Unbounded symmetrysets

Unbounded symmetrysets R Z_pⁿ are introduced which, in the presence of a Jacobi condition, are classified and can be written as R = Z_p^n₁ + + Z_pⁿ_k (inner direct sum), where n_i n for all i = 1, , k. The properties of these unbounded symmetrysets are easily verified for the set R of roots of Witt Lie algebras. This paper is a step in the direction of classifying simple Lie algebras of characteristic, p, by studying their rootsystems.     

On coercivity and irregularity for some nonlinear degenerate elliptic systems

We study the ‘universal’ strong coercivity problem for variational integrals of degenerate p-Laplacian type by mixing finitely many homogenous systems. We establish the equivalence between universal p-coercivity and a generalized notion of p-quasiconvex extreme points. We then give sufficient conditions and counterexamples for universal coercivity. In the case of noncoercive systems we give examples showing that the corresponding variational integral may have infinitely many non-trivial minimizers in W ₀^1,p which are nowhere C ¹ on their supports. We also give examples of universally p-coercive variational integrals in W ₀^1,p for p ⩾ with L ^∞ coefficients for which uniqueminimizers under affine boundary conditions are nowhere C ¹.      

On some new paranormed euler sequence spaces and Euler Core

In this paper, the sequence spaces e0^τ（u, p） and ec^τ（u, p） of non-absolute type which are the generalization of the Maddox sequence spaces have been introduced and it is proved that the spaces e0^τ（u,p） and ec^τ（u,p） are linearly isomorphic to spaces co（p） and c（p）, respectively. Furthermore, the α-, β- and γ-duals of the spaces 0^τ（u,p） and ec^τ（u,p） have been computed and their bases have been constructed and some topological properties of these spaces have been investigated. Besides this, the class of matrices （e0^τ）（u, p） ： μ） has been characterized, where μ is one of the sequence spaces l∞, c and co and derives the other characterizations for the special cases of μ. In the last section, Euler Core of a complex-valued sequence has been introduced, and we prove some inclusion theorems related to this new type of core.     

Inversion of a theorem on action of analytic functions and multiplicative properties of some subclasses of the hardy space H∞

In this paper, function spaces V∩l _A ^p (w) are considered in the context of their multiplicative structure. The space V is determined by conditions on the values of a function in a disk (for example, C_A,Lip _Aα). We denote by l _A ^p (w) the space of power series such that their Taylor coefficients are p-summable with weight w. For an analytic function Φ acting in a space of this type, we prove the following alternative: either Φ″(z)≡0, or the space is a Banach algebra with respect to pointwise multiplication. For a wide class of weights w, we establish the continuity of the identity embeddingmult(V∩l _A ^p (w))↪multl _A ^p . An estimate for the l^p-multiplicative norm of random polynomials is found. This estimate can be considered as an extension of the known result by Salem-Zygmund. Bibliography: 10 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 232, 1996, pp. 50–72. Translated by S. Shimorin.     

Dunford-Pettis like properties on tensor products

In this paper we study equivalent formulations of the DP^? P_p (1 < p < ∞). We show that X has the DP^? P_p if and only if every weakly-p-Cauchy sequence in X is a limited subset of X. We give su?cient conditions on Banach spaces X and Y so that the projective tensor product X ?_π Y, the dual (X ?_? Y)^? of their injective tensor product, and the bidual (X ?_π Y)^?? of their projective tensor product, do not have the DP P_p, 1 < p < ∞. We also show that in some cases, the projective and the injective tensor products of two spaces do not have the DP^? P_p, 1 < p < ∞.     

