Compactness Theorem Research Articles

Hilbert spaces and low-lying zeros of L-functions

Generalizing previous work of Iwaniec, Luo, and Sarnak (2000), we use information from one-level density theorems to estimate the proportion of non-vanishing of L-functions in a family at a low-lying height on the critical line (measured by the analytic conductor). To solve the Fourier optimization problems that arise, we provide a unified framework based on the theory of reproducing kernel Hilbert spaces of entire functions (there is one such space associated to each symmetry type). Explicit expressions for the reproducing kernels are given. We also revisit the problem of estimating the height of the first low-lying zero in a family, considered by Hughes and Rudnick (2003) and Bernard (2015). We solve the associated Fourier optimization problem in this setting by establishing a connection to the theory of de Branges spaces of entire functions and using the explicit reproducing kernels. In an appendix, we study the related problem of determining the sharp embeddings between the Hilbert spaces associated to the five symmetry types and the classical Paley-Wiener space.

ω1-strongly compact cardinals and normality

We present more applications of the recently introduced ω1-strongly compact cardinals in the context of either consistency or reflection results in General Topology, focusing on issues related to normality. In particular, we show that such large cardinal notion provides a new upper bound for the consistency strength of the statement “All normal Moore spaces are metrizable” (NMSC). The proof uses random forcing, as in the original consistency proof of NMSC due to Nykos-Kunen-Solovay (see Fleissner [10]). We establish a compactness theorem for normality (i.e., reflection of non-normality) in the realm of first countable spaces, using the least ω1-strongly compact cardinal, as well as two more similar compactness results on related topological properties. We finish the paper by combining the techniques of reflection and forcing to show that our new upper bound for the consistency strength of NMSC can be also obtained via Cohen forcing, using some arguments from Dow-Tall-Weiss [6].

Vibration Isolation and Noise Reduction Method Based on Phononic Crystal.

Phononic crystal is a new kind of sound insulation material, which has an elastic wave gap. When the elastic wave falls within the band gap, it will attenuate strongly in phononic crystal, and its attenuation degree is far greater than the predicted value of the mass density theorem. In this paper, the practical application of phononic crystals in low frequency sound insulation is taken as the breakthrough point. Firstly, the theory of phononic crystal band gap generation is calculated and analyzed, and the band structure of one-dimensional two-component Bragg scattering phononic crystals is calculated. Hypermesh is used to build the model, and the sound insulation performance of phononic crystals is simulated and analyzed through Nastran. A sound isolation test platform was built for the local resonant phononic crystal samples to verify its sound isolation ability.

A compactness result for the div-curl system with inhomogeneous mixed boundary conditions for bounded Lipschitz domains and some applications

For a bounded Lipschitz domain with Lipschitz interface we show the following compactness theorem: Any mathsf {L}_{}^{2}-bounded sequence of vector fields with mathsf {L}_{}^{2}-bounded rotations and mathsf {L}_{}^{2}-bounded divergences as well as mathsf {L}_{}^{2}-bounded tangential traces on one part of the boundary and mathsf {L}_{}^{2}-bounded normal traces on the other part of the boundary, contains a strongly mathsf {L}_{}^{2}-convergent subsequence. This generalises recent results for homogeneous mixed boundary conditions in Bauer et al. (SIAM J Math Anal 48(4):2912-2943, 2016) Bauer et al. (in: Maxwell’s Equations: Analysis and Numerics (Radon Series on Computational and Applied Mathematics 24), De Gruyter, pp. 77-104, 2019). As applications we present a related Friedrichs/Poincaré type estimate, a div-curl lemma, and show that the Maxwell operator with mixed tangential and impedance boundary conditions (Robin type boundary conditions) has compact resolvents.

The Density of Borel Sets

Using a novel approach, we develop a result analogous to the Lebesgue density theorem for Borel sets using the notion of category: If $B\subset \lbrack 0,1]$ is a Borel set, then there exists a first category set $S\subset B$ with the property that for every $x\in B-S$ there exists $\varepsilon >0$ such that $B\cap (x-\varepsilon ,x+\varepsilon )$ is a residual subset of $(x-\varepsilon ,x+\varepsilon )$.

The Dirichlet problem for Orlicz-Sobolev mappings between metric spaces

In this paper, we solve the Dirichlet problem for Orlicz-Sobolev maps between singular metric spaces that extends the corresponding result of Guo et al. [arXiv 2021]. As an intermediate step, we develop a version of the Rellich-Kondrachov compactness theorem for Orlicz-Sobolev mappings between metric spaces that extends a previous result of Guo and Wenger [Comm. Anal. Geom. 2020]. Another crucial ingredient is an Orlicz-Sobolev extension of the trace theory for metric valued Sobolev maps developed by Korevaar and Schoen [Comm. Anal. Geom. 1993].

