On the trigonometric polynomials of Fejér and Young

The trigonometric polynomials of Fejér and Young are defined by $S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}}$S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}} and $C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}$C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}, respectively. We prove that the inequality $\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}}$\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}} holds for all n ≥ 2 and x ∈ (0, π). The lower bound is sharp.     

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

Stability of symmetric closed characteristics on symmetric compact convex hypersurfaces in ℝ<Superscript>2<Emphasis Type="Italic">n</Emphasis></Superscript> under a pinching condition

In this paper, let Σ R2n be a symmetric compact convex hypersurface which is ( r, R )- pinched with R/r (5/3)1/2 . Then Σ carries at least two elliptic symmetric closed characteristics; moreover, Σ carries at least E [ n-1/2 ] + E [ n-1/3 ] non-hyperbolic symmetric closed characteristics.     

Existence and Comparison Results for Non-Uniformly Parabolic Problems

In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem

Jacob’s ladders,the structure of the Hardy-Littlewood integral and some new class of nonlinear integral equations

In this paper we obtain new formulae for short and microscopic parts of the Hardy-Littlewood integral, and the first asymptotic formula for the sixth-order expression $\left| {\zeta \left( {\tfrac{1} {2} + i\phi _1 \left( t \right)} \right)} \right|^4 \left| {\zeta \left( {\tfrac{1} {2} + it} \right)} \right|^2$\left| {\zeta \left( {\tfrac{1} {2} + i\phi _1 \left( t \right)} \right)} \right|^4 \left| {\zeta \left( {\tfrac{1} {2} + it} \right)} \right|^2. These formulae cannot be obtained in the theories of Balasubramanian, Heath-Brown and Ivić.     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

Statistical ensemble error bounds for homogenized microheterogeneous solids

Typically, in order to characterize the homogenized effective macroscopic response of new materials possessing random heterogeneous microstructure, a relation between averages $ {\left\langle {\user2{ \sigma }} \right\rangle }_{\Omega } = \mathbb{E}^{*} :{\left\langle {\user2{ \epsilon }} \right\rangle }_{\Omega } $ {\left\langle {\user2{ \sigma }} \right\rangle }_{\Omega } = \mathbb{E}^{*} :{\left\langle {\user2{ \epsilon }} \right\rangle }_{\Omega } is sought, where $ {\left\langle \cdot \right\rangle }_{\Omega } {\mathop = \limits^{{\text{def}}} }\tfrac{1} {{|\Omega |}}{\int_\Omega { \cdot \,d} }\Omega , $ {\left\langle \cdot \right\rangle }_{\Omega } {\mathop = \limits^{{\text{def}}} }\tfrac{1} {{|\Omega |}}{\int_\Omega { \cdot \,d} }\Omega , and where \varvecs\varvec{\sigma} and \varvece\varvec{\epsilon} are the stress and strain tensor fields within a statistically representative volume element (SRVE) of volume |$ \mathbb{E}^{*} , $ \mathbb{E}^{*} , is known as the effective property, and is the elasticity tensor used in usual macroscale analyses. In order to generate homogenized responses computationally, a series of detailed boundary value representations resolving the heterogeneous microstructure, posed over the SRVEs domain, must be solved. This requires an enormous numerical effort that can overwhelm most computational facilities. A natural way of generating an approximation to the SRVEs response is by first computing the response of smaller (subrepresentative) samples, each with a different random realization of the microstructural type under investigation, and then to ensemble average the results afterwards. Compared to a direct simulation of an SRVE, testing many small samples is a computationally inexpensive process since the number of floating point operations is greatly reduced, as well as the fact that the samples responses can be computed trivially in parallel. However, there is an inherent error in this process. Clearly the populations ensemble average is not the SRVEs response. However, as shown in this work, the moments on the distribution of the population can be used to generate rigorous upper and lower error bounds on the quality of the ensemble-generated response. Two-sided bounds are given on the SRVE response in terms of the ensemble average, its standard deviation and its skewness.     

