<Emphasis Type="Italic">W</Emphasis><Superscript>1,<Emphasis Type="Italic">p</Emphasis></Superscript>(R<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)-boundedness for maximal functions

In this paper, we get W ^1,p (R ⁿ )-boundedness for tangential maximal function and nontangential maximal function, which improves J.Kinnunen, P.Lindqvist and Tananka’s results. Supported by the key Academic Discipline of Zhejiang Province of China under Grant No.2005 and the Zhejiang Provincial Natural Science Foundation of China.     

On <Emphasis Type="Italic">U</Emphasis>-orthodox semigroups

As a generalization of an orthodox semigroup in the class of regular semigroups, a type W semigroup was first investigated by El-Qallali and Fountain. As an analogy of the type W semigroups in the class of abundant semigroups, we introduce the U-orthodox semigroups. It is shown that the maximum congruence μ contained in on U-orthodox semigroups can be determined. As a consequence, we show that a U-orthodox semigroup S can be expressed by the spined product of a Hall semigroup W _U and a V-ample semigroup (T,V). This theorem not only generalizes a known result of Hall-Yamada for orthodox semigroups but also generalizes another known result of El-Qallali and Fountain for type W semigroups. This work was supported by National Natural Science Foundation of China (Grant No. 10671151) and Natural Science Foundation of Shaanxi Province (Grant No. SJ08A06), and partially by UGC (HK) (Grant No. 2160123)     

A new proof of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the parabolic polyharmonic equations

In this paper we obtain local L ^p estimates for the parabolic polyharmonic equations by a straightforward approach. Yao was supported by the Innovation Foundation of Shanghai University (Grant No. A10-0101-08-905), Shanghai Leading Academic Discipline Project (Grant No. J50101) and Key Disciplines of Shanghai Municipality (Grant No. S30104). Zhou was supported by the National Basic Research Program of China (Grant No. 2006CB705700), National Natural Science Foundation of China (Grant No. 60532080), and the Key Project of Chinese Ministry of Education (Grant No. 306017)     

Tracial Limit of C^＊-algebras

A new limit of C^*-algebras, the tracial limit, is introduced in this paper. We show that a separable simple C^*-algebra A is a tracial limit of C^*-algebras in Z-(k) if and only if A has tracial topological rank no more than k. We present several known results using the notion of tracial limits.     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-mixed intersection bodies

In this paper the author first introduce a new concept of L _p -dual mixed volumes of star bodies which extends the classical dual mixed volumes. Moreover, we extend the notions of L _p intersection body to L _p -mixed intersection body. Inequalities for L _p -dual mixed volumes of L _p -mixed intersection bodies are established and the results established here provide new estimates for these type of inequalities. This work was supported by the Natural Science Foundation of Zhejiang Province of China (Grant No. Y605065) and the Foundation of the Education Department of Zhejiang Province of China (Grant No. 20050392)     

<Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> discrepancy of generalized two-dimensional Hammersley point sets

We determine the L _p discrepancy of the two-dimensional Hammersley point set in base b. These formulas show that the L _p discrepancy of the Hammersley point set is not of best possible order with respect to the general (best possible) lower bound on L _p discrepancies due to Roth and Schmidt. To overcome this disadvantage we introduce permutations in the construction of the Hammersley point set and show that there always exist permutations such that the L _p discrepancy of the generalized Hammersley point set is of best possible order. For the L ₂ discrepancy such permutations are given explicitly. F.P. is supported by the Austrian Science Foundation (FWF), Project S9609, that is part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.     

Minimal Non—Commutative n—Insertive Rings

Let n be an integer with |n| > 1. If p is the smallest prime factor of |n|, we prove that a minimal non-commutative n-insertive ring contains n ⁴ elements and these rings have five (2p+4) isomorphic classes for p = 2 (p ≠ 2). This research is supported by the National Natural Science Foundation of China, and the Scientific Research Foundation for “Bai-Qian-Wan” Project, Fujian Province of China     

Modulus hyperinvariant subspaces for quasinilpotent operators at a non-zero positive vector on <Emphasis Type="Italic">l</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

In 1993, Y. A. Abramovich, C. D. Aliprantis and O. Burkinshaw showed that every continuous operator with modulus on an l_p-space (1 ≤ p < ∞) whose modulus commutes with a non-zero positive operator T on l_p that is quasinilpotent at a non-zero positive vector x₀ has a non-trivial invariant closed subspace. In this paper, it is proved that if is a collection of continuous operators with moduli on l_p that is finitely modulus-quasinilpotent at a non-zero positive vector x ₀ then and its right modulus sub-commutant have a common non-trivial invariant closed subspace. In particular, all continuous operators with moduli on l _p whose moduli commute with a non-zero positive operator I on l _p that is quasinilpotent at a non-zero positive vector x ₀ have a common non-trivial invariant closed subspace, so that all positive operators on l _p which commute with a non-zero positive operator S on l _p that is quasinilpotent at a non-zero positive vector x ₀ have a common non-trivial invariant closed subspace. This research was supported by the Natural Science Foundation of Hunan Province of P. R. China (04JJ6004), the Foundation of Education Department of Hunan Province of P. R. China (04C002) and the Natural Science Foundation of P. R. China (10671147). Received: 4 December 2005 Revised: 19 June 2006     

Zero-density estimates for automorphic <Emphasis Type="Italic">L</Emphasis>-functions

In this paper we prove zero-density estimates of the large sieve type for the automorphic L-function L（s, f × χ）, where f is a holomorphic cusp form and χ（mod q） is a primitive character.     

