On context patterns associated with concept lattices

In this paper, we consider the following reconstruction problem: Given two ordered sets (G, ) and (M, ) representing join- and meet-irreducible elements, respectively together with three relationsJ,, onG×M modelling comparability (gm) and maximal noncomparability with respect tog (gm, butgm*) and with respect tom (gm, butgm*). We determine necessary and sufficient conditions for the existence of a finite latticeL and injections :GJ(L) and :MM(L) such that the given order relations and the abstract relations coincide with the one induced by the latticeL.     

Fast Preconditioned Iterative Methods for Convolution-Type Integral Equations

We consider solving the Fredholm integral equation of the second kind with the piecewise smooth displacement kernel x(t) + _j=1 ^m µ_j x(t – t _j) + ₀ k(t – s)x(s) ds = g(t), 0 t , where t _j (–, ), for 1 j m. The direct application of the quadrature rule to the above integral equation leads to a non-Toeplitz and an underdetermined matrix system. The aim of this paper is to propose a numerical scheme to approximate the integral equation such that the discretization matrix system is the sum of a Toeplitz matrix and a matrix of rank two. We apply the preconditioned conjugate gradient method with Toeplitz-like matrices as preconditioners to solve the resulting discretization system. Numerical examples are given to illustrate the fast convergence of the PCG method and the accuracy of the computed solutions.     

Einschliessungsaussagen für Differentialoperatoren vierter Ordnung und ein Verfahren zur Berechnung von Schrankenfunktionen

For a linear fourth order ordinary differential operator M we study Range Domain Implications (RDI). Let C_o [O,1] be positive; we show under what conditions there exists a C_O[O,1] such that the following RDI holds: Mu(x) (x) (0x1) u(x) (0x1). In particular we provide a numerical procedure to calculate .RDI are used to obtain error estimations and to solve related nonlinear problems.The basic idea to prove RDI is to split M into a product of second order differential operators which are easier to handle. For the general case that there exists no global splitting the concept of a local splitting is introduced.

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The author would like to thank the European Research Office of the United States Army for their kind interest.     

ALTERNATION THEORY IN APPROXIMATION BY POLYNOMIALS HAVING BOUNDED COEFFICIENTS

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Method for symmetrization and estimation of solutions of the Neumann problem for the equation of a porous medium in domains with noncompact boundary for infinitely increasing time

We consider the initial boundary-value Neumann problem for the equation of a porous medium in a domain with noncompact boundary. By using a symmetrization method, we obtain exactL _p-estimates, 1p, for solutions as t.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 147–157, February, 1995.     

Acceleration of Superlinearly Convergent Sequences

A method analogous to the Aitken extrapolation is proposed to accelerate the convergence of sequences of real or complex numbers with asymptotic behavior where e _i is the error of the ith element of the sequence, d _j1 for all 1jr, and d_j > 1. The R-order of the resulting sequences is computed using methods which are also of independent interest.     

The existence of some resolvable block designs with divisibility into groups

This paper proves the existence of resolvable block designs with divisibility into groups GD(v; k, m; ₁, ₂) without repeated blocks and with arbitrary parameters such that ₁ = k, (v–1)/(k–1) ₂ v^k–2 (and also ₁ k/2, (v–1)/(2(k–1)) ₂ v^k–2 in case k is even) k 4 andp=1 (mod k–1), k < p for each prime divisor p of number v. As a corollary, the existence of a resolvable BIB-design (v, k, ) without repeated blocks is deduced with X = k (and also with = k/2 in case of even k) k , where a is a natural number if k is a prime power and=1 if k is a composite number.Translated from Matematicheskie Zametki, Vol. 19, No. 4, pp. 623–634, April, 1976.     

On the eigenvalues of the matrix pencilA+μB

Summary The number of independent invariants ofn×n matricesA, B and their products on which the eigenvalues () of the matrix pencilA+B depend is determined by means of the theory of algebraic invariants and combinatorial analysis. Formulas are displayed for coefficients for the calculation of () forn5.

