On relations for the multiple <Emphasis Type="Italic">q</Emphasis>-zeta values

We prove a new relation for the multiple q-zeta values (MqZV’s). It is a q-analogue of the Ohno-Zagier relation for the multiple zeta values (MZV’s). We discuss the problem of determining the dimension of the space spanned by MqZV’s over ℚ, and present an application to MZV. The first author is supported by Grant-in-Aid for Young Scientists (B) No. 17740026 and the second author is supported by Grant-in-Aid for Young Scientists (B) No. 17740089.     

<Emphasis Type="Italic">G</Emphasis>-covering subgroup systems for the classes of supersoluble and nilpotent groups

LetF be a class of groups andG a group. We call a set Σ of subgroups ofG aG-covering subgroup system for the classF (or directly aF-covering subgroup system ofG) ifG ∈F whenever every subgroup in Σ is inF. In this paper, we provide some nontrivial sets of subgroups of a finite groupG which are simultaneouslyG-covering subgroup systems for the classes of supersoluble and nilpotent groups. Research of the first author is supported by the NNSF of China (Grant No. 10171086) and QLGCF of Jiangsu Province and a Croucher Fellowship of Hong Kong. Research of the second author is partially supported by a UGC (HK) grant #2060176 (2001/2002).     

On the existence of <Emphasis Type="Italic">F</Emphasis>-crystals

Let (N,F) be an F-isocrystal, with associated Newton vector ν in . To any lattice M in N (an F-crystal) is associated its Hodge vector in . By Mazur's inequality we have . We show that, conversely, for any with , there exists a lattice M in N such that . We also give variants of this existence theorem for symplectic F-isocrystals, and for periodic lattice chains. Received: March 1, 2002     

On the value distribution of <Emphasis Type="Italic">ƒ</Emphasis> + <Emphasis Type="Italic">a</Emphasis>(<Emphasis Type="Italic">ƒ</Emphasis>′)<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Let ƒ be a transcendental meromorphic function, a a nonzero finite complex number, and n ⩾ 2 a positive integer. Then ƒ + a(ƒ′) ⁿ assumes every complex value infinitely often. This answers a question of Ye for n = 2. A related normality criterion is also given. This work was supported by the National Natural Science Foundation of China (Grant No. 10771076), the Natural Science Foundation of Guangdong Province, China (Grant No. 07006700) and by the German-Israeli Foundation for Scientific Research and Development (Grant No. G-809-234.6/2003)     

On the Edgeworth expansion and the <Emphasis Type="Italic">M</Emphasis> out of <Emphasis Type="Italic">N</Emphasis> bootstrap accuracy for a Studentized trimmed mean

We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur’s type representation for an M out of N bootstrap quantile. Our results supplement previous work on (Studentized) trimmed means by Hall and Padmanabhan [13], Bickel and Sakov [7], and Gribkova and Helmers [11].      

Bloch constant of holomorphic mappings on the unit polydisk of ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we give a definition of Bloch mappings defined in the unit polydisk D ⁿ , which generalizes the concept of Bloch functions defined in the unit disk D. It is known that Bloch theorem fails unless we have some restrictive assumption on holomorphic mappings in several complex variables. We shall establish the corresponding distortion theorems for subfamilies β(K) and β _loc(K) of Bloch mappings defined in the polydisk D ⁿ , which extend the distortion theorems of Liu and Minda to higher dimensions. As an application, we obtain lower and upper bounds of Bloch constants for various subfamilies of Bloch mappings defined in D ⁿ . In particular, our results reduce to the classical results of Ahlfors and Landau when n = 1. This work was supported by the National Natural Science Foundation of China (Grant No. 10571164) and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (Grant No. 20050358052)     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> solvability of the stationary stokes problem in domains with conical singularity in any dimension

