We show that OD(12;1,1,1,9) is the only orhtogonal design of order 12 constructible from 16 circulant matrices.

Following the main idea if MVI the inverse parabolic problem of coefficient identification from over-posed data is imbedded into a quadratic functional from governed equation. The necessary condition for minimization of the functional (the Euler-Lagrange equations) comprise a fourth-order in space and second-order in time elliptic boundary value problem for the temperature and a second-order in space ODE for the unknown coefficient.

The imbedding problem is well-posed for redundant data at boundaries and possesses a unique solution which means that when the imbedding functional is zero, the over-posed data is consistent and the solution of the imbedding problem coincides with the sought solution of the inverse problem.

A difference scheme of splitting type is employed and featuring examples are elaborated numerically. The numerical results confirm that the solution of the imbedding problem coincides with the direct simulation of the original problem within the truncation error O(t²+h²).

This is joint work with D. Panario (Carleton University) and S. Q. Wang (University of Lethbridge).

We give a characterization of permutation polynomials over a finite field F_q based on their coefficients. As a result, explicit conditions for a polynomial to be a permutation polynomial are given when q=2, 3, 5. Moreover, permutation binomials over F_q are completely described. New classes of permutation binomials and polynomials can be constructed accordingly.

We study the cohomology H^*(A)=Ext_A^*(k,k) of a locally finite, connected, cocommutative Hopf algebra A over k=F_p. Specifically, we are interested in those algebras A for which H^*(A) is generated as an algebra by H¹(A) and H²(A). We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras F®A® B with F monogenic and B semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for A to be semi-Koszul. Special attention is given to the case in which A is the restricted universal enveloping algebra of the Lie algebra obtained from the mod-p lower central series of a p-group. We show that the algebras arising in this way from extensions by Z/(p) of an abelian p-group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 p-groups, and it is shown that these are all semi-Koszul for p ³ 5.

Two vectors v, w on Z_g are vector independent if for all pairs (a, b) Î Z_g ×Z_g there is a position i in the vectors where (a,b) = (v_i,w_i). A covering array on a graph G, denoted CA(G,g), is an array on Z_g with |V(G)| rows and the property that any two rows which correspond to adjacent vertices in G are vector independent. We define a family of graphs G(n,g) whose vertices are length n vectors on Z_g and edges join two vertices if they are vector independent. A graph G has a covering array with n columns if and only if there is a homomorphism from G to G(n,g). Hence, finding a covering array on a graph is similar to finding the chromatic number, the r chromatic number or the star chromatic number of a graph.

The vector space of symmetric functions possesses a well-known scalar product which has a variety of combinatorial applications. This scalar product has been used to express generating functions of combinatorial objects, and often one can determine conditions for when these series are D-finite. That is, when do such series satisfy a differential equation with polynomial coefficients? We extend earlier work by providing algorithms that compute the differential equations these generating functions satisfy.

This talk will outline the fundamentals of the so called holonomic approach to combinatorics. The principal algorithms compute the differential equations satisfied by images of (series of) symmetric functions under different maps; Notably the aforementioned scalar product, the tensor (or Kronecker) product (linked to the computation of certain characters) and various q-analogs. The algorithms use Groebner bases in non-commutative algebras, and are implemented in Maple.

Finally, we will present a host of examples from enumerative combinatorics well suited to this approach: regular graphs, plane partitions, latin squares, Young tableaux,...