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Exact estimates with respect to the order of magnitude are obtained for the ortho-projective and linear diameters of the classes B _p,?? ^r periodic functions of several variables in the spaces L _q , 1 ?? p, q ?? ??. The order of magnitude of the best approximation is established in the space Leo of the classes B _??,?? ^r of periodic functions of two variables with trigonometric polynomials with harmonics from a hyperbolic cross.     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

Inversion complexity of self-correcting circuits for a certain sequence of boolean functions

The asymptotics L _k ^? (f ₂ ⁿ ) ?? n min{k+1, p} is obtained for the sequence of Boolean functions $f_2^n \left( {x_1 , \ldots ,x_n } \right) = \mathop \vee \limits_{1 \leqslant i < j \leqslant n}$ for any fixed k, p ?? 1 and growing n, here L _k ^? (f ₂ ⁿ ) is the inversion complexity of realization of the function f ₂ ⁿ by k-self-correcting circuits of functional elements in the basis B = {&, ?}, p is the weight of a reliable invertor.     

The Z \to c\bar c \to \gamma \gamma *, Z \to b\bar b \to \gamma \gamma * triangle diagrams and the Z → γψ, Z → γγ decays

We expounded an approach for studying the Z ?? ??? and Z ?? ???? decay based on the sum rules for the $Z \to c\bar c \to \gamma \gamma *$ and $Z \to b\bar b \to \gamma \gamma *$ amplitudes and their derivatives. We calculate the branching ratios of the Z ?? ??? and Z ?? ???? decays under different suppositions about the saturation of the sum rules. We find the lower bounds of ?? _?? BR(Z ?? ???) = 1.95 · 10 ^?7 and ?? _?? BR(Z ?? ????) = 7.23 · 10^?7 and discuss deviations from the lower bounds including the possibility of BR[Z ?? ??J/??(1S)] ?? BR[Z ?? ????(1S)] ?? 10 ^?6 , which is probably measurable at the LHC. Moreover, we calculate the angle distributions in the Z ?? ??? and Z ?? ???? decays.     

Caricature of hydrodynamics for lattice dynamics

We consider the lattice dynamics in the half-space, with zero boundary condition. The initial data are supposed to be random function. We introduce the family of initial measures {?? ₀ ^? , ? > 0} depending on a small scaling parameter ?. We assume that the measures ?? ₀ ^? are locally homogeneous for space translations of order much less than ? ^?1 and nonhomogeneous for translations of order ? ^?1. Moreover, the covariance of ?? ₀ ^? decreases with distance uniformly in ?. Given ?? ?? ? / 0, r ?? ? ₊ ^d , and ?? > 0, we consider the distributions of random solution in the time moments t = ??/? _?? and at lattice points close to [r/?] ?? ? ₊ ^d . Themain goal is to study the asymptotic behavior of these distributions as ? ?? 0 and to derive the limit hydrodynamic equations of the Euler or Navier-Stokes type.     

A geometric interpretation of the transition density of a symmetric Lévy process

We study for a class of symmetric Lévy processes with state space R n the transition density pt（x） in terms of two one-parameter families of metrics, （dt）t>0 and （δt）t>0. The first family of metrics describes the diagonal term pt（0）; it is induced by the characteristic exponent ψ of the Lévy process by dt（x, y） = ^1/2tψ（x-y）. The second and new family of metrics δt relates to ^1/2tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density: pt（x） = pt（0）e- δ2t （x,0） where pt（0） corresponds to a volume term related to tψ and where an "exponential" decay is governed by δ2t . This gives a complete and new geometric, intrinsic interpretation of pt（x）.     

Fixed points of automorphisms preserving the length of words in free solvable groups

Let ?? be an automorphism of prime order p of the free group F _n . Suppose ?? has no fixed points and preserves the length of words. By ?? :=??? ^(m) we denote the automorphism of the free solvable group ${F_{n}/F_n^{(m)} }$ induced by ??. We show that every fixed point of ?? has the form ${cc^{\sigma} \ldots c^{\sigma^{p-1}}}$ , where ${c\in F_n^{(m-1)}/F_n^{(m)}}$ . This is a generalization of some known results, including the Macedo??ska?CSolitar Theorem [10].     

Extension of functions from Sobolev spaces

By definition, the domain Ω ??ⁿ belongs to the class EW _p ^l if there exists a continuous linear extension operator . An example is given of a domain Ω ??² with compact closure and Jordan boundary, having the following properties: (1) The curve ?Ω is not a quasicircle, has finite length and is Lipschitz in a neighborhood of any of its points except one. (2) Ω ε EW _p ¹ for p<2. and Ω ? EW _p ¹ for p?2. (3) for p>2 and for p?2.     

