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Most of the known continued fraction expansions of special functions are limit periodic. This means that the classical approximants S n (0) are normally not the best ones to use for approximations. In this paper we suggest a number of approximants S n (w n ) which converge faster. The estimation of the improvement and bounds for the error |f - Sn(wn)| (which we still call the truncation error) are mainly obtained by means of Thron's parabola sequence theorem and the oval sequence theorem.

iF(y)=( where G c R* is a bounded domain, y: G -» R , y'(x) = (dyydx), and F: M -> R is continuous. Here M denotes the linear space of real N X k matrices. We suppose throughout that K > 2, N > 2. In [7] F is called T-conυex if there exists a convex function /, defined on R, r = (t) ~ 1, such that F(p) = f(τ(p)) for all/? e M, where τ(p) denotes the minors of p of all orders j , 1 y uniformly on G with supx χGG\yj(x) ~ yj(x)\ < C < oo for ally. (Equivalently, if G has sufficiently regular boundary then IF is lsc if and only if IF is sequentially weak* lower semicontinuous on the Sobolev space W(G; R).) A consequence of [7? Theorem 3.6] is that IF lsc implies F polyconvex; that this conclusion is false was pointed out implicitly by Morrey [4, p. 26]. Morrey's remark is based on an example due to Terpstra [8] of a quadratic form

A subset A of a topological space is said to be semi-open if there exists an open set U such that U C A C Cl(U) where Cl(U) denotes the closure of U. The class of semi-open sets of a given topological space (X, J) is denoted S.O. (X, J). In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if J and V* are two topologies for a set X such that S.O.(X, 3) C S.O.(X, J*), then 'J C P. In a corollary to this theorem, Levine states that if S.O.(X, if) = S.O.(X,Jf*), then _T= f*. An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subse,ts can arise from functions, and points out some of the implications of two topologies being related in this manner. In [6] Norman Levine defines a set A in a topological space X to be semi-open if there exists an open set U such that U C A C Cl (U), where Cl(U) denotes the closure of U. The class of semi-open sets for a given topological space (X, i) is denoted S.O. (X, sT). Levine states in Theorem 10 of [6] that if Jf and * are two topologies for a set X such that S.O. (X, i) C S.O. (X, J9 then iT C *. In a corollary to this theorem, Levine states that if S.O. (X, ) = S.O. (X, iT*), then Jf = -" The following example which is due to S. Gene Crossley and S. K. Hildebrand [1, Example 1.1] shows the above-mentioned theorem and its corollary are false. Example. Let X = la, b, cl, J; = t0, sal, la, bl, la, cl, XI, ;* = 10, sal, la, bl, XI. An exhaustion of all possibilities shows that S.O. (X, ) = S.O. (x, 5j*). Crossley and Hildebrand [3] defined two topologies iT and T* on a set X to be semi-correspondent if S.O. (X, J) = S.O. (X, 5f*). It is shown in [3] that semi-correspondence is an equivalence relation on the collection of Received by the editors March 3, 1974. AMS (MOS) subject classifications (1970). Primary 54B99; Secondary 54C10.

Submitted by Vitor Silverio Rodrigues (vitorsrodrigues@reitoria.unesp.br) on 2014-05-27T11:25:28Z No. of bitstreams: 0Bitstream added on 2014-05-27T14:41:41Z : No. of bitstreams: 1 2-s2.0-79951846250.pdf: 494002 bytes, checksum: 56b6ee8beddda3e7dae971355d44a19f (MD5) Made available in DSpace on 2014-05-27T11:25:28Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-03-01 Item merged in doublecheck by Felipe Arakaki (arakaki@reitoria.unesp.br) on 2015-12-11T17:28:11Z Item was identical to item(s): 71803, 21370 at handle(s): http://hdl.handle.net/11449/72321, http://hdl.handle.net/11449/21804 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) The simplest necessary conditions for an entire function ψ(x) =∞ ∑ k=0 γk xk/k! to be in the Laguerre-Pólya class are the Turán inequalities γ2 k- γk+1γk-1 ≥ 0. These are in fact necessary and sufficient conditions for the second degree generalized Jensen polynomials associated with ψ to be hyperbolic. The higher order Turán inequalities 4(γ2 n - γn-1γn+1)(γ2n +1 - γnγn+2) - (γnγn+1 - γn-1γn+2) 2 ≥ 0 are also necessary conditions for a function of the above form to belong to the Laguerre-Pólya class. In fact, these two sets of inequalities guarantee that the third degree generalized Jensen polynomials are hyperbolic. Pólya conjectured in 1927 and Csordas, Norfolk and Varga proved in 1986 that the Turán inequalities hold for the coefficients of the Riemann ψ-function. In this short paper, we prove that the higher order Turán inequalities also hold for the ψ-function, establishing the hyperbolicity of the associated generalized Jensen polynomials of degree three. © 2010 American Mathematical Society. Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP Departamento de matemática Aplicada IMECC UNICAMP, 13083-859 Campinas, SP Departamento de Ciências de Computação e Estatística IBILCE, Universidade Estadual Paulista, 15054-000 São José do Rio Preto, SP FAPESP: 03/01874-2 FAPESP: 06/60420-0 CNPq: 305622/2009-9 CAPES: DGU-160

In this paper, we study the following critical weighted elliptic problem:\begin{equation*}\begin{cases}-\text{div}(|x|^\theta\nabla u) =(N^\prime+\tau)(N^\prime-2)|x|^\ell u^{p_\tau}, \text{in}\ \mathbb{R}^N,\\ u>0, \text{in}\ \mathbb{R}^N,\end{cases}\end{equation*} where $N^\prime=N+\theta>2,$ $\tau=\ell-\theta>-2,$ $ p_\tau=\frac{N^\prime+2+2\tau}{N^\prime-2}.$ Du and Guo (2015) showed the above problem possesses a family of finite Morse index solutions. Our main aim in this paper is to give its exact characterization of Morse index, which depends only on $N,\tau$ and $\theta$.

Let $H_1$, $H_2$, $H_3$ be real Hilbert spaces, let $A:H_1\rightarrow H_3$, $B:H_2\rightarrow H_3$ be two bounded linear operators. The general split common fixed-point problem under consideration in this paper is to $$\text{find}\ \ x \in \cap_{i=1}^p F(U_i),\ \ y \in \cap_{j=1}^r F(T_j)\ \ \text{such that}\ \ Ax = By,\eqno{(1)}$$ where $p$, $r\geq 1$ are integers, $U_i:H_1\rightarrow H_1$ $(1\leq i\leq p)$ and $T_j:H_2\rightarrow H_2$ $(1\leq j\leq r)$ are quasi-nonexpansive mappings with nonempty common fixed-point sets $\cap_{i=1}^pF(U_i)=\cap_{i=1}^p\{x\in H_1:U_ix=x\}$ and $\cap_{j=1}^rF(T_j)=\cap_{j=1}^r\{x\in H_2:T_jx=x\}$. Note that, the above problem (1) allows asymmetric and partial relations between the variables $x$ and $y$. If $H_2=H_3$ and $B=I$, then the general split common fixed-point problem (1) reduces to the general split common fixed-point problem proposed by Censor and Segal $\cite{C}$. In this paper, we introduce simultaneous parallel and cyclic algorithms for the general split common fixed-point problems (1). We introduce a way of selecting the stepsizes such that the implementation of our algorithms does not need any prior information about the operator norms. We prove the weak convergence of the proposed algorithms and apply the proposed algorithms to the multiple-set split feasibility problems. Our results improve and extend the corresponding results announced by many others.

