Gao’s conjecture on zerosum sequences

In this paper, we shall address three closely-related conjectures due to van Emde Boas, W D Gao and Kemnitz on zero-sum problems on Z_p ⊗ Z_p. We prove a number of results including a proof of the conjecture of Gao for the primep = 7 (Theorem 3.1). The conjecture of Kemnitz is also proved (Propositions 4.6, 4.9, 4.10) for many classes of sequences.     

Remarks on some zero-sum problems

Let Z denote the ring of integers and for a prime p and positive integers r and d, let f_r(P, d) denote the smallest positive integer such that given any sequence of f_r(p, d) elements in (Z/pZ(^d, there exists a subsequence of (rp) elements whose sum is zero in (Z/pZ(^d. That f₁(p, 1) = 2p − 1, is a classical result due to Erdős, Ginzburg and Ziv. Whereas the determination of the exact value of f₁(p, 2) has resisted the attacks of many well known mathematicians, we shall see that exact values of f_r(p, 1) for r ≥ 1 can be easily obtained from the above mentioned theorem of Erdős, Ginzburg and Ziv and those of f_r(p, 2) for r ≥ 2 can be established by the existing techniques developed by Alon, Dubiner and Rónyai in connection with obtaining good upper bounds for f₁(p, 2). We shall also take this opportunity to describe some of the early results in the introduction.     

Efficient p-adic Cell Decompositions for Univariate Polynomials

Cell decompositions are constructed for polynomials f(x)Z_p[x] of degree n, such that n<p, using O(n²) cells. When f is square-free this yields a polynomial-time algorithm for counting and approximating roots in Z_p. These results extend to give a polynomial-time algorithm in the bit model for fZ[x].     

Torsion-Free Space Groups, 2-Cohomology, and Scott Modules

For a subgroupCof orderpof a finite groupG, we find the summandMof thep-adic permutation module ind_C^GZ_psuch thatH²(G, M)≠0, and determine whenMis the Scott module. This is applied to the study of torsion-free space groups.     

Duadic Z4-Codes

The structure of abelian Z₄-codes (and more generally Z_p^m-codes) is studied. The approach is spectral: discrete Fourier transform and idempotents. A criterion for self-duality is derived. An arithmetic test on the length for the existence of nontrivial abelian self-dual codes is derived. A natural generalization of both the supplemented quadratic residue codes and the binary duadic codes is introduced. Isodual abelian Z₄ codes are considered, constructed, and used to produce 4-modular lattices.     

Un théorème de Harris et Segal pour un anneau d'entiers algébriques exceptionnel

Let K be a number field, O_K the ring of the integers of K, ℓ a prime integer and Z_(ℓ) the localisation of Z at ℓ. Harris and Segal [4] proved that there exists infinitely many primes p of O_K such that the natural morphism K_i(O_K) ⊗ Z_(ℓ) → K_i(O_K /p) ⊗ Z_(ℓ) in algebraic K-theory is split surjective for i > 0, except if ℓ = 1 and K is exceptional. In this Note, we prove that the Harris-Segal theorem is still true for ℓ = 2 in the exceptional case, if we replace algebraic K-theory by orthogonal K-theory defined by Karoubi [5]. Thanks to [3], we can then determine a direct summand of the 2-torsion of KO_n(O_K).     

Conjectures de Greenberg et extensions¶pro-p-libres d'un corps de nombres

For an algebraic number field k and a prime number p (if p=2, we assume that μ₄⊂k), we study the maximal rank ρ _p of a free pro-p-extension of k. This problem is related to deep conjectures of Greenberg in Iwasawa theory. We give different equivalent formulations of these conjectures and we apply them to show that, essentially, ρ _k =r ₂(k)+1 if and only if k is a so-called p-rational field. Received: 29 April 1999 / Revised version: 31 January 2000     

The complexity of minimum difference cover

The complexity of searching minimum difference covers, both in Z⁺ and in Z_n, is studied. We prove that these two optimization problems are NP-hard. To obtain this result, we characterize those sets—called extrema—having themselves plus zero as minimum difference cover. Such a combinatorial characterization enables us to show that testing whether sets are not extrema, both in Z⁺ and in Z_n, is NP-complete. However, for these two decision problems we exhibit pseudo-polynomial time algorithms.     

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

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, (0δ1) as 0, and for

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as .     

