On the parity of the class number of an imaginary abelian field of conductor 2<Superscript><Emphasis Type="Italic">a</Emphasis></Superscript><Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">b</Emphasis></Superscript>

Let p be an odd prime number, and p^n₀{p^{n_0}} the highest power of p dividing 2 ^p−1 − 1. Let K_n=Q(z_pⁿ⁺¹){K_n={\bf Q}(\zeta_{p^{n+1}})} and L_n,j=K_n⁺(z_2^j+2){L_{n,j}=K_n^+(\zeta_{2^{j+2}})} for j ≥ 0. Let h_n^*{h_n^*} be the relative class number of K _n , and h _n,j the class number of L _n,j , respectively. Let n be an integer with n ≥ n ₀. We prove that if the ratio h_n^*/h_n-1^*{h_n^*/h_{n-1}^*} is odd, then h _n,j /h _n−1,j is odd for any j ≥ 0.     

Arithmetical properties of the number of t-core partitions

Let Λ={λ ₁≥⋅⋅⋅≥λ _s ≥1} be a partition of an integer n. Then the Ferrers-Young diagram of Λ is an array of nodes with λ _i nodes in the ith row. Let λ _j ′ denote the number of nodes in column j in the Ferrers-Young diagram of Λ. The hook number of the (i,j) node in the Ferrers-Young diagram of Λ is denoted by H(i,j):=λ _i +λ _j ′−i−j+1. A partition of n is called a t-core partition of n if none of the hook numbers is a multiple of t. The number of t-core partitions of n is denoted by a(t;n). In the present paper, some congruences and distribution properties of the number of 2 ^t -core partitions of n are obtained. A simple convolution identity for t-cores is also given.      

The generating function for traces of singular moduli and an application to Borcherds products

Let p be a prime and f(z)=∑ n a(n)q n be a weakly holomorphic modular function for \varGamma 0*(p2)\varGamma _{0}^{*}(p^{2}) with a(0)=0. We use Bruinier and Funke’s work to find the generating series of modular traces of f(z) as Jacobi forms. And as an application we construct Borcherds products related to the Hauptmoduln jp2*j_{p^{2}}^{*} for genus zero groups \varGamma 0*(p2)\varGamma _{0}^{*}(p^{2}).     

A local version of a result of gamlen and gaudet

LetX _D ^p be the span of the Haar function {h _j: J ε D} inL ^p (1 <p < ∞) endowed withL ^p norm. Then for any finite setD, the spacesX _D ^p andl _*D ^p areK _p-isomorphic whereK _p depends onp only.     

Solving the knapsack problem via -transform

Given vectors and , we consider the (unbounded) knapsack optimization problem . We compute the minimum value p^* using techniques from complex analysis, namely Cauchy's Residue Theorem to integrate a function in , and the -transform of an appropriate function related to . The computational complexity depends on s ∑_j=1ⁿa_j, not on the magnitude of b as in dynamic programming based approaches. We also completely characterize the number of solutions with value less than p, as a function of p.     

Three-term recurrence relations with matrix coefficients. The completely indefinite case

In the space {ol _p ² of vector sequences, we consider the symmetric operatorL generated by the expression (lu)j:=Bj ^uj+1+Aj ^uj+ _B ^j−1/* uj−1, whereu−1 = 0,u ₀,u ₁, … ∈ ℂ ^p ,A _j andB _j arep × p matrices with entries from ℂ,A _j ^* =A_j, and the inversesB _j ⁻¹ (j = 0, 1, …) exist. We state a necessary and sufficient condition for the deficiency numbers of the operatorL to be maximal; this corresponds to the completely indefinite case for the expressionl. Tests for incomplete indefiniteness and complete indefiniteness forl in terms of the coefficientsA _j andB _j are derived. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 709–716, May, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00333.     