The paper deals with a new object of study --- hyperbolic Hurwitz zeta function, which is given in the right \(\alpha\)-semiplane \( \alpha = \sigma + it \), \( \sigma> 1 \) by the equality $$ \zeta_H(\alpha; d, b) = \sum_{m \in \mathbb Z} \left(\, \overline{dm + b} \, \right)^{-\alpha}, $$ where \( d \neq0 \) and \( b \) --- any real number. Hyperbolic Hurwitz zeta function \( \zeta_H (\alpha; d, b) \), when \( \left\| \frac {b} {d} \right\|> 0 \) coincides with the hyperbolic zeta function of shifted one-dimensional lattice \( \zeta_H (\Lambda (d, b) | \alpha) \). The importance of this class of one-dimensional lattices is due to the fact that each Cartesian lattice is represented as a union of a finite number of Cartesian products of one-dimensional shifted lattices of the form \( \Lambda (d, b) = d \mathbb{Z} + b \). Cartesian products of one-dimensional shifted lattices are in substance shifted diagonal lattices, for which in this paper the simplest form of a functional equation for the hyperbolic zeta function of such lattices is given. The connection of the hyperbolic Hurwitz zeta function with the Hurwitz zeta function \( \zeta^* (\alpha; b)\) periodized by parameter \(b\) and with the ordinary Hurwitz zeta function \( \zeta (\alpha; b) \) is studied. New integral representations for these zeta functions and an analytic continuation to the left of the line \( \alpha = 1 + it \) are obtained. All considered hyperbolic zeta functions of lattices form an important class of Dirichlet series directly related to the development of the number-theoretical method in approximate analysis. For the study of such series the use of Abel's theorem is efficient, which gives an integral representation through improper integrals. Integration by parts of these improper integrals leads to improper integrals with Bernoulli polynomials, which are also studied in this paper.

This paper studies the approximation of the Hilbert transform f ? = H f of continuous functions f with continuous conjugate f ? based on a finite number of samples. It is known that every sequence { H N f } N ? N which approximates f ? from samples of f diverges (weakly) with respect to the uniform norm. This paper conjectures that all of these approximation sequences even contain no convergent subsequence. A property which is termed strong divergence.The conjecture is supported by two results. First it is proven that the sequence of the sampled conjugate Fejer means diverges strongly. Second, it is shown that for every sample based approximation method { H N } N ? N there are functions f such that ? H N f ? ∞ exceeds any given bound for any given number of consecutive indices N .As an application, the later result is used to investigate a problem associated with a question of Ul'yanov on Fourier series which is related to the possibility to construct adaptive approximation methods to determine the Hilbert transform from sampled data. This paper shows that no such approximation method with a finite search horizon exists.

The packing constant is an important and interesting geometric parameter of Banach spaces. Inspired by the packing constant for Orlicz sequence spaces, the main purpose of this paper is calculating the Kottman constant and the packing constant of the Cesaro-Orlicz sequence spaces \(({\text{ce}}{{\text{s}}_\varphi })\) defined by an Orlicz function φ equipped with the Luxemburg norm. In order to compute the constants, the paper gives two formulas. On the base of these formulas one can easily obtain the packing constant for the Cesaro sequence space cesp and some other sequence spaces. Finally, a new constant \(\widetilde D\)(X), which seems to be relevant to the packing constant, is given.

This paper deals with the two-species chemotaxis system where Ω is a bounded domain in RN with smooth boundary ∂Ω, N∈N; h,Xi are functions satisfying some conditions. Global existence and asymptotic stability of solutions to the above system were established under some conditions [11]. The main purpose of the present paper is to improve smallness conditions for chemotactic effect deriving asymptotic stability and to give the convergence rate in stabilization which cannot be attained in the previous work.

In this expository paper, we consider the Hardy-Schr\"odinger operator $-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega}, and describe how the location of the singularity 0, be it in the interior of \Omega or on its boundary, affects its analytical properties. We compare the two settings by considering the optimal Hardy, Sobolev, and the Caffarelli-Kohn-Nirenberg inequalities. The latter rewrites: $C(\int_{\Omega}\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_{\Omega} |\nabla u|^2dx-\gamma \int_{\Omega}\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(\Omega)$, where \gamma, Comment: Expository paper. 48 pages

In this paper we complete the study of the following elliptic equation driven by a general non-local integrodifferential operator $$\mathcal {L}_K$$ $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} \mathcal {L}_K u+\lambda u+|u|^{2^*-2}u+f(x, u)=0 &{} \hbox {in } \Omega \\ u=0 &{} \hbox {in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$ that was started by Servadei and Valdinoci (Commun Pure Appl Anal 12(6):2445–2464, 2013). Here $$s\in (0,1),\, \Omega $$ is an open bounded set of $${\mathbb {R}}^n,\, n>2s$$ , with continuous boundary, $$\lambda $$ is a positive real parameter, $$2^*=2n/(n-2s)$$ is a fractional critical Sobolev exponent and $$f$$ is a lower order perturbation of the critical power $$|u|^{2^*-2}u$$ , while $$\mathcal {L}_K$$ is the integrodifferential operator defined as $$\begin{aligned} \mathcal {L}_Ku(x)= \int _{{\mathbb {R}}^n}\left( u(x+y)+u(x-y)-2u(x)\right) K(y)\,dy, \quad x\in {\mathbb {R}}^n. \end{aligned}$$ Under suitable growth condition on $$f$$ , we show that this problem admits non-trivial solutions for any positive parameter $$\lambda $$ . This existence theorem extends some results obtained in [15, 19, 20]. In the model case, that is when $$K(x)=|x|^{-(n+2s)}$$ (this gives rise to the fractional Laplace operator $$-(-\Delta )^s$$ ), the existence result proved along the paper may be read as the non-local fractional counterpart of the one obtained in [12] (see also [9]) in the framework of the classical Laplace equation with critical nonlinearities.

