Regularizing and decay rate estimates for solutions to the Cauchy problem of the Debye–Hückel system

In this paper we establish some regularizing and decay rate estimates for mild solutions of the Debye–Hückel system. We prove that if the initial data belong to the critical Lebesgue space L^\fracn2(\mathbbRⁿ){L^{\frac{n}{2}}(\mathbb{R}^{n})} , then the L ^q -norm ( \fracn2 ￡ q ￡ ￥{\frac{n}{2} \leq q \leq \infty}) of the βth order spatial derivative of mild solutions are majorized by K₁(K₂|b|)^|b|t^{-\frac|b|2-1+\fracn2q}{K_{1}(K_{2}|\beta|)^{|\beta|}t^{-\frac{|\beta|}{2}-1+\frac{n}{2q}}} for some constants K ₁ and K ₂. These estimates particularly imply that mild solutions are analytic in the space variable, and provide decay estimates in the time variable for higher-order derivatives of mild solutions. We also prove that similar estimates also hold for mild solutions whose initial data belong to the critical homogeneous Besov space [(B)\dot]^-2+\fracnp_p,￥(\mathbbRⁿ){\dot{B}^{-2+\frac{n}{p}}_{p,\infty}(\mathbb{R}^n)} ( \fracn2 < p < n{\frac{n}{2} < p < n}).     

A Hyperplane Inequality for Measures of Convex Bodies in ? n , n≤4

Let 2≤n≤4. We show that for an arbitrary measure μ with even continuous density in ℝ ⁿ and any origin-symmetric convex body K in ℝ ⁿ ,

Spectral decomposition and Gelfand’s theorem

We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set $\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\}We consider the spectral decomposition of A, the generator of a polynomially bounded n-times integrated group whose spectrum set s(A)={il_k;k ? \mathbb\mathbbZ^*}\sigma(A)=\{i\lambda_{k};k\in\mathbb{\mathbb{Z}}^{*}\} is discrete and satisfies ?\frac1|l_k|^ld_kⁿ < ￥\sum \frac{1}{|\lambda_{k}|^{\ell}\delta_{k}^{n}}<\infty , where ℓ is a nonnegative integer and d_k=min(\frac|l_k+1-l_k|2,\frac|l_k-1-l_k|2)\delta _{k}=\min(\frac{|\lambda_{k+1}-\lambda _{k}|}{2},\frac{|\lambda _{k-1}-\lambda _{k}|}{2}) . In this case, Theorem 3, we show by using Gelfand’s Theorem that there exists a family of projectors (P_k)_{k ? \mathbb\mathbbZ^*}(P_{k})_{k\in\mathbb{\mathbb{Z}}^{*}} such that, for any x∈D(A ^n+ℓ ), the decomposition ∑P _k x=x holds.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

An Exceptional Set in the Ergodic Theory of Expanding Maps on Manifolds

Under a general hypothesis an expanding map T of a Riemannian manifold M is known to preserve a measure equivalent to the Liouville measure on that manifold. As a consequence of this and Birkhoff’s pointwise ergodic theorem, the orbits of almost all points on the manifold are asymptotically distributed with regard to this Liouville measure. Let T be Lipschitz of class τ for some τ in (0,1], let Ω(x) denote the forward orbit closure of x and for a positive real number δ and let E(x₀, δ) denote the set of points x in M such that the distance from x₀ to Ω is at least δ. Let dim A denote the Hausdorff dimension of the set A. In this paper we prove a result which implies that there is a constant C(T) > 0 such that dimE(x₀,d) 3 dimM - \fracC(T)|logd| \dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\vert\!\log \delta \vert} if τ = 1 and dimE(x₀,d) 3 dimM - \fracC(T)log|logd|\dim E(x_0,\delta) \ge \dim M - \frac{C(T)}{\log \vert \log \delta \vert} if τ < 1. This gives a quantitative converse to the above asymptotic distribution phenomenon. The result we prove is of sufficient generality that a similar result for expanding hyperbolic rational maps of degree not less than two follows as a special case.     

