On Chebyshev quadrature for ultraspherical weight functions

It is well-known that for the ultraspherical weight function (1-x²)^λ-1/2 there exist no Chebyshev quadrature formulae in the strict sense having n nodes, where n is sufficiently large and λ>0, whereas on the other hand for λ=0 every Gaussian quadrature formulae is a Chebyshev formula in the strict sense. In this paper we study the open question of Chebyshev quadrature for λ <0. It is shown that there exists no Chebyshev quadrature formula in the strict sense having more than two nodes for λ≤λ₀=-.30056... (for definition of λ₀ see (1.8) below). Moreover, results are obtained for Chebyshev-type formulae and Chebyshev formulae of closed type. For the remaining values of λ (λ₀<λ<0) numerical results are given.     

Combinatorial properties of noninteger vertices of a polytope in a three-index axial assignment problem

It is proved that, for any r ∈ { 2n, 2n + 1,…, 3n−2} and only for such r, the polytope of a three-index axial assignment problem of order n, n ≥ 2, contains completely r-noninteger vertices (r-CNVs), i.e., vertices such that all their positive components are fractional and their number equals r. For each r ∈ {2n, 2n + 1,…, 3n −2}, all the types of r-CNVs are characterized and the combinatorial properties of completely r-noninteger vertices of the polytope are studied. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 33–44, January–February 2007.     

Algorithms for solving systems of linear diophantine equations in residue rings

Algorithms are proposed that construct the basis of the set of solutions to a system of homogeneous or inhomogeneous linear Diophantine equations in a residue ring modulo n when the prime factors of n are known. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 27–40, November–December 2007.     

An asymptotic estimate of the error of eigenvalues in the method of finite elements for the sturm-liouville problem

A scheme of the second order of accuracy for the Sturm—Liouville problem is constructed by the method of finite elements with the use of a special basic system of compact functions. The convergence of the method of finite elements is proved. An exact formula for estimating errors of eigenvalues is obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 28–36, March–Apni, 2000.     

On the numerical solution of nonlinear systems of Volterra integro-differential equations with delay arguments

Particular cases of nonlinear systems of delay Volterra integro-differential equations (denoted by DVIDEs) with constant delay τ > 0, arise in mathematical modelling of ‘predator–prey’ dynamics in Ecology. In this paper, we give an analysis of the global convergence and local superconvergence properties of piecewise polynomial collocation for systems of this type. Then, from the perspective of applied mathematics, we consider the Volterra’s integro-differential system of ‘predator–prey’ dynamics arising in Ecology. We analyze the numerical issues of the introduced collocation method applied to the ‘predator–prey’ system and confirm that we can achieve the expected theoretical orders of convergence.      

Computational investigation of elliptic systems with discontinuous coefficients

The Dirichlet problem is considered for a linear elliptic system of the second order with discontinuous coefficients in a rectangle Ω. For this problem, a difference scheme of the order of accuracy is constructed in a net norm W ₂ ¹ (ω), provided that the solution of the original problem belongs to the space W ₂ ^k (Ω_α), α= 1,2, k=2,3, in each of the subdomains into which the rectangle is divided by the curve of discontinuity of coefficients. This work was supported by the State Foundation for Basic Research. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 157–161, July–August, 2000.     

A generalization of the Jensen inequality

A new approach for estimating functions and functionals based on the concept of function superconvexity and a generalization of the Jensen inequality is proposed. A new concept is introduced and the associated analytical results are substantiated. Some nontrivial estimates of functions are derived. Translated from Kibemetika i Sistemnyi Analiz, No. 2, pp. 89–99, March–April, 2000.     

Stability condition for m/d/l retrial queuing system with a limited waiting time

A stability condition is derived for a queueing system with a Poisson input with parameter λ and constant service time τ. If the virtual waiting time is less than a constant a, then the demand can be serviced; otherwise, it is repeated in exponentially distributed time or is lost with a probability q >- 0. Translated from Kibemetika i Sisternnyi Anaiii. No. 2, pp. 184–186, March–April, 2000.     

Generation of root functionals of a system of polynomials

A root functional is a functional that is defined over a polynomial ring and annihilates the ideal of polynomials. The concept of a root functional is a generalization of the concept of a root to the case of multiple roots. This article considers a bilinear operation of generation of root functionals for a system of polynomials in variables, which allows one to construct a root functional from two root functionals. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 22–46, January–February 2008.     

Simulation of Motion of a Collection of Rigid Bodies,Taking into Account Their Collisions and Joints

Existing methods of simulation of motion of rigid bodies are considered that take into account collisions and joints of such bodies. A new fast universal fixed-step-size algorithm of speed constraints is proposed. This algorithm combines all the advantages of well-known algorithms and is deprived of their drawbacks. Using a modified Dantzig method, the upper time estimate equal to O(4k) is obtained for the algorithm. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 29–38, July–August 2005.     

Calculation of the number of complete mappings for permutations

The computation time for counting “good” permutations rapidly grows as the length of permutations increases. The paper presents algorithms for enumeration of “good” permutations. Algorithms reducing twice the number of “good” permutations that should be counted are considered along with the algorithm employing the concept of weight of a “good” permutation. Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 106–110, March–April, 2000.     