In this paper, we study right half-plane harmonic mappingsf 0 and f, where f 0 is xed and f is such that its dilatation of a conformalautomorphism of the unit disk. We obtain a su cient condition for theconvolution of such mappings to be convex in the direction of the realaxis. The result of the paper is a generalization of the result of by Li andPonnusamy [11], which itself originates from a problem posed by Dor etal. in [7]. 1. IntroductionLet D = fz2C : jzj

In the paper [4, Theorem 1] Gat, Goginava and the author proved that the maximal operator σκ,∗ of Marcinkiewicz-Fejer means of Walsh-Kaczmarz-Fourier series, is bounded from the dyadic Hardy space Hp into the space Lp for p > 2/3 . Moreover, Goginava and the author showed that σκ,∗ is not bounded from the Hardy space H2/3 to the space L2/3 [6, Theorem 1]. The main aim of this paper is to show that the maximal operator σκ,∗ f := supn∈P |σκ n f | log3/2(n+1) , is bounded from the Hardy space H2/3 into the space L2/3. Moreover, we prove that the order of deviant behavior of the n th Walsh-Kacmarz-Marcinkiewicz-Fejer mean is exactly log3/2(n+1) in the endpoint p = 2/3 . Mathematics subject classification (2010): 42C10.

This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \[\label{*} \left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates or converges to zero, provided that the second order equation \[\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation (\(\ast\)) oscillates if equation (\(\ast\ast\)) is nonoscillatory. We also establish results for the oscillation of equation (\(\ast\)) when equation (\(\ast\ast\)) is oscillatory.

In this paper, we establish some new Ostrowski type inequalities for s logarithmically convex functions. Some applications of our results to P.D.F.’s and in numerical integration are given. 1. INTRODUCTION Let f : I [0;1] ! R be a di¤erentiable mapping on I , the interior of the interval I, such that f 0 2 L [a; b] where a; b 2 I with a < b. If jf 0 (x)j M , then the following inequality holds (see [2]): (1.1) f(x) 1 b a Z b a f(u)du M b a " (x a) + (b x) 2 # : This inequality is well known in the literature as the Ostrowski inequality. For some results which generalize, improve and extend the inequality (1.1) see ([2]-[6]) and the references therein. Let us recall some known de…nitions and results which we will use in this paper. The function f : I R ! [0;1) is said to be log convex or multiplicatively convex if log t is convex, or, equivalenly, if for all x; y 2 I and t 2 [0; 1] ; one has the inequality (See [7], p.7): f (tx+ (1 t) y) [f (x)] + [f (y)] t) : In [1], Akdemir and Tunc were introduced the class of s logarithmically convex functions in the …rst sense as the following: De…nition 1. A function f : I R0 ! R+ is said to be s logarithmically convex in the …rst sense if (1.2) f ( x+ y) [f (x)] s [f (y)] s for some s 2 (0; 1], where x; y 2 I and s + s = 1: In [8], authors introduced the class of s logarithmically convex functions in the second sense as the following: De…nition 2. A function f : I R0 ! R+ is said to be s logarithmically convex in the second sense if (1.3) f (tx+ (1 t) y) [f (x)] s [f (y)] (1 t) 2000 Mathematics Subject Classi…cation. 26D10, 26A15, 26A16, 26A51. Key words and phrases. Hadamard’s inequality, s geometrically convex functions . 1 2 MEVLUT TUNC AND AHMET OCAK AKDEMIR for some s 2 (0; 1], where x; y 2 I and t 2 [0; 1]. Clearly, when taking s = 1 in De…nition 1.2 or De…nition 1.3, then f becomes the standard logarithmically convex function on I. The main purpose of this paper is to establish some new Ostrowski’s type inequalities for s logarithmically convex functions. We also give some applications to P.D.F.’s and to midpoint formula. 2. THE NEW RESULTS In order to prove our main results, we will use following Lemma which was used by Alomari and Darus (see [3]): Lemma 1. Let f : I R ! R, be a di¤erentiable mapping on I where a; b 2 I, with a < b. Let f 0 2 L[a; b]; then the following equality holds

We consider the equation $$ - (r(x)y'(x))' + q(x)y(x) = f(x),x \in \mathbb{R}$$ (*) where f ∈ Lp(ℝ), p ∈ (1,∞) and $$r > 0,q \geqslant 0,\frac{1} {r} \in L_1^{loc} (\mathbb{R}),q \in L_1^{loc} (\mathbb{R})$$ , $$\mathop {\lim }\limits_{|d| \to \infty } \int_{x - d}^x {\frac{{dt}} {{r(t)}}} \cdot \int_{x - d}^x {q(t)dt = \infty } $$ . In an earlier paper, we obtained a criterion for correct solvability of (*) in Lp(ℝ), p ∈ (1,∞). In this criterion, we use values of some auxiliary implicit functions in the coefficients r and q of equation (*). Unfortunately, it is usually impossible to compute values of these functions. In the present paper we obtain sharp by order, two-sided estimates (an estimate of a function f(x) for x ∈ (a, b) through a function g(x) is sharp by order if c−1|g(x)| ⩽ |f(x)| ⩽ c|g(x)|, x ∈ (a, b), c = const) of auxiliary functions, which guarantee efficient study of the problem of correct solvability of (*) in Lp(ℝ), p ∈ (1,∞).

In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, G is a compact simple Lie group, and $\xi \in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^{*}$ and the flag manifold structure of $\mathrm{Ad} (G) \xi$ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo-Riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where $\xi$ is elliptic and $G$ is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.

The Euler zeta function is defined by ζ E (s)=P ∞n=1(−1) n−1s .The purpose of this paper is to find formulas of the Euler zeta func-tion’s values. In this paper, for s ∈ N we find the recurrence formula ofζ E (2s) using the Fourier series. Also we find the recurrence formula ofP ∞n=1(−1) n−1 (2n−1) 2s−1 , where s ≥ 2(∈ N). 1. IntroductionThe Euler zeta function is defined by ζ E (s) =P ∞n=1(−1) n n s (see [3, 4]). Inthis paper we investigate the recurrence formula of the Euler zeta function fors = 2n with Fourier series. By this result we can find ζ E (2n) for all n ∈ N.For s ∈ C, the Riemann zeta function or the Euler-Riemann zeta function,ζ(s) is defined byζ(s) =X ∞n=1 1n s (s ∈ C), (see [5, 6])which converges when the real part of s is greater than 1. R. Ap´ery provedthat the number ζ(3) is irrational. But it is still an open problem to proveirrationality of ζ(2k +1) for the long time.As well known special values, for any positive even number 2n,ζ(2n) = (−1) n+1 B 2n (2π) 2n 2(2n)!, (see [1])where B

In this paper, we consider the (L, 1) state-dependent reflecting random walk (RW) on the half line, which is an RW allowing jumps to the left at a maximal size L. For this model, we provide an explicit criterion for (positive) recurrence and an explicit expression for the stationary distribution. As an application, we prove the geometric tail asymptotic behavior of the stationary distribution under certain conditions. The main tool employed in the paper is the intrinsic branching structure within the (L, 1)-random walk.

