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).     

Transcendental values of the digamma function

Let ψ(x) denote the digamma function, that is, the logarithmic derivative of Euler's Γ-function. Let q be a positive integer greater than 1 and γ denote Euler's constant. We show that all the numbers

     

Transcendental submanifolds of Rn

In this paper we show how the restriction of the complex algebraic cycles to real part of a complex algebraic set is related to the real algebraic cycles of the real part. As a corollary we give examples of smooth submanifolds of a Euclidean space which can not be isotoped to real parts of complex nonsingular subvarieties in the corresponding projective space. Dedicated to memory of Mario Raimondo     

Preface of the Special Issue Error Bounds,Regularity, and Nonsmooth Optimization

Set-Valued and Variational Analysis -     

Transcendental divisors and their critical functions

 In this paper we try to generalize the notion of a minimal polynomial of an algebraic number to a class of transcendental elements from where is the completion of the algebraic closure of ℚ in ℂ, relative to the spectral norm on : ([PPP], [PPZ1], [PPZ2], [PPZ3]). Received: 2 July 2002 / Revised version: 13 January 2003 Published online: 20 March 2003 Mathematics Subject Classification (2000): 11R99     

Transcendental solutions of diophantine equations

Orlov State Pedagogical Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 22, No. 4, pp. 78–79, October–December, 1988.     

On Nondeterminism,Enumeration Reducibility and Polynomial Bounds

Enumeration reducibility is a notion of relative computability between sets of natural numbers where only positive information about the sets is used or produced. Extending e-reducibility to partial functions characterises relative computability between partial functions. We define a polynomial time enumeration reducibility that retains the character of enumeration reducibility and show that it is equivalent to conjunctive non-deterministic polynomial time reducibility. We define the polynomial time e-degrees as the equivalence classes under this reducibility and investigate their structure on the recursive sets, showing in particular that the pe-degrees of the computable sets are dense and do not form a lattice, but that minimal pairs exist. We define a jump operator and use it to produce a characterisation of the polynomial hierarchy.     

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.     

Affine Density,Frame Bounds,and the Admissibility Condition for Wavelet Frames

For a large class of irregular wavelet frames we derive a fundamental relationship between the affine density of the set of indices, the frame bounds, and the admissibility constant of the wavelet. Several implications of this theorem are studied. For instance, this result reveals one reason why wavelet systems do not display a Nyquist phenomenon analogous to Gabor systems, a question asked in Daubechies' Ten Lectures book. It also implies that the affine density of the set of indices associated with a tight wavelet frame has to be uniform. Finally, we show that affine density conditions can even be used to characterize the existence of wavelet frames, thus serving, in particular, as sufficient conditions.     

Heat Bounds and the Blowtorch Theorem

We study driven systems with possible population inversion and we give optimal bounds on the relative occupations in terms of released heat. A precise meaning to Landauer’s blowtorch theorem (Phys Rev A 12:636–638, 1975) is obtained stating that nonequilibrium occupations are essentially modified by kinetic effects. Towards very low temperatures we apply a Freidlin-Wentzel type analysis for continuous time Markov jump processes. It leads to a definition of dominant states in terms of both heat and escape rates.     

数学船型中的超越方程组求解

数学船型是船舶型线的数学表示,在缺乏原型船的条件下可自动生成船舶型线,从而实现数字化设计,满足总体概要设计与虚拟采办的要求.数学船型归为一类特殊的超越方程组.利用分离变量与分层搜索技术,求解超越方程组,并以椭圆形潜艇的型线设计为例,验证了方法的有效性.     

Transcendental division algebras and simple Noetherian rings

In this paper we prove the following theorem: Let D be a division ring with center the field k, and let k (x ₁, …, x_n) denote the rational function field in n variables over k. If D contains a maximal subfield which has transcendence degree at least n over k, then D ⊗_k k (x₁, …, x_n) is a simple Noetherian domain of Krull and global dimensions n. Rather surprisingly, the preceding result can be used to determine the maximum transcendence degrees of the commutative subalgebras of several classically studied division rings. Using the theorem we prove, for example, that in the division ring of quotients of the Weyl algebra,A _n, every maximal subfield has transcendence degree at mostn over the center.     

复合函数-差分方程的亚纯解

利用Nevanlinna值分布理论,研究了两类复合函数-差分方程超越解的性质,超越解的特征函数,非积分计数函数及其存在性,并相应地给出估计.     

超越方程的优化解法

针对粒子群算法局部搜索能力差,后期收敛速度慢等缺点,提出了一种改进的粒子群算法,该算法是在粒子群算法后期加入拟牛顿方法,充分发挥了粒子群算法的全局搜索性和拟牛顿法的局部精细搜索性,从而克服了粒子群算法的不足,把超越方程转化为函数优化的问题,利用该算法求解,数值实验结果表明,算法有较高的收敛速度和求解精度。