Una nuova rappresentazione asintotica dei polinomi ultrasferici

In this paper we obtain a new asymptotic formula for the ultraspherical polynomialP _n ^(λ) (x), asn→∞, with an error term which isO (n ^λ−5 ) uniformly in the interval −1+δ≤x≤1−δ,δ>0. Very accurate approximations for the zeros ofP _n ^(λ) (x) are also derived from the preceding formula.

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Lavoro eseguito nell'ambito del Gruppo Nazionale per l'Informatica Matematica del C. N. R..     

Estimation of the length of reset words for automata with simple idempotents

A quadratic upper bound on the length of a minimal reset word is obtained for finite automata with simple idempotents. Each input symbol of the automata considered induces a transformation that is an idempotent with the unit defect or a bijection on the set of states. This bound is only twice as large as the well-known lower bound of this length. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 32–39, May–June, 2000.     

Complexity of boolean functions in a class of canonical polarized polynomials

Relationships for the maximum of a minimal number of summands among all canonical polarized polynomials of a Boolean function are obtained. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 179–183, May–June, 2000.     

Relative rate of change of a function

A quantity is examined that characterizes the relative deviation of a derivative of a function from a specified reference value. This quantity is shown to be interpreted as a derivative of a function in some field constructed using a new operation. Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 160–165, July–August, 1999.     

Boolean problem of sequential minimization of moduli of linear functions and stability theorems

A multicriterion Boolean problem of lexicographic optimization with partial criteria represented by moduli (absolute values) of linear functions is considered. Five types of stability of a set of lexicographic optima under “small” variations in the parameters of a vector criterion are investigated. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 178–187, September–October 2007.     

Invariance theorems for a class of systems of random nonlinear equations over an arbitrary finite ring with left unity

A class of homogenous systems of random nonlinear equations over an arbitrary finite ring with left unity is considered. The author analyzes the invariance boundaries for limit factorial moments of nonzero solutions, the limit distribution of the number of nonzero solutions, and the geometrical structure of the set of nonzero solutions of the system as the number of unknowns tends to infinity. Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 106–115, November–December 2008.     

Effects of planar inlet plenums on the hydrodynamically developing flows in rectangular microchannels of complementary aspect ratios

The effects of planar inlet plenum geometry on the developing flow fields in two rectangular microchannels of reciprocal aspect ratios (H/W ∼2.75 and ∼0.40) were investigated for Re _D =  1–100 using micro particle image velocimetry (μPIV). These two microchannels were made by a precision sawing and silicon microfabrication techniques. Both the velocity profiles and centerline velocity developments were clearly resolved and extracted along the axial distance from μPIV results. The entrance lengths were found from the centerline velocities using a decaying exponential fitting function where the centerline velocity reaches 99% of the fully developed centerline velocity. The proposed fitting function showed excellent agreement with the experimental results. The planar plenum was shown to cause an upstream predevelopment resulting in the significant reductions in the entrance lengths. Two entrance length correlations were proposed in the forms of Atkinson et al.’s (AIChE J 15:548–553, 1969) and Chen’s (J Fluids Eng 95:153–158, 1973) correlations. The proposed entrance length correlations showed that acquired constant portion and slope of the entrance lengths showed 23–27 and 70–81% reductions for H/W =  2.75 while the entrance length correlation for H/W =  0.40 showed 69–73% increase and 41–63% decrease in the constant portion and slope, respectively.     

Multicast key distribution schemes based on Stinson-Van Trung designs

The efficiency characteristics of multicast key distribution schemes based on (v, b, r, λ)-designs are investigated. The stability and connectivity of such designs are estimated. These estimates generalize and enhance the well-known estimates for the efficiency characteristics of multicast key distribution schemes based on incomplete block designs. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 100–111, May–June 2007.     

On the Cardinality Spectrum and the Number of Latin Bitrades of Order 3

By a (Latin) unitrade of order k, we call a subset of vertices of the Hamming graph H(n, k) that intersects any maximal clique at either 0 or 2 vertices. A bitrade is a bipartite unitrade, i.e., a unitrade that can be split into two independent subsets. We study the cardinality spectrum of bitrades in the Hamming graph H(n, k) with k = 3 (ternary hypercube) and the growth of the number of such bitrades as n grows. In particular, we determine all possible small (up to 2.5·2n) and large (from 14·3n−3) cardinalities of bitrades of dimension n and prove that the cardinality of a bitrade takes only values equivalent to 0 or 2n modulo 3 (this result can be treated in terms of a ternary Reed-Muller type code). A part of the results are valid for an arbitrary k. For k = 3 and n → ∞ we prove that the number of nonequivalent bitrades is not less than 2(2/3−o(1))n and not greater than $$2^{\alpha^n}$$, α < 2 (the growth order of the double logarithm of this number remains unknown). Alternatively, the studied set Bn of bitrades of order 3 can be defined as follows: B0 consists of three rationals - 1, 0, 1; Bn consists of ordered triples (a, b, c) of elements from Bn−1 such that a + b + c = 0.