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators T�, where � = F +G∗, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space L2(D,Cn). We also show some necessary conditions for the hyponormality of TF+Gwith F + G∗ 2 h∞ Mn×n on L2(D,Cn). Let D and T be the open unit disk and unit circle in the complex plane C re- spectively and dA be the normalized Lebesgue area measure on D. h ∞ denotes the space of all bounded harmonic functions on D. L ∞ (D,dA) and L 2 (D,dA) denote the space of essential bounded measurable functions and the space of the square integrable functions on D with respect to dA, respectively. The Bergman space L 2 consists of all analytic functions in L 2 (D,dA). We denote the space of vector valued square integrable functions on D by L 2 (D,C n ) = L 2 (D,dA)⊗C n and the vector valued Bergman space on D by L 2(D,C n ) = L 2 ⊗ C n , respec- tively, where ⊗ denotes the Hilbert space tensor product. In this paper F T denotes the transpose of the matrix F and G ∗ denotes the adjoint of the matrix G.

We will show that two functions in the extended Selberg class satisfying the same functional equation must be identically equal if they have sufficiently many common zeros. This paper concerns the question of how L-functions are determined by their zeros. L-functions are Dirichlet series with the Riemann zeta function ζ(s) = ∑∞ n=1 1 ns as the prototype and are important objects in number theory. The Selberg class S of L-functions is the set of all Dirichlet series L(s) = ∑∞ n=1 a(n) ns of a complex variable s = σ + it with a(1) = 1, satisfying the following axioms (see [7]): (i) (Dirichlet series) For σ > 1, L(s) is an absolutely convergent Dirichlet series. (ii) (Analytic continuation) There is a nonnegative integer k such that (s − 1)L(s) is an entire function of finite order. (iii) (Functional equation) L satisfies a functional equation of type ΛL(s) = ωΛL(1− s), where ΛL(s) = L(s)Q s ∏K j=1 Γ(λjs+μj) with positive real numbers Q, λj and with complex numbers μj , ω with Reμj ≥ 0 and |ω| = 1. (iv) (Ramanujan hypothesis) a(n) n for every e > 0; (v) (Euler product) logL(s) = ∑∞ n=1 b(n) ns , where b(n) = 0 unless n is a positive power of a prime and b(n) n for some θ < 12 . The degree dL of an L-function L is defined to be dL = 2 ∑K j=1 λj , where K, λj are the numbers in axiom (iii). The Selberg class includes the Riemann zeta-function ζ and essentially those Dirichlet series where one might expect the analogue of the Riemann hypothesis. At the same time, there are a whole host of interesting Dirichlet series not possessing a Euler product (see e.g. [3], [8]). Throughout the paper, all L-functions are assumed to be functions from the extended Selberg class of those only satisfying the axioms (i)-(iii) (see [3]). Thus, the results obtained in the present paper particularly apply to L-functions in the Selberg class. Received by the editors October 5, 2011 and, in revised form, February 12, 2012. 2010 Mathematics Subject Classification. Primary 11M36, 30D30.

The main purpose of this paper is to study the generalized Hilbert operator {equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt {equation*} acting on the weighted Bergman space $A^p_\om$, where the weight function $\om$ belongs to the class $\R$ of regular radial weights and satisfies the Muckenhoupt type condition {equation}\label{Mpconditionaabstract} \sup_{0\le r, Comment: This paper has been accepted for publication in Advances in Mathematics

For a Banach space E and its bidual space E'', the function k(H) defined on bounded subsets H of E measures how far H is from being σ(E,E')-relatively compact in E. This concept, introduced independently by Granero, and Cascales et al., has been used to study a quantitative version of Krein¿s theorem for Banach spaces E and spaces Cp(K) over compact K. In the present paper, a quantitative version of Krein¿s theorem on convex envelopes coH of weakly compact sets H is proved for Fréchet spaces, i.e. metrizable and complete locally convex spaces. For a Fréchet space E, the above function k(H) has been defined in thisi paper by menas of d(h,E) is the natural distance of h to E in the bidual E''. The main result of the paper is the following theorem: For a bounded set H in a Fréchet space E, the following inequality holds k(coH) < (2^(n+1) − 2)k(H) + 1/2^n for all n ∈ N. Consequently, this yields also the following formula k(coH) ≤ (k(H))^(1/2))(3-2(k(H)^(1/2))). Hence coH is weakly relatively compact provided H is weakly relatively compact in E. This extends a quantitative version of Krein¿s theorem for Banach spaces (obtained by Fabian, Hajek, Montesinos, Zizler, Cascales, Marciszewski, and Raja) to the class of Fréchet spaces. We also define and discuss two other measures of weak non-compactness lk(H) and k'(H) for a Fréchet space and provide two quantitative versions of Krein¿s theorem for both functions., The research was supported for C. Angosto by the project MTM2008-05396 of the Spanish Ministry of Science and Innovation, for J. Kakol by National Center of Science, Poland, Grant No. N N201 605340, and for M. Lopez-Pellicer by the project MTM2010-12374-E (complementary action) of the Spanish Ministry of Science and Innovation.

In this paper, we introduce the q-analogue of p-adic log gamma functions with weight alpha. Moreover, we give a relationship be- tween weighted p-adic q-log gamma functions and q-extension of Genocchi and Euler numbers with weight alpha. Assume that p is a fixed odd prime number. Throughout this paper Z, Zp, Qp and Cp will denote the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp, respectively. Also we denote N � = N ( {0} and exp(x) = e x . Let vp : Cp ! Q ( {1} (Q is the field of rational numbers) denote the p-adic valuation of Cp normalized so that vp (p) = 1. The absolute value on Cp will be denoted as |·|, and |x| p = p v p(x)

In this paper, we define the diamond Marcel Riesz operator of order (α, β) on the function f by M (f) = Kα,β ∗ f, where Kα,β is diamond kernel of Marcel Riesz, α, β ∈ C, the symbol ∗ designates the convolution, and f ∈ S, S is the Schwartz space of functions. Our purpose of this paper is to obtain the operator N (α,β) = [ M (α,β) ]−1 such that if M (α,β)(f) = φ, then N (α,β)φ = f. Our results generalize formulae appearing in A. Kananthai [On the convolutions of the diamond kernel of Marcel Riesz, Applied Mathematics and Computation, 114(2000), 95 − 101]. Mathematics Subject Classification: 46F10, 46F12