Some estimates of the normal approximation for ��-mixing random variables

Let ?? _n be a ??-mixing sequence of real random variables such that $ \mathbb{E}{\xi_n} = 0 $ , and let Y be a standard normal random variable. Write S _n = ?? ₁ + · · · + ?? _n and consider the normalized sums Z _n = S _n /B _n , where $ B_n^2 = \mathbb{E}S_n^2 $ . Assume that a thrice differentiable function $ h:\mathbb{R} \to \mathbb{R} $ satisfies $ {\sup_{x \in \mathbb{R}}}\left| {{h^s}(x)} \right| < \infty $ . We obtain upper bounds for $ {\Delta_n} = \left| {\mathbb{E}h\left( {{Z_n}} \right) - \mathbb{E}h(Y)} \right| $ in terms of Lyapunov fractions with explicit constants (see Theorem 1). In a particular case, the obtained upper bound of ?? _n is of order O(n ^?1/2). We note that the ??-mixing coefficients ??(r) are defined between the ??past?? and ??future.?? To prove the results, we apply the Bentkus approach.     

Interpolation in certain Hilbert spaces of analytic functions

In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(z_k)} _k=1 ^∞ :f∈B} on the choice of the sequence Z={z_k} _k=1 ^∞ of distinct points of the unit disk [6].     

HERBRAND’s Fundamental Theorem in the Eyes of JEAN VAN HEIJENOORT

Using Heijenoort??s unpublished generalized rules of quantification, we discuss the proof of Herbrand??s Fundamental Theorem in the form of Heijenoort??s correction of Herbrand??s ??False Lemma?? and present a didactic example. Although we are mainly concerned with the inner structure of Herbrand??s Fundamental Theorem and the questions of its quality and its depth, we also discuss the outer questions of its historical context and why Bernays called it ??the central theorem of predicate logic?? and considered the form of its expression to be ??concise and felicitous??.     

Uniform approximation of Poisson integrals of functions from the class H ω by de la Vallée Poussin sums

We obtain asymptotic equalities for least upper bounds of deviations in the uniform metric of de la Vallée Poussin sums on the sets C _?? ^q H _?? of Poisson integrals of functions from the class H _?? generated by convex upwards moduli of continuity ??(t) which satisfy the condition ??(t)/t ?? ?? as t ?? 0. As an implication, a solution of the Kolmogorov-Nikol??skii problem for de la Vallée Poussin sums on the sets of Poisson integrals of functions belonging to Lipschitz classes H ^??, 0 < ?? < 1, is obtained.     

Root System of a Perturbation of a Selfadjoint Operator with Discrete Spectrum

We analyze the perturbations T?+?B of a selfadjoint operator T in a Hilbert space H with discrete spectrum ${\{ t_k\}, T \phi_k = t_k \phi_k}$ . In particular, if t _k+1 ? t _k ?? ck ^{?? - 1}, ?? > 1/2 and ${\| B \phi_k \| = o(k^{\alpha - 1})}$ then the system of root vectors of T?+?B, eventually eigenvectors of geometric multiplicity 1, is an unconditional basis in H (Theorem 6). Under the assumptions ${t_{k+p} - t_k \geq d > 0, \forall k}$ (with d and p fixed) and ${\| B \phi_k \| \rightarrow 0}$ a Riesz system {P _k } of projections on invariant subspaces of T?+?B, Rank P _k ?? p, is constructed (Theorem 3).     

The annihilator of Bergman space

LetD be a bounded plane domain (with some smoothness requirements on its boundary). LetB _p(D), 1≤p<∞, be the Bergmanp-space ofD. In a previous paper we showed that the “natural projection”P, involving the Bergman kernel forD, is a bounded projection fromL _p(D) ontoB _p(D), 1<p<∞. With this we have the decompositionL _p(D)=B _p(D)⊕B _q ^⊥ (D,p ^–1+q ^–=1, 1<p< ∞. Here, we show that the annihilatorB _q ^⊥ (D) is the space of allL _p-complex derivatives of functions belonging to Sobolev space and which vanish on the boundary ofD. This extends a result of Schiffer for the casep=2. We also study certain operators onL _p(D). Especially, we show that , whereI is the identity operator and ? is an operator involving the adjoint of the Bergman kernel. Other relationships relevant toB _q ^⊥ (D) are studied.     

