $c\in C$ such that $Tc\in [c,\rightarrow )_{G}$, then $T$ has a fixed point provided for each $a\in C$, $[a,a]_{G}$ has the fixed point property for nonexpansive mappings. In particular, it gives an essential generalization of the Dehaish-Khamsi theorem concerning partial orders in complete uniformly convex hyperbolic metric spaces. Some counterparts of this result for modular spaces, and for commutative families of mappings are given too.

The high-order approximation of SPDEs with multiplicative noise via amplitude equations

The aim of this paper is to investigate the high-order approximation of a class of SPDEs with cubic nonlinearity driven by multiplicative noise with the help of the amplitude equations. The highlight of our work is the provision of approximate solutions with enhanced accuracy. Precisely, previous researches primarily concentrated on deriving approximate solutions via the first-order amplitude equations. However, this paper constructs approximate solutions by utilizing both first-order and second-order amplitude equations. And, we rigorously prove that such approximate solutions enjoy improved convergence property. To further illustrate our demonstration intuitively, we apply our main theorem to stochastic Allen–Cahn equation and present a numerical analysis.

A generalization of Alperin Fusion theorem and its applications

Let F be a saturated fusion system on a finite p-group S, and let P be a strongly F-closed subgroup of S. We define the concept “F-essential subgroups with respect to P” which are some proper subgroups of P satisfying some technical conditions and show that an F-isomorphism between subgroups of P can be factorized by some automorphisms of P and F-essential subgroups with respect to P. When P is taken to be equal to S, the Alperin-Goldschmidt fusion theorem can be obtained as a special case. We also show that P⊴F if and only if there is no F-essential subgroup with respect to P. The following definition is made: A p-group P is strongly resistant in saturated fusion systems if P⊴F whenever there is an over p-group S and a saturated fusion system F on S such that P is strongly F-closed. It is shown that several classes of p-groups are strongly resistant, which appears as our third main theorem. We also give a new necessary and sufficient criterion for a strongly F-closed subgroup to be normal in F. These results are obtained as a consequence of developing a theory of quasi and semi-saturated fusion systems, which seems to be interesting in its own right.

Shifts of finite type on locally finite groups

Abstract In this work we prove that every shift of finite type (SFT), sofic shift, and strongly irreducible shift on locally finite groups has strong dynamical properties. These properties include that every sofic shift is an SFT, every SFT is strongly irreducible, every strongly irreducible shift is an SFT, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that if every SFT on a group is strongly irreducible, or if every sofic shift is an SFT, then the group must be locally finite, and this extends to all of the properties we explore. These results are collected in two main theorems which characterize the local finiteness of groups by purely dynamical properties. In pursuit of these results, we present a formal construction of free extension shifts on a group G, which takes a shift on a subgroup H of G, and naturally extends it to a shift on all of G.

Corrigendum to “Non-linear singularity formation for circular vortex sheets”

This paper serves as a short corrigendum, describing an error in the derivation of the linearized equation. The main theorems remain correct, but they are based upon an incorrect linearization of the Birkhoff-Rott equation and hence are not directly applicable to the original physical problem.

Degree Factors with Red-Blue Coloring of Regular Graphs

Recently, motivated to control a distribution of the vertices having specified degree in a degree factor, the authors introduced a new problem in [Graphs Combin. 39 (2023) #85], which is a degree factor problem of graphs whose vertices are colored with red or blue. In this paper, we continue its research on regular graphs. Among some results, our main theorem is the following:
 Let $a$, $b$ and $k$ be integers with $1\leq a\leq k\leq b\leq k+a+1$, and let $r$ be a sufficiently large integer compared to $a$, $b$ and $k$. Let $G$ be an $r$-regular graph. We arbitrarily color every vertex of $G$ with red or blue so that no two red vertices are adjacent. Then $G$ has a factor $F$ such that $\deg_{F}(x)\in \{a,b\}$ for every red vertex $x$ and $\deg_{F}(y)\in \{k,k+1\}$ for every blue vertex $y$.

