The three-dimensional origin of the classifying algebra

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra.     

Hamiltonicity and pancyclicity of cartesian products of graphs

The cartesian product of a graph G with K₂ is called a prism over G. We extend known conditions for hamiltonicity and pancyclicity of the prism over a graph G to the cartesian product of G with paths, cycles, cliques and general graphs. In particular we give results involving cubic graphs and almost claw-free graphs.We also prove the following: Let G and H be two connected graphs. Let both G and H have a 2-factor. If Δ(G)≤g^′(H) and Δ(H)≤g^′(G) (we denote by g^′(F) the length of a shortest cycle in a 2-factor of a graph F taken over all 2-factorization of F), then G□H is hamiltonian.     

An algebraic foundation for factoring linear boundary problems

Motivated by boundary problems for linear differential equations, we define an abstract boundary problem as a pair consisting of a surjective linear map (“differential operator”) and an orthogonally closed subspace of the dual space (“boundary conditions”). Defining the composition of boundary problems corresponding to their Green’s operators in reverse order, we characterize and construct all factorizations of a boundary problem from a given factorization of the defining operator. For the case of ordinary differential equations, the main results can be made algorithmic. We conclude with a factorization of a boundary problem for the wave equation. This work was supported by the Austrian Science Fund (FWF) under the SFB grant F1322.     

Convolution factorability of bilinear maps and integral representations

In this paper we consider a special class of continuous bilinear operators acting in a product of Banach algebras of integrable functions with convolution product. In the literature, these bilinear operators are called ‘zero product preserving’, and they may be considered as a generalization of Lamperti operators. We prove a factorization theorem for this class, which establishes that each zero product preserving bilinear operator factors through a subalgebra of absolutely integrable functions. We obtain also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces, as the Dunford–Pettis property or the Schur property and we give integral representations by some concavity properties of operators. Finally, we give some applications for integral transforms, and an integral representation for Hilbert–Schmidt operators.     

MCD-finite Domains and Ascent of IDF Property in Polynomial Extensions

An integral domain is said to have the IDF property when every non-zero element of it has only a finite number of non-associate irreducible divisors. A counterexample has already been found showing that the IDF property does not necessarily ascend in polynomial extensions. In this paper, we introduce a new class of integral domains, called MCD-finite domains, and show that for any domain D, D[X] is an IDF domain if and only if D is both IDF and MCD-finite. This result entails all the previously known sufficient conditions for the ascent of the IDF property. Our new characterization of polynomial domains with the IDF property enables us to use a different construction and build another counterexample which strengthen the previously known result on this matter.     

On the enumeration of polynomials with prescribed factorization pattern

We use generating functions over group rings to count polynomials over finite fields with the first few coefficients and a factorization pattern prescribed. In particular, we obtain different exact formulas for the number of monic n-smooth polynomials of degree m over a finite field, as well as the number of monic n-smooth polynomials of degree m with the prescribed trace coefficient.     

Prediction of incompressible flow separation with the approximate factorization technique

This paper describes one application of the approximate factorization technique to the solution of incompressible steady viscous flow problems in two dimensions. The velocity-pressure formulation of the Navier-Stokes equations written in curvilinear non-orthogonal co-ordinates is adopted. The continuity equation is replaced with one equation for the pressure by means of the artificial compressibility concept to obtain a system parabolic in time. The resulting equations are discretized in space with centred finite differences, and the steady state solution obtained by a time-marching ADI method requiring to solve 3 x 3 block tridiagonal linear systems. An optimized fourth-order artificial dissipation is introduced to damp the numerical instabilities of the artificial compressibility equation and ensure convergence. The resulting solver is applied to the prediction of a wide variety of internal flows, including both streamlined boundaries and sharp corners, and fast convergence and good results obtained for all the configurations investigated.     

A stable method for solving certain constrained least squares problems

This paper presents a feasible descent algorithm for solving certain constrained least squares problems. These problems are specially structured quadratic programming problems with positive semidefinite Hessian matrices that are allowed to be singular. The algorithm generates a finite sequence of subproblems that are solved using the numerically stable technique of orthogonal factorization with reorthogonalization and Given's transformation updating.This material is based upon work supported by the National Science Foundation under Grant No. MCS 78-06716 and by the International Institute for Applied Systems Analysis.     

A practicable steepest-edge simplex algorithm

It is shown that suitable recurrences may be used in order to implement in practice the steepest-edge simplex linear programming algorithm. In this algorithm each iteration is along an edge of the polytope of feasible solutions on which the objective function decreases most rapidly with respect to distance in the space of all the variables. Results of computer comparisons on medium-scale problems indicate that the resulting algorithm requires less iterations but about the same overall time as the algorithm of Harris [8], which may be regarded as approximating the steepest-edge algorithm. Both show a worthwhile advantage over the standard algorithm.