Classical Theorem Research Articles

A local Sobolev inequality on Ricci flow and its applications

In this article, we prove a local Sobolev inequality for complete Ricci flows. Our main result is that the local ν-functional of a disk on a Ricci flow depends only on the Nash entropy based at the center of the disk, and consequently depends only on the volume of the disk. Furthermore, we introduce some applications of this local Sobolev inequality. These applications reveal the way in which the local geometry evolves along Ricci flow. In particular, we show that several classical theorems related to Perelman's monotonicity formula can be derived from our results.

The universal approximation theorem for complex-valued neural networks

We generalize the classical universal approximation theorem for neural networks to the case of complex-valued neural networks. Precisely, we consider feedforward networks with a complex activation function σ:C→C in which each neuron performs the operation CN→C,z↦σ(b+wTz) with weights w∈CN and a bias b∈C, and with σ applied componentwise. We completely characterize those activation functions σ for which the associated complex networks have the universal approximation property, meaning that they can uniformly approximate any continuous function on any compact subset of Cd arbitrarily well. Unlike the classical case of real networks, the set of “good activation functions”—which give rise to networks with the universal approximation property—differs significantly depending on whether one considers deep networks or shallow networks: For deep networks with at least two hidden layers, the universal approximation property holds as long as σ is neither a polynomial, a holomorphic function, or an antiholomorphic function. Shallow networks, on the other hand, are universal if and only if the real part or the imaginary part of σ is not a polyharmonic function.

Motives of moduli spaces of bundles on curves via variation of stability and flips

AbstractWe study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall‐crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall‐crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulae for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

Base matrices of various heights

AbstractA classical theorem of Balcar, Pelant, and Simon says that there is a base matrix of height ${\mathfrak h}$ , where ${\mathfrak h}$ is the distributivity number of ${\cal P} (\omega ) / {\mathrm {fin}}$ . We show that if the continuum ${\mathfrak c}$ is regular, then there is a base matrix of height ${\mathfrak c}$ , and that there are base matrices of any regular uncountable height $\leq {\mathfrak c}$ in the Cohen and random models. This answers questions of Fischer, Koelbing, and Wohofsky.

1/L2 corrected soft photon theorem from a CFT3 Ward identity

Classical soft theorems applied to probe scattering processes on AdS4 spacetimes predict the existence of perturbative 1/L2 corrections to the soft photon and soft graviton factors of asymptotically flat spacetimes. In this paper, we establish that the 1/L2 corrected soft photon theorem can be derived from a large N CFT3 Ward identity. We derive a perturbed soft photon mode operator on a flat spacetime patch in global AdS4 in terms of an integrated expression of the boundary CFT current. Using the same in the CFT3 Ward identity, we recover the 1/L2 corrected soft photon factor derived from classical soft theorems.

The Convolution Theorem Involving Windowed Free Metaplectic Transform

The convolution product is widely used in many fields, such as signal processing, numerical analysis and so on; however, the convolution theorem in the domain of the windowed metaplectic transformation (WFMT) has not been studied. The primary goal of this paper is to give the convolution theorem of WFMT. Firstly, we review the definitions of the FMT and WFMT and give the inversion formula of the WFMT and the relationship between the FMT and WFMT. Then, according to the form of the classical convolution theorem and the convolution operator of the FMT, the convolution theorem in the domain of the WFMT is given. Finally, we prove the existence theorems of the proposed convolution theorem.

Kähler-Ricci flow on rational homogeneous varieties

In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the Kähler-Ricci flow with homogeneous initial condition. This description enables us to compute explicitly the maximal existence time for any solution starting at a homogeneous Kähler metric and obtain explicit upper and lower bounds for several geometric quantities along the flow, including curvatures, volume, diameter, and the first non-zero eigenvalue of the Laplacian. As an application of our main result, we investigate the relationship between certain numerical invariants associated with ample divisors and numerical invariants arising from solutions of the Kähler-Ricci flow in the homogeneous setting.

On the $q$-Analogue of Pólya's Theorem

We answer a question posed by Michael Aissen in 1979 about the $q$-analogue of a classical theorem of George Pólya (1922) on the algebraicity of (generalized) diagonals of bivariate rational power series. In particular, we prove that the answer to Aissen's question, in which he considers $q$ as a variable, is negative in general. Moreover, we show that the answer is positive if and only if $q$ is a root of unity.

The Differentiation Lemma and the Reynolds Transport Theorem for submanifolds with corners

The Reynolds Transport Theorem, colloquially known as “differentiation under the integral sign”, is a central tool of applied mathematics, finding application in a variety of disciplines such as fluid dynamics, quantum mechanics, and statistical physics. In this work, we state and prove generalizations thereof to submanifolds with corners evolving in a manifold via the flow of a smooth time-independent or time-dependent vector field. Thereby we close a practically important gap in the mathematical literature, as related works require various “boundedness conditions” on domain or integrand that are cumbersome to satisfy in common modeling situations. By considering manifolds with corners, a generalization of manifolds and manifolds with boundary, this work constitutes a step towards a unified treatment of classical integral theorems for the “unbounded case” for which the boundary of the evolving set can exhibit some irregularity.

On the quasi-Figiel problem and extension of ε-isometry on unit sphere of [formula omitted] space

This paper aims to study the extension of a bijective ε-isometry and the existence of the Figiel operator of an isometric embedding between two unit spheres of Banach spaces. In the first part, we introduce and study the quasi-Figiel problem about a quasi-isometric embedding between unit spheres of two Banach spaces. Consequently, a quasi-anti-Lipschitz type inequality is obtained when the domain space is l∞n. Based on this quasi-anti-Lipschitz type inequality, the corresponding quasi-anti-Lipschitz type inequality is also obtained for an ε-isometric embedding defined on the unit sphere of an L∞,1+ space, i.e., a Banach space whose finite-dimensional subspaces can be any close to the finite-dimensional subspaces of l∞ in the sense of the Banach-Mazur distance. As an application of the quasi-anti-Lipschitz type inequality, we show that every bijective ε-isometry between the unit spheres of an L∞,1+ space and another Banach space can be extended to a bijective 5ε-isometry between their corresponding unit balls. In particular, this implies that every L∞,1+ space admits the Mazur-Ulam property. Furthermore, in this paper, some attempts are also made to generalize the classical Figiel's theorem to the local case with respect to an isometric embedding between unit spheres.

Free gs-Monoidal Categories and Free Markov Categories

Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams (=term graphs) as a combinatorial data structure. We formulate the appropriate 2-categorical universal property based on ideas of Walters and prove that our categories satisfy it. We expect our free categories to be relevant for computer implementations and we also argue that they can be used as statistical causal models generalizing Bayesian networks.

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.