In this paper, we are concerned with the existence of positive solutions of the following nonlinear elliptic system involving critical Hardy-Sobolev exponent \begin{equation*}\label{eq:1}(*) \left\{ \begin{array}{lll} -\Delta u= \frac{2\alpha}{\alpha+\beta}\frac{u^{\alpha-1}v^\beta}{|x|^s}-\lambda u^p, & \quad {\rm in}\quad \Omega,\\[2mm] -\Delta v= \frac{2\beta}{\alpha+\beta}\frac{u^\alpha v^{\beta-1}}{|x|^s}-\lambda v^p, & \quad {\rm in}\quad \Omega,\\[2mm] u\gt 0, v\gt 0, &\quad {\rm in}\quad \Omega,\\[2mm] u=v=0, &\quad {\rm on}\quad \partial\Omega, \end{array} \right. \end{equation*} where \(N\geq 4\) and \(\Omega\) is a \(C^1\) bounded domain in \(\mathbb{R}^N\) with \(0\in\partial\Omega\). \(0\lt s \lt 2\), \(\alpha+\beta=2^*(s)=\frac{2(N-s)}{N-2}\), \(\alpha,\beta\gt 1\), \(\lambda\gt 0\) and \(1 \lt p\lt \frac{N+2}{N-2}\). The case when 0 belongs to the boundary of \(\Omega\) is closely related to the mean curvature at the origin on the boundary. We show in this paper that problem \((*)\) possesses at least a positive solution.

The well known partition function $p(n)$ has a long research history, where $p(n)$ denotes the number of solutions of the equation $n = a_1 + \cdots + a_k$ with integers $1 \leq a_1 \leq \cdots \leq a_k$. In this paper, we investigate a new partition function. For any real number $r \gt 1$, let $p_r(n)$ be the number of solutions of the equation $n = \lfloor \sqrt[r]{a_1} \rfloor + \cdots + \lfloor \sqrt[r]{a_k} \rfloor$ with integers $1 \leq a_1 \leq \cdots \leq a_k$, where $\lfloor x \rfloor$ denotes the greatest integer not exceeding $x$. In this paper, it is proved that $\exp(c_1 n^{r/(r+1)}) \leq p_r(n) \leq \exp(c_2n^{r/(r+1)})$ for two positive constants $c_1$ and $c_2$ (depending only $r$).

In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k, 0 or 1 , by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m=5,7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1, m as sum of row terms of corresponding triangle. This version might be out of date, proceed by the link to see actual version., 16 pages, 8 figures, 2 tables, typos revised, added missing references, results generalized and shortened, derivations detailed.

In this paper, we consider sharp estimates of integral functionals $\int_0^{2\pi } {\phi (L|Lf_n (t)|)dt} $ for functions φ defined on the semiaxis (0, ∞) and operators L on the set T n of real trigonometric polynomials f n of order n ≥ 1 by the uniform norm $\left\| {f_n } \right\|_{C_{2\pi } } $ of the polynomials. We also consider similar problems for algebraic polynomials on the unit circle of the complex plane. P. Erdos, A. Calderon, G. Klein, L. V. Taikov, and others investigated such inequalities. In particular, we show that, for 0 ≤ q < ∞, the sharp inequality $\left\| {D^\alpha f_n } \right\|_{L_q } \leqslant n^\alpha \left\| {\cos t} \right\|_{L_q } \left\| {f_n } \right\|_\infty $ holds on the set T n , n ≥ 1, for the Weyl fractional derivatives D α f n of order α ≥ 1. For q = ∞ (α ≥ 1), this fact was proved by P.I. Lizorkin (1965). For 1 ≤ q < ∞ and positive integer α, the inequality was proved by L.V. Taikov (1965); however, in this case, the inequality follows from results of an earlier paper by A. P. Calderon and G. Klein (1951).

In the present paper, we study conditions under which the metric projection of a polyhedral Banach space X onto a closed subspace is Hausdorff lower or upper semicontinuous. For example, we prove that if X satisfies (*) (a geometric property stronger than polyhedrality) and Y@?X is any proximinal subspace, then the metric projection P"Y is Hausdorff continuous and Y is strongly proximinal (i.e., if {y"n}@?Y, x@?X and @?y"n-x@?->dist(x,Y), then dist(y"n,P"Y(x))->0). One of the main results of a different nature is the following: if X satisfies (*) and Y@?X is a closed subspace of finite codimension, then the following conditions are equivalent: (a) Y is strongly proximinal; (b) Y is proximinal; (c) each element of Y^@? attains its norm. Moreover, in this case the quotient X/Y is polyhedral. The final part of the paper contains examples illustrating the importance of some hypotheses in our main results.

Let (X, d) be a compact metric space and let denote the space of all finite signed Borel measures on X. Define by I(μ) = ∫X∫Xd(x, y) dμ(x)dμ(y), and set , where μ ranges over the collection of signed measures in of total mass 1. This paper, with an earlier and a subsequent paper [Peter Nickolas and Reinhard Wolf, Distance geometry in quasihypermetric spaces. I and III], investigates the geometric constant M(X) and its relationship to the metric properties of X and the functional-analytic properties of a certain subspace of when equipped with a natural semi-inner product. Using the work of the earlier paper, this paper explores measures which attain the supremum defining M(X), sequences of measures which approximate the supremum when the supremum is not attained and conditions implying or equivalent to the finiteness of M(X). © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The main purpose of this paper is two-fold. On one hand, we prove a sharpercovering lemma in Euclidean space R n for all n ≥ 2 (see Theorem 1.5). On the otherhand, we apply this covering lemma to improve existing results for BMO and volumeestimates of nodal sets for eigenfunctions u satisfying 4u + λu = 0 on n-dimensionalRiemannian manifolds when λ is large (see Theorems 1.7, 1.8). We also improve theBMO estimates for the function q = |∇u| 2 + λn u 2 (see Theorem 1.10). Our coveringlemma sharpens substantially earlier results and is fairly close to the optimal one we canexpect (Conjecture 1.6). 1. IntroductionLet M be a smooth, compact and connected Riemannian manifold without bound-ary. Let 4 denote the Laplacian on M. Assume throughout this paper that u is thesolution to 4u + λu = 0, λ > 1, i.e., u is an eigenfunction with eigenvalue λ. Thenodal set N of u is defined to be the set of points x ∈ M where u(x) = 0. Then out-side the singular set S = {x|u(x) = 0,∇u(x) = 0}, N is a regular (n−1)-dimensionalsubmanifold of M. The main focus of the current paper concerns a geometric cover-ing lemma in the Euclidean space R

Let f be a continuous function defined on Ω:=[0,1]N which depends on only l coordinate variables, \(f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}})\). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i1,…,il are known to us, then by asking for the values of f at m=Ll uniformly spaced points, we could recover f to the accuracy |g|Lip1L−1 in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i1,…,il are not known to us. A prototypical result of this paper is that by asking for C(l)Ll(log 2N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|Lip1L−1 when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance e (which is not known) by functions of l variables.