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Zusammenfassung Wir bestimmen die Anzahl der unabhängigen Invarianten dern×n MatrizenA, B und ihrer Produkte, von denen die Eigenwerte () der MatrixbüschelA+B abhängen, mittels der Theorie der algebraischen Invarianten und mittels kombinatorischer Analyse. Formeln für Koeffizienten zur Berechnung von () werden angegeben fürn5.

     

Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

Connected and alternating vectors: Polyhedra and algorithms

Given a graphG = (V, E), leta ^S, S L, be the edge set incidence vectors of its nontrivial connected subgraphs.The extreme points of = {x R ^E: a^sx |V(S)| - |S|, S L} are shown to be integer 0/± 1 and characterized. They are the alternating vectorsb ^k, k K, ofG. WhenG is a tree, the extreme points ofB 0,b ^kx 1,k K} are shown to be the connected vectors ofG together with the origin. For the four LP's associated with andA, good algorithms are given and total dual integrality of andA proven.On leave from Swiss Federal Institute of Technology, Zurich.     

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.     

On TDP permutations

A permutation :i| _i , 0i<n is called a TDP permutation ifi–a _i j–a _j (modn) fori j. This paper finds all TDP permutations forn15, discusses the method for generating TDP permutations, and finally by applying MLE method obtains a formula for estimating the number of TDP permutations forn> 15.Project supported by National Natural Science Foundation of China.     

Best Linear Methods of Approximation of Functions of the Hardy Class H p

We determine the exact value of the best linear polynomial approximation of a unit ball of the Hardy space H _p, 1 p , on concentric circles , 0 < 1, in the uniform metric. We construct the best linear method of approximation and prove the uniqueness of this method.     

The problem of irregular perfect systems of sets of iterated differences

Fors2, the set of iterated differences associated with the prescribed integersa(s, j), 1js, is the set {a(i, j): 1jis} wherea(i–1,j)=|a(i, j)–a(i, j+1)|, general problem raised by work of Kreweras and Loeb concerns the existence of partitions of runs of consecutive integers into full sets of iterated differences. In the regular case, where all the sets of iterated differences have the same valencys, it is known that such partitions do not exist at least fors>8. We find here that the problem is more challenging in the case where the sets have different valencies.     

Legitimate colorings of projective planes

For a projective plane _n of ordern, let( _n ) denote the minimum numberk, so that there is a coloring of the points of _n ink colors such that no two distinct lines contain precisely the same number of points of each color. Answering a question of A. Rosa, we show that for all sufficiently largen, 5 ( _n ) 8 for every projective plane _n of ordern. Research supported in part by Allon Fellowship and by a grant from the United States Israel Binational Science Foundation     

Errors in approximate solutions of Cauchy's problem for a first-order quasilinear equation

The proximity is investigated of the solution of Cauchy's problem for the equation u _t +((u))_x= u _xx ((u) > 0) to the solution of Cauchy's problem for the equation u_t+ ((u))_x= 0, when the solution of the latter problem has a finite number of lines of discontinuity in the strip 0 t T. It is proved that, everywhere outside a fixed neighborhood of the lines of discontinuity, we have |u–u| C, where the constant C is independent of. Similar inequalities are derived for the first derivatives of u–u.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 309–320, September, 1970.In conclusion we express our gratitude to L. A. Chudov for his valuable advice concerning this work.     

Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

An observation concerning Ryser's conjecture

If in anr-partiter-graphH everyk edges intersect (wherek>2), then (H)[(r–1)/(k–1)].     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.     

On the ω-limit Set Dichotomy of Cooperating Kolmogorov Systems

The authors study the -limit set dichotomy of the Kolmogorov systems _i=x_i f _i(x)x _i0, 1in with the cooperative and irreducible hypotheses and obtain the quasiconvergence almost everywhere when n=3, which gives an affirmative answer to the open problem by Smith [9, p.72] in the case of n=3.