The Loewner–Kufarev energy and foliations by Weil–Petersson quasicircles

AbstractWe study foliations by chord–arc Jordan curves of the twice punctured Riemann sphere using the Loewner–Kufarev equation. We associate to such a foliation a function on the plane that describes the “local winding” along each leaf. Our main theorem is that this function has finite Dirichlet energy if and only if the Loewner driving measure has finite Loewner–Kufarev energy, defined by whenever is of the form , and set to otherwise. Moreover, if either of these two energies is finite, they are equal up to a constant factor, and in this case, the foliation leaves are Weil–Petersson quasicircles. This duality between energies has several consequences. The first is that the Loewner–Kufarev energy is reversible, that is, invariant under inversion and time reversal of the foliation. Furthermore, the Loewner energy of a Jordan curve can be expressed using the minimal Loewner–Kufarev energy of those measures that generate the curve as a leaf. This provides a new and quantitative characterization of Weil–Petersson quasicircles. Finally, we consider conformal distortion of the foliation and show that the Loewner–Kufarev energy satisfies an exact transformation law involving the Schwarzian derivative. The proof of our main theorem uses an isometry between the Dirichlet energy space on the unit disc and that we construct using Hadamard's variational formula expressed by means of the Loewner–Kufarev equation. Our results are related to ‐parameter duality and large deviations of Schramm–Loewner evolutions coupled with Gaussian random fields.

A regularization method for solving the G-variational inequality problem and fixed-point problems in Hilbert spaces endowed with graphs

This article considers and investigates a variational inequality problem and fixed-point problems in real Hilbert spaces endowed with graphs. A regularization method is proposed for solving a G-variational inequality problem and a common fixed-point problem of a finite family of G-nonexpansive mappings in the framework of Hilbert spaces endowed with graphs, which extends the work of Tiammee et al. (Fixed Point Theory Appl. 187, 2015) and Kangtunyakarn, A. (Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112:437–448, 2018). Under certain conditions, a strong convergence theorem of the proposed method is proved. Finally, we provide numerical examples to support our main theorem. The numerical examples show that the speed of the proposed method is better than some recent existing methods in the literature.

Nonlocal critical growth elliptic problems with jumping nonlinearities

In this paper we study a nonlocal critical growth elliptic problem driven by the fractional Laplacian in the presence of a jumping nonlinearity. By using variational and topological methods and applying some new linking theorems recently proved by Perera and Sportelli in [19], we prove the existence of a nontrivial solution for the problem under consideration.The results we obtain here are the nonlocal counterparts of the ones obtained in [19] in the context of a local equation. Due to the nonlocal nature of our problem, some additional difficulties arise, and the arguments employed in the local setting need to be improved or reconceived. In fact, the proofs of our main theorems require some refined techniques and new regularity results for weak solutions of nonlocal problems that are of independent interest.We would like to point out that our results are specifically for a nonlocal problem with the fractional operator in integral form. However, we do not exclude the possibility that our results may have a counterpart for the spectral operator studied in [27]. Since nonlocal operators in integral form are being widely investigated in the current literature, especially in connection with geometric problems, we have restricted ourselves to elliptic equations driven by a fractional operator in integral form here.

Simple Hurwitz groups and eta invariant

A Hurwitz group is a conformal automorphism group of a compact Riemann surface with precisely $84(g - 1)$ automorphisms, where $g$ is the genus of the surface. Our starting point is a result on the smallest Hurwitz group $\mathrm{PSL}(2,\mathbb{F}_{7})$ which is the automorphism group of the Klein surface. In this paper, we generalize it to various classes of simple Hurwitz groups and discuss a relationship between the surface symmetry and spectral asymmetry for compact Riemann surfaces. To be more precise, we show that the reducibility of an element of a simple Hurwitz group is equivalent to the vanishing of the $\eta$-invariant of the corresponding mapping torus. Several wide classes of simple Hurwitz groups which include the alternating group, the Chevalley group and the Monster, which is the largest sporadic simple group, satisfy our main theorem.

