Compactness Theorem Research Articles

UNIFORM CONTINUITY OF FRACTIONAL ORDER INTEGRAL OF FRACTAL INTERPOLATION FUNCTION

It is the purpose of this paper that uniform continuity of fractional order integral of fractal interpolation function (FIF) is researched. First, FIF generated by affine transformation is constructed. Second, the combination between Riemann–Liouville fractional order integral and FIF is defined. Finally, the continuity and the uniform continuity of fractional order integral of FIF are proved by compactness theorem of numbers sequence or finite covering theorem. The result shows that fractional order integral of FIF is uniformly continuous on a closed interval.

On a Class of Lacunary Almost Newman Polynomials Modulo P and Density Theorems

Abstract The reduction modulo p of a family of lacunary integer polynomials, associated with the dynamical zeta function ζβ (z)of the β-shift, for β> 1 close to one, is investigated. We briefly recall how this family is correlated to the problem of Lehmer. A variety of questions is raised about their numbers of zeroes in 𝔽 p and their factorizations, via Kronecker’s Average Value Theorem (viewed as an analog of classical Theorems of Uniform Distribution Theory). These questions are partially answered using results of Schinzel, revisited by Sawin, Shusterman and Stoll, and density theorems (Frobenius, Chebotarev, Serre, Rosen). These questions arise from the search for the existence of integer polynomials of Mahler measure > 1 less than the smallest Salem number 1.176280. Explicit connection with modular forms (or modular representations) of the numbers of zeroes of these polynomials in 𝔽 p is obtained in a few cases. In general it is expected since it must exist according to the Langlands program.

On the hat guessing number of a planar graph class

The hat guessing number is a graph invariant based on a hat guessing game introduced by Winkler. Using a new vertex decomposition argument involving an edge density theorem of Erdős for hypergraphs, we show that the hat guessing number of all outerplanar graphs is less than 2125000. We also define the class of layered planar graphs, which contains outerplanar graphs, and we show that every layered planar graph has bounded hat guessing number.

Lebesgue’s density theorem and definable selectors for ideals

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which makes the ideal of countable sets satisfy an analogue to the density theorem. The proofs of the positive results use only elementary combinatorics of trees, while the negative results rely on forcing arguments.

Primes in the Chebotarev density theorem for all number fields (with an Appendix by Andrew Fiori)

Let L/K be any Galois extension of number fields such that L≠Q, and let C be a conjugacy class in Gal(L/K). We show that there exists a degree-one unramified prime p of K such that σp=C and Np≤dLB with B=310. This improves upon B=12577 obtained by Ahn and Kwon by making the argument of Lagarias, Montgomery, and Odlyzko explicit. Our improvements come from using the following key ideas: new weights, an optimized analysis on the location of the potential exceptional zeros for ζL(s), and a new version of Turán's power sum method which gives a stronger Deuring-Heilbronn phenomenon for ζL(s). We also use Fiori's numerical verification for number fields up to a certain discriminant height. Other results include a lower bound for the number of unramified primes p of K such that σp=C.

A Compactness Theorem for Stable Flat SL(2, ℂ) Connections on 3-Folds

A Compactness Theorem for Stable Flat SL(2, ℂ) Connections on 3-Folds

A compactness theorem for locally homogeneous spaces

We prove the existence and uniqueness of geometric models of local isometry classes of locally homogeneous spaces with sectional curvature $|\operatorname{sec}|\leq 1$. Moreover, we show that the set of geometric models is compact in the pointed $\mathcal{C}^{1,\alpha}$-topology.

Paraconsistent Metatheory: New Proofs with Old Tools

This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic(s) can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-capture’ versions of standard theorems, given suitable restrictions and background assumptions; but the shift to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems.

A Zero Density Estimate for Dedekind Zeta Functions

Abstract Given a nontrivial finite group $G$, we prove the first zero density estimate for families of Dedekind zeta functions associated to Galois extensions $K/{\mathbb {Q}}$ with $\textrm {Gal}(K/{\mathbb {Q}})\cong G$ that does not rely on unproven progress towards the strong form of Artin’s conjecture. We use this to remove the hypothesis of the strong Artin conjecture from the work of Pierce, Turnage-Butterbaugh, and Wood on the average error in the Chebotarev density theorem and $\ell $-torsion in ideal class groups.

