A construction method for generalized Room squares

A Generalized Room Square (GRS) of ordern and degreek is an \(\left( {\begin{array}{*{20}c} {n - 1} \\ {k - 1} \\ \end{array} } \right) \times \left( {\begin{array}{*{20}c} {n - 1} \\ {k - 1} \\ \end{array} } \right)\) array of which each cell is either empty or contains an unorderedk-tuple of a setS, |S|=n, such that each row and each column of the array contains each element ofS exactly once and the array contains each unorderedk-tuple exactly once. A method of generating the unordered triples on the setS=GF(q) ? {∞} is given, 3 ∣ (q ∣ 1). This method is used to construct GRS's of appropriate ordern and degree 3, for alln<50.     

An infinite class of generalized room squares

A generalized Room square $\text{G}$ of order n and degree k is an ${}^{\mathrm{n?1}}{}^{\mathrm{k?1}}\text{×}{}^{\mathrm{n?1}}{}^{\mathrm{k?1}}$ array, each cell of which is either empty or contains an unordered k-tuple of a set S, |S| = n, such that each row and each column of the array contains each element of S exactly once and $\text{G}$ contains each unordered k-tuple of S exactly once. Using a class of Steiner systems and a generalized Room square of order 18 and degree 3 constructed by ad hoc methods, an infinite class of degree 3 squares is constructed.     

A generalized successive overrelaxation method for least squares problems

In this paper a new iterative method is given for solving large sparse least squares problems and computing the minimum norm solution to underdetermined consistent linear systems. The new scheme is called the generalized successive overrelaxation (GSOR) method and is shown to be convergent ifA is full column rank. The GSOR method involves a parameter ρ and an auxiliary matrixP. One can choose matrix P so that the GSOR method only involves matrix and vector operations; therefore the GSOR method is suitable for parallel computations. Besides, the GSOR method can be combined with preconditioning techniques, and therefore can be expected to be more effective. This author's work was supported by Natural Science Foundation of Liaoning Province, China.     

Preconditioned conjugate gradient method for generalized least squares problems

A variant of the preconditioned conjugate gradient method to solve generalized least squares problems is presented. If the problem is min (Ax − b)^TW⁻¹(Ax − b) with A ∈ R^m×n and W ∈ R^m×m symmetric and positive definite, the method needs only a preconditioner A₁ ∈ R^n×n, but not the inverse of matrix W or of any of its submatrices. Freund's comparison result for regular least squares problems is extended to generalized least squares problems. An error bound is also given.     

Pairwise “Orthogonal generalized room squares” and equidistant permutation arrays

A generalized Room square S(r, λ; v) is an r × r array such that every cell in the array contains a subset of a v-set V. This subset could of course be the empty set. The array has the property that every element of V is contained precisely once in every row and column and that any two distinct elements of V are contained in precisely λ common cells. In this paper we define pairwise orthogonal generalized Room squares and give a construction for these using finite projective geometries. This is another generalization of the concept of pairwise orthogonal latin squares. We use these orthogonal arrays to construct permutations having a constant Hamming distance.     

A note on generalized Room squares

Using the properties of the Steiner system on 24 points a generalized Room square of degree 4 and order 24 is constructed. Results on a proposed alternative method for constructing generalized Room squares are given which use the notion of a (2, 4, k) array, introduced here.     

A construction for doubly pandiagonal magic squares

In this note, a doubly magic rectangle is introduced to construct a doubly pandiagonal magic square. A product construction for doubly magic rectangles is also presented. Infinite classes of doubly pandiagonal magic squares are then obtained, and an answer to problem 22 of [G. Abe, Unsolved problems on magic squares, Discrete Math. 127 (1994) 3] is given.     

A generalized sewing construction for polytopes

We describe a new technique for constructing convex polytopes—a generalization of Shemer’s sewing construction for simplicial neighborly polytopes that has been modified to allow the creation of nonsimplicial polytopes as well. We show that Bisztriczky’s ordinary polytopes can be constructed in this manner, and we also construct several infinite families of polytopes. We consider bounds on the flag f-vectors of 4-polytopes that can be inductively constructed by generalized sewing starting from the 4-simplex.     

A negative-norm least squares method for Reissner-Mindlin plates

In this paper a least squares method, using the minus one norm developed by Bramble, Lazarov, and Pasciak, is introduced to approximate the solution of the Reissner-Mindlin plate problem with small parameter , the thickness of the plate. The reformulation of Brezzi and Fortin is employed to prevent locking. Taking advantage of the least squares approach, we use only continuous finite elements for all the unknowns. In particular, we may use continuous linear finite elements. The difficulty of satisfying the inf-sup condition is overcome by the introduction of a stabilization term into the least squares bilinear form, which is very cheap computationally. It is proved that the error of the discrete solution is optimal with respect to regularity and uniform with respect to the parameter . Apart from the simplicity of the elements, the stability theorem gives a natural block diagonal preconditioner of the resulting least squares system. For each diagonal block, one only needs a preconditioner for a second order elliptic problem.

     

Perturbation theory for generalized and constrained linear least squares

The perturbation analysis of weighted and constrained rank‐deficient linear least squares is difficult without the use of the augmented system of equations. In this paper a general form of the augmented system is used to get simple perturbation identities and perturbation bounds for the general linear least squares problem both for the full‐rank and rank‐deficient problem. Perturbation identities for the rank‐deficient weighted and constrained case are found as a special case. Interesting perturbation bounds and condition numbers are derived that may be useful when considering the stability of a solution of the rank‐deficient general least squares problem. Copyright © 2000 John Wiley & Sons, Ltd.     

A weighted least squares method for scattered data fitting

In this paper, we present a weighted least squares method to fit scattered data with noise. Existence and uniqueness of a solution are proved and an error bound is derived. The numerical experiments illustrate that our weighted least squares method has better performance than the traditional least squares method in case of noisy data.     

A simple method for constructing doubly diagonalized Latin squares

A Latin square of order n we call doubly diagonalized if both its diagonals consist of n distinct symbols. In this paper a new method to construct such squares for any order is given.     

A tripling construction for mutually orthogonal symmetric hamiltonian double Latin squares

We provide two new constructions for pairs of mutually orthogonal symmetric hamiltonian double Latin squares. The first is a tripling construction, and the second is derived from known constructions of hamilton cycle decompositions of when is prime.     

A generalized quasilinearization method for telegraph system

In this note, we build a maximum principle for the doubly periodic solutions of telegraph system. And the generalized quasilinearization technique is applied to obtain a monotone sequence of iterates converging uniformly and quadratically to a solution of a coupled telegraph system with doubly periodic boundary condition.