Analytic Ax–Schanuel for semi-abelian varieties and Nevanlinna theory

Let $A$ be a semi-abelian variety with an exponential map $\exp : \mathop{\mathrm{Lie}}(A) \to A$. The purpose of this paper is to explore Nevanlinna theory of the entire curve $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f := (\exp f, f) : \mathbf{C} \to A \times \mathop{\mathrm{Lie}}(A)$ associated with an entire curve $f : \mathbf{C} \to \mathop{\mathrm{Lie}}(A)$. Firstly we give a Nevanlinna theoretic proof to the analytic Ax–Schanuel Theorem for semi-abelian varieties, which was proved by J. Ax 1972 in the case of formal power series (Ax–Schanuel Theorem). We assume some non-degeneracy condition for $f$ such that the elements of the vector-valued function $f(z) - f(0) \in \mathop{\mathrm{Lie}}(A) \cong \mathbf{C}^n$ are $\mathbf{Q}$-linearly independent in the case of $A = (\mathbf{C}^{*})^{n}$. Our proof is based on the Log Bloch–Ochiai Theorem and a key estimate which we show. Our next aim is to establish a Second Main Theorem for $\hskip1pt{\widehat{\mathrm{exp}}}\hskip2pt f$ and its $k$-jet lifts with truncated counting functions at level one. We give some applications to a problem of a type raised by S. Lang and the unicity. The results clarify a relationship between the problems of Ax–Schanuel type and Nevanlinna theory.

Common Best Proximity Point Theorems for Generalized Dominating with Graphs and Applications in Differential Equations

In this paper, we initiate a concept of graph-proximal functions. Furthermore, we give a notion of being generalized Geraghty dominating for a pair of mappings. This permits us to establish the existence of and unique results for a common best proximity point of complete metric space. Additionally, we give a concrete example and corollaries related to the main theorem. In particular, we apply our main results to the case of metric spaces equipped with a reflexive binary relation. Finally, we demonstrate the existence of a solution to boundary value problems of particular second-order differential equations.

An analysis of the convergence problem of a function of hexagonal Fourier series in generalized Hölder norm using Hausdorff operator

In the present paper, we obtain the results on the degree of convergence of an ‐periodic continuous function in generalized Hölder spaces using Hausdorff operator with monotonically non‐decreasing and monotonically non‐increasing rows of its hexagonal Fourier series. Some important corollaries are also deduced from our main theorems. Applications of main theorems are also obtained in this paper.

Boundedness of the p-primary torsion of the Brauer group of an abelian variety

We prove that the $p^\infty$-torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a ‘flat Tate conjecture’ for divisors. We also study other geometric Galois-invariant $p^\infty$-torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely $p$-divisible. We explain how the existence of these $p$-divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic $p$.

Invertible bases and root vectors for analytic matrix-valued functions

We revisit the concept of a minimal basis through the lens of the theory of modules over a commutative ring $R$. We first review the conditions for the existence of a basis for submodules of $R^n$ where $R$ is a Bézout domain. Then, we define the concept of invertible basis of a submodule of $R^n,$ and when $R$ is an elementary divisor domain, we link it to the Main Theorem of G. D. Forney Jr. [SIAM J. Control, 13:493-520, 1975]. Over an elementary divisor domain, the submodules admitting an invertible basis are precisely the free pure submodules of $R^n$. As an application, we let $\Omega \subseteq \mathbb{C}$ be either a connected compact set or a connected open set, and we specialize to $R=\mathcal{A}(\Omega)$, the ring of functions that are analytic on $\Omega$. We show that, for any matrix $A(z) \in \mathcal{A}(\Omega)^{m \times n}$, $\ker A(z) \cap \mathcal{A}(\Omega)^n$ is a free $\mathcal{A}(\Omega)$-module and admits an invertible basis, or equivalently a basis that is full rank upon evaluation at any $\lambda \in \Omega$. Finally, given $\lambda \in \Omega$, we use invertible bases to define and study maximal sets of root vectors at $\lambda$ for $A(z)$. This in particular allows us to define eigenvectors also for analytic matrices that do not have full column rank.