Vanishing theorem for sheaves of microfunctions at the boundary on cr–manifolds

Let X be a complex analytic manifold. Consider S?M?Xreal analytic submonifolds with codium ^R _MS=1,and let ω be a connected component of M\S. Let p∈S X_MT_M ^*X where T^* _Xdenotes the conormal bundle to M in X, and denote by ν(p) the complex radial Euler field at p. Denote by μ*(Ox) (for * = M, ω) the microlocalization of the sheaf of holomorphic functions along *.

Under the assumption dim^R(T_pT_M ^*X? ν(p)) = 1, a theorem of vanishing for the cohomology groups H^jμM(Ox)p is proved in [K-S 1, Prop. 11.3.1], j being related to the number of positive and negative eigenvalue for the Levi form of M.

Under the hypothesis dim^R(T_pT_S ^*X∩ν(p))=1, a similar result is proved here for the cohomology groups of the complex of microfunctions at the boundary μω(Ox).Stating this result in terms of regularity at the boundary for CR–hyperfunctions a local Bochner–type theorem is then obtained.     

On an Approximate Solution of the Dirichlet Problem for the Generalized Laplacian

For an arbitrary differential operator P of order p on an open set X ? R ⁿ, the Laplacian is defined by Δ = P*P. It is an elliptic differential operator of order 2p provided the symbol mapping of P is injective. Let O be a relatively compact domain in X with smooth boundary, and B_j(j = 0…,p — 1) be a Dirichlet system of order p ? 1 on ?O. By {C_j} we denote the Dirichlet system on ?O adjoint for {B_j} with respect to the Green formula for P. The Hardy space H²(O) is defined to consist of all the solutions f of Δf = 0 in O of finite order of growth near the boundary such that the weak boundary values of the expression {B_jf} and {C_j(Pf)} belong to the Lebesgue space L²(?O). Then the Dirichlet problem consists of finding a solution f ? H²(O) with prescribed data {B_jf} on ?O. We develop the classical Fischer-Riesz equations method to derive a solvability condition of the Dirichlet problem as well as an approximate formula for solutions.     

Residuality of Sets of Hypercyclic Operators

We prove a criterion that guarantees that in a given class of operators the set of hypercyclic ones is residual. We also prove the existence of quasinilpotent Volterra composition operators, V_j{V_\varphi} , such that both V_j{V_\varphi} and V_j^*{V_\varphi^\star} are supercyclic and both I + V_j{I + V_\varphi} and I + V_j^*{I + V_\varphi^\star} are hypercyclic.     

Symmetrization of the symbol in Banach algebras generated by idempotents

It is known [2] that a Banach algebra generated by 2N idempotents with relations (1), (2) is an algebra with matrix symbol of order 2N. This symbol is completely and explicitly defined via the spectrums of two (indicator) elements (3) and (5). But in the case when is a c^*-algebra, the symbol constructed in [2] is notsymmetric. Moreover in order to construct asymmetric symbol, one needs some additional information about the algebra , even for the case whenp _j=p _j ^* (j=1, ..., 2N). These additional conditions and the explicit form ofsymmetric symbol are described in this paper.     

The existence and regularity of multiple solutions for a class of infinitely degenerate elliptic equations

Let X = (X₁, …, X_m) be an infinitely degenerate system of vector fields. We study the existence and regularity of multiple solutions of the Dirichlet problem for a class of semi‐linear infinitely degenerate elliptic operators associated with the sum of square operator δ_X = ∑_{j = 1}^m X_j^* X_j (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A note on "More Operator Versions of the Schwarz Inequality"

It is shown that for any (n + 1)-positive (possibly non-linear) map and any bounded linear operators A _i ,i = 1,¨,n we have [(A _i ^* A _j )] _{i,j = 1} ^*[(A _i )^*(A _j )] _{i,j = 1} ^*, and that the statement is false if "(n + 1)-positive" is replaced by "n-positive". This resolves an issue raised by Bhatia and Davis in relation to a Schwartz inequality which can be regarded as a non-commutative variance-covariance inequality [2]     

