The Real Part of the Character Table of a Finite Group

In the ordinary character table of a finite group G, the values of the real valued irreducible characters on the real conjugacy classes form a sub-table which is square by Brauer's permutation lemma. We call this table the real part of the character table of G. Unlike the ordinary character table, viewed as a square matrix the real part of the character table is often singular. We present some results linking nonsingularity of this table to other properties of G.     

THE MAXIMAL NUMBER OF I_π-CHARACTERS OVER NORMAL SUBGROUPS

Let G be a π-separable group for a set of primes, let N be a normal subgroup of G, and let θ be an Iπ-character(i.e., irreducible π-partial character) of N. We obtain a necessary and suffcient condition for the number of Iπ-characters of G over θ to take the possible maximum |G : N |π. Some applications are given.     

Lower Bounds For Multidimensional Zero Sums

Let f(n, d) denote the least integer such that any choice of f(n, d) elements in contains a subset of size n whose sum is zero. Harborth proved that (n-1)2 ^d +1 f(n,d) (n-1)n ^d +1. The upper bound was improved by Alon and Dubiner to c _d n. It is known that f(n-1) = 2n-1 and Reiher proved that f(n-2) = 4n-3. Only for n = 3 it was known that f(n,d) > (n-1)2 ^d +1, so that it seemed possible that for a fixed dimension, but a sufficiently large prime p, the lower bound might determine the true value of f(p,d). In this note we show that this is not the case. In fact, for all odd n 3 and d 3 we show that .     

关于Wallis不等式的上界和下界

利用Wallis公式对两个已知的不等式进行了改进,相应地,我们得到了两个更加精细的结果.     

伪Smarandache函数的上下界

对于正整数n,设Z(n)=min{m|m∈N,1/2m(m+1)≡0(modn)},称为n的伪Smarandache函数.设r是正整数.根据广义Ramanujan-Nagell方程的结果,运用初等数论方法证明了下列结果:i)1/2(-1+(8n+1)≤Z(n)≤2n-1.ii)当r≠1,2,3或5时,Z(2~r+1)≥1/2(-1+(2~(r+3)·5+41)).iii)当r≠1,2,3,4或12时,Z(2~r-1)≥1/2(-1+(2~(r+3)·3-23).     

Upper Bounds on Character Sums with Rational Function Entries

We obtain formulac and estimates for character sums of the type S(x,f,p^m)=∑x=1^p^mx(f(f)),where p^m is a prime power with m≥2,x is a multiplicative character(mod p^m),and f=f1/f2 is a rational function over Z.In particular,if p is odd,d=deg(f1) deg(f2) and d^*=max(deg(f1),deg(f2)) then we obtain |S(x,f,p^m)|≤(d-1)p^m(1-1/d^*) for any non-constant f (mod p) and primitive character x.For p=2 an extra factor of 2√2 is needed.     

Groups with One or Two Super-Brauer Character Theories

A super-Brauer character theory of a group G and a prime p is a pair consisting of a partition of the irreducible p-Brauer characters and a partition of the p-regular elements of G that satisfy certain properties.We classify the groups and primes that have exactly one super-Brauer character theory.We discuss the groups with exactly two super-Brauer character theories.     

Sharp Upper and Lower Bounds for Basket Options

Given a basket option on two or more assets in a one‐period static hedging setting, the paper considers the problem of maximizing and minimizing the basket option price subject to the constraints of known option prices on the component stocks and consistency with forward prices and treat it as an optimization problem. Sharp upper bounds are derived for the general n‐asset case and sharp lower bounds for the two‐asset case, both in closed forms, of the price of the basket option. In the case n = 2 examples are given of discrete distributions attaining the bounds. Hedge ratios are also derived for optimal sub and super replicating portfolios consisting of the options on the individual underlying stocks and the stocks themselves.     

On Lower Bounds for the L 1-Norm of Exponential Sums

In the paper a method for finding lower bounds of the L ₁-norm of some exponential sums is described.     

关于A-H上下界的几个结果

关于算术平均和调和平均的差的估计,利用最值压缩定理,给出了其四个新的上下界.     

加倍测度空间中的热核估计

该文讨论了在加倍测度度量空间中,热核估计的上界与下界的关系.若相应的狄氏型满足局部性条件,则可以由近对角下界估计推出上对角估计.和此前这类问题的研究工作相比,该文给出了更加一般性的结果.     

Projective characters of the infinite generalized symmetric group

We consider the infinite generalized symmetric group S(∞)?? _m ^∞ , introduce its covering $\tilde B_m $ , and describe all indecomposable characters on the group $\tilde B_m $ .     

Optimal Upper and Lower Bounds for the Upper Tails of Compound Poisson Processes

A compound Poisson process is of the form where Z, Z ₁, Z ₂, are arbitrary i.i.d. random variables and N is an independent Poisson random variable with parameter . This paper identifies the degree of precision that can be achieved when using exponential bounds together with a single truncation to approximate . The truncation level introduced depends only on and Z and not on the overall exceedance level a.     

一类具有上下界的均衡问题

本文获得一类具有上、下界的抽象均衡问题解的存在性条件, 回答了Isac, Sehgal和Singh提出的一个公开问题,进一步研究了具上、下界及约束条件的抽象均 衡问题,获得了更为深刻的结果. 最后,把上述公开问题推广到两个空间的情形.     

Upper and Lower Bounds on the Dimension of Superspline Spaces

Upper and lower bounds are provided on the dimension of bivariate polynomial superspline spaces which are defined by enforcing smoothness conditions across the interior edges of the underlying triangulation. The results generalize known bounds for classical spline spaces. As an example of the usefulness of such bounds, we show how they can be applied to analyze a new macroelement.     

Endopermutation Scott modules,slash indecomposability and saturated fusion systems

This paper introduces the notion of Frobenius-friendly modules which is a generalisation of p-permutation modules and we obtain the slash functors for these modules. Generalisations of Scott modules and Brauer indecomposability, which are called endopermutation Scott modules and slash indecomposability respectively, are given by using the slash functors. We also describe slash indecomposability in terms of saturated fusion systems, which is a generalisation of the case of Brauer indecomposability.     

m-Reducibility with Upper and Lower Bounds for the Reducing Functions

We study pairs $\left( {\mathfrak{T}^1 ,\mathfrak{T}^0 } \right)$ of classes of nondecreasing total one-place arithmetic functions that specify reflexive and transitive binary relations $$\left\{ {\left( {A,B} \right)\left| {A,B \subseteq N\& \left( {\exists g.r,fh} \right)\left( {\exists f_1 \in \mathfrak{T}^0 } \right)\left[ {A \leqslant _m^h B\& f_0 \underline \triangleleft h\underline \triangleleft f_1 } \right]} \right.} \right\}$$ . (Here $k\underline \triangleleft {\kern 1pt} {\kern 1pt} {\kern 1pt} l$ means that the function l majorizes the function k almost everywhere.) Criteria for reflexivity and transitivity of such relations are established. Evidence of extensive branching of the arising system of bounded m-reducibilities is obtained. We construct examples of such reducibilities that essentially differ from the standard m-reducibility in the structure of systems of undecidability degrees that they generate and in the question of completeness of sets.     

关于非负矩阵Perron特征值的上、下界

本文通过构造一可逆矩阵,对一类非负矩阵A进行若干次简单的相似变换,便可同时得到矩阵A之Perron特征值的较好的上、下界.