Embeddings of weighted sobolev spaces and generalized Caffarelli-Kohn-Nirenberg inequalities

We characterize all the real numbers a, b, c and 1 ?? p, q, r < ?? such that the weighted Sobolev space $$W_{\{ a,b\} }^{\{ q,q\} }({R^N}\backslash \{ 0\} ): = \{ u \in L_{loc}^1({R^N}\backslash \{ 0\} ):{\left| x \right|^{a/q}} \in {L^q}({R^{N),}}{\left| x \right|^{b/p}}\nabla u \in {({L^p}({R^N}))^N}\} $$ is continuously embedded into $${L^r}({R^N};{\left| x \right|^c}dx): = \{ u \in L_{loc}^1({R^N}\backslash \{ 0\} ):{\left| x \right|^{c/r}}u \in {L^r}({R^N})\} $$ with norm ??·?? _c,r . It turns out that, except when N ?? 2 and a = c = b ? p = ?N, such an embedding is equivalent to the multiplicative inequality $${\left\| u \right\|_{c,r}} \le C\left\| {\nabla u} \right\|_{b,p}^\theta \left\| u \right\|_{a,q}^{1 - \theta }$$ for some suitable ?? ?? [0, 1], which is often but not always unique. If a, b, c > ?N, then C ₀ ^?? (? ^N ) ? W _{a,b} ^(q,p) (? ^N {0}) ?? L ^r (? ^N ; |x| ^c dx) and such inequalities for u ?? C ₀ ^?? (? ^N ) are the well-known Caffarelli-Kohn-Nirenberg inequalities; but their generalization to W _{a,b} ^(q,p) (? ^N {0}) cannot be proved by a denseness argument. Without the assumption a, b, c > ?N, the inequalities are essentially new, even when u ?? C ₀ ^?? (? ^N {0}), although a few special cases are known, most notably the Hardy-type inequalities when p = q. In a different direction, the embedding theorem easily yields a generalization when the weights |x| ^a , |x| ^b and |x| ^c are replaced with more general weights w _a ,w _b and w _c , respectively, having multiple power-like singularities at finite distance and at infinity.     

Convergence of H 1-Galerkin mixed finite element method for parabolic problems with reduced regularity on initial data

We study the convergence of H ¹-Galerkin mixed finite element method for parabolic problems in one space dimension. Both semi-discrete and fully discrete schemes are analyzed assuming less regularity on initial data. More precisely, for the spatially discrete scheme, error estimates of order \(\mathcal{O}\) (h ² t ^?1/2) for positive time are established assuming the initial function p ₀ ∈ H ²(Ω) ∩ H ₀ ¹ (Ω). Further, we use energy technique together with parabolic duality argument to derive error estimates of order \(\mathcal{O}\) (h ² t ^?1) when p ₀ is only in H ₀ ¹ (Ω). A discrete-in-time backward Euler method is analyzed and almost optimal order error bounds are established.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Heat equation with a singular potential on the boundary and the Kato inequality

This paper is concerned with the heat equation in the half-space ? ₊ ^N with the singular potential function on the boundary, (P) $\left\{ \begin{gathered} \frac{\partial } {{\partial t}}u - \Delta u = 0\operatorname{in} \mathbb{R}_ + ^N \times (0,T), \hfill \\ \frac{\partial } {{\partial x_N }}u + \frac{\omega } {{|x|}}u = 0on\partial \mathbb{R}_ + ^N \times (0,T), \hfill \\ u(x,0) = u_0 (x) \geqslant ()0in\mathbb{R}_ + ^N , \hfill \\ \end{gathered} \right. $ where N ?? 3, ?? > 0, 0 < T ?? ??, and u ₀ ?? C ₀(? ₊ ^N ). We prove the existence of a threshold number ?? _N for the existence and the nonexistence of positive solutions of (P), which is characterized as the best constant of the Kato inequality in ? ₊ ^N .     

The bounds of restricted isometry constants for low rank matrices recovery

This paper discusses conditions under which the solution of linear system with minimal Schatten-p norm, 0 < p ? 1, is also the lowest-rank solution of this linear system. To study this problem, an important tool is the restricted isometry constant (RIC). Some papers provided the upper bounds of RIC to guarantee that the nuclear-norm minimization stably recovers a low-rank matrix. For example, Fazel improved the upper bounds to δ _4r ^A < 0.558 and δ _3r ^A < 0.4721, respectively. Recently, the upper bounds of RIC can be improved to δ _2r ^A < 0.307. In fact, by using some methods, the upper bounds of RIC can be improved to δ _2r ^A < 0.4931 and δ _2r ^A < 0.309. In this paper, we focus on the lower bounds of RIC, we show that there exists linear maps A with δ _2r ^A > 1/√2 or δ _r ^A > 1/3 for which nuclear norm recovery fail on some matrix with rank at most r. These results indicate that there is only a little limited room for improving the upper bounds for δ _2r ^A and δ _r ^A . Furthermore, we also discuss the upper bound of restricted isometry constant associated with linear maps A for Schatten p (0 < p < 1) quasi norm minimization problem.