On Full and Faithful Kan Extensions

The paper introduces some mild requirements under which a Kan extension is a full embedding that preserves factorization systems and colimits of -directed diagrams. The existence of such Kan extensions is then used to establish conditions equivalent to algebraic universality of concrete categories such as varieties of algebras of any given similarity type.     

Solvability of a type of one-dimensional pseudodifferential operators

The paper suggests an approach to one-dimensional pseudodifferential equations of nonnegative order, whose symbols are of the form A ₁(ξ) + th(kx + ω)A ₂(ξ). The method is based on reduction of the considered pseudodifferential equation to an integral equation. Some integral representations of solutions are found.     

The generalized triangular decomposition

Given a complex matrix , we consider the decomposition , where is upper triangular and and have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where is an upper triangular matrix with the eigenvalues of on the diagonal. We show that any diagonal for can be achieved that satisfies Weyl's multiplicative majorization conditions:

where is the rank of , is the -th largest singular value of , and is the -th largest (in magnitude) diagonal element of . Given a vector which satisfies Weyl's conditions, we call the decomposition , where is upper triangular with prescribed diagonal , the generalized triangular decomposition (GTD). A direct (nonrecursive) algorithm is developed for computing the GTD. This algorithm starts with the SVD and applies a series of permutations and Givens rotations to obtain the GTD. The numerical stability of the GTD update step is established. The GTD can be used to optimize the power utilization of a communication channel, while taking into account quality of service requirements for subchannels. Another application of the GTD is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values.

     

Algebraic approach and coherent states for a relativistic quantum particle in cosmic string spacetime

We study a relativistic quantum particle in cosmic string spacetime in the presence of a magnetic field and a Coulomb-type scalar potential. It is shown that the radial part of this problem possesses the su(1,1)$su(1,1)$ symmetry. We obtain the energy spectrum and eigenfunctions of this problem by using two algebraic methods: the Schrödinger factorization and the tilting transformation. Finally, we give the explicit form of the relativistic coherent states for this problem.     

Hermite type ladders in q-Weyl algebras

In this paper we show that the q-deformed commutator for a ladder is related to the usual commutator of the modified ladder. In addition, we consider a family of ladders in the (q, h)-Weyl algebra, which gives a factorization of certain second order difference equations.     

Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices

This report may be considered as a non-trivial extension of an unpublished report by William Kahan (Accurate Eigenvalues of a symmetric tri-diagonal matrix, Technical Report CS 41, Computer Science Department, Stanford University, 1966). His interplay between matrix theory and computer arithmetic led to the development of algorithms for computing accurate eigenvalues and singular values. His report is generally considered as the precursor for the development of IEEE standard 754 for binary arithmetic. This standard has been universally adopted by virtually all PC, workstation and midrange hardware manufactures and tens of billions of such machines have been produced. Now we use the features in this standard to improve the original algorithm.In this paper, we describe an algorithm in floating-point arithmetic to compute the exact inertia of a real symmetric (shifted) tridiagonal matrix. The inertia, denoted by the integer triplet (π, ν, ζ), is defined as the number of positive, negative and zero eigenvalues of a real symmetric (or complex Hermitian) matrix and the adjective exact refers to the eigenvalues computed in exact arithmetic. This requires the floating-point computation of the diagonal matrix D of the LDL^t factorization of the shifted tridiagonal matrix T − τI with +∞ and −∞ rounding modes defined in IEEE 754 standard. We are not aware of any other algorithm which gives the exact answer to a numerical problem when implemented in floating-point arithmetic in standard working precisions. The guaranteed intervals for eigenvalues are obtained by bisection or multisection with this exact inertia information. Similarly, using the Golub-Kahan form, guaranteed intervals for singular values of bidiagonal matrices can be computed. The diameter of the eigenvalue (singular value) intervals depends on the number of shifts with inconsistent inertia in two rounding modes. Our algorithm not only guarantees the accuracy of the solutions but is also consistent across different IEEE 754 standard compliant architectures. The unprecedented accuracy provided by our algorithms could be also used to debug and validate standard floating-point algorithms for computation of eigenvalues (singular values). Accurate eigenvalues (singular values) are also required by certain algorithms to compute accurate eigenvectors (singular vectors).We demonstrate the accuracy of our algorithms by using standard matrix examples. For the Wilkinson matrix, the eigenvalues (in IEEE double precision) are very accurate with an (open) interval diameter of 6 ulps (units of the last place held of the mantissa) for one of the eigenvalues and lesser (down to 2 ulps) for others. These results are consistent across many architectures including Intel, AMD, SGI and DEC Alpha. However, by enabling IEEE double extended precision arithmetic in Intel/AMD 32-bit architectures at no extra computational cost, the (open) interval diameters were reduced to one ulp, which is the best possible solution for this problem. We have also computed the eigenvalues of a tridiagonal matrix which manifests in Gauss-Laguerre quadrature and the results are extremely good in double extended precision but less so in double precision. To demonstrate the accuracy of computed singular values, we have also computed the eigenvalues of the Kac₃₀ matrix, which is the Golub-Kahan form of a bidiagonal matrix. The tridiagonal matrix has known integer eigenvalues. The bidiagonal Cholesky factor of the Gauss-Laguerre tridiagonal is also included in the singular value study.     

Elasticity in certain block monoids via the Euclidean table

This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤ _n , S) where S = (hereafter denoted ℬ _a (n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬ _a (n). In Section 3 we use the Euclidean table to compute the elasticity of ℬ _a (n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬ _a (n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬ _a (p). Part of this work was completed while the second author was on an Academic Leave granted by the Trinity University Faculty Development Committee.     

Doubly transitive 2‐factorizations

Let be a 2‐factorization of the complete graph K_v admitting an automorphism group G acting doubly transitively on the set of vertices. The vertex‐set V(K_v) can then be identified with the point‐set of AG(n, p) and each 2‐factor of is the union of p‐cycles which are obtained from a parallel class of lines of AG(n, p) in a suitable manner, the group G being a subgroup of A G L(n, p) in this case. The proof relies on the classification of 2‐(v, k, 1) designs admitting a doubly transitive automorphism group. The same conclusion holds even if G is only assumed to act doubly homogeneously. © 2006 Wiley Periodicals, Inc. J Combin Designs