Convergence problem of Schrödinger equation in Fourier-Lebesgue spaces with rough data and random data

In this paper, we consider the convergence problem of Schrödinger equation. Firstly, we show the almost everywhere pointwise convergence of Schrödinger equation in Fourier-Lebesgue spaces H ^ 1 p , p 2 ( R ) ( 4 ≤ p > ∞ ) , \hat {H}^{\frac {1}{p},\frac {p}{2}}(\mathbf {R})(4\leq p>\infty ), H ^ 3 s 1 p , 2 p 3 ( R 2 ) ( s 1 > 1 3 , 3 ≤ p > ∞ ) , \hat {H}^{\frac {3 s_{1}}{p},\frac {2p}{3}}(\mathbf {R}^{2})(s_{1}>\frac {1}{3},3\leq p>\infty ), H ^ 2 s 2 p , p ( R n ) ( s 2 > n 2 ( n + 1 ) , 2 ≤ p > ∞ , n ≥ 3 ) \hat {H}^{\frac {2 s_{2}}{p},p}(\mathbf {R}^{n})(s_{2}>\frac {n}{2(n+1)},2\leq p>\infty ,n\geq 3) with rough data. Secondly, we show that the maximal function estimate related to one dimensional Schrödinger equation can fail with data in H ^ s , p 2 ( R ) ( s > 1 p ) \hat {H}^{s,\frac {p}{2}}(\mathbf {R})(s>\frac {1}{p}) . Finally, we show the stochastic continuity of Schrödinger equation with random data in L ^ r ( R n ) ( 2 ≤ r > ∞ ) \hat {L}^{r}(\mathbf {R}^{n})(2\leq r>\infty ) almost surely. The main ingredients are maximal function estimates and density theorem in Fourier-Lebesgue spaces as well as some large deviation estimates.

An abstract approach to approximation in spaces of pseudocontinuable functions

We provide an abstract approach to approximation with a wide range of regularity classes X X in spaces of pseudocontinuable functions K ϑ p K^p_\vartheta , where ϑ \vartheta is an inner function and p > 0 p>0 . More precisely, we demonstrate a general principle, attributed to A. Aleksandrov, which asserts that if a certain linear manifold X X is dense in K ϑ q K^{q}_\vartheta for some q > 0 q>0 , then X X is in fact dense in K ϑ p K^p_{\vartheta } for all p > 0 p>0 . Moreover, for a rich class of Banach spaces of analytic functions X X , we describe the precise mechanism that determines when X X is dense in a certain space of pseudocontinuable functions. As a consequence, we obtain an extension of Aleksandrov’s density theorem to the class of analytic functions with uniformly convergent Taylor series.

QUASI-ISOMETRIC EMBEDDINGS INAPPROXIMABLE BY ANOSOV REPRESENTATIONS

AbstractWe construct examples of quasi-isometric embeddings of word hyperbolic groups into $\mathsf {SL}(d,\mathbb {R})$ for $d \geq 4$ which are not limits of Anosov representations into $\mathsf {SL}(d,\mathbb {R})$ . As a consequence, we conclude that an analogue of the density theorem for $\mathsf {PSL}(2,\mathbb {C})$ does not hold for $\mathsf {SL}(d,\mathbb {R})$ when $d \geq 4$ .

Global weak solution to the degenerate quantum Navier‐Stokes‐Maxwell equations with damping term

AbstractIn this paper we investigate the global existence of the finite energy weak solution to the isentropic quantum Navier‐Stokes‐Maxwell equations with density‐dependent viscosity and the damping term in three‐dimensional periodic domain. This model is composed of the isentropic compressible Navier‐Stokes‐Maxwell equation with density‐dependent viscosity, the quantum Bohm potential and a damping term. This model can be used to describe the motion of the particle in the plasma under the electric–magnetic field. The main techniques are the Faedo‐Galerkin method, some imbedding inequalities and some compactness theorems.

