A method for efficiently computing the number of codewords of fixed weights in linear codes

The problem of computing the number of codewords of weights not exceeding a given integer in linear codes over a finite field is considered. An efficient method for solving this problem is proposed and discussed in detail. It builds and uses a sequence of different generator matrices, as many as possible, so that the identity matrix takes disjoint places in them. The efficiency of the method is achieved by optimizations in three main directions: (1) the number of the generated codewords, (2) the check whether a given codeword is generated more than once, and (3) the operations for generating and computing these codewords. Since the considered problem generalizes the well-known problems “Weight Distribution” and “Minimum Distance”, their efficient solutions are considered as applications of the algorithms from the method.     

A variant of the Reynolds operator

Let be a linearly reductive group over a field , and let be a -algebra with a rational action of . Given rational --modules and , we define for the induced -action on Hom a generalized Reynolds operator, which exists even if the action on Hom is not rational. Given an -module homomorphism , it produces, in a natural way, an -module homomorphism which is -equivariant. We use this generalized Reynolds operator to study properties of rational - modules. In particular, we prove that if is invariantly generated (i.e. ), then is a projective (resp. flat) -module provided that is a projective (resp. flat) -module. We also give a criterion whether an -projective (or -flat) rational --module is extended from an -module.

     

A Characterization of the Riesz Probability Distribution

Bobecka and Wesolowski (Studia Math. 152:147–160, [2002]) have shown that, in the Olkin and Rubin characterization of the Wishart distribution (see Casalis and Letac in Ann. Stat. 24:763–786, [1996]), when we use the division algorithm defined by the quadratic representation and replace the property of invariance by the existence of twice differentiable densities, we still have a characterization of the Wishart distribution. In the present work, we show that when we use the division algorithm defined by the Cholesky decomposition, we get a characterization of the Riesz distribution.     

Complex variables in junior high school: the role and potential impact of an outreach mathematician

Outreach mathematicians are college faculty who are trainedin mathematics but who undertake an active role in improvingprimary and secondary education. This role is examined througha study where an outreach mathematician introduced the conceptof complex variables to junior high school students in the UnitedStates with the goal of stimulating their interest in mathematicsand improving their algebra skills. Comparison of pre- and post-testresults showed that ninth-grade students displayed a significantchange in algebraic skills while the eighth-grade students madelittle progress. The outreach mathematician lacked some awarenessof the eighth-grade students’ foundational backgroundand motivation. This illustrates the importance of working moreclosely with the participating teacher, who understands betterthe curriculum and the students’ background knowledge,levels of maturity and levels of motivation.     

Separations of first and second order theories in bounded arithmetic

We prove that PTC_N(n) (the polynomial time closure of the nonstandard natural number n in the model N of S₂.) cannot be a model of U¹₂. This implies that there exists a first order sentence of bounded arithmetic which is provable in U¹₂ but does not hold in PTC_N(n).     

The transition between the Stokes equations and the Reynolds equation: A mathematical proof

The Reynolds equation is used to calculate the pressure distribution in a thin layer of lubricant film between two surfaces. Using the asymptotic expansion in the Stokes equations, we show the existence of singular perturbation phenomena whenever the two surfaces are in relative motion. We prove that the Reynolds equation is an approximation of the Stokes equations and that the kind of convergence is strongly related with the boundary conditions on the velocity field.     

From how to why: A quest for the common mathematical meanings behind two different division algorithms

During their education cycle in mathematics, students are exposed to algorithms as early as primary school. Several studies show how students frequently learn to perform these algorithms without controlling the mathematical meanings behind them. On the other hand, several National Standards have highlighted the need to construct meanings in mathematics from the first cycle of education. In this paper we focus on division algorithms, investigating to what extent 6th graders can be guided to understanding the whys behind an algorithm, through the comparison of two different algorithms for integer division. Our results suggest, on the one hand, that “it could work!”, and on the other hand, that the exposure to different algorithms for the same mathematical operation seems particularly significant for bringing out the whys behind such algorithms, as well as for capturing the difference between a mathematical operation and algorithms for calculating the result of such an operation.     

中国引进外资与本币供应量之间数量关系

引进外资是影响中国货币供应量的重要因素之一，两者之间存在一定的数量关系。根据近年来的数据进行实证分析其结果表明：引进外资对中国通货膨胀有一定的影响，今后中国引进外资工作仍要慎行     

On robust identification in the square and king grids

Fault diagnosis of multiprocessor systems gives the motivation for robust identifying codes. We provide robust identifying codes for the square and king grids. Often we are able to find optimal such codes.     

Factorizations of tropical and sign polynomials

In this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call tropical polynomials and sign polynomials, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials.     

Smooth numbers and the norms of arithmetic Dirichlet convolutions

In this paper we consider the problem of exactly evaluating the p-norm of a linear operator linked with arithmetic Dirichlet convolutions. We prove that a simply derived upper bound for this norm is actually attained for several different classes of arithmetic functions including completely multiplicative functions, but not for certain multiplicative functions. Our proof depends fundamentally on the asymptotic distribution properties of smooth numbers.     

Weighted sampling and reconstruction in weighted reproducing kernel spaces

In this paper, we discuss sampling and reconstruction of signals in the weighted reproducing kernel space associated with an idempotent integral operator. We show that any signal in that space can be stably reconstructed from its weighted samples taken on a relatively-separated set with sufficiently small gap. We also develop an iterative reconstruction algorithm for the reconstruction of a signal from its weighted samples taken on a relatively-separated set with sufficiently small gap.     

A Criterion for the Triviality of G(D) and Its Applications to the Multiplicative Structure of D

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*/Nrd _D/F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK ₁(D) = 1 and F ^*2 = F ^*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated     

Promoting symmetric weight selection in data envelopment analysis: A penalty function approach

Traditionally, data envelopment analysis models assume total flexibility in weight selection, though this assumption can lead to several variables being ignored in determining the efficiency score. Existing methods constrain weight selection to a predefined range, thus removing possible feasible solutions. As such, in this paper we propose the symmetric weight assignment technique (SWAT) that does not affect feasibility and rewards decision making units (DMUs) that make a symmetric selection of weights. This allows for a method of weight restrictions that does not require preference constraints on the variables. Moreover, we show that the SWAT method may be used to differentiate among efficient DMUs.     

A simplified version of the DEA cost efficiency model

This short paper deals with a current cost efficiency model and decreases the number of its constraints as well as its variables, which leads to a strong reduction in the computational requirements.