A constructive method of solving the Liapounov equation for complex matrices

Summary This paper describes a method of solving the Liapounov equation (1)HM+M ^* H=2D, M in upper Hessenberg form,D diagonal. Initialising the first row of the matrixA arbitrarily, one can find (by solving equations with one unknown) the unknown elements ofA such that (2)AM+M ^* A ^* =2F, whereA differs from a Hermitian matrix only in that its diagonal elements need not be real.F is a diagonal matrix which is uniquely determined by the first row ofA. By solving Eq. (2) for several initial values one may generate several matricesA andF (in the most unfavourable case 2n–1A's andF's are needed) and superpose them to getn linearly independent Hermitian matricesH _j andD _j respectively for whichH _j M+M ^* H _j =2D _j is valid. Then one can solve the real system to obtain the solution of Eq. (1).This work was performed under the terms of the agreement on association between the Max-Planck-Institut für Plasmaphysik and Euratom.     

General Wentzell Boundary Conditions and Analytic Semigroups on W 1, p (0, 1)

We are concerned with the analyticity of the (C ₀) semigroups generated by the realizations of the Laplacian Δu:=u″ in the spaces C[0, 1] and W ^{1, p} (0, 1) with the general Wentzell boundary conditions Δu(j)+β ^ju″(j)+γ _ju(j)=0 for j=0,1. Here 1<p<∞ and β _j , γ _j are arbitrary complex numbers for j=0,1.     

Asymptotic Behaviour of Best lp-Approximations from Affi ne Subspaces

In this paper we consider the problem of best approximation in ℓ_pⁿ, 1<p∞. If h_p, 1<p<∞, denotes the best ℓ_p-approximation of the element h ⁿ from a proper affine subspace K of ⁿ, hK, then lim_p→∞h_p=h_∞^*, where h_∞^* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to prove that for all r there are α_j ⁿ, 1jr, such that

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, with γ_p^{(r) n} and γ_p^(r)= (p^−r−1).     

A Load-Balanced Network with Two Servers

A load-balanced network with two queues Q ₁ and Q ₂ is considered. Each queue receives a Poisson stream of customers at rate _i , i=1,2. In addition, a Poisson stream of rate arrives to the system; the customers from this stream join the shorter of two queues. After being served in the ith queue, i=1,2, customers leave the system with probability 1–p _i ^*, join the jth queue with probability p(i,j), j=1,2, and choose the shortest of two queues with probability p(i,{1,2}). We establish necessary and sufficient conditions for stability of the system.     

On the Coefficients and Zeros of a Polynomial

Letp(z)=1+∑ⁿ_j=1 b_jz^jbe a complex polynomial. Two theorems on the coefficients and zeros ofp(z) are proved in this paper.     

Small prime solutions of a pair of linear equations in five prime variables

Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ （λ= 1, 2） has solutions in primes pj satisfying pj 〈〈 （｜b1｜＋｜b2｜＋1） maxλ,j ｜αλj｜^2318＋ε. This improves the exponent 79680 without assuming the additional condition of the second author＇s.     

Sums of almost equal prime cubes

We prove that almost all integers N satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 5; 6; 7; 8, i.e., N = p ₁³ + ... + p _j ³ with |p _i − (N/j)^1/3| ≦ $ N^{1/3 - \delta _j + \varepsilon } $ N^{1/3 - \delta _j + \varepsilon } (1 ≦ i ≦ j), for δ _j = 1/45; 1/30; 1/25; 2/45, respectively.     

Chevet's theorem for stable processes II

LetB be a separable Banach space and let {:||1} denote the unit ball ofB ^*. LetX be a symmetricp-stableB-valued random variable and let {X _j } _j=1 ⁿ be i.i.d. copies ofX. LetB ₁ be a finite-dimensional Banach space with a symmetric unconditional basis {y _j } _j=1 ⁿ . An upper bound is obtained for that improves the one given by Giné, Marcus and Zinn [J. Functional Anal. 63, 47–73 (1985)].