改进的R.Nevanlinna第二基本定理

设ｆ（ｚ）是超越亚纯函数，ｎ为任一非负整数，ε为任意正数，ａ＾（ｉ）ｒ＝（ｖ＝１，２，…，ｑｉ，ｉ＝０，１，…，ｎ）均为有穷非零复数，满足ｍｉｎ／１≤ｖ１＜ｖ２≤ｑｉ｜ａ＾（ｉ）ｖ１－ａ＾（ｉ）ｖ２｜≥δ＞０，ｉ＝０，１，…，ｎ。若ｆ（０）≠０，∞，П＾ｎｉ＝１ｆ（ｉ）（０）≠０，则对０＜ｒ＜＋∞有ｍ（ｒ，ｆ）＋ｍ（ｒ，１／ｆ）＋Σ＾ｕｉ＝０Σ＾ｑｉｒ＝１ｍ（ｒ，ａ＾（ｉ）ｖ，ｆ＾（ｉ）≤（２＋ε     

图Kn—E（k0P3（∪↑r↓i=1kiPqi—1））的色唯一性

记δｎ＝Σ↓ｋ≤ｎ（＾ｋｎ－ｋ），在本文中证明了：Ａ↓ｒ∈Ｎ，若Ａ↓∈Ｎ，若Ａ↓∈｛１，２，…，ｒ｝，ｑｉ（〉５）都是素数，并且［（δｑｉ－１－１）！＋１］／δｑｉ－１是正整数，则图簇Ｋｎ－Ｅ（ｋ０Ｐ３∪ｋ１Ｐｑ１－１∪…∪ｋｒＰｑｒ－１）是色唯一的，推广了文［１］的结果。     

一组互相关联的不等式命题

大家知道，由ｎ元均值不等式可方便地得到如下一个不等式：设ａｉ∈Ｒ＋（ｉ＝１，２，…，ｎ，ｎ≥２），则∑ｎｉ＝１ａｉ∑ｎｉ＝１１ａｉ≥ｎ２；（１）不等式（１）相当有用，对它作适当代换，可引出一组互相关联的不等式命题；首先，对（１）作代换（Ｓ－ａ１，Ｓ－ａ２，…，Ｓ－ａｎ）→（ａ１，ａ２，…，ａｎ），其中Ｓ＝∑ｎｉ＝１ａｉ，得命题１　设ａｉ∈Ｒ＋（ｉ＝１，２，…，ｎ，ｎ≥２），∑ｎｉ＝１ａｉ＝Ｓ，则∑ｎｉ＝１１Ｓ－ａｉ≥ｎ２（ｎ－１）Ｓ　；（２）证明　由（１），∑ｎｉ＝１（Ｓ－ａｉ）∑ｎｉ＝１１Ｓ－…     

变换图的直径及Brualdi猜想

设Ｒ＝（ｒ１，ｒ２…ｒｍ）及 Ｓ＝（Ｓ１，Ｓ２，…，Ｓｎ）为两个正整数向量，满足Σｍｉ＝１ｒｉ＝Σｎｊ＝１ｓｊ＝ Ｋ．记Ｇ（Ｒ，Ｓ）为（０，１）－矩阵类 Ｕ（Ｒ，Ｓ）的变换图．Ｂｒｕａｌｄｉ在文山中给出了 Ｇ（Ｒ，Ｓ）的直径厂（Ｇ（Ｒ，Ｓ））的一个上界：ｍｎ／２－１，并猜想Ｄ（Ｇ（Ｒ，Ｓ））≤ｍｎ／４．本文通过对有向图围长的研究得到了Ｄ（Ｇ（Ｒ，Ｓ））的一个新的上界：１／２ｍｎ－１／６ｔ（ｔ－１）（４ｔ＋１），其中Ｔ＝　　．     