Existence and multiplicity of solutions to Dirichlet problem for semilinear subelliptic equation with a free perturbation

This paper is concerned with the existence and multiplicity results for the semilinear subelliptic equation with free perturbation term. By using the degenerate Rellich-Kondrachov compact embedding theorem, the precise lower bound estimate of Dirichlet eigenvalues for the finitely degenerate elliptic operator and minimax method, we obtain the existence and multiplicity of weak solutions for the problem.

Polynomial effective density in quotients of $${\mathbb {H}}^3$$ and $${\mathbb {H}}^2\times {\mathbb {H}}^2$$

We prove effective density theorems, with a polynomial error rate, for orbits of the upper triangular subgroup of \({\mathrm{SL}}_2({\mathbb {R}})\) in arithmetic quotients of \({\mathrm{SL}}_2({\mathbb {C}})\) and \({\mathrm{SL}}_2({\mathbb {R}})\times {\mathrm{SL}}_2({\mathbb {R}})\). The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.

Global dynamics of stochastic tidal equations

The current work deals with the global dynamics of 2D stochastic tidal equations in a highly heterogeneous environment. With the help of the stochastic version of the sigma-convergence method in conjunction with the Prokhorov and Skorokhod compactness theorems, we prove that the dynamics at the macroscopic level is of the same type at the microscopic level, but this time with non oscillating parameters. We also prove a corrector-type result.

A foliated Hitchin-Kobayashi correspondence

We prove an analogue of the Hitchin-Kobayashi correspondence for compact, oriented, taut Riemannian foliated manifolds with transverse Hermitian structure. In particular, our Hitchin-Kobayashi theorem holds on any compact Sasakian manifold. We define the notion of stability for foliated Hermitian vector bundles with transverse holomorphic structure and prove that such bundles admit a basic Hermitian-Einstein connection if and only if they are polystable. Our proof is obtained by adapting the proof by Uhlenbeck and Yau to the foliated setting. We relate the transverse Hermitian-Einstein equations to higher dimensional instanton equations and in particular we look at the relation to higher contact instantons on Sasakian manifolds. For foliations of complex codimension 1, we obtain a transverse Narasimhan-Seshadri theorem. We also demonstrate that the weak Uhlenbeck compactness theorem fails in general for basic connections on a foliated bundle. This shows that not every result in gauge theory carries over to the foliated setting.

Density of smooth functions in Musielak–Orlicz spaces

We provide necessary and sufficient conditions for the space of smooth functions with compact supports $$C^\infty _c(\Omega )$$ to be dense in Musielak–Orlicz spaces $$L^\Phi (\Omega )$$ where $$\Omega $$ is an open subset of $${\mathbb {R}}^d$$ . In particular, we prove that if $$\Phi $$ satisfies condition $$\Delta _2$$ , the closure of $$C^\infty _c(\Omega )\cap L^\Phi (\Omega )$$ is equal to $$L^\Phi (\Omega )$$ if and only if the measure of singular points of $$\Phi $$ is equal to zero. This extends the earlier density theorems proved under the assumption of local integrability of $$\Phi $$ , which implies that the measure of the singular points of $$\Phi $$ is zero. As a corollary we obtain analogous results for Musielak–Orlicz spaces generated by double phase functional and we recover the well-known result for variable exponent Lebesgue spaces.

A compactness theorem for SO(3) anti-self-dual equation with translation symmetry

Motivated by the Atiyah–Floer conjecture, we consider SO(3) anti-self-dual instantons on the product of the real line and a three-manifold with cylindrical end. We prove a Gromov–Uhlenbeck type compactness theorem, namely, any sequence of such instantons with uniform energy bound has a subsequence converging to a type of singular objects which may have both instanton and holomorphic curve components. In particular, we prove that holomorphic curves that appear in the compactification must satisfy the Lagrangian boundary condition, a claim which has been long believed in the literature. This result is the first step towards constructing a natural bounding cochain proposed by Fukaya for the SO(3) Atiyah–Floer conjecture.

An algorithm to recognize regular singular Mahler systems

This paper is devoted to the study of the analytic properties of Mahler systems at 0 0 . We give an effective characterisation of Mahler systems that are regular singular at 0 0 , that is, systems which are equivalent to constant ones. Similar characterisations already exist for differential and ( q q -)difference systems but they do not apply in the Mahler case. This work fills in the gap by giving an algorithm which decides whether or not a Mahler system is regular singular at 0 0 . In particular, it gives an effective characterisation of Mahler systems to which an analog of Schlesinger’s density theorem applies.

