Classical Theorem Research Articles

Boas Type and Titchmarsh Type Theorems for Generalized Fourier-Bessel Transform

Let $$n\in \mathbb Z_{+}=\{0,1,\dots \}$$ , $$\nu >-1/2$$ . For a function f in the space $$L^{1}_{n,\nu }(\mathbb R_{+})$$ of special kind, we consider the generalized Fourier-Bessel transform $$\mathcal F_{n,\nu }(f)$$ . We prove a Boas type result about necessary and sufficient conditions for f to belong to the generalized uniform Lipschitz classes in terms of $$\mathcal F_{n,\nu }(f)$$ . Also an analogue of classical Titchmarsh theorem describing Lipschits classes in $$L^{2}$$ space with power weight is established in a more general setting.

Trotter–Kato approximations of semilinear stochastic evolution equations in Hilbert spaces

This article deeply studies a class of semilinear stochastic evolution equations in real separable Hilbert spaces. The main goal here is to consider Trotter–Kato approximations for mild solutions of such equations. After proving the existence and uniqueness of mild solutions, the Trotter–Kato approximation system is introduced and studied in detail. Based on this, the weak convergence of probability measures induced by mild solutions of Trotter–Kato approximation equations and the approximation error estimator are established. Finally, as an application, the classical limit theorem about the parameter dependence of this kind of equations is exhibited. To deal with the general additive diffusion term, which not only depends on the state but also depends on the probability distribution of the solution process at that time, we adopt some specific techniques of measure theory and random functional analysis.

Structure theorems for actions of homeomorphism groups

We give general classification and structure theorems for actions of groups of homeomorphisms and diffeomorphisms on manifolds, reminiscent of classical results for actions of (locally) compact groups. This gives a negative answer to Ghys’s extension problem for diffeomorphisms of manifolds with boundary, as well as a classification of all homomorphisms Homeo0(M)→Homeo0(N) when dim(M)=dim(N) (and related results for diffeomorphisms), and a complete classification of actions of Homeo0(S1) on surfaces. This resolves many problems in a program initiated by Ghys, and gives definitive answers to conjectures of Militon and Hurtado and a question of Rubin.

Further improvements to the Chevalley-Warning theorems

We study lower bound estimates for the number of solutions of systems of equations over finite fields. Heath-Brown improved the lower bounds given by the classical Chevalley-Warning Theorems by excluding systems of equations whose solutions form an affine space. We improve each of Heath-Brown's results and demonstrate sharpness in several cases.

Localized versions of extremal problems

We generalize several classical theorems in extremal combinatorics by replacing a global constraint with an inequality which holds for all objects in a given class. In particular we obtain generalizations of Turán’s theorem, the Erdős-Gallai theorem, the LYM-inequality, the Erdős-Ko-Rado theorem and the Erdős-Szekeres theorem on sequences.

How to tighten the control set?

We examine the problem of reducing the control set in a dynamical system so that the solution set and the attainable sets remain essentially unchanged. We cite some classical results to exhibit the problem and promote a set-valued approach. A necessary condition is formulated by using a concept of set-valued derivative, which can be regarded as an extension of the classical Relaxation theorem.

Tame fundamental groups of pure pairs and Abhyankar’s lemma

Let $(R,\mathfrak{m}, k)$ be a strictly local normal $k$-domain of positive characteristic and $P$ be a prime divisor on $X=\text{Spec } R$. We study the Galois category of finite covers over $X$ that are at worst tamely ramified over $P$ in the sense of Grothendieck--Murre. Assuming that $(X,P)$ is a purely $F$-regular pair, our main result is that every Galois cover $f \: Y \to X$ in that Galois category satisfies that $\bigl(f^{-1}(P)\bigr)_{\text{red}}$ is a prime divisor. We shall explain why this should be thought as a (partial) generalization of a classical theorem due to S.S.~Abhyankar regarding the \'etale-local structure of tamely ramified covers between normal schemes with respect to a divisor with normal crossings. Additionally, we investigate the formal consequences this result has on the structure of the fundamental group representing the Galois category. We also obtain a characteristic zero analog by reduction to positive characteristics following Bhatt--Gabber--Olsson's methods.