Bubbling of quasiregular maps

We give a version of Gromov’s compactness theorem for pseudoholomorphic curves in the case of quasiregular mappings between closed manifolds. More precisely we show that, given \(K\ge 1\) and \(D\ge 1\), any sequence \((f_n :M \rightarrow N)\) of K-quasiregular mappings of degree D between closed Riemannian d-manifolds has a subsequence which converges to a K-quasiregular mapping \(f:X\rightarrow N\) of degree D on a nodal d-manifold X.

Optimal Local Approximation Spaces for Parabolic Problems

We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in space and time. To construct local approximation spaces we consider a compact transfer operator that acts on the space of local solutions and covers the full time dimension. The optimal local spaces are then given by the left singular vectors of the transfer operator. To prove compactness of the latter we combine a suitable parabolic Caccioppoli inequality with the compactness theorem of Aubin--Lions. In contrast to the elliptic setting [I. Babuška and R. Lipton, Multiscale Model. Simul., 9 (2011), pp. 373--406] we need an additional regularity result to combine the two results. Furthermore, we employ the generalized finite element method to couple local spaces and construct an approximation of the global solution. Since our approach yields reduced space-time bases, the computation of the global approximation does not require a time stepping method and is thus computationally efficient. Moreover, we derive rigorous local and global a priori error bounds. In detail, we bound the global approximation error in a graph norm by the local errors in the $L^2(H^1)$-norm, noting that the space the transfer operator maps to is equipped with this norm. Numerical experiments demonstrate an exponential decay of the singular values of the transfer operator and the local and global approximation errors for problems with high contrast or multiscale structure regarding space and time.

On compact classes of Beltrami solutions and Dirichlet problem

This article is devoted to the study of the problem of compactness of solutions of the differential Beltrami equation with degeneration. We study the case when the complex characteristic of the equations satisfies the constraints of integral type. In this case, we have proved a theorem on the compactness of the class of homeomorphic solutions of the Beltrami equation, which satisfy the hydrodynamic normalization condition at infinity. Another important result is the compactness theorem for the class of open discrete solutions of the Dirichlet problem for the Beltrami equations in the Jordan domain, the imaginary part of which vanishes at some predetermined inner point.

A note on extrapolation of compactness

This note is devoted to the study of Hytönen’s extrapolation theorem of compactness on weighted Lebesgue spaces. Two criteria of compactness of linear operators in the two-weight setting are obtained. As applications, we obtain two-weight compactness of commutators of Calderón-Zygmund operators, fractional integrals and bilinear Calderón-Zygmund operators.

The density theorem for projective representations via twisted group von Neumann algebras

We consider converses to the density theorem for square-integrable, irreducible, projective, unitary group representations restricted to lattices using the dimension theory of Hilbert modules over twisted group von Neumann algebras. We show that the restriction of such a σ-projective unitary representation π of a unimodular, second-countable group G to a lattice Γ extends to a Hilbert module over the twisted group von Neumann algebra of (Γ,σ). We then compute the center-valued von Neumann dimension of this Hilbert module. For abelian groups with 2-cocycle satisfying Kleppner's condition, we show that the center-valued von Neumann dimension reduces to the scalar value dπvol(G/Γ), where dπ is the formal dimension of π and vol(G/Γ) is the covolume of Γ in G. We apply our results to characterize the existence of multiwindow super frames and Riesz sequences associated to π and Γ. In particular, we characterize when a lattice in the time-frequency plane of a second-countable, locally compact abelian group admits a Gabor frame or Gabor Riesz sequence.

On geometrically finite degenerations II: convergence and divergence

In this paper, we study quasi post-critically finite degenerations for rational maps. We construct limits for such degenerations as geometrically finite rational maps on a finite tree of Riemann spheres. We prove the boundedness for such degenerations of hyperbolic rational maps with Sierpinski carpet Julia set and give criteria for the convergence for quasi-Blaschke products Q B d \mathcal {QB}_d , making progress towards the analogues of Thurston’s compactness theorem for acylindrical 3 3 -manifold and the double limit theorem for quasi-Fuchsian groups in complex dynamics. In the appendix, we apply such convergence results to show the existence of certain polynomial matings.

On the density theorem of Halász and Turán

On the density theorem of Halász and Turán