Random subgraphs of finite graphs: I. The scaling window under the triangle condition

We study random subgraphs of an arbitrary finite connected transitive graph ?? obtained by independently deleting edges with probability 1 ? p. Let V be the number of vertices in ??, and let Ω be their degree. We define the critical threshold p_c = p_c (??, λ) to be the value of p for which the expected cluster size of a fixed vertex attains the value λV^1/3, where λ is fixed and positive. We show that, for any such model, there is a phase transition at p_c analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold p_c. In particular, we show that the largest cluster inside a scaling window of size |p ? p_c| = Θ(Ω^?1V^?1/3) is of size Θ(V^2/3), while, below this scaling window, it is much smaller, of order O(?^?2 log(V?³)), with ? = Ω(p_c ? p). We also obtain an upper bound O(Ω(p ? p_c)V) for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order Θ(Ω(p ? p_c)). Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the n‐cube and certain Hamming cubes, as well as the spread‐out n‐dimensional torus for n > 6. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

On the subspaces ofL p (p>2) spanned by sequences of independent random variables

Let 2<p<∞. The Banach space spanned by a sequence of independent random variables inL ^p , each of mean zero, is shown to be isomorphic tol ²,l ^p ,l ²⊕l ^p , or a new spaceX _p , and the linear topological properties ofX _p are investigated. It is proved thatX _p is isomorphic to a complemented subspace ofL ^p and another uncomplemented subspace ofL ^p , whence there exists an uncomplemented subspace ofl ^p isomorphic tol ^p . It is also proved thatX _p is not isomorphic to the previously knownℒ _p spaces. The work for this research was partially supported by the National Science Foundation GP-12997.     

Weak Limits for Multivariate Random Sums

Let {Xi, i1} be a sequence of i.i.d. random vectors inRd, and letνp, 0<p<1, be a positive, integer valued random variable, independent ofXis. Theν-stable distributions are the weak limits of properly normalized random sums ∑νpi=1 Xiasνp ∞ andpνp ν. We study the properties ofν-stable laws through their representation via stable laws. In particular, we estimate their tail probabilities and provide conditions for finiteness of their moments.     

关于环Fpm+uFpm+u2Fpm上循环码的注记

本文研究了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码分类.通过建立环F_p^m+uF_p^m+u²F_p^m到环F_p^m+uF_p^m的同态,给出了环F_p^m+uF_p^m+u²F_p^m上长度为p^s的循环码的新分类方法.应用这种方法,得到了环F_p^m+uF_p^m+u²F_p^m长度为p^s的循环码的码词数.     

Nonnegative functions in weighted hardy spaces

Let W be a nonnegative summable function whose logarithm is also summable with respect to the Lebesgue measure on the unit circle. For 0?<?p?<?∞ , H^p (W) denotes a weighted Hardy space on the unit circle. When W?≡?1, H ^p(W) is the usual Hardy space H^p . We are interested in H^p ( W)₊ the set of all nonnegative functions in H^p ( W). If p?≥?1/2, H^p ₊ consists of constant functions. However H^p ( W)₊ contains a nonconstant nonnegative function for some weight W. In this paper, if p?≥?1/2 we determine W and describe H^p ( W)₊ when the linear span of H^p ( W)₊ is of finite dimension. Moreover we show that the linear span of H^p (W)₊ is of infinite dimension for arbitrary weight W when 0?<?p?<?1/2.     

Cyclic Codes over the Integers Modulop

The purpose of this paper is twofold. First, we generalize the results of Pless and Qian and those of Pless, Solé, and Qian for cyclic ₄-codes to cyclic _p^m-codes. Second, we establish connections between this new development and the results on cyclic _p^m-codes obtained by Calderbank and Sloane. We produce generators for the cyclic _p^m-codes which are analogs to those for cyclic ₄-codes. We show that these may be used to produce a single generator for such codes. In particular, this proves that the ringR_n= _p^m[x]/(xⁿ− 1) is principal, a result that had been previously announced with an incorrect proof. Generators for dual codes of cyclic _p^m-codes are produced from the generators of the corresponding cyclic _p^m-codes. In addition, we also obtain generators for the cyclicp^m-ary codes induced from the idempotent generators for cyclicp-ary codes.     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups II

Let p ₁, p ₂, p ₃ be primes. This is the second article in a series of three on the (p ₁, p ₂, p ₃)-generation of the finite projective special unitary and linear groups PSU₃(p ⁿ ), PSL₃(p ⁿ ), where we say a noncyclic group is (p ₁, p ₂, p ₃)-generated if it is a homomorphic image of the triangle group T _{p 1, p 2, p 3} . This paper is concerned with the case where p ₁ = 2 and p ₂ = p ₃. We determine for any prime p ₂ the prime powers p ⁿ such that PSU₃(p ⁿ ) (respectively, PSL₃(p ⁿ )) is a quotient of T = T _{2, p 2, p 2} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU₃(p ⁿ )) (respectively, Hom(T, PSL₃(p ⁿ ))) is surjective as p ⁿ tends to infinity.     