关于МОФОНЪКО不等式的精确性

证明了如下定理：设Φ（ｚ）＝Σ↑ｎ↓ｕ＝１∏↑ｔ↓ｉ＝１ｆｉ＾ａ＾ｕｉ０…（ｆｉ＾（ｋｉ））＾ａｉｋｉ＾ｕ／Σ↑ｍ↓ｖ＝１∏↑ｔ↓ｉ＝１ｆｉ＾ｂ＾ｖｉ０…（ｆｉ＾（ｋｉ））＾ｂｉｋｉ＾ｖ其中ｆｉ（ｚ），（１≤ｉ≤ｔ）是亚纯函数，ａｉｊ＾ｕ，ｂｉｊ＾ｖ为非负整数，则有Ｔ（ｒ，Φ）≤Σ↑ｔ↓ｉ＝１｛［Ｓｉ＋ｏ（１）］ｍ（ｒ，ｆｉ）＋［△ｉ＝ｕｉ＋ｏ（１）］Ｎ（ｒ，ｆｉ）＋（△ｉ－Ｓｉ）Ｎ↑－（ｒ，ｆｉ     

截断和的弱大数律

投｛Ｘｎ，ｎ≥１｝ｉ．ｉ．ｄ．，Ｘｎ，１≤Ｘｎ，２≤…≤Ｘｎ，ｎ是Ｘ１，Ｘ２，…，Ｘｎ的次序统计量．对非负整数ｋ，ｒ，ｋ＋ｒ≤ｎ，令．本文研究当ｋ＝ｋｎ，ｒ＝ｒｎ满足ｍｉｎ（ｋ，ｒ）→∞，ｍａｘ（ｋ，ｒ）→０时截断和Ｓｎ（ｋ，ｒ）的弱大数律．设βｎ＞０，Ｃｎ∈Ｒ，文中给出了依概率收敛的充要条件．     

图K_n-E(k_0P_3(∪ri=1k_iP(q_i-1)))的色唯一性

记δｎ＝ｋ≤ｎｋｎ－ｋ，在本文中证明了：ｒ∈Ｎ，若ｉ∈｛１，２，…，ｒ｝，ｑｉ（＞５）都是素数，并且［（δｑｉ－１－１）！＋１］／δｑｉ－１是正整数，则图簇Ｋｎ－Ｅｋ０Ｐ３∪ｋ１Ｐｑ１－１∪ｋ２Ｐｑ２－１∪…∪ｋｒＰｑｒ－１是色唯一的，推广了文［１］的结果     

数学奥林匹克讲与练(19)

例题讲解１４５．给定实数ａ１，…，ａｍ；ｂ１，…，ｂｎ及正数ｐ１，…，ｐｍ；ｑ１，…，ｑｎ，组成一个ｍ×ｎ的数表，其位于第ｉ行、第ｊ列的一个数为ａｉ＋ｂｊｐｉ＋ｑｊ（ｉ＝１，２，…，ｍ；ｊ＝１，２，…，ｎ）．求证：在此数表中存在一个数，它不小于与其同行的任意一个数，又不大于与其同列的任意一个数．证法１　我们要在ｍ×ｎ的数表中找出一数ｘ＝ａｋ＋ｂｌｐｋ＋ｑｌ，满足条件ａｋ＋ｂｊｐｋ＋ｑｊ≤ｘ　（１≤ｊ≤ｎ）ａｉ＋ｂｌｐｉ＋ｂｌ≥ｘ　（１≤ｉ≤ｍ）（１）将两式变形为ｐｋｘ－ａｋ≥－ｑｊｘ＋…     

贪婪t－相交有限序列族

设Ｆ为有限序列族，对ａ＝（ａ１，ａ２，…，ａｎ）∈Ｆ，ａｉ为整数且０≤ａｉ≤ｓｉ（整数），记ｓ（ａ）＝｛ｊ｜１≤ｊ≤ｎ，ａｊ＞０｝，ｓ（Ｆ）＝｛ｓ（ａ）｜ａ∈Ｆ｝，及Ａ｛１，２，…，ｎ｝时Ｗ（Ａ）＝Пｉ∈Ａｓｉ．称Ｆ为贪婪ｔ－相交，如对任何ａ，ｂ∈Ｆ，至少有ｔ个ａｉ，ｂｉ＞０，且Ｗ（Ａ）≥Ｗ（（｛１，２，…，ｎ｝－Ａ）＋Ｂ）对任何Ａ∈Ｓ（Ｆ）及ＢＡ（｜Ｂ｜＝ｔ－１）成立．本文得到当ｓ１＞ｓ２＞…＞ｓｎ时的最大贪婪ｔ－相交有限序列族．     