Even though there is a formula for h 0 ( X,O X ( nK X )) for acanonical threefold X , it is not easy to compute h 0 ( X,O X ( nK X )) be-cause the formula has a term due to singularities. In this paper, we finda way to control the term due to singularities. We show nonvanishing ofplurigenus for the case when the index r in the singularity type 1 r (1 ,− 1 ,b )is sufficiently large. Throughout this paper X is assumed to be a projective threefold with onlycanonical singularities and an ample canonical divisor K X over the complexnumber field C, i.e., a canonical threefold.It is well known that H 0 ( X,O X ( mK X )) does not vanish and generates abirational map for a sufficiently large m . When X is a surface of general type, H 0 ( X,O X ( mK X )) does not vanish for m ≥ 2 and H 0 ( X,O X ( mK X )) generatesa birational map for m ≥ 5.In a case of a threefold of general type, M. Reid and A. R. Fletcher de-scribed the formula for χ ( O X ( nK X )) (see [1]). Combining the formula for χ

In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation $$ (r(t)(y(t) - p(t)y(t - \tau ))')' + q(t)G(y(h(t)) = f(t) $$ has a positive and bounded solution, where q, h, f ∈ C ([0, ∞), ℝ) such that q(t) ≥ 0, but ≢ 0, h(t) ≤ t, h(t) → ∞ as t → ∞, r ∈ C (1) ([0, ∞), (0, ∞)), p ∈ C (2) [0, ∞), ℝ), G ∈ C(ℝ, ℝ) and τ ∈ ℝ+. In our work r(t) ≡ 1 is admissible and neither we assume G is non-decreasing, xG(x) > 0 for x ≠ 0, nor we take G is Lipschitzian. Hence the results of this paper improve many recent results.

We give an explicit Krasnoselski–Mann type method for finding common solutions of the following system of equilibrium and hierarchical fixed points: where C is a closed convex subset of a Hilbert space H, G:C×C→ℝ is an equilibrium function, T:C→C is a nonexpansive mapping with Fix(T) its set of fixed points and f:C→C is a ρ-contraction. Our algorithm is constructed and proved using the idea of the paper of [Y. Yao and Y.-C. Liou, ‘Weak and strong convergence of Krasnosel’skiĭ–Mann iteration for hierarchical fixed point problems’, Inverse Problems24 (2008), 501–508], in which only the variational inequality problem of finding hierarchically a fixed point of a nonexpansive mapping T with respect to a ρ-contraction f was considered. The paper follows the lines of research of corresponding results of Moudafi and Théra.

We give a multimensional version of the p-adic heat equation, and show that its fundamental solution is the transition density of a Markov process. 1. Introduction. In recent years p adic analysis has received a lot of attention due to its applications in mathematical physics, see e.g. [1], [2], [4], [5], [16], [17], [18], [22], [28] and references therein. One motivation comes from statistical physics, in particular in connection with models describing relaxation in glasses, macromolecules, and proteins. It has been proposed that the non exponential nature of those relaxations is a consequence of a hierarchical structure of the state space which can in turn be put in connection with p adic structures ([4], [5], [22]). In [4] was demostrated that the p-adic analysis is a natural basis for the construction of a wide variety of models of ultrametric di¤usion constrained by hierarchical energy landscapes. To each of these models is associated a stochastic equation (the master equation). In several cases this equation is a p-adic parabolic equation of type: > : @u(x;t) @t + a(Au)(x; t) = f(x; t); x 2 Q n p ; t 2 (0; T ]; u(x; 0) = '(x); (1.1) where a is a positive constant, A is pseudo-di¤erential operator, and Qp is the …eld of p-adic numbers. The simplest case occurs when n = 1 and A is the Vladimirov operator: (D ') (x) = F 1 !x j j p Fx! '(x) ; > 0; where F is the Fourier transform. The fundamental solution of (1.1) is density transition of a timeand space-homogeneous Markov process, that is consider the p adic counterpart of the Brownian motion (see [18], [28]). It is relevant to mention that in the case n = 1, the fundamental solution of (1.1) when A = D (also called the p adic heat kernel) has been studied extensively, see e.g. [6], [11], [12], [14], [18], [28]. A natural problem is to study the initial value problem (1.1) in the n-dimensional case. Recently, the second author considered Cauchy’s problem (1.1) when (A') (x) = F 1 !x jf ( )j p Fx! '(x) ; > 0; 2000 Mathematics Subject Classi…cation. Primary 35R60, 60J25; Secondary 47S10, 35S99. Key words and phrases. Parabolic equations, Markov processes, p-adic numbers, ultrametric di¤usion. 1 2 J. J. RODRIGUEZ-VEGA AND W. A. ZUNIGA-GALINDO here f ( ) is an elliptic homogeneous polynomial in n variables, and the datum ' is a locally constant and integrable function. Under these hypotheses it was established the existence of a unique solution to Cauchy’s problem (1.1). In addition, the fundamental solution is a transition density of a Markov process with space state Qp (see [29]). In this paper we study Cauchy’s problem (1.1) when A is the Taibleson pseudodi¤erential operator which is de…ned as follows: D T' (x) = F 1 !x max 1 i n j ijp Fx! '(x) ! ; > 0: (1.2) Recently Albeverio, Khrennikov, and Shelkovich studied D T in the context of the Lizorkin spaces [3]. We prove existence and uniqueness of the Cauchy problem (1.1-1.2) in spaces of increasing functions introduced by Kochubei in [19], see Theorem 1. We also associate a Markov processes to equation the fundamental solution (see Theorem 2). These results constitute an extension of the corresponding results in [18], [28]. Let us explain the connection between the results of this paper and those of [29]. There are in…nitely many homogeneous polynomial functions satisfying jf ( )jp = max 1 i n j ijp d ; for any 2 Qp ; here d denotes degree of f (c.f. Lemmas 14-15). Hence the pseudo-di¤erential operators considered here are a subclass of the ones considered in [29]. However, the function spaces for the solutions and initial data are completely di¤erent. In this paper the initial datum and the solution to Cauchy problem (1.1-1.2) are not necessarily bounded, neither integrable, but in [29] are. Finally, our results can be extended to operators of the form (A') (x) = a0(x; t)(D T')(x) + n X k=1 ak(x; t)(D k T ')(x) + b(x; t)'(x); (1.3) > 1, 0 < 1 < : : : < n < ; where the ak(x; t) ,and b(x; t) are bounded continuous functions, using the techniques presented in [18]-[20]. These results will appear later elsewhere. 2. Preliminary Results As general reference for p-adic analysis we refer the reader to [25] and [28]. The …eld of p-adic numbers Qp is de…ned as the completion of the …eld of rational numbers Q with respect to the non-Archimedean p-adic norm j jp which is de…ned as follows: j0jp = 0; if x 2 Q , x = p ab with a, b integers coprime to p, then jxjp = p . The integer = (x) is called the p-adic order of x, and it will be denoted as ord (x). We use the same symbol, j jp, for the p-adic norm on Qp. We extend the p-adic norm to Qp as follows: kxkp := max 1 i n jxijp , for x = (x1; : : : ; xn) 2 Q n p . Note that kxkp = p min1 i nford(xi)g. p-ADIC PARABOLIC EQUATIONS AND ULTRAMETRIC DIFFUSION 3 Any p-adic number x 6= 0 has a unique expansion of the form

