Geometric construction of association schemes from non-degenerate quadrics

LetF _q be a finite field withq elements, whereq is a power of an odd prime. In this paper, we assume that δ=0,1 or 2 and consider a projective spacePG(2ν+δ,F _q ), partitioned into an affine spaceAG(2ν+δ,F _q ) of dimension 2ν+δ and a hyperplaneℋ=PG(2ν+δ−1,F _q ) of dimension 2ν+δ−1 at infinity. The points of the hyperplaneℋ are next partitioned into three subsets. A pair of pointsa andb of the affine space is defined to belong to classi if the line meets the subseti of ℋ. Finally, we derive a family of three-class association schemes, and compute their parameters. This project is supported by the National Natural Science Foundation of China (No. 19571024).     

Characterization of properly optimal elements with variable ordering structures

In vector optimization with a variable ordering structure, the partial ordering defined by a convex cone is replaced by a whole family of convex cones, one associated with each element of the space. In recent publications, it was started to develop a comprehensive theory for these vector optimization problems. Thereby, also notions of proper efficiency were generalized to variable ordering structures. In this paper, we study the relation between several types of proper optimality. We give scalarization results based on new functionals defined by elements from the dual cones which allow complete characterizations also in the nonconvex case.     

Characterizations of wreath products of one-class association schemes

In this paper we show that the wreath product of one-class association schemes is characterized by the algebraic structure of its Bose-Mesner algebra.     

On higher indicators of regular association schemes

In the present paper, we will define the higher Frobenius–Schur indicators and the higher indicators of association schemes as a generalization of those of finite groups. The higher indicators of any association scheme are always positive rational numbers. Especially, for any positive integer $n$, the $n$th indicator of any regular association scheme is the number of relations such that its strong girth divides $n$. Thus, all higher indicators of any regular association scheme are natural numbers, and the sequence of the indicators is periodic. We will show that the converses of these facts are also true for finite exponent association schemes. Finally, we introduce a family of infinite exponent association schemes all higher indicators of which are natural numbers and the sequence of the indicators of which is periodic.     

On IP-graphs of association schemes and applications to group theory

Let G be a group acting transitively on a set X such that all subdegrees are finite. Isaacs and Praeger (1993) [5] studied the common divisor graph of (G,X). For a group G and its subgroup A, based on the results in Isaacs and Praeger (1993) [5], Kaplan (1997) [6] proved that if A is stable in G and the common divisor graph of (A,G) has two components, then G has a nice structure. Motivated by the notion of the common divisor graph of (G,X), Camina (2008) [3] introduced the concept of the IP-graph of a naturally valenced association scheme. The common divisor graph of (G,X) is the IP-graph of the association scheme arising from the action of G on X. Xu (2009) [8] studied the properties of the IP-graph of an arbitrary naturally valenced association scheme, and generalized the main results in Isaacs and Praeger (1993) [5] and Camina (2008) [3]. In this paper we first prove that if the IP-graph of a naturally valenced association scheme (X,S) is stable and has two components (not including the trivial component whose only vertex is 1), then S has a closed subset T such that the thin residue O?(T) and the quotient scheme (X/O?(T),S//O?(T)) have very nice properties. Then for an association scheme (X,S) and a closed subset T of S such that S//T is an association scheme on X/T, we study the relations between the closed subsets of S and those of S//T. Applying these results to schurian schemes and common divisor graphs of groups, we obtain the results of Kaplan [6] as direct consequences.     

变序结构局部弱非控点的二阶刻画

引进了一种二阶切导数,借助该切导数给出了变序结构集值优化问题取得局部弱非控点的二阶最优性必要条件.在某种特殊情况下,给出了一阶最优性条件.通过修正的Dubovitskij-Miljutin切锥导出的约束规格,给出了两个集值映射之和的二阶相依切导数的关系式,进一步得到目标函数与变锥函数的二阶相依切导数分开形式的最优性必要条件.     

Geometric construction of some families of two-class and three-class association schemes from nondegenerate quadrics in characteristic two

1.IntroductionThetheoryoflinearspacesintriteprojectivegeometryhasbeenusedbyseveralauthorsinconstructingBIBandPBIBdesigns.BoseI21firstusedthepropertiesofquadricsurfaCesinfiniteprojectivegeometryoftwoandthreedimensionsforconstr-netingexperimelltaldesigns.D.K.Ray-Chaudhurils]usedthegeometryofquadricstoconstructseveralseriesofPBIBdesignswithtwoassociateclasses.I.M.Chakravartila]usednondegenerateanddegenerateHebotianvarietiestoconstructsomefamiliesoftwo-classandthree-classassociationschemes…     

Optimality conditions for vector optimization problems with variable ordering structures

Our main concern in this article are concepts of nondominatedness w.r.t. a variable ordering structure introduced by Yu [P.L. Yu, Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), pp. 319–377]. Our studies are motivated by some recent applications e.g. in medical image registration. Restricting ourselves to the case when the values of a cone-valued map defining the ordering structure are Bishop–Phelps cones, we obtain for the first time scalarizing functionals for nondominated elements, Fermat rule, Lagrange multiplier rule and duality results for a single- or set-valued vector optimization problem with a variable ordering structure.     

Characters of table algebras and applications to association schemes

Hanaki [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195-201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363-369] generalized many properties of characters of finite groups to characters of association schemes. In this paper we show that many of these properties also hold for table algebras. Our approach is not to generalize the proofs in [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195-201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363-369] to table algebras, but to prove many stronger properties, and then obtain results in [A. Hanaki, Representations of association schemes and their factor schemes, Graphs Combin. 19 (2003) 195-201; A. Hanaki, Characters of association schemes and normal closed subsets, Graphs Combin. 19 (2003) 363-369] as direct consequences.     

Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure

In this paper, we define two new concepts of efficiency for vector optimization with variable ordering structure, namely the sharp and robust efficiencies, and we study their connections with classical concepts of efficiency in vector optimization. Then, we get necessary optimality conditions for them using Fréchet and Mordukhovich calculus coupled with the Gerstewitz’s (Tammer’s) scalarizing functional and openness results for set-valued maps.     

Generalized wreath products of table algebras determined by their character tables and applications to association schemes

Structures of table algebras whose character tables have a zero submatrix were studied in Blau and Xu (2014) and Chen and Xu (2018). In this paper we continue the research in this direction, and investigate the conditions under which the character table with a zero submatrix yields a generalized wreath product of table algebras. Applications to association schemes are also discussed.     

A coradiant based scalarization to characterize approximate solutions of vector optimization problems with variable ordering structures

This paper investigates some properties of approximate efficiency in variable ordering structures where the variable ordering structure is given by a special set valued map. We characterize $\epsilon$-minimal and $\epsilon$- nondominated elements as approximate solutions of a multiobjective optimization problem with a variable ordering structure and give necessary and sufficient conditions for these solutions, via scalarization.     

Classification of skew‐Hadamard matrices of order 32 and association schemes of order 31

Using a backtracking algorithm along with an essential change to the rows of representatives of known 13 710 027 equivalence classes of Hadamard matrices of order 32, we make an exhaustive computer search feasible and show that there are exactly 6662 inequivalent skew‐Hadamard matrices of order 32. Two skew‐Hadamard matrices are considered SH ‐equivalent if they are similar by a signed permutation matrix. We determine that there are precisely 7227 skew‐Hadamard matrices of order 32 up to SH ‐equivalence. This partly settles a problem posed by Kim and Solé. As a consequence, we provide the classification of association schemes of order 31.     

带有变量选择过程的分类模型误差分析

偏倚一方差分析方法是在模型选择过程中权衡模型对现有样本解释程度和未知样本估计准确度的分析方法,目的是使选定的模型检验误差尽量小.在分类或回归过程中进行有效的变量筛选可以获得更准确的模型表达,但也会因此带来一定误差.提出"选择误差"的概念,用于刻画带有变量选择的分类问题中由于变量的某种选择方法所引起的误差.将分类问题的误差分解为偏倚—方差—选择误差进行研究,考察偏倚、方差和选择误差对分类问题的总误差所产生的影响.     

一类变结构动态系统的非光滑最优性条件

研究了一类事件驱动的变结构动态系统的非光滑最优性条件. 通过引入一个新的时间变量, 将变结构动态系统的最优性问题转化为古典动态系统的最优性问题. 基于广义微分和古典动态系统的最优性理论, 得到了该系统的Frechet上微分形式的必要性条件, 推广了已有文献的相关结论. 结果表明, 在系统的连续运行过程中, 控制变量、协态变量和状态变量满足最小值原理和协态方程. 在系统的运行模型发生改变时, 协态变量产生一定的跳跃, 哈密尔顿函数连续. 最后通过一个算例说明了该结论的有效性.     

The partial adjustment valuation approach with dynamic and variable speeds of adjustment to evaluating and measuring the business value of information technology

In this paper we develop the partial adjustment valuation approach in which the speeds of (partial) adjustment are assumed to be dynamic and variable, rather than fixed or constant, to assessing the value of information technology (IT). The speeds of adjustment are a function of a set of macroeconomic and/or microeconomic variables, observed and unobserved and, hence, become time-varying or dynamic and variable over time. The approach is illustrated by a practical application. The results imply that the constant speeds of adjustment may overestimate or underestimate the actual speeds of adjustment and, accordingly, may miscalculate the values of performance metrics. Thus, the partial adjustment valuation approach with dynamic and variable speeds of adjustment is more realistic and, more importantly, captures the changing patterns and trends of the adjustment speeds and the performance measures as well. As such, the partial adjustment valuation approach with constant speeds of adjustment fails to adequately explain the dynamic production process of a decision making unit. The empirical evidence also conflicts with the lopsided view that the productivity paradox does not exist in developed countries.     

Exact solutions of nonlinear evolution equations with variable coefficients using exp-function method

In this paper, we apply the exp-function method to construct generalized solitary and periodic solutions of nonlinear evolution equations with variable coefficients. The proposed technique is tested on the Zakharov-Kuznetsov and (2+1)-dimensional Broer-Kaup equations with variable coefficients. These equations play a very important role in mathematical physics and engineering sciences. The suggested algorithm is quite efficient and is practically well suited for use in these problems. Obtained results clearly indicate the reliability and efficiency of the proposed exp-function method.     

Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions

In this paper, the uniform stabilization of the Cauchy-Ventcel problem with variable coefficients is considered, and the uniform energy decay rate for the problem is established by Riemannian geometry methods.     

Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions

In this paper, we consider a viscoelastic wave equation of variable coefficients in the presence of past history with nonlinear damping and delay in the internal feedback and dynamic boundary conditions. Under suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumption on the behavior of the relaxation function at infinity by Riemannian geometry method and Lyapunov functional method.     

Modelling and optimization of reservoirs for long-term regulation with variable head using functional analysis

In this paper, functional analysis and minimum norm formulation are applied to maximize the total benefits from two hydro reservoirs. The hydroelectric power generation is treated as a nonlinear function; water head variation and stochasticity of the river flows are included. The resulting problem has a nonlinear objective function and linear constraints. The proposed method is computationally efficient, compared to previous techniques. Numerical results are presented for widely different water conditions for an actual system in operation.This work was supported by the National Research Council of Canada, Grant No. A4146. The authors wish to thank B. C. Hydro for providing the reservoir data.