Universal approach to the Takesaki-Takai $\gamma $-duality for crossed products

In this article, we generalize and simplify the proof of the Takesaki-Takai $\gamma $-duality theorem. Assume a morphism \textbf{\textit{$\omega \; :\; G\to Aut\left({\rm A}\right)$}} is a projective representation of the locally compact Abel group \textbf{\textit{$G$}} in \textbf{\textit{$Aut\left({\rm A}\right)$}}, mapping $\gamma \; :\; G\to G$ is continuous, and $\left({\rm A},\; G,\; \omega \right)$ is a dynamic system then there exists isomorphism \[\Upsilon \; :\; Env_{\hat{\omega }} {}^{\gamma } \left(L^{1} \left(\hat{G},\; Env_{\omega } {}^{\gamma } \left(L^{1} \left(G,\; {\rm A}\right)\right)\right)\right)\to {\rm A}\otimes LK\left(L^{2} \left(G\right)\right) \] which is the equivariant for the double dual action \[\hat{\hat{\omega }}\; :\; G\to Aut\left(Env_{\hat{\omega }} {}^{\gamma } \left(L^{1} \left(\hat{G},\; Env_{\omega } {}^{\gamma } \left(L^{1} \left(G,\; {\rm A}\right)\right)\right)\right)\right).\] These results deepen our understanding of the representation theory and are especially interesting given their possible applications to problems of the quantum theory.     

Some Tauberian conditions for Cesàro summability method

Let u = (u _n ) be a sequence of real numbers whose generator sequence is Cesàro summable to a finite number. We prove that (u _n ) is slowly oscillating if the sequence of Cesàro means of (ω _n ^(m−1)(u)) is increasing and the following two conditions are hold:

Factorizations that Involve Ramanujan＇s Function k（q） = r（q）r2（q2）

In the “lost notebook”, Ramanujan recorded infinite product expansions for

A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals

The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type

A sharp form of an embedding into multiple exponential spaces

Let Θ be a bounded open set in ℝ ⁿ , n ⩾ 2. In a well-known paper Indiana Univ. Math. J., 20, 1077–1092 (1971) Moser found the smallest value of K such that

On some problems involving Hardy’s function

Some problems involving the classical Hardy function

On the Fon-Der-Flaass interpretation of extremal examples for Turán’s (3, 4)-problem

Fon-Der-Flaass (1988) presented a general construction that converts an arbitrary [(C)\vec]₄\vec C_4 -free oriented graph Γ into a Turán (3, 4)-graph. He observed that all Turán-Brown-Kostochka examples result from his construction, and proved the lower bound $\tfrac{4} {9} $\tfrac{4} {9} (1 − o(1)) on the edge density of any Turán (3, 4)-graph obtainable in this way. In this paper we establish the optimal bound $\tfrac{3} {7} $\tfrac{3} {7} (1 − o(1)) on the edge density of any Turán (3, 4)-graph resulting from the Fon-Der-Flaass construction under any of the following assumptions on the undirected graph G underlying the oriented graph Γ: (i) G is complete multipartite; (ii) the edge density of G is not less than $\tfrac{2} {3} - \varepsilon $\tfrac{2} {3} - \varepsilon for some absolute constant ε > 0. We are also able to improve Fon-Der-Flaass’s bound to $\tfrac{7} {{16}} $\tfrac{7} {{16}} (1 − o(1)) without any extra assumptions on Γ.     

On a variant of Giuga numbers

In this paper, we characterize the odd positive integers n satisfying the congruence∑n -1 j=1 j n-1/2 ≡ 0 (mod n). We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8.     

Infinitely many low- and high-energy solutions for a class of elliptic equations with variable exponent

This paper is concerned with the $p(x)$-Laplacian equation of the form $$ \left\{\begin{array}{ll} -\Delta_{p(x)} u=Q(x)|u|^{r(x)-2}u, &\mbox{in}\ \Omega,\u=0, &\mbox{on}\ \partial \Omega, \end{array}\right. \eqno{0.1} $$ where $\Omega\subset\R^N$ is a smooth bounded domain, $1p^+$ and $Q: \overline{\Omega}\to\R$ is a nonnegative continuous function. We prove that (0.1) has infinitely many small solutions and infinitely many large solutions by using the Clark''s theorem and the symmetric mountain pass lemma.     

Integers without large prime factors in short intervals: Conditional results

Under the Riemann hypothesis and the conjecture that the order of growth of the argument of ζ(1/2 + it) is bounded by $\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)}$\left( {\log t} \right)^{\frac{1} {2} + o\left( 1 \right)} , we show that for any given α > 0 the interval $(X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ]$(X,X + \sqrt X (\log X)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + o\left( 1 \right)} ] contains an integer having no prime factor exceeding X ^α for all X sufficiently large.     

On the hierarchies of higher order mKdV and KdV equations

The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $ \hat H_s^r \left( \mathbb{R} \right) $ \hat H_s^r \left( \mathbb{R} \right) defined by the norm

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u|t ≥ 2u + |v|q,|v|t ≥ 2v + |u|p in S = Rn × R+ with p,q > 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n4 ≤ max(ppq+11,pqq+11).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.