For convex bodies, chord power integrals were introduced and studied in several papers (see [3], [6], [14], [15], etc.). The aim of this article is to study them further, that is, we establish the BrunnMinkowski-type inequalities and get the upper bound for chord power integrals of convex bodies. Finally, we get the famous Zhang projection inequality as a corollary. Here, it is deserved to mention that we make use of a completely distinct method, that is using the theory of inclusion measure, to establish the inequality. 1. Preliminaries The setting for this paper is n-dimensional Euclidean space R. We will denote by convex figure a compact convex subset of R, and by convex body a convex figure with nonempty interior. Let Sn−1 denote the unit sphere centered at the origin o in R, and write αn−1 for the (n − 1)-dimensional volume of Sn−1. Let Bn be the closed unit ball in R, write ωn for the n-dimensional volume of Bn. Note that ωn = 2π n 2 nΓ(n2 ) , αn−1 = nωn. By a direction, we mean a unit vector, that is, an element of Sn−1. If u is a direction, we denote by u⊥ the (n − 1)-dimensional subspace orthogonal to u and by lu the line through the origin parallel to u. Denote by AGi,n the affine Grassmann manifold of i-dimensional planes in R. It is a homogeneous space under the action of the motion group G(n) (see [7], p.199). Let dξk be the normalized invariant measure of AGi,n whose restriction to the Grassmann manifold Gi,n is the invariant probability measure. Let ξ1 be a random line intersecting K. Then vol1(K ∩ ξ1) is the chord length of the intersection K ∩ ξ1. The chord power integrals of K are defined by Iλ(K) = 2αn−1 n ∫ ξ1∈AG1,n vol1(K ∩ ξ1)dξ1, 0 ≤ λ < ∞. Received November 30, 2006; Revised April 5, 2007. 2000 Mathematics Subject Classification. 52A40.

Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space H1 L(R n ) and a BMO L (R n ) space associated with the operator L were introduced and studied. In this paper we define a class of H p L (R n ) spaces associated with the operator L for a range of p < 1 acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical H p (R n ) spaces. We then establish a duality theorem between the H p L (R n ) spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on H p L (R n ) and give the inclusion between the classical H p (R n ) spaces and the H p L (R n ) spaces associated with operators.

This paper has two parts. In the first part, we study shift coordinates on a sphere $S$ equipped with three distinguished points and a triangulation whose vertices are the distinguished points.These coordinates parametrize a space $\widetilde{\mathcal {T}}(S)$ that we call an {\it unfolded Teichmüller space}. This space contains Teichmüller spaces of the sphere with $\mathfrak{b}$ boundary components and $\mathfrak{p}$ cusps (which we call generalized pairs of pants), for all possible values of $\mathfrak{b}$ and $\mathfrak{p}$ satisfying $\mathfrak{b}+\mathfrak{p}=3$. The parametrization of $\widetilde{\mathcal {T}}(S)$ by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in $\mathbb{R}^3$. Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with $\mathfrak{b}$ boundary components and $\mathfrak{p}$ cusps, for fixed $\mathfrak{b}$ and $\mathfrak{p}$, the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on $\widetilde{\mathcal {T}}(S)$ whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space $\widetilde{\mathcal {T}}(S)$. Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2.

In this paper, we study scalar difference-differential equations of neutral type of general form $$\sum\limits_{j = 0}^m {\int_0^h {u^{(j)} (t - \theta )d\sigma _j (\theta ) = 0,t > h,} } $$ where the σj(θ) are functions of bounded variation. For the solutions of this equation, we obtain the following estimate: $$\left\| {u(t)} \right\|W_2^m (T,T + h) \leqslant CT^{q - 1} e^{\kappa T} \left\| {u(t)} \right\|W_2^m (0,h),$$ where C is a constant independent of u0(t) and the values of q and ℋ are determined by the properties of the characteristic determinant of this equation. Earlier, this estimate was proved for equations of less general form. For example, for piecewise constant functions σj(θ) or for the case in which the function σm(θ) has jumps at both points θ = 0 and θ = h. In the present paper, this estimate is obtained under the only condition that σm(θ) experiences a jump at the point θ = 0; this condition is necessary for the correct solvability of the initial-value problem.

In this paper, necessary and sufficient conditions for the oscillation and asymptotic behaviour of solutions of the second order neutral delay differential equation (NDDE) $$\left[ {r(t)(y(t) - p(t)y(t - \tau ))'} \right]^\prime + q(t)G(y(h(t))) = 0$$ are obtained, where q, h ∈ C([0, ∞), ℝ) such that q(t) ≥ 0, r ∈ C (1) ([0, ∞), (0, ∞)), p ∈ C ([0, ∞), ℝ), G ∈ C (ℝ, ℝ) and τ ∈ ℝ+. Since the results of this paper hold when r(t) ≡ 1 and G(u) ≡ u, therefore it extends, generalizes and improves some known results.

Philippe et al. [9], [10] introduced two distinct time-varying mutually invertible fractionally integrated filters A(d), B(d) depending on an arbitrary sequence d = (dt)t∈ℤ of real numbers; if the parameter sequence is constant dt ≡ d, then both filters A(d) and B(d) reduce to the usual fractional integration operator (1 − L)−d. They also studied partial sums limits of filtered white noise nonstationary processes A(d)et and B(d)et for certain classes of deterministic sequences d. The present paper discusses the randomly fractionally integrated stationary processes XtA = A(d)et and XtB = B(d)et by assuming that d = (dt, t ∈ ℤ) is a random iid sequence, independent of the noise (et). In the case where the mean \(\bar d = \mathbb{E}d_0 \in \left( {0,1/2} \right)\), we show that large sample properties of XA and XB are similar to FARIMA(0, \(\bar d\), 0) process; in particular, their partial sums converge to a fractional Brownian motion with parameter \(\bar d + (1/2)\). The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions h(XtA) of a randomly fractionally integrated process XtA with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of h. For the special case of a constant deterministic sequence dt, this reduces to the standard Hermite rank used in Dobrushin and Major [2].

