Reducing systems of linear differential equations to a passive form

In this paper, an application of the Riquer-Thomas-Janet theory is described for the problem of transforming a system of partial differential equations into a passive form, i.e., to a special form which contains explicitly both the equations of the initial system and all their nontrivial differential consequences. This special representation of a system markedly facilitates the subsequent integration of the corresponding differential equations. In this paper, the modern approach to the indicated problem is presented. This is the approach adopted in the Knuth-Bendix procedure [13] for critical-pair/completion and then Buchberger's algorithm for completion of polynomial ideal bases [13] (or, alternatively, for the construction of Groebner bases for ideals in a differential operator ring [14]). The algorithm of reduction to the passive form for linear system of partial differential equations and its implementation in the algorithmic language REFAL, as well as the capabilities of the corresponding program, are outlined. Examples illustrating the power and efficiency of the system are presented.     

Point approximation of a space-homogeneous transport equation

Summary We present an approximation method of a space-homogeneous transport equation which we prove is convergent. The method is very promising for numerical computation. Comparison of a numerical computation with an exact solution is given for the Master equation.     

Dielectric and optical properties of polymer-liquid crystal composite

Dielectric properties of polymer-liquid crystal mixture, having constituent polymer, poly-butyl methacrylate (PBMA) and liquid crystal, cholesteryl nonanoate, are reported as a function of frequency and temperature. The measurement has been done in a temperature range of 300-375 K and frequency range of 100 Hz-10 MHz. The dielectric permittivity and dielectric loss shows significant changes with the addition of polymer molecules in liquid crystal. The significant feature of composite formation is that the pure liquid crystal and polymer do not show dielectric relaxation in the frequency range covered, while the composite shows relaxation peak at a particular frequency. The optical transmittance of pure liquid crystal and composite has also been measured and compared.     

Optimal tristance anticodes in certain graphs

For z₁,z₂,z₃∈Zⁿ, the tristance d₃(z₁,z₂,z₃) is a generalization of the L₁-distance on Zⁿ to a quantity that reflects the relative dispersion of three points rather than two. A tristance anticodeA_d of diameter d is a subset of Zⁿ with the property that d₃(z₁,z₂,z₃)?d for all z₁,z₂,z₃∈A_d. An anticode is optimal if it has the largest possible cardinality for its diameter d. We determine the cardinality and completely classify the optimal tristance anticodes in Z² for all diameters d?1. We then generalize this result to two related distance models: a different distance structure on Z² where d(z₁,z₂)=1 if z₁,z₂ are adjacent either horizontally, vertically, or diagonally, and the distance structure obtained when Z² is replaced by the hexagonal lattice A₂. We also investigate optimal tristance anticodes in Z³ and optimal quadristance anticodes in Z², and provide bounds on their cardinality. We conclude with a brief discussion of the applications of our results to multi-dimensional interleaving schemes and to connectivity loci in the game of Go.     

Absolutely indecomposable symmetric matrices

Let A be a symmetric matrix of size n×n with entries in some (commutative) field K. We study the possibility of decomposing A into two blocks by conjugation by an orthogonal matrix T∈Mat_n(K). We say that A is absolutely indecomposable if it is indecomposable over every extension of the base field. If K is formally real then every symmetric matrix A diagonalizes orthogonally over the real closure of K. Assume that K is a not formally real and of level s. We prove that in Mat_n(K) there exist symmetric, absolutely indecomposable matrices iff n is congruent to 0, 1 or −1 modulo 2s.     

Influence functions and efficiencies of the canonical correlation and vector estimates based on scatter and shape matrices

In this paper, the influence functions and limiting distributions of the canonical correlations and coefficients based on affine equivariant scatter matrices are developed for elliptically symmetric distributions. General formulas for limiting variances and covariances of the canonical correlations and canonical vectors based on scatter matrices are obtained. Also the use of the so-called shape matrices in canonical analysis is investigated. The scatter and shape matrices based on the affine equivariant Sign Covariance Matrix as well as the Tyler's shape matrix serve as examples. Their finite sample and limiting efficiencies are compared to those of the Minimum Covariance Determinant estimators and S-estimator through theoretical and simulation studies. The theory is illustrated by an example.     

Symmetries and conservation laws of partial differential equations: Basic notions and results

The main notions and results which are necessary for finding higher symmetries and conservation laws for general systems of partial differential equations are given. These constitute the starting point for the subsequent papers of this volume. Some problems are also discussed.     

Sharp estimates in terms of moduli of smoothness for compound cubature rules

Summary In 1980 Dahmen-DeVore-Scherer introduced a modulus of continuity which turns out to reflect invariance properties of compound cubature rules effectively. Accordingly, sharp error bounds are derived, the existence of relevant counterexamples being a consequence of a quantitative resonance principle, established previously.     

On the density of the space of continuous and uniformly continuous functions

For X a metrizable space and (Y,ρ) a metric space, with Y pathwise connected, we compute the density of (C(X,(Y,ρ)),σ)—the space of all continuous functions from X to (Y,ρ), endowed with the supremum metric σ. Also, for (X,d) a metric space and (Y,‖⋅‖) a normed space, we compute the density of (UC((X,d),(Y,ρ)),σ) (the space of all uniformly continuous functions from (X,d) to (Y,ρ), where ρ is the metric induced on Y by ‖⋅‖). We also prove that the latter result extends only partially to the case where (Y,ρ) is an arbitrary pathwise connected metric space.To carry such an investigation out, the notions of generalized compact and generalized totally bounded metric space, introduced by the author and A. Barbati in a former paper, turn out to play a crucial rôle. Moreover, we show that the first-mentioned concept provides a precise characterization of those metrizable spaces which attain their extent.