Random and Deterministic Triangle Generation of Three-Dimensional Classical Groups III

Let p₁, p₂, p₃ be primes. This is the final paper in a series of three on the (p₁, p₂, p₃)-generation of the finite projective special unitary and linear groups PSU ₃(pⁿ), PSL ₃(pⁿ), where we say a noncyclic group is (p₁, p₂, p₃)-generated if it is a homomorphic image of the triangle group T_{p1, p2, p3} . This article is concerned with the case where p₁ = 2 and p₂ ≠ p₃. We determine for any primes p₂ ≠ p₃ the prime powers pⁿ such that PSU ₃(pⁿ) (respectively, PSL ₃(pⁿ)) is a quotient of T = T_{2, p2, p3} . We also derive the limit of the probability that a randomly chosen homomorphism in Hom(T, PSU ₃(pⁿ)) (respectively, Hom(T, PSL ₃(pⁿ))) is surjective as pⁿ tends to infinity.     

Об отделимости множе ств гиперплоскостью вL p

LetA and be two arbitrary sets in the real spaceL _p, 1p<. Sufficient conditions are obtained for their strict separability by a hyperplane, in terms of the distance between the setsd(A,B) _p=inf{x-y_p,xA,yB} and their diametersd(A) _p, d(B)_p, whered(A) _p=sup{x-y_p; x,yA}. In particular, it is proved that if in an infinite-demensional spaceL _p we haved ^r(A,B)_p>2^–r+1(d^r(A)_p+d^r(B)_p), r=min{p, p(p–1)^–1}, then there is a hyperplane which separatesA andB. On the other hand, the conditiond ^r(A,B)_p=2^–r+1(d^r(A)_p+d^r(B)_p) does not guarantee strict separability. Earlier these results where obtained by V. L. Dol'nikov for the case of Euclidean spaces.     

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.     

Algebraic surfaces of general type withK 2=2p g−1,p g≥5

This paper mainly deals with minimal algebraic surfaces of general type withK ²=2p _g–1. We prove that forp _g7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces withp _g=5, 6. Lastly we show that whenp _g13 all these surfaces are simply-connected.     

Modes and moments of unimodal distributions

Summary For a unimodal distribution relations of its modea with its absolute momentβ _p and central absolute momentγ _p of orderp are considered. The best constantA _p andB _p are given for the inequalities |a|≦A _p β _p ^1/p (p>0) and |a−m|≦B _p γ _p ^1/p (p≧1) wherem is the mean. the results follow from discussion of more general moments.     

A survey of orthogonal arrays of strength two

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Codes over p-adic Numbers and Finite Rings Invariant under the Full Affine Group

Codes over p-adic numbers and over integers modulo p^d of block length p^m invariant under the full affine group AGL_m(F_p) are described.     

Carleson measures and the BMO space on the p ‐adic vector space

For a prime number p, let Q _p be the p ‐adic field and let Q _p ^d denote a vector space over Q _p which consists of all d ‐tuples of Q _p . Then we study the p ‐adic version of the Calderón–Zygmund decomposition, Carleson measures on the vector space Q _p ^{d +1} and the space BMO ( Q _p ^d ) of functions of bounded mean oscillation on Q _p ^d . In particular, it turns out that the operator norms of various oncoming operators are independent of the dimension d and the prime number p, which is one of the big differences from that of the Euclidean case. Interestingly, the independence of the dimension d and p makes it possible to develop Harmonic Analysis on the infinite dimensional p ‐adic vector space as the importance had already been pointed out in the Euclidean case (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)