Sharpening the triple diagonal form

Let R be a Euclidean domain with quotient field F of characteristic not equaling 2. Jacobi showed that every symmetric R-matrix is congruent over R to a matrix in triple diagonal form. Since it is generally not possible to fully diagonalize these matrices, it is of importance to gain as much control as possible of this triple diagonal form. This paper focuses on controlling the off-diagonal elements.     

Lie triple derivation of the Lie algebra of strictly upper triangular matrix over a commutative ring

Let N(n,R) be the nilpotent Lie algebra consisting of all strictly upper triangular n×n matrices over a 2-torsionfree commutative ring R with identity 1. In this paper, we prove that any Lie triple derivation of N(n,R) can be uniquely decomposited as a sum of an inner triple derivation, diagonal triple derivation, central triple derivation and extremal triple derivation for n≧6. In the cases 1≦n≦5, the results are trivial.     

On Generalized Jordan Triple (α, β)*-Derivations and Related Mappings

Let R be a 2-torsion free semiprime *-ring and let α, β be surjective endomorphisms of R. The aim of the paper is to show that every generalized Jordan triple (α, β)*-derivation on R is a generalized Jordan (α, β)*-derivation. This result makes it possible to prove that every generalized Jordan triple (α, β)*-derivation on a semisimple H*- algebra is a generalized Jordan (α, β)*-derivation. Finally, we prove that every Jordan triple left α*-centralizer on a 2-torsion free semiprime ring is a Jordan left α*-centralizer.     

Normal dilatation of triangular matrices

LetR be a (real or complex) triangular matrix of ordern, say, an upper triangular matrix. Is it true that there exists a normaln×n matrixA whose upper triangle coincides with the upper triangle ofR? The answer to this question is “yes” and is obvious in the following cases: (1)R is real; (2)R is a complex matrix with a real or a pure imaginary main diagonal, and moreover, all the diagonal entries ofR belong to a straight line. The answer is also in the affirmative (although it is not so obvious) for any matrixR of order 2. However, even forn=3 this problem remains unsolved. In this paper it is shown that the answer is in the affirmative also for 3×3 matrices.     

Matrix equivalence and isomorphism of modules

It is well known that if A and B are n × m matrices over a ring R, then coker A ? coker B does not imply A and B are equivalent. An elementary proof is given that the implication does hold if 1 is in the stable range of R. Furthermore, for certain R (including commutative rings), if A is block diagonal and B is block upper triangular with the same diagonal blocks as A, then coker A ? coker B implies A and B are equivalent under a special equivalence. This extends results of Roth and Gustafson. As a corollary, a theorem on decomposition of modules is obtained.     

An extension of a theorem of Frobenius and Stickelberger to modules of projective dimension one over a factorial domain

Let R be a Cohen–Macaulay ring. A quasi-Gorenstein R-module is an R-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Cohen–Macaulay factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative factorial Cohen–Macaulay domain.     

Resonance in equations with a diagonal field

Let A be a real square matrix, and let J?$\text{R}$ be an interval not containing an eigenvalue of A. Is A–D nonsingular for all diagonal matrices D with entries d_i ∈J? This holds if A is symmetric, but is not true in general. We prove a necessary condition and indicate implications for an equation with a diagonal field.     

A dynamical systems result on asymptotic integration of linear differential systems

We are interested in the asymptotic integration of linear differential systems of the form x′=[Λ(t)+R(t)]x, where Λ is diagonal and R∈L^p[t₀,∞) for p∈[1,2]. Our dichotomy condition is in terms of the spectrum of the omega-limit set ω_Λ. Our results include examples that are not covered by the Hartman-Wintner theorem.     

Full field algebras, operads and tensor categories

We study the operadic and categorical formulations of (conformal) full field algebras. In particular, we show that a grading-restricted R×R-graded full field algebra is equivalent to an algebra over a partial operad constructed from spheres with punctures and local coordinates. This result is generalized to conformal full field algebras over VL⊗VR, where VL and VR are two vertex operator algebras satisfying certain finiteness and reductivity conditions. We also study the geometry interpretation of conformal full field algebras over VL⊗VR equipped with a nondegenerate invariant bilinear form. By assuming slightly stronger conditions on VL and VR, we show that a conformal full field algebra over VL⊗VR equipped with a nondegenerate invariant bilinear form exactly corresponds to a commutative Frobenius algebra with a trivial twist in the category of VL⊗VR-modules. The so-called diagonal constructions [Y.-Z. Huang, L. Kong, Full field algebras, arXiv: math.QA/0511328] of conformal full field algebras are given in tensor-categorical language.     

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on Rn is n+1, thus complementing a recent result due to Feldman.     

Notes on Jordan (σ, τ)*-derivations and Jordan triple (σ, τ)*-derivations

Let R be a 2-torsion free semiprime *-ring, σ, τ two epimorphisms of R and f, d : R → R two additive mappings. In this paper we prove the following results: (i) d is a Jordan (σ, τ)*-derivation if and only if d is a Jordan triple (σ, τ)*-derivation. (ii) f is a generalized Jordan (σ, τ)*-derivation if and only if f is a generalized Jordan triple (σ, τ)*-derivation.     

Reducing complex matrices to condensed forms by unitary congruence transformations

We show that any n × n conjugate-normal matrix can be brought by a unitary congruence transformation to block-tridiagonal form with the orders of the consecutive diagonal blocks not exceeding 1, 2, 3, ..., respectively. The proof is constructive; namely, a finite process is described that implements the reduction to the desired form. Sufficient conditions are indicated for the orders of the diagonal blocks to stabilize. In this case, the condensed form is a band matrix.     

Determinantal formulae for matrices with sparse inverses,II: asymmetric zero patterns

In an earlier paper, formulae for det A as a ratio of products of principal minors of A were exhibited, for any given symmetric zero-pattern of A⁻¹. These formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of A⁻¹. However, just as the determinant of a diagonal and of a triangular matrix are both the product of the diagonal entries, the symmetry of the zero pattern is not essential for these formulae. We describe here how analogous formulae for det A may be obtained in the asymmetric-zero-pattern case by introducing a directed spanning tree. We also examine the converse question of determining all possible zero patterns of A⁻