We consider the Dirichlet boundary value problem for the Stokes operator with L ^p data in any dimension in domains with conical singularity (not necessarily a Lipschitz graph). We establish the solvability of the problem for all p ∈ (2 − ε, ∞] and also in C(D) for the data in C( [`(D)] ) C\left( {\overline D } \right) . Bibliography: 14 titles. In memory of Michael Sh. Birman     

A generalization of Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> by using atoms

Let X = （X, d,μ） The purpose of this paper is to be a space of homogeneous type in the sense of Coifman and Weiss. generalize the definition of Hardy space H^P（X） and prove that the generalized Hardy spaces have the same property as H^P（X）. Our definition includes a kind of Hardy- Orlicz spaces and a kind of Hardy spaces with variable exponent. The results are new even for the R^n case. Let （X, δ, μ） be the normalized space of （X, d, μ） in the sense of Macias and Segovia. We also study the relations of our function spaces for （X, d, μ） and （X, δ,μ）.     

On the word problem for the free Burnside semigroups satisfying <Emphasis Type="Italic">x</Emphasis><Superscript>2</Superscript> = <Emphasis Type="Italic">x</Emphasis><Superscript>3</Superscript>

We study the word problem for the free Burnside semigroups satisfying x ² = x ³. For any k > 2, we reduce this problem for the k-generated free Burnside semigroup B(2, 1, k) to the word problem for the two-generated semigroup B(2, 1, 2). Furthermore, if every element of B(2, 1, 2) is a regular language, then every element of B(2, 1, k) appears to be a regular language as well. Thus, Brzozowski’s conjecture holds for the semigroup B(2, 1, k) if and only if it holds for B(2, 1, 2).     

On the Nevanlinna characteristic of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">z</Emphasis>+<Emphasis Type="Italic">η</Emphasis>) and difference equations in the complex plane

We investigate the growth of the Nevanlinna characteristic of f(z+η) for a fixed η∈C in this paper. In particular, we obtain a precise asymptotic relation between T(r,f(z+η)) and T(r,f), which is only true for finite order meromorphic functions. We have also obtained the proximity function and pointwise estimates of f(z+η)/f(z) which is a discrete version of the classical logarithmic derivative estimates of f(z). We apply these results to give new growth estimates of meromorphic solutions to higher order linear difference equations. This also allows us to solve an old problem of Whittaker (Interpolatory Function Theory, Cambridge University Press, Cambridge, 1935) concerning a first order difference equation. We show by giving a number of examples that all of our results are best possible in certain senses. Finally, we give a direct proof of a result in Ablowitz, Halburd and Herbst (Nonlinearity 13:889–905, 2000) concerning integrable difference equations. This research was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China (HKUST6135/01P). The second author was also partially supported by the National Natural Science Foundation of China (Grant No. 10501044) and the HKUST PDF Matching Fund.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the uniform norm of solutions of quasilinear SPDE's

In this paper we prove L^p estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE.     

A linear programming perspective on the Frankl—Rödl—Pippenger theorem

For H a hypergraph on vertex set V and t: H → R ⁺, set. α(t) = max {Σ {t(A): x, y ϵ A ϵ H}: x, y ϵ V, x ≠ y} We give a simple proof, based on an argument of Pippenger and Spencer [19] of the following rather general statement. Theorem 1.5. Fix k and l. Let H be a k-bounded hypergraph and t a fractional matching of H. Let, furthermore, c₁, …, c_l: H → R ⁺, each satisfying $ \sum_{A\in {\cal H}}\ c^2_i (A)t(A) \lt o\left(\left(\sum_{A\in {\cal H}}\ c_i(a)t(A)\right)^2\right) $ Then there is matching M of H satisfying $ \sum_{A\in {\cal M}} \ c_i(A) \sim c_i ({\cal H}) \kern4em \hbox{ for } i=1,\dots,l, $ limits taken as α( t)→0. This contains some previously announced results of the author which interpreted the theorem of the title as a statement about the relation between integer and linear programs and extended it in this vein. © 1996 John Wiley & Sons, Inc.     