A remark on the transformation formula of theta functions associated to positive definite quadratic forms

Let L be a positive Z-lattice with level N = cd, (c, d) = 1. Then the Fourier expansion at cusp $\text{1}\text{d}$ of the theta function associated to L is a theta function associated to L₁, where a lattice L₁ is defined by Z_pL₁ = Z_pL for $\text{p?c}$, Z_pL₁ = the dual of Z_pL for p | c.     

Integral Kirillov model

We prove that an irreducible cuspidal Q̄_ℓ-representation of GL(n, Q_p) with a central character with values in Z̄_ℓ^* has a unique Z̄_ℓ-integral structure, given by the Kirillov Z̄_ℓ-representation.     

Van der Put basis and <Emphasis Type="Italic">p</Emphasis>-adic dynamics

We consider a function g: Z _p → Z _p and its the van der Put series. Then we get a criteria of Haar’s measure preserving compatible p-adic functions which, actually, need not be uniformly differentiable modulo p. This is used to study ergodicity of p-adic dynamical systems [2, 16].     

An addition theorem for the elementary Abelian group of type (p,p)

In this paper we prove that every element in the finite Abelian groupZ _p ×Z _p ,p>3,p prime, can be written as a sum over a subset of the setA, whereA is any set of non-zero elements ofZ _p ×Z _p with |A|=2p–2.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

On Group Connectivity of Graphs

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero Z ₃-flow and Jaeger et al. [Group connectivity of graphs–a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165-182] further conjectured that every 5-edge-connected graph is Z ₃-connected. These two conjectures are in general open and few results are known so far. A weaker version of Tutte’s conjecture states that every 4-edge-connected graph with each edge contained in a circuit of length at most 3 admits a nowhere-zero Z ₃-flow. Devos proposed a stronger version problem by asking if every such graph is Z ₃-connected. In this paper, we first answer this later question in negative and get an infinite family of such graphs which are not Z ₃-connected. Moreover, motivated by these graphs, we prove that every 6-edge-connected graph whose edge set is an edge disjoint union of circuits of length at most 3 is Z ₃-connected. It is a partial result to Jaeger’s Z ₃-connectivity conjecture. Received: May 23, 2006. Final version received: January 13, 2008     

Z-cyclic triplewhist tournaments—The noncompatible case,part I

Two odd primes odd, are said to be noncompatible if b₁ ≠ b₂. For all noncompatible (ordered) pairs of primes (p₁, p₂) such that p_i ≡ p_i < 200, i = 1,2 we establish the existence of Z-cyclic triplewhist tournaments on 3p₁ p₂ + 1 players. It is believed that these results are the first examples of such tournaments, indeed the first examples of Z-cyclic whist tournaments for such players. In Part 2 we extend the results of this study and establish the existence of Z-cyclic triplewhist tournaments on players for all α₁ ≥ 1, α₂ ≥ 1 and p₁, p₂ as described above. © 1997 John Wiley & Sons, Inc.     

On the construction of Wold decomposition for multivariate stationary processes

A method to construct the Wold decomposition for multivariate stationary stochastic processes x_k, k Z, is presented. The method is based on orthogonal decompositions for x_k, k Z, obtained by forming orthogonal projections of x_k, k Z, onto its component processes , k Z, j = 1, …, q. The method does not give a complete solution to the Wold decomposition problem.     

Construction of free subgroups in the group of units of modular group algebras

Let KGbe the group algebra of a p¹ -group Gover a field Kof characteristic p > 0, and let U(KG)be its group of units. If KGcontains a nontrivial bicyclic unit and if Kis not algebraic over its prime field, then we prove that the free product Z_p? Z_p? Z_pcan be embedded in U(KG).     

On minimal decomposition of p-adic polynomial dynamical systems

A polynomial of degree ?2 with coefficients in the ring of p-adic numbers Zp is studied as a dynamical system on Zp. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on Z₂, we exhibit all its minimal subsystems.     

CirculantGH(p 2; Z p ) exist for all primesp

The only known circulant ordinary Hadamard matrix is developed from the initial row-1, 1, 1, 1. Letp be a prime, and letZ _p denote the cyclic group of orderp. In this paper, we construct circulantGH(p ²;Z _p ) for all primesp. Whenp is odd, this result also extends the earlier result that there exist circulantGH(p;Z _p ) for all odd primesp. Other families ofGH-matrices which are developed modulo a group are discussed.