In this paper, we consider the boundary value problem $$y^{\prime\prime}_{j} + \lambda^{2}y_{j} = \sum^{n}_{k = 1} V_{jk} (x) y_{k}, \quad x \in{\mathbb R}_{+} : = ( 0, \infty),$$$$y^{\prime}_{j} (0) + (\alpha_{0} + \alpha_{1} \lambda+ \alpha_{2} \lambda^{2}) y_{j} (0) = 0, \quad j = 1, 2, \ldots, n,$$ where λ is the spectral parameter and $V (x) = \Vert V_{jk} (x)\Vert^{n}_{1}$ is a Hermitian matrix such that $$\int^{\infty}_{x} t\vert V (t)\vert dt < \infty, \quad x \in {\mathbb R}_{+}$$ and αi ∈ ℂ, i = 0, 1, 2, with α2 ≠ 0. In this paper, we investigate the eigenvalues and spectral singularities of L. In particular, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, under the Naimark and Pavlov conditions.

Let ( R , m ) (R,\mathfrak {m}) be an n n -dimensional regular local ring, essentially of finite type over a field of characteristic zero. Given an m \mathfrak {m} -primary ideal a \mathfrak {a} of R R , the relationship between the singularities of the scheme defined by a \mathfrak {a} and those defined by the multiplier ideals J ( a c ) \mathcal {J}(\mathfrak {a}^c) , with c c varying in Q + \mathbb {Q}_+ , are quantified in this paper by showing that the Samuel multiplicity of a \mathfrak {a} satisfies e ( a ) ≥ ( n + k ) n / c n e(\mathfrak {a}) \ge (n+k)^n/c^n whenever J ( a c ) ⊆ m k + 1 \mathcal {J}(\mathfrak {a}^c) \subseteq \mathfrak {m}^{k+1} . This formula generalizes an inequality on log canonical thresholds previously obtained by Ein, Mustaţǎ and the author of this paper. A refined inequality is also shown to hold for small dimensions, and similar results valid for a generalization of test ideals in positive characteristics are presented.

In the present paper, we give a description of the derived category of coherent sheaves on a Fano threefold of index 1 and degree 12 (the variety V 12). It can easily be shown that if X is a V 12 variety, then its derived category contains an exceptional pair of vector bundles ( $$\mathcal{U},\mathcal{O}_X$$ ), where $$\mathcal{O}_X$$ is the trivial bundle, and $$\mathcal{U}$$ is the Mukai bundle of rank 5 (which induces the embedding X → Gr(5, 10)). The orthogonal subcategory $$\mathcal{A}_X = ^ \bot \left\langle {\mathcal{U},\mathcal{O}} \right\rangle \subset \mathcal{D}^b (X)$$ can be treated as the nontrivial part of the derived category of X. The main result of the present paper is the construction of the category equivalence $$\mathcal{A}_X \cong \mathcal{D}^b (C^ \vee )$$ , where C ∨ is the curve of genus 7 which can be canonically associated to X according to the results due to Iliev and Markushevich. In the construction of the equivalence, we make use of the geometric results due to Iliev and Markushevich, as well as the Bondal and Orlov results about derived categories. As an application, we prove that the Fano surface of X (which is the surface parametrizing conics on X) is isomorphicto S 2 C ∨, the symmetric square of the corresponding curve of genus 7.

We provide a criterion for midpoint locally uniformly rotund renormability of normed spaces involving the class of -slicely continuous maps, recently introduced by Molto, Orihuela, Troyanski and Valdiva in 2003. As a consequence of this result, we obtain a theorem of G. Alexandrov concerning the three space problem for midpoint locally uniformly rotund renormings of Banach spaces. A normed space (X,k · k) (or its norm) is said to be midpoint locally uniformly rotund if for every x 2 X and every sequence (xn)n in X such that kxn + xk!kxk and kxn xk!kxk we have kxnk!0. Recall also that X is locally uniformly rotund if for every x 2 X and every sequence (xn)n X such that lim n kxnk = kxk and lim n kxn +xk = 2kxk we have lim n kxn xk = 0, and that X is strictly convex or rotund (R for short) if x = y whenever x and y are elements of X such that kxk = kyk = (x + y)/2 . It is clear that locally uniformly rotund ) midpoint locally uniformly rotund and that midpoint locally uniformly rotund ) R. In the paper [5], devoted to the renorming of spaces of continuous functions on trees, R. Haydon provides the first example (the only known to date) of midpoint locally uniformly rotund space with no equivalent locally uniformly rotund renorming. There, he also shows that for every tree , the existence of an equivalent strictly convex norm on C() implies midpoint locally uniformly rotund renormability on this space. This coincidence is not true in general: an example of strictly convexifiable space without midpoint locally uniformly rotund renorming is ‘1 (see [2, 6]). Our aim in this paper is to provide a criterion for midpoint locally uniformly rotund renorming of spaces that have images in midpoint locally uniformly rotund spaces through special non linear maps. These are the -slicely continuous maps recently introduced in [11], where a non linear transfer technique for locally uniformly rotund renormability has been developed.

and proved that, for q = 2, 3, . . . , the relation A(1/q) = 1/q holds, the extremal function is φ1/q(x) , and hence derived the asymptotics A(h) = h+O(h) , h → 0 . A. Yu. Popov conjectured that, as h → 0 , the stronger asymptotics A(h) = h+O(h) is valid. Further progress in the Turan problem was made in the papers of Gorbachev and Manoshina [2– 6]. They proved Popov’s conjecture and reduced the Turan problem for rational h to a finitedimensional extremal problem, and evaluated A(h) for h = 2/q, 3/q, 4/q, p/(2p+ 1) . In this paper, the Turan problem is solved for all rational h . We need some results from [6]. Suppose that p, q ∈ N , (p, q) = 1, p ≤ q/2 . Consider the following linear programming problem.

In this paper, we introduce the SP-integral and the SPfi-integral deflned on an interval in the n-dimensional Euclidean space R n . We also investigate the relationship between these two integrals. It is well known (3) that the Perron integral deflned on an interval of the real line R by major and minor functions which are not assumed to be continuous is equivalent to the one deflned by continuous major and minor functions and that the strong Perron integral deflned on an interval of R by strong major and minor functions is equivalent to the McShane integral. In this paper, we introduce Perron-type integrals deflned on an inter- val of the n-dimensional Euclidean spaceR n using the strong major and minor functions, and investigate the relationship between these integrals. We shall call it the strong Perron integral, or brie∞y SP-integral. For a subset E of the n-dimensional Euclidean spaceR n , the Lebesgue measure of E is denoted by jEj. For a point x = (x1;x2;¢¢¢ ;xn) 2R n , the norm of x is kxk = max1•in jxij and the --neighborhood U(x;-) of x is an open cube centered at x with sides equal to 2-. For an interval I = (a1;b1)£(a2;b2)£¢¢¢(an;bn) ofR n with ai fi(fi 2 (0;1)), then the interval I is said to be fi-regular.