Density and Positive Mass Theorems for Initial Data Sets with Boundary

We prove a harmonic asymptotics density theorem for asymptotically flat initial data sets with compact boundary that satisfy the dominant energy condition. We use this to settle the spacetime positive mass theorem, with rigidity, for initial data sets with apparent horizon boundary in dimensions less than $8$ without a spin assumption.

The Optimal Classical Continuous Control Quaternary Vector of Quaternary Nonlinear Hyperbolic Boundary Value Problem

This work is concerned with studying the optimal classical continuous control quaternary vector problem. It is consisted of; the quaternary nonlinear hyperbolic boundary value problem and the cost functional. At first, the weak form of the quaternary nonlinear hyperbolic boundary value problem is obtained. Then under suitable hypotheses, the existence theorem of a unique state quaternary vector solution for the weak form where the classical continuous control quaternary vector is considered known is stated and demonstrated by employing the method of Galerkin and the compactness theorem. In addition, the continuity operator between the state quaternary vector solution of the weak form and the corresponding classical continuous control quaternary vector is demonstrated in three different infinite dimensional spaces (Hilbert spaces). Furthermore, with suitable hypotheses, the existence theorem of an optimal classical continuous control quaternary vector dominated by the weak form of the quaternary nonlinear hyperbolic boundary value problem is stated and demonstrated.

Positive logics

Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation, there is an obvious extension of first order logic with the two model theoretic properties mentioned, namely existential second order logic. We show that existential second order logic has a whole family of proper extensions satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. Furthermore, we show that in the context of negation-less logics, positive logics, as we call them, there is no strongest extension of first order logic with the Compactness Theorem and the Downward Löwenheim-Skolem Theorem.

A compactness Theorem for functions on Poisson point clouds

In this work we show a compactness Theorem for discrete functions on Poisson point clouds. We consider sequences with equibounded non-local p-Dirichlet energy: the novelty consists in the intermediate-interaction regime at which the non-local energy is computed.

Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domains

Let pi _{alpha } be a holomorphic discrete series representation of a connected semi-simple Lie group G with finite center, acting on a weighted Bergman space A^2_{alpha } (Omega ) on a bounded symmetric domain Omega , of formal dimension d_{pi _{alpha }} > 0. It is shown that if the Bergman kernel k^{(alpha )}_z is a cyclic vector for the restriction pi _{alpha } |_{Gamma } to a lattice Gamma le G (resp. (pi _{alpha } (gamma ) k^{(alpha )}_z)_{gamma in Gamma } is a frame for A^2_{alpha }(Omega )), then {{,mathrm{vol},}}(G/Gamma ) d_{pi _{alpha }} le |Gamma _z|^{-1}. The estimate {{,mathrm{vol},}}(G/Gamma ) d_{pi _{alpha }} ge |Gamma _z|^{-1} holds for k^{(alpha )}_z being a p_z-separating vector (resp. (pi _{alpha } (gamma ) k^{(alpha )}_z)_{gamma in Gamma / Gamma _z} being a Riesz sequence in A^2_{alpha } (Omega )). These estimates improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for G ={mathrm {PSU}}(1, 1).

Abductive Reasoning in Intuitionistic Propositional Logic via Theorem Synthesis

AbstractWith help of a compact Prolog-based theorem prover for Intuitionistic Propositional Logic, we synthesize minimal assumptions under which a given formula formula becomes a theorem. After applying our synthesis algorithm to cover basic abductive reasoning mechanisms, we synthesize conjunctions of literals that mimic rows of truth tables in classical or intermediate logics and we abduce conditional hypotheses that turn the theorems of classical or intermediate logics into theorems in intuitionistic logic. One step further, we generalize our abductive reasoning mechanism to synthesize more expressive sequent premises using a minimal set of canonical formulas, to which arbitrary formulas in the calculus can be reduced while preserving their provability. Organized as a self-contained literate Prolog program, the paper supports interactive exploration of its content and ensures full replicability of our results.

Polynomials over ring of integers of global fields that have roots modulo every finite indexed subgroup

A polynomial with coefficients in the ring of integers OK of a global field K is called intersective if it has a root modulo every finite-indexed subgroup of OK. We prove two criteria for a polynomial f(x)∈OK[x] to be intersective. One of these criteria is in terms of the Galois group of the splitting field of the polynomial, whereas the second criterion is verifiable entirely in terms of constants which depend upon K and the polynomial f. The proofs use the theory of global field extensions and upper bound on the least prime ideal in the Chebotarev density theorem.