厄米特矩阵的迹的几点性质

应用矩阵Ａ＝（ａｉｊ）∈Ｃｎ×ｎ的弗罗伯尼范数ＡＦ和谱范数ＡＳ，研究厄米特矩阵的迹的性质，得到几个结论：Ｔｒ（ＡＢ）＝∑ｎｉ＝１λｉ∑ｎｊ＝１ｔｉｊμｊ（λｉ，μｊ分别为Ａ，Ｂ的特征值，０≤ｔｉｊ≤１，且∑ｎｉ＝１ｔｉｊ＝１，ｊ＝１，２，…，ｎ）；Ｔｒ（ＡＢ）≤Ｔｒ（Ａ）ＢＳ；Ｔｒ（ＡＢ）Ｈ（ＡＢ）］≤Ｔｒ（ＡＨＡ）［ｍａｘ１≤ｉ≤ｎλｉ］２（λｉ是Ｂ的特征值）等．     

On a class of self-similar processes with stationary increments in higher order Wiener chaoses

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local Hölder regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener–Itô integrals.     

A REPRESENTATION FORMULA RELATED TO SCHR(O)DINGER OPERATORS

Schr(o)dinger operator is a central subject in the mathematical study of quantum mechanics.Consider the Schrodinger operator H = -△ V on R, where △ = d2/dx2 and the potential function V is real valued. In Fourier analysis, it is well-known that a square integrable function admits an expansion with exponentials as eigenfunctions of -△. A natural conjecture is that an L2 function admits a similar expansion in terms of "eigenfunctions" of H, a perturbation of the Laplacian (see [7], Ch. Ⅺ and the notes), under certain condition on V.     

On a class of Riemann surfaces

It is considered the class of Riemann surfaces with dimT1 = 0, where T1 is a subclass of exact harmonic forms which is one of the factors in the orthogonal decomposition of the spaceΩH of harmonic forms of the surface, namely The surfaces in the class OHD and the class of planar surfaces satisfy dimT1 = 0. A.Pfluger posed the question whether there might exist other surfaces outside those two classes. Here it is shown that in the case of finite genus g, we should look for a surface S with dimT1 = 0 among the surfaces of the form Sg\K , where Sg is a closed surface of genus g and K a compact set of positive harmonic measure with perfect components and very irregular boundary.     

CONTENTS OF VOLUME 30

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Instructions for contributions

正Applied Mathematics-A Journal of Chinese Universities,Series B(Appl.Math.J.Chinese Univ.,Ser.B)is a comprehensive applied mathematics journal jointly sponsored by Zhejiang University,China Society for Industrial and Applied Mathematics,and Springer-Verlag.It is a quarterly journal with     

Journal of Mathematical Research with Applications Guide for Authors

正Journal overview:Journal of Mathematical Research with Applications(JMRA),formerly Journal of Mathematical Research and Exposition(JMRE)created in 1981,one of the transactions of China Society for Industrial and Applied Mathematics,is a home for original research papers of the highest quality in all areas of mathematics with applications.The target audience comprises:pure and applied mathematicians,graduate students in broad fields of sciences and technology,scientists and engineers interested in mathematics.     

Staircase transportation problems with superadditive rewards and cumulative capacities

A cumulative-capacitated transportation problem is studied. The supply nodes and demand nodes are each chains. Shipments from a supply node to a demand node are possible only if the pair lies in a sublattice, or equivalently, in a staircase disjoint union of rectangles, of the product of the two chains. There are (lattice) superadditive upper bounds on the cumulative flows in all leading subrectangles of each rectangle. It is shown that there is a greatest cumulative flow formed by the natural generalization of the South-West Corner Rule that respects cumulative-flow capacities; it has maximum reward when the rewards are (lattice) superadditive; it is integer if the supplies, demands and capacities are integer; and it can be calculated myopically in linear time. The result is specialized to earlier work of Hoeffding (1940), Fréchet (1951), Lorentz (1953), Hoffman (1963) and Barnes and Hoffman (1985). Applications are given to extreme constrained bivariate distributions, optimal distribution with limited one-way product substitution and, generalizing results of Derman and Klein (1958), optimal sales with age-dependent rewards and capacities.To our friend, Philip Wolfe, with admiration and affection, on the occasion of his 65th birthday.Research was supported respectively by the IBM T.J. Watson and IBM Almaden Research Centers and is a minor revision of the IBM Research Report [6].