Growth rate for beta-expansions

Let β > 1 and let m > β be an integer. Each x ? I_b:=[0,\fracm-1b-1]{x\in I_\beta:=[0,\frac{m-1}{\beta-1}]} can be represented in the form

Spectral properties of higher order anharmonic oscillators

We discuss spectral properties of the selfadjoint operator - \fracd²dt² + ( \fract^{k + 1}k + 1 - a )² \begin{gathered} - \frac{{{d^2}}}{{d{t^2}}} + {\left( {\frac{{{t^{k + 1}}}}{{k + 1}} - \alpha } \right)^2} \hfill \\ \hfill \\ \end{gathered} in L ²(ℝ) for odd integers k. We prove that the minimum over α of the ground state energy of this operator is attained at a unique point which tends to zero as k tends to infinity. We also show that the minimum is nondegenerate. These questions arise naturally in the spectral analysis of Schr?dinger operators with magnetic field. Bibliography: 13 titles. Illustrations: 2 figures.     

Pan-H-Linked Graphs

Let H be a multigraph, possibly containing loops. An H-subdivision is any simple graph obtained by replacing the edges of H with paths of arbitrary length. Let H be an arbitrary multigraph of order k, size m, n ₀(H) isolated vertices and n ₁(H) vertices of degree one. In Gould and Whalen (Graphs Comb. 23:165–182, 2007) it was shown that if G is a simple graph of order n containing an H-subdivision H{\mathcal{H}} and d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}}, then G contains a spanning H-subdivision with the same ground set as H{\mathcal{H}} . As a corollary to this result, the authors were able to obtain Dirac’s famed theorem on hamiltonian graphs; namely that if G is a graph of order n ≥ 3 with d(G) 3 \fracn2{\delta(G)\ge\frac{n}{2}} , then G is hamiltonian. Bondy (J. Comb. Theory Ser. B 11:80–84, 1971) extended Dirac’s theorem by showing that if G satisfied the condition d(G) 3 \fracn2{\delta(G) \ge \frac{n}{2}} then G was either pancyclic or a complete bipartite graph. In this paper, we extend the result from Gould and Whalen (Graphs Comb. 23:165–182, 2007) in a similar manner. An H-subdivision H{\mathcal{H}} in G is 1-extendible if there exists an H-subdivision H^*{\mathcal{H}^{*}} with the same ground set as H{\mathcal{H}} and |H^*| = |H| + 1{|\mathcal{H}^{*}| = |\mathcal{H}| + 1} . If every H-subdivision in G is 1-extendible, then G is pan-H-linked. We demonstrate that if H is sufficiently dense and G is a graph of large enough order n such that d(G) 3 \fracn+m-k+n₁(H)+2n₀(H)2{\delta(G) \ge \frac{n+m-k+n_1(H)+2n_0(H)}{2}} , then G is pan-H-linked. This result is sharp.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.     

Arithmetic mean of differences of Dedekind sums

Recently, Girstmair and Schoissengeier studied the asymptotic behavior of the arithmetic mean of Dedekind sums \frac1j(N) ? _{0 ￡ m < Ngcd(m,N)=1} |S(m,N)|\frac{1}{\varphi(N)} \sum_{\mathop{\mathop{ 0 \le m< N}}\limits_{\gcd(m,N)=1}} \vert S(m,N)\vert , as N → ∞. In this paper we consider the arithmetic mean of weighted differences of Dedekind sums in the form A_h(Q)=\frac1?_{\fracaq ? FQ}h(\fracaq) ×?_{\fracaq ? FQ}h(\fracaq) |s(a^￠,q^￠)-s(a,q)|A_{h}(Q)=\frac{1}{\sum_{\frac{a}{q} \in {\cal F}_{Q}}h\left(\frac{a}{q}\right)} \times \sum_{\frac{a}{q} \in {\cal F}_{\!Q}}h\left(\frac{a}{q}\right) \vert s(a^{\prime},q^{\prime})-s(a,q)\vert , where h:[0,1] ? \Bbb Ch:[0,1] \rightarrow {\Bbb C} is a continuous function with ò₀¹ h(t)  d t 1 0\int_0^1 h(t) \, {\rm d} t \ne 0 , \fracaq{\frac{a}{q}} runs over F_Q{\cal F}_{\!Q} , the set of Farey fractions of order Q in the unit interval [0,1] and \fracaq < \fraca^￠q^￠{\frac{a}{q}}<\frac{a^{\prime}}{q^{\prime}} are consecutive elements of F_Q{\cal F}_{\!Q} . We show that the limit lim _Q→∞ A _h (Q) exists and is independent of h.     

