A mean value theorem on the binary Goldbach problem and its application

We study the binary Goldbach problem with one prime number in a given residue class, and obtain a mean value theorem. As an application, we prove that for almost all sufficiently large even integers n satisfying n ≢ 2(mod 6), the equation p ₁ + p ₂ = n is solvable in prime variables p ₁, p ₂ such that p ₁ + 2 = P ₃, and for every sufficiently large odd integer satisfying ≢ 1(mod 6), the equation p ₁ + p ₂ + p ₃ = is solvable in prime variables p ₁, p ₂, p ₃ such that p ₁ + 2 = P ₂, p ₂ + 2 = P ₃. Here P _k denotes any integer with no more than k prime factors, counted according to multiplicity.     

On the structure ofp-zero-sum free sequences and its application to a variant of Erdös-Ginzburg-Ziv theorem

Letp be any odd prime number. Letk be any positive integer such that . LetS = (a ₁,a ₂,...,a _2p−k ) be any sequence in ℤ_p such that there is no subsequence of lengthp of S whose sum is zero in ℤ_p. Then we prove that we can arrange the sequence S as follows:

Extensions of thep-adic fields generated by the roots of a single polynomial in two variables

One of the generalizations of the Hilbert's Irreducibility Theorem states that if F(X, Y) is irreducible in Q _p [X, Y] then, for almost every n N, F(n, Y) is irreducible in Q _p [Y]. Let E{p, d} be the set of all the extensions of Q _p of degree d. We shall prove that for each subset S E{p, d}there exist polynomials F(X, Y) such that the set of extensions of Q _p generated by the roots of F (n, Y)is exactly S. Moreover we shall prove that all the functions f are induced by polynomials F(X) Z[X].M. S. C. N. 11S05, (11S15).The author would like to thank Prof. Roberto Dvornicich for several useful conversations and advices.     

The noncentral Wishart as an exponential family, and its moments

While the noncentral Wishart distribution is generally introduced as the distribution of the random symmetric matrix where Y₁,…,Y_n are independent Gaussian rows in with the same covariance, the present paper starts from a slightly more general definition, following the extension of the chi-square distribution to the gamma distribution. We denote by γ(p,a;σ) this general noncentral Wishart distribution: the real number p is called the shape parameter, the positive definite matrix σ of order k is called the shape parameter and the semi-positive definite matrix a of order k is such that the matrix ω=σaσ is called the noncentrality parameter. This paper considers three problems: the derivation of an explicit formula for the expectation of when Xγ(p,a,σ) and h₁,…,h_m are arbitrary symmetric matrices of order k, the estimation of the parameters (a,σ) by a method different from that of Alam and Mitra [K. Alam, A. Mitra, On estimated the scale and noncentrality matrices of a Wishart distribution, Sankhyā, Series B 52 (1990) 133–143] and the determination of the set of acceptable p’s as already done by Gindikin and Shanbag for the ordinary Wishart distribution γ(p,0,σ).     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

Hua’s theorem for five almost equal prime squares

Let p_i, 1≦ i ≦ 5, be prime numbers. It is proved that every sufficiently large integer N that satisfies N ≡ 5(mod 24) can be written as N = p²₁ + p²₂ + p²₃ + p²₄ + p²₅, where Received: 10 February 2005     

Low Rank Approximation of a Hankel Matrix by Structured Total Least Norm

The structure preserving rank reduction problem arises in many important applications. The singular value decomposition (SVD), while giving the closest low rank approximation to a given matrix in matrix L ₂ norm and Frobenius norm, may not be appropriate for these applications since it does not preserve the given structure. We present a new method for structure preserving low rank approximation of a matrix, which is based on Structured Total Least Norm (STLN). The STLN is an efficient method for obtaining an approximate solution to an overdetermined linear system AX B, preserving the given linear structure in the perturbation [E F] such that (A + E)X = B + F. The approximate solution can be obtained to minimize the perturbation [E F] in the L _p norm, where p = 1, 2, or . An algorithm is described for Hankel structure preserving low rank approximation using STLN with L _p norm. Computational results are presented, which show performances of the STLN based method for L ₁ and L ₂ norms for reduced rank approximation for Hankel matrices.     

Uniform-fast-gerade Funktionen mit vorgegebenen Werten,II

For a given sequence n₁ < n₂ < ... of integers satisfying and for a given convergent sequence of complex numbers {a_j}, it was shown in [4] that there is a uniformly-almost-even function assuming the values f(n_j) = a_j. For the proof, Gelfands theory of commutative Banach algebras and Tietzes extension theorem were used. In [3] an incomplete proof [by elementary means] of this result was given.¹⁾The aim of this note is to give some results which can be proved by the method from [3].¹⁾The first-named author is grateful to the second author for pointing out a missing case in the above-mentioned proof.Received: 7 November 2002     

Perturbation Analysis for the Eigenvalue Problem of a Formal Product of Matrices

We study the perturbation theory for the eigenvalue problem of a formal matrix product A ₁ ^{s 1} ··· A _p ^{s p}, where all A _k are square and s _k {–1, 1}. We generalize the classical perturbation results for matrices and matrix pencils to perturbation results for generalized deflating subspaces and eigenvalues of such formal matrix products. As an application we then extend the structured perturbation theory for the eigenvalue problem of Hamiltonian matrices to Hamiltonian/skew-Hamiltonian pencils.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

Lower Bounds For Multidimensional Zero Sums

Let f(n, d) denote the least integer such that any choice of f(n, d) elements in contains a subset of size n whose sum is zero. Harborth proved that (n-1)2 ^d +1 f(n,d) (n-1)n ^d +1. The upper bound was improved by Alon and Dubiner to c _d n. It is known that f(n-1) = 2n-1 and Reiher proved that f(n-2) = 4n-3. Only for n = 3 it was known that f(n,d) > (n-1)2 ^d +1, so that it seemed possible that for a fixed dimension, but a sufficiently large prime p, the lower bound might determine the true value of f(p,d). In this note we show that this is not the case. In fact, for all odd n 3 and d 3 we show that .     