Design in the service of topology and the classification theorem

Topology is considered an advanced field in mathematics, and it might seem off-putting to people with no previous experience in mathematics. The classification theorem, which lies within the field of algebraic topology, is fascinating, but understanding it requires extensive mathematical knowledge. In this manuscript, we present a modular object that provides a glimpse into the construction of surfaces as defined by the classification theorem. This allows any user to explore different options and discover possible surfaces with three holes or fewer. This tactile experience of exploration grants the user intuition about the meaning of the theorem, without needing the rigorous proof; thus, it simplifies the depth of the classification theorem and its respective proof for non-mathematical audiences. We strive to accurately illustrate these esoteric concepts by giving them a form and to describe the modeling process of creating a tangible, touchable sculpture designed to intrigue and invite simple understanding. We believe that this playful object will be exciting for a variety of communities: university students, graduate students, teachers, professional mathematicians, and especially designers, who can use it to produce new tools.

The Steinberg representation is irreducible

We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.

Combinatorial classification of (±1)-skew projective spaces

AbstractThe non-commutative projective scheme $\operatorname{\mathsf{Proj_{nc}}} S$ of a $(\pm 1)$-skew polynomial algebra S in n variables is considered to be a $(\pm 1)$-skew projective space of dimension n − 1. In this paper, using combinatorial methods, we give a classification theorem for $(\pm 1)$-skew projective spaces. Specifically, among other equivalences, we prove that $(\pm 1)$-skew projective spaces $\operatorname{\mathsf{Proj_{nc}}} S$ and $\operatorname{\mathsf{Proj_{nc}}} S^{\prime}$ are isomorphic if and only if certain graphs associated with S and Sʹ are switching (or mutation) equivalent. We also discuss invariants of $(\pm 1)$-skew projective spaces from a combinatorial point of view.

Finding popular branchings in vertex-weighted directed graphs

Popular matchings have been intensively studied recently as a relaxed concept of stable matchings. By applying the concept of popular matchings to branchings in directed graphs, Kavitha et al. introduced popular branchings. Let G=(VG,EG) be a directed graph, where each vertex has preferences over its incoming arcs, and B and B′ be branchings in G. A vertex v∈VG prefers B to B′ if v prefers its incoming arc of B to that of B′, where having an arbitrary incoming arc is preferred to having no incoming arc. We say that B is more popular than B′ if the number of vertices preferring B to B′ is greater than the number of vertices preferring B to B′. A branching B is called a popular branching if there is no branching more popular than B. Kavitha et al. proposed an algorithm for finding a popular branching when the preferences of each vertex are given by a strict partial order. The correctness of this algorithm is proved by utilizing classical theorems on the duality of weighted arborescences. In this paper, we generalize popular branchings to weighted popular branchings in vertex-weighted directed graphs, in the same manner as weighted popular matchings by Mestre. We give an algorithm for finding a weighted popular branching for the case where the preferences of each vertex are given by a total preorder and the weights satisfy certain conditions. Our algorithm is based on Kavitha et al.'s algorithm, while it includes elaborated procedures resulting from the vertex-weights. Its correctness also builds upon the duality of weighted arborescences.

Pushdown Automata and Context-Free Grammars in Bisimulation Semantics

The Turing machine models an old-fashioned computer, that does not interact with the user or with other computers, and only does batch processing. Therefore, we came up with a Reactive Turing Machine that does not have these shortcomings. In the Reactive Turing Machine, transitions have labels to give a notion of interactivity. In the resulting process graph, we use bisimilarity instead of language equivalence. Subsequently, we considered other classical theorems and notions from automata theory and formal languages theory. In this paper, we consider the classical theorem of the correspondence between pushdown automata and context-free grammars. By changing the process operator of sequential composition to a sequencing operator with intermediate acceptance, we get a better correspondence in our setting. We find that the missing ingredient to recover the full correspondence is the addition of a notion of state awareness.

Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition

Abstract We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

Recognition of Affine-Equivalent Polyhedra by Their Natural Developments

The classical Cauchy rigidity theorem for convex polytopes reads that if two convex polytopes have isometric developments then they are congruent. In other words, we can decide whether two convex polyhedra are isometric or not by only using their developments. We study a similar problem of whether it is possible to understand that two convex polyhedra in Euclidean 3-space are affine-equivalent by only using their developments.

Relative VGIT and an Application to Degenerations of Hilbert Schemes

We generalize the classical semicontinuity theorem for GIT (semi)stable loci under variations of linearizations to a relative situation of an equivariant projective morphism X→S over an affine base S. As an application to moduli problems, we consider degenerations of Hilbert schemes and give a conceptual interpretation of the (semi)stable loci of the degeneration families constructed in [GHH19].

Envy-free division via configuration spaces

The classical approach to envy-free division and equilibrium problems arising in mathematical economics typically relies on Knaster-Kuratowski-Mazurkiewicz theorem, Sperner's lemma or some extension involving mapping degree. We propose a different and relatively novel approach where the emphasis is on configuration spaces and equivariant topology, originally developed for applications in discrete and computational geometry (Tverberg type problems, necklace splitting problem, etc.). We illustrate the method by proving several counterparts and extensions of the classical envy-free division theorem of David Gale, where the emphasis is on preferences allowing the players to choose degenerate pieces of the cake.

Borsuk-Ulam theorems for elementary abelian 2-groups

Let $G$ be a fct Lie group and let $U$ and $V$ be finite-dimensional real $G$-modules with $V^G=0$. A theorem of Marzantowicz, de Mattos and dos Santos estimates the covering dimension of the zero-set of a $G$-map from the unit sphere in $U$ to $V$ when $G$ is an elementary abelian $p$-group for some prime $p$ or a torus. In this note, the classical Borsuk-Ulam theorem will be used to give a refinement of their result estimating the dimension of that part of the zero-set on which an elementary abelian $p$-group $G$ acts freely or a torus $G$ acts with finite isotropy groups. The methods also provide an easy answer to a question raised in \cite{DM}.

A discrete four vertex theorem for hyperbolic polygons

There are many four vertex type theorems appearing in the literature, coming in both smooth and discrete flavors. The most familiar of these is the classical theorem in differential geometry, which states that the curvature function of a simple smooth closed curve in the plane has at least four extreme values. This theorem admits a natural discretization to Euclidean polygons due to O. Musin. In this article we adapt the techniques of Musin and prove a discrete four vertex theorem for convex hyperbolic polygons.

Boundedness of functions in fractional Orlicz–Sobolev spaces

A necessary and sufficient condition for fractional Orlicz–Sobolev spaces to be continuously embedded into L∞(Rn) is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into L∞(Rn) fails. They also augment a classical embedding theorem for standard fractional Sobolev spaces.

Nonlinear function-on-scalar regression via functional universal approximation.

We consider general nonlinear function-on-scalar (FOS) regression models, where the functional response depends on multiple scalar predictors in a general unknown nonlinear form. Existing methods either assume specific model forms (e.g., additive models) or directly estimate the nonlinear function in a space with dimension equal to the number of scalar predictors, which can only be applied to models with a few scalar predictors. To overcome these shortcomings, motivated by the classic universal approximation theorem used in neural networks, we develop a functional universal approximation theorem which can be used to approximate general nonlinear FOS maps and can be easily adopted into the framework of functional data analysis. With this theorem and utilizing smoothness regularity, we develop a novel method to fit the general nonlinear FOS regression model and make predictions. Our new method does not make any specific assumption on the model forms, and it avoids the direct estimation of nonlinear functions in a space with dimension equal to the number of scalar predictors. By estimating a sequence of bivariate functions, our method can be applied to models with a relatively large number of scalar predictors. The good performance of the proposed method is demonstrated by empirical studies on various simulated and realdatasets.