RESEARCH ANNOUNCEMENTS：Morrey Regularity of the Solution to the Diri

To study the local regularity of solutions to second orderelliptic partial differential equations, Morrey in ［1］ introduced somefunction spaces, which are called the Morrey spaces today. Since then, manymathematicians have studied regularities of solutions to some kinds of secondorder elliptic equations in Morrey spaces.     

RESEARCH ANNOUNCEMENTS——Strong Discreteness of Subgroups of SL（2, Гn）

As in [1,6], let Гn be the n-dimensional Clifford group, i.e., the set of all elements in n-dimensional Clifford algebra An, which can be expressed as a finite product of non-zero-vertors and SL(2, Гn) denote the group of n-dimensional Clifford matrices A=(α b c d)which satisfy:     

RESEARCH ANNOUNCEMENTS Strong Discreteness of Subgroups of SL(2, Γ_n)

As in [1,6], let Г_n be the n-dimensional Clifford group, i.e., the set of all elements in ndimensional Clifford algebra An, which can be expressed as a finite product of non-zero-vectors and SL(2, Г_n) denote the group of n-dimensional Clifford matrices which satisfy:     

RESEARCH ANNOUNCEMENTS On a Generalization of the c-numerical Range

Let Cn×n be the set of　n×n complex matrices and (?)n　the set of　orthonormal　n-tuples ofvectors in Cn.For a vector c in Cn　and a matrix A in Cn×n,the c-numerical range of　A is theset     

RESEARCH ANNOUNCEMENTS──On the Maps of Deleted Product

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RESEARCH ANNOUNCEMENTS——The Construction of Eigenvalue Problem Equivalent to Multivariate Polynomial System and the Groebner Basis

In this paper we will show that one cau build up a joint eigenvalue problem eq-uivalent to the. given system. By this way, finding the solutions of the given systemis equivalent to finding all eigenvalues and eigenvectors of one matrix or matrix pen-cil. For the special case that the system has finite isolated solutions, we can obtainall solutions through computing the eigenvalues and eigenvectors of a matrix whichcan Le obtained by Gauss-Jordan elimination. Furthermore, we also find that one canget Groebner Basis for the ideal geuerated by the given system iu this way. For any polynomial f(x)∈K[x_1,x_2,…,x_n],f(x) can be written as     

The Greatest Lower Bound of a Boros–Moll Sequence

Mathematical Notes - The Boros–Moll polynomials $$P_m(a)$$ arise in the evaluation of a quartic integral. In the past few years, there has been some remarkable research on the properties of...     

On the Lower Bound of the Minimum Area of Convex Lattice Polygons

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RESEARCH ANNOUNCEMENTS On the Fractional Calculus of a Type of Weierstrass Functions

1 IntroductionIn recent years, fractals have shown important applications in many fields. [1,2] and [3] have done some excellent initial and conclusion work on fractal and it's mathematical foundations. However, a fractal function: a type of Weierstrass functions defined byW(t) = (?), 0 < a < 1, A > 1, (1)because of it's special fractal properties, [1, 2, 4, 5] have given some detailed discussion about it's graph, fractal dimension, etc.Fractional calculus attract more and more attention recently. In this field, [6, 7] have done some important and detailed work. The fundamental conception is     

A Lower Bound for the Differences of Powers of Linear Operators

Let T be a bounded linear operator in a Banach space, with σ（T）={1}. In 1983, Esterle-Berkani＇ s conjecture was proposed for the decay of differences （I - T） T^n as follows： Eitheror lim inf （n→∞（n＋1）｜｜（I-T）T^n｜｜≥1/e or T = I. We prove this claim and discuss some of its consequences.     

