On the pivot structure for the weighing matrix W(12,11)

In the present article we concentrate our study on the growth problem for the weighing matrix W(12,11) and show that the unique W(12,11) has three pivot structures. An improved algorithm for extending a k × k (0,+,-) matrix to a W(n,n-1), if possible, has been developed to simplify the proof. For the implementation of the algorithm special emphasis is given to the notions of data structures and parallel processing.     

Continuous time collocation methods for Volterra-Fredholm integral equations

Summary Integral equations of mixed Volterra-Fredholm type arise in various physical and biological problems. In the present paper we study continuous time collocation, time discretization and their global and discrete convergence properties.     

Symmetries and conservation laws of Navier-Stokes equations

All the symmetries and conservation laws of Navier-Stokes equations are calculated.     

Extendible elements of the alternating groups

An element σ of A_n, the Alternating group of degree n, is extendible in S_n, the Symmetric group of degree n, if there exists a subgroup H of S_n but not A_n whose intersection with A_n is the cyclic group generated by σ. A simple number-theoretic criterion, in terms of the cycle-decomposition, for an element of A_n to be extendible in S_n is given here.     

Congruences,equational theories and lattice representations

We survey results concerning the representations of lattices as lattices of congruences and as lattices of equational theories. Recent results and open problems will be mentioned.To László Fuchs on the occasion of his 70th birthday     

Elliptic semiplanes and group divisible designs with orthogonal resolutions

In this paper we prove that there exists an elliptic semiplaneS(v, k, m) withk –m 2 if and only if there exists a group divisible design GDD _k ((k –m)(k – 1);k –m; 0, 1) withm pairwise orthogonal resolutions. As an example of this theorem, we construct an elliptic semiplaneW(45, 7, 3) and show thatW is isomorphic to the elliptic semiplaneS(45, 7, 3) given by R. D. Baker.     

A priori-Schranken für diskrete Lösungen monotoner Operatorgleichungen und Anwendungen

Summary Operator equationsTu=f are approximated by Galerkin's method, whereT is a monotone operator in the sense of Browder and Minty, so that existence results are available in a reflexive Banach spaceX. In a normed spaceY error estimates are established, which require a priori bounds for the discrete solutionsu _h in the norm of a suitable space . Sufficient conditions for the uniform boundedness u _{h Z} =O(1) ash0 are proved. Well-known error estimates in [3] for the special caseX=Y=Z are generalized by this. The theory is applied to quasilinear elliptic boundary value problems of order 2m in a bounded domain . The approximating subspaces are finite element spaces. Especially the caseX=W ₀ ^{m, p} (), 1<p<,Y=W ₀ ^{m. 2} (),Z=W ₀ ^{m. max (2,p)} ()W^m, () is treated. Some examples for 1<p<2 are considered. Forp2 a refined technique is introduced in the author's paper [7].

     

Incomplete idempotent Schröder quasigroups and related packing designs

Summary In this paper it is proved that, for any positive integern 2, 3 (mod 4),n 7, there exists an incomplete idempotent Schröder quasigroup with one hole of size two IISQ(n, 2) except forn = 10. It is also proved that for any positive integern 0, 1 (mod 4), there exists an idempotent Schröder quasigroup ISQ(n) except forn = 5 and 9. These results completely determine the spectrum of ISQ(n) and provide an application to the packing of a class of edge-coloured block designs.Research supported by NSERC grant A-5320.Research supported by NSFC grant 19231060-2.     

Zero cycles on conic fibrations and a conjecture of Bloch

Let X be a smooth projective surface over a number field k. Let (CH₀(X)) denote the Chow group of zero-cyles modulo rational equivalence on X. Let CH₀(X) be the subgroup of CH ₀(X) consisting of classes which vanish when going over to an arbitrary completion of k. Bloch put forward a conjecture asserting that this group is isomorphic to the Tate-Shafarevich group of a certain Galois module atttached to X. In this paper, we disprove this general conjecture. We produce a conic bundle X over an elliptic curve, for which the group (CH₀(X) is not zero, but the Galois-theoretic Tate-Shafarevich group vanishes.     

On Segal-Wilson's construction for the τ-functions of the constrained KP hierarchies

In this Letter, we study the constrained KP hierarchies by employing Segal-Wilson's theory on the -functions of the KP hierarchy. We first describe the elements of the Grassmannian which correspond to solutions of the constrained KP hierarchy, and then we show how to construct its rational and soliton solutions from these elements of the Grassmannian.