A Property of <Emphasis Type="Italic">g</Emphasis>-Expectation

This paper proves that, under the hypothesis g(t, 0, 0) ≡ 0 and some natural assumptions, the generator g of a backward stochastic differential equation can be uniquely determined by the corresponding g-expectations with all terminal conditions. The main result of this paper also confirms and extends Peng Shige‘s conjecture.     

Small Hankel Operators on the Dirichlet-Type Spaces on the Unit Ball of <Emphasis Type="Bold">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper the small Hankel operators on the Dirichlet-type spaces Dp on the unit ball of C^n are considered. A similar result to that of the one-dimensional setting is given, which characterizes the boundedness of the small Hankel operators on Dp.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> approximation capability of RBF neural networks

L ^p approximation capability of radial basis function (RBF) neural networks is investigated. If g: R ₊¹ → R ¹ and ∈ L _loc ^p (R ⁿ ) with 1 ≤ p < ∞, then the RBF neural networks with g as the activation function can approximate any given function in L ^p (K) with any accuracy for any compact set K in R ⁿ , if and only if g(x) is not an even polynomial. Partly supported by the National Natural Science Foundation of China (10471017)     

A Converse Comparison Theorem for <Emphasis Type="Italic">g</Emphasis>-Expectations

With the help of a limit property of solutions of backward stochastic differential equations(BSDEs),this paper establishes a converse comparison theorem for deterministic generators g of BSDEs under the assumption g(t, y, 0) ≡0.     

On <Emphasis Type="Italic">K</Emphasis>-groups of operator algebra on the 1-shift space

In this paper we discuss the K-groups of Wiener algebra ;W. For the 1-shift space XGM2,We obtain a characterization of Fredholm operators on X^nGM2 for all n ∈ N. We also calculate the K-groups of operator algebra on the 1-shift space XGM2.     

A Relationship Between the Conditional <Emphasis Type="Italic">g</Emphasis>-Evaluation System and the Generator <Emphasis Type="Italic">g</Emphasis> and Its Applications

In this paper, under the most elementary conditions on a backward stochastic differential equation （BSDE for short） introduced by Peng, a new relationship between the conditional g-evaluation system and the generator g of BSDE is obtained in the sense of ＂process＂, based on some recent results of Jiang. Moreover, as applications, two converse comparison theorems and two uniqueness theorems on the generators of BSDEs are proved.     

Generation and Propagation of Interfaces for <Emphasis Type="Italic">p</Emphasis>-Laplacian Equations

In this paper, we consider the generation and propagation of interfaces for p-Laplacian equations with the derivative of a bi-stable potential.     

On the McKay quivers and <Emphasis Type="Italic">m</Emphasis>-Cartan matrices

In this paper, we introduce the m-Cartan matrix and observe that some properties of the quadratic form associated to the Cartan matrix of an Euclidean diagram can be generalized to the m-Cartan matrix of a McKay quiver. We also describe the McKay quiver for a finite abelian subgroup of a special linear group. This work was supported by National Natural Science Foundation of China (Grant No. 10671061) and the Research Foundation for Doctor Programme (Grant No. 200505042004)     

On <Emphasis Type="Italic">S</Emphasis>-quasinormal and <Emphasis Type="Italic">c</Emphasis>-normal subgroups of a finite group

Let be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G. Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No. 0249001). Corresponding author. Supported in part by the Natural Science Foundation of China (10571181), NSF of Guangdong Province (06023728) and ARF(GDEI).     

The Cauchy Boundary Value Problems on Closed Piecewise Smooth Manifolds in <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Suppose that D is a bounded domain with a piecewise C^1 smooth boundary in C^n. Let ψ∈C^1 α(δD). By using the Hadamard principal value of the higher order singular integral and solid angle coefficient method of points on the boundary, we give the Plemelj formula of the higher order singular integral with the Boehner-Martinelli kernel, which has integral density ψ. Moreover, by means of the Plemelj formula and methods of complex partial differential equations, we discuss the corresponding Cauehy boundary value problem with the Boehner-Martinelli kernel on a closed piecewise smooth manifold and obtain its unique branch complex harmonic solution.     

Congruences Concerning Values of the Modular Functions <Emphasis Type="Italic">j</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The j-function j(z) = q⁻¹+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j_m(ζ) = (j(ζ) – 744)∨T₀(m), where T₀(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL₂(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function. 2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30 The author thanks the National Science Foundation for their generous support.