Ultra <Emphasis Type="Italic">LI</Emphasis>-ideals in lattice implication algebras and <Emphasis Type="Italic">MTL</Emphasis>-algebras

A mistake concerning the ultra LI-ideal of a lattice implication algebra is pointed out, and some new sufficient and necessary conditions for an LI-ideal to be an ultra LI-ideal are given. Moreover, the notion of an LI-ideal is extended to MTL-algebras, the notions of a (prime, ultra, obstinate, Boolean) LI-ideal and an ILI-ideal of an MTL-algebra are introduced, some important examples are given, and the following notions are proved to be equivalent in MTL-algebra: (1) prime proper LI-ideal and Boolean LI-ideal, (2) prime proper LI-ideal and ILI-ideal, (3) proper obstinate LI-ideal, (4) ultra LI-ideal. This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant No. Y605389) and K. C. Wong Magna Fund in Ningbo University.     

Maximal subrings and <Emphasis Type="Italic">E</Emphasis>-groups

For a finite group G, let E(G) denote the near-ring of functions generated by the semigroup, End(G), of endomorphisms of G. We characterize when E(G) is maximal as a subnear-ring of M ₀(G). A group G is an E-group if E(G) is a ring. We give a new characterization of finite E-groups and investigate the problem of determining, for finite E-groups, when E(G) is maximal as a ring in M₀(G). Received: 26 June 2006     

On the evaluation at (<Emphasis Type="Italic">j</Emphasis>, <Emphasis Type="Italic">j</Emphasis><Superscript>2</Superscript>) of the Tutte polynomial of a ternary matroid

F. Jaeger has shown that up to a ± sign the evaluation at (j, j ²) of the Tutte polynomial of a ternary matroid can be expressed in terms of the dimension of the bicycle space of a representation over GF(3). We give a short algebraic proof of this result, which moreover yields the exact value of ±, a problem left open in Jaeger's paper. It follows that the computation of t(j, j ²) is of polynomial complexity for a ternary matroid.E. Gioan: C.N.R.S., MontpellierM. Las Vergnas: C.N.R.S., Paris     

On the integrability of the <Emphasis Type="Italic">n</Emphasis>-centre problem

It is known that for n3 centres and positive energies the n-centre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy. Here we consider the energies above some threshold and show: Whereas for arbitrary g>1 independent integrals of Gevrey class g exist, no real-analytic (that is, Gevrey class 1) independent integral exists.Mathematics Subject Classification (2000): 70F10, 37J30, 37J35, 37N05, 70F15, 70H06, 81U10     

On perfect Γ‐decompositions of the complete graph

Generalizing the well‐known concept of an i‐perfect cycle system, Pasotti [Pasotti, in press, Australas J Combin] defined a Γ‐decomposition (Γ‐factorization) of a complete graph K_v to be i‐perfect if for every edge [x, y] of K_v there is exactly one block of the decomposition (factor of the factorization) in which x and y have distance i. In particular, a Γ‐decomposition (Γ‐factorization) of K_v that is i‐perfect for any i not exceeding the diameter of a connected graph Γ will be said a Steiner (Kirkman) Γ‐system of order v. In this article we first observe that as a consequence of the deep theory on decompositions of edge‐colored graphs developed by Lamken and Wilson [Lamken and Wilson, 2000, J Combin Theory Ser A 89, 149–200], there are infinitely many values of v for which there exists an i‐perfect Γ‐decomposition of K_v provided that Γ is an i‐equidistance graph, namely a graph such that the number of pairs of vertices at distance i is equal to the number of its edges. Then we give some concrete direct constructions for elementary abelian Steiner Γ‐systems with Γ the wheel on 8 vertices or a circulant graph, and for elementary abelian 2‐perfect cube‐factorizations. We also present some recursive constructions and some results on 2‐transitive Kirkman Γ‐systems. © 2008 Wiley Periodicals, Inc. J Combin Designs 17: 197–209, 2009