Weight distribution of some reducible cyclic codes

Let q=p^m where p is an odd prime, m3, k1 and gcd(k,m)=1. Let Tr be the trace mapping from to and . In this paper we determine the value distribution of following two kinds of exponential sums

and

where is the canonical additive character of . As an application, we determine the weight distribution of the cyclic codes and over with parity-check polynomial h₂(x)h₃(x) and h₁(x)h₂(x)h₃(x), respectively, where h₁(x), h₂(x) and h₃(x) are the minimal polynomials of π⁻¹, π⁻² and π^−(pk+1) over , respectively, for a primitive element π of .     

基于改进二次多项式拟合的布里渊频移快速高精度提取算法

为了在保证测量准确性的基础上提高基于布里渊散射的光纤分布式传感的实时性,对布里渊频移的快速、高精度提高算法进行了研究。实现了基于二次多项式拟合的布里渊频移提取算法和典型的基于洛伦兹、高斯、伪Voigt和Voigt模型的算法,采用光时域反射计(BOTDR)实测了一段长光纤上的布里渊谱,采用以上算法提取了对应的布里渊频移。计算结果表明,二次多项式拟合算法的计算速度明显快于以上经典算法,其计算耗时仅分别为以上经典算法的1.15%,1.80%,1.51%和0.51%,但计算误差明显大于经典算法,影响了其实际应用。以上结果与对应数值产生布里渊谱的计算结果吻合。为了提高该算法的计算准确性,系统研究了扫频范围、扫频点数、信噪比、线宽和扫频范围偏差对基于二次多项式的布里渊频移提取准确性的影响。结果表明:当扫频点数固定时随扫频范围增加布里渊频移误差先减少到最小值后逐渐增加,扫频点数固定时最佳扫频范围为1个线宽;扫频范围不变时随扫频点数和信噪比的增加布里渊频移误差分别成幂和指数规律减少;扫频范围与线宽比值不变及扫频点数不变时随线宽增加布里渊频移误差线性增大;随扫频范围偏差增加误差逐渐增大,实际用于拟合的谱信号尽量围绕布里渊频移左右对称。根据以上研究结果提出了一种用于布里渊频移快速提取的改进二次多项式拟合算法,该算法从测量得到布里渊谱中截取1倍线宽且关于最大增益对称的谱信号用于后续拟合,较之经典的谱拟合算法,改进算法不仅能大幅提高计算速度且计算准确性与经典算法相似。采用数值产生及实测布里渊谱的计算结果验证了所提出算法的有效性。提出的算法不仅能有效提高基于布里渊散射的光纤分布式传感的实时性。     

Non-surjectivity of the Clifford invariant map

The question whether there exists a commutative ringA for which there is an element in the 2-torsion of the Brauer group not represented by a Clifford algebra was raised by Alex Hahn. Such an example is constructed in this paper and is arrived at using certain results of Parimala-Sridharan and Parimala-Scharlau which are also reviewed here. Dedicated to the memory of Professor K G Ramanathan     

Uniqueness of Limit Cycle for the Quadratic Systems with Weak Saddle and Focus

It is proved that the quadratic system with a weak saddle has at most one limit cycle, and that if this system has a separatrix cycle passing through the weak saddle, then the stability of the separatrix cycle is contrary to that of the singular point surrounded by it.     

Rank-1 codimension one singularities of positive quadratic differential forms

We complete the study of first-order structural stability at singular points of positive quadratic differencial forms on two manifolds. For this, we consider the generic 1-parameter bifurcation of a D₂₃-singular point. This situation consists in having, before the bifurcation, two locally stable singular points (one of type D₂ and the other of type D₃) which collapse at the D₂₃-singular point when the bifurcation parameter is reached, and afterwards disappear. In local (x,y)-coordinates, such a point appears at the origin of a planar differential equation of the form with (b²-ac)(x,y)?0, such that

(1)

the first jet of the map (a,b,c) at the origin is T₁(a,b,c)(0,0)=(y,0,-y) and

(2)

     

Ulam stability problem for generalized A-quadratic mappings

It is the purpose of this paper to investigate the stability problem of Ulam for an approximate mapping for the following generalized quadratic functional equation:

     

Asymptotics and Bounds for Multivariate Gaussian Tails

Let {X_n, n 1} be a sequence of centered Gaussian random vectors in , d 2. In this paper we obtain asymptotic expansions (n ) of the tail probability P{X_n >t_n} with t_n a threshold with at least one component tending to infinity. Upper and lower bounds for this tail probability and asymptotics of discrete boundary crossings of Brownian Bridge are further discussed.     

A simplicial decomposition algorithm for solving the variational inequality formulation of the general traffic assignment problem for large scale networks

Summary The class of simplicial decomposition methods has been shown to constitute efficient tools for the solution of the variational inequality formulation of the general traffic assignment problem. This paper presents a particular implementation of such an algorithm, with emphasis on its ability to solve large scale problems efficiently. The convergence of the algorithm is monitored by the primal gap function, which arises naturally in simplicial decomposition schemes. The gap function also serves as an instrument for maintaining a reasonable subproblem size, through its use in column dropping criteria. The small dimension and special structure of the subproblems also allows for the use of very efficient algorithms; several algorithms in the class of linearization methods are presented. When restricting the number of retained extremal flows in a simplicial decomposition scheme, the number of major iterations tends to increase. For large networks the shortest path calculations, leading to new extremal flow generation, require a large amount of the total computation time. A special study is therefore made in order to choose the most efficient extremal flow generation technique. Computational results on symmetric problems are presented for networks of some large cities, and on asymmetric problems for some of the networks used in the literature. Computational results for bimodal models of some large cities leading to asymmetric problems are also discussed.     

Some Diophantine equations of the form

Let be a prime. It is shown that each of the two Diophantine equations or has integral solutions.

     

A strongly polynomial algorithm for minimum convex separable quadratic cost flow problems on two-terminal series—parallel networks

We present strongly polynomial algorithms to find rational and integer flow vectors that minimize a convex separable quadratic cost function on two-terminal series—parallel graphs.