A characterisation result on a particular class of non-weighted minihypers

We present a characterisation of {e₁ (q+1)+e₀,e₁ ;n,q}{\{\epsilon_1 (q+1)+\epsilon_0,\epsilon_1 ;n,q\}} -minihypers, q square, q = p ^h , p > 3 prime, h ≥ 2, q ≥ 1217, for e₀ + e₁ < \fracq^7/122-\fracq^1/42{\epsilon_0 + \epsilon_1 < \frac{q^{7/12}}{2}-\frac{q^{1/4}}{2}}. This improves a characterisation result of Ferret and Storme (Des Codes Cryptogr 25(2): 143–162, 2002), involving more Baer subgeometries contained in the minihyper.     

On the directions problem in <Emphasis Type="Italic">AG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)

We prove that if q = p ^h , p a prime, do not exist sets U í AG(n,q){U {\subseteq} AG(n,q)}, with |U| = q ^k and 1 < k < n, determining N directions where

Univalent Functions, <Emphasis Type="Italic">VMOA</Emphasis> and Related Spaces

This paper is concerned mainly with the logarithmic Bloch space ℬ_log which consists of those functions f which are analytic in the unit disc \mathbbD{\mathbb{D}} and satisfy sup_{|z| < 1}(1-|z|)log\frac11-|z||f^￠(z)| < ￥\sup_{\vert z\vert <1}(1-\vert z\vert )\log\frac{1}{1-\vert z\vert}\vert f^{\prime}(z)\vert <\infty , and the analytic Besov spaces B ^p , 1≤p<∞. They are all subspaces of the space VMOA. We study the relation between these spaces, paying special attention to the membership of univalent functions in them. We give explicit examples of:

Criteria of exponential decay for eigenfunctions of some classes of integral operators

We study sufficient conditions for exponential decay at infinity for eigenfunctions of a class of integral equations in unbounded domains in ℝ ⁿ . We consider integral operators K whose kernels have the form

Improved Decay for Solutions to the Linear Wave Equation on a Schwarzschild Black Hole

We prove that sufficiently regular solutions to the wave equation ${\square_g\phi=0}We prove that sufficiently regular solutions to the wave equation \square_gf = 0{\square_g\phi=0} on the exterior of the Schwarzschild black hole obey the estimates |f| ￡ C_d v₊^-\frac32+d{|\phi|\leq C_\delta v_+^{-\frac{3}{2}+\delta}} and |?_tf| ￡ C_d v₊^-2+d{|\partial_t\phi|\leq C_{\delta} v_+^{-2+\delta}} on a compact region of r, including inside the black hole region. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite r and along the event horizon.     

Smooth Universal Taylor Series

Let W í \Bbb C\Omega \subseteq {\Bbb C} be a simply connected domain in \Bbb C{\Bbb C} , such that {￥} è[ \Bbb C \[`(W)]]\{\infty\} \cup [ {\Bbb C} \setminus \bar{\Omega}] is connected. If g is holomorphic in Ω and every derivative of g extends continuously on [`(W)]\bar{\Omega} , then we write g ∈ A^∞ (Ω). For g ∈ A^∞ (Ω) and z ? [`(W)]\zeta \in \bar{\Omega} we denote S_N (g,z)(z) = ?^N_l=0\fracg^(l) (z)l ! (z-z)^lS_N (g,\zeta )(z)= \sum^{N}_{l=0}\frac{g^{(l)} (\zeta )}{l !} (z-\zeta )^l . We prove the existence of a function f ∈ A^∞(Ω), such that the following hold:

Roots of Ehrhart polynomials arising from graphs

Several polytopes arise from finite graphs. For edge and symmetric edge polytopes, in particular, exhaustive computation of the Ehrhart polynomials not merely supports the conjecture of Beck et al. that all roots α of Ehrhart polynomials of polytopes of dimension D satisfy −D≤Re(α)≤D−1, but also reveals some interesting phenomena for each type of polytope. Here we present two new conjectures: (1) the roots of the Ehrhart polynomial of an edge polytope for a complete multipartite graph of order d lie in the circle |z+\fracd4| ￡ \fracd4|z+\frac{d}{4}| \le \frac{d}{4} or are negative integers, and (2) a Gorenstein Fano polytope of dimension D has the roots of its Ehrhart polynomial in the narrower strip -\fracD2 ￡ Re(a) ￡ \fracD2-1-\frac{D}{2} \leq \mathrm{Re}(\alpha) \leq \frac{D}{2}-1. Some rigorous results to support them are obtained as well as for the original conjecture. The root distribution of Ehrhart polynomials of each type of polytope is plotted in figures.     

The Ground State and the Long-Time Evolution in the CMC Einstein Flow

Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)³Vol_g(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by V_inf=(\frac-16Y(S))^\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g _i , K _i )} satisfying: (i) k _i  = −3, (ii) V_iˉ V_inf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q ₀((g _i , K _i )) ≤ Λ, where Q ₀ is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)²g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E₁)\tilde]=E₁(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E₁=Q₀+Q₁{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ V_inf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.     

Inequalities for irregular Gabor frames

In [C.K. Chui and X.L. Shi, Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal., 24 (1993), 263–277], the authors proved that if {e^imbxg(x-na): m,n ? \Bbb Z}\{e^{imbx}g(x-na): m,n\in{\Bbb Z}\} is a Gabor frame for L²(\Bbb R)L^2({\Bbb R}) with frame bounds A and B, then the following two inequalities hold: A ￡ \frac2pb?_{n ? \Bbb Z}|g(x-na)|² ￡ B,     a.e.A\le \frac{2\pi}{b}\sum_{n\in{\Bbb Z}}\vert g(x-na)\vert^2\le B, \quad a.e. and A ￡ \frac1a?_{m ? \Bbb Z}|[^(g)](w-mb)|² ￡ B,     a.e.A\le \frac{1}{a}\sum_{m\in{\Bbb Z}}\vert \hat{g}(\omega-mb)\vert^2\le B, \quad a.e. . In this paper, we show that similar inequalities hold for multi-generated irregular Gabor frames of the form è_{1 ￡ k ￡ r}{e^{iáx, l?}g_k(x-m): m ? D_k, l ? L_k }\bigcup_{1\le k\le r}\{e^{i\langle x, \lambda\rangle}g_{k}(x-\mu):\, \mu\in \Delta_k, \lambda\in\Lambda_k \} , where Δ _k and Λ _k are arbitrary sequences of points in \Bbb R^d{\Bbb R}^d and g_k ? L²(\Bbb R^d)g_k\in{L^2{(\Bbb R}^d)} , 1 ≤ k ≤ r.     

Remarks on a sine polynomial

Let n ≥ 0 be an integer. Then we have for ${x\in(0,\pi)}Let n ≥ 0 be an integer. Then we have for x ? (0,p){x\in(0,\pi)} :

?k=0n (( 2n+1) || (n-k ))\fracsin((2k+1)x)2k+1 ￡ \frac8n  n!(2n+1)!!.\sum_{k=0}^n { 2n+1 \choose n-k }\frac{\sin((2k+1)x)}{2k+1}\leq\frac{8^n \, n!}{(2n+1)!!}.