On linear equations with prime variables of special type

Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. It is proved that for every sufficiently large odd integer , the equation p₁+p₂+p₃=n is solvable in prime variables p₁,p₂,p₃ such that p₁+2=P₂, , and for almost all sufficiently large even integer , the equation p₁+p₂=n is solvable in prime variables p₁,p₂ such that p₁+2=P₂.     

Fractals related to Pascal's triangle

A precise definition of a fractalF _p ^{r 1} derived from Pascal's triangle modulop ^r (p prime) is given. The number of nonzero terms in the firstp ^s lines of Pascal's triangle modulop ^r is computed. From this result the Hausdorff dimension and Hausdorff measure ofF _p ^{r 1} are deduced. The nonself-similarty ofF _p ^{r 1},r2, is also discussed.     

Alternating Sign Matrices and Some Deformations of Weyl's Denominator Formulas

An alternating sign matrix is a square matrix whose entries are 1, 0, or –1, and which satisfies certain conditions. Permutation matrices are alternating sign matrices. In this paper, we use the (generalized) Littlewood's formulas to expand the products and 2 as sums indexed by sets of alternating sign matrices invariant under a 180° rotation. If we put t = 1, these expansion formulas reduce to the Weyl's denominator formulas for the root systems of type B _n and C _n. A similar deformation of the denominator formula for type D _n is also given.     

Limit theorems for radial random walks on p × q‐matrices as p tends to infinity

For a fixed probability measure ν ∈ M¹([0, ∞[) and any dimension $ p\in {\mathbb N}$ there is a unique radial probability measure $ \nu_p\in M^1({\mathbb R}^p)$ with ν as its radial part. In this paper we study the limit behavior of ‖S^p_n‖₂ for the associated radial random walks (S_n)_n≥0 on $ {\mathbb R}^p$ whenever n, p tend to ∞ in some coupled way. In particular, weak and strong laws of large numbers as well as a large deviation principle are presented. In fact, we shall derive these results in a higher rank setting, where $ {\mathbb R}^p$ is replaced by the space of p × q matrices and [0, ∞[ by the cone Π_q of positive semidefinite matrices. All proofs are based on the fact that in this general setting the (S^p_k)_k≥0 form Markov chains on Π_q whose transition probabilities are given in terms Bessel functions J_μ of matrix argument with an index μ depending on p. The limit theorems then follow from new asymptotic results for the J_μ as μ → ∞. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

A construction of minimal asymptotic bases

Jia and Nathanson [2] give a simple and explicit construction of minimal asymptotic bases of orderh for everyh2. They constructed minimal asymptotic bases from partition of N by means of powers of 2. In this paper, we extend the results of that paper to asymptotic bases constructed from partitions of N by means ofg-adic representations forg2. Corollary 3 shows that given partition N=W ₀W ₁...W _h–1 such that eachW _i contains infinitely many pairs of consecutive integers we can construct a minimal asymptotic bases of orderh in infinitely many way.     

Symmetric square L-values and dihedral congruences for cusp forms

Let be a prime, and k=(p+1)/2. In this paper we prove that two things happen if and only if the class number . One is the non-integrality at p of a certain trace of normalised critical values of symmetric square L-functions, of cuspidal Hecke eigenforms of level one and weight k. The other is the existence of such a form g whose Hecke eigenvalues satisfy “dihedral” congruences modulo a divisor of p (e.g. p=23, k=12, g=Δ). We use the Bloch-Kato conjecture to link these two phenomena, using the Galois interpretation of the congruences to produce global torsion elements which contribute to the denominator of the conjectural formula for an L-value. When , the trace turns out always to be a p-adic unit.     

Admissible inference rules for polymodal logicS5 n C

We study polymodal logics with n modal connectives □₁,...,□_n, each of which satisfies the axioms of S5 and, moreover, obeys the commutativity laws . The following results are proved: (1) the logic S5_nC is not locally finite; (2) the inference rule A(p₁, …, p_m)/B(p₁, …, p_m) is not admissible in , and on a one-element model ∉, there exists a valuation of variables p₁, …, p_m, such that ∉ ⊪ A. Supported by RFFR grant No. 96-01-00228. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 483–493, September–October, 1997.     

Partitions of bipartite numbers

Letp _j(m, n) be the number of partitions of (m, n) into at mostj parts. We prove Landman et al.'s conjecture: for allj andn, p _j(x, 2n–x) is a maximum whenx-n. More generally we prove that for all positive integersm, n andj, p _j(n, m)=p_j(m, n)p_j(m–1, n+1) ifmn.     

The 2-primary torsion on elliptic curves in the Z p -extensions of Q

Let E be an elliptic curve over Q and p a prime number. Denote by Q_p,∞ the Z_p-extension of Q. In this paper, we show that if p≠3, then where E(Q_p,∞)₍₂₎ is the 2-primary part of the group E(Q_p,∞) of Q_p,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q_2,∞)₍₂₎ in terms of E(Q)₍₂₎; in case p≥5 (or in case p=3 and E(Q)₍₂₎≠{O}), we show that E(Q_p,∞)₍₂₎=E(Q)₍₂₎.