A Fréchet-Optimal Strengthening of the Dawson-Sankoff Lower Bound

This paper proposes a lower bound for the probability that at least one out of arbitrary events occurs. The information used consists of the first- and second- degree Bonferroni summations in conjunction with and , where is the probability that exactly one event occurs and is the probability that all events occur. We prove that the proposed bound is a Fréchet optimal lower bound, which is a criterion difficult to achieve in general. The two additional non-negative terms used in the proposed bound make it at least as good as the Dawson–Sankoff lower bound, a Fréchet optimal degree two lower bound using the first- and second- degree Bonferroni summations only. A numerical example is presented to illustrate that in some cases, the improvement can be substantial. *Part of the work of this author was done as Visiting Scholar, 2004, at the University of Sydney.     

RESEARCH ANNOUNCEMENTS：On the Fractional Calculus of a Type of Weierstrass Functions

I Introduction In recent years, fractals have shown important applications in many fields. [1, 2] and [3] havedone some excellent initial and conclusion work on fractal and it's mathematical foundations.However, a fractal function: a type of Weierstrass functions defined bybecause of it's special fractal properties, [1,2, 4, 5] have given some detailed discussion about it'sgraph, fractal dimension, etc.     

RESEARCH ANNOUNCEMENTS—On a Generalization of the c—numerical Range

Let Cn×n be the set of n × n complex matrices and An the set of orthonormal n-tuples of vectors in Cn. For a vector c in Cn and a matrix A in Cn×n, the c-numerical range of A is the set Wc(A)={n∑i=1 Ci(Axi,xi):(x1,…xn)∈∧n} When c = (1,0,…,0), Wc(A) is reduced to the classical numerical range W(A) (see [1]). For the classical numerical range and its generalizations, one may see the survey article[2].     

RESEARCH ANNOUNCEMENTS The C~1-Admissible Approximation for Lipschitz Fun-ctions and the Hamiltonian Inclusions

In this paper,C~(1-0)(D,R)denotes the set of the locally Lipschitz continuousfunctions from D to R,For f∈C~(1-0)(D,R)and x∈D,f(x)denotes the Clarke’sgeneralized gradient of f at x. Definition 1 Let D be an open subset of R~n and f∈C~(1-0)(D,R).Suppose thatε:D→R is a positive function.If a function g∈C~1(D,R)and satisfies     

RESEARCH ANNOUNCEMENTS The New Necessary and Sufficient Conditions for the Uniformly Continuous and Continuous of Duality Mappings

In[1],Ma shaoqin gave the necessary and sufficient condition for the uniformlycontinuous and continuous of duality mappings.In the note,We give very simplecriteria for the uniformly continuous and continuous of duality mappings.And by[1],the necessary and sufficient conditions for the norm of a real Banach Space to beuniformly Frechet differentiable and Frechet differentiable are described.     

RESEARCH ANNOUNCEMENTS The Comparison Theorem for the Bergman and Kobayashi Metrics on a Class of Pseudoconvex Domains

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RESEARCH ANNOUNCEMENTS Einstein-Khler Metric on Cartan-Hartogs Domain of the First Type

S. Y. Cheng and S. T. Yau showed in [CY] that any C2 bounded pseudoconvex domain in C?has a complete Einstein-Kahler metric with constant negative Ricci curvature. N. Mok and S. T. Yau[MY] have extended this result to arbitrary bounded pseudoconvex domain in Cn. Complete Einstein-Kahler metric with Explicit form, however, is only known in the case of homogeneous domain.     

RESEARCH ANNOUNCEMENTS On the Least Pair of Quadratic Non-Residues (mod p)

For each odd prime p,let a(p) denote the least positive residue class n(mod p),so that both n and n+1 are quadratic non-residues.The previously best knownestimate for a(p) obtained by Elliott is as follows     

RESEARCH ANNOUNCEMENTS Large Deviations for the Boundary Crossing Probabilities of Some Random Fields

Let,1)P拜denote the Probability】nea8Urewhiehmakes YI,YZ,…1 .1 .d.=歹)·Forlo,0<亡。<忿:<1。Then as”乞~。。p,,‘r毛尹,‘1,一誉“、(”,“{邓z(‘石,》士”l(‘全l一l)士一l名1一t- X2)(一。:)[·(·+丁)l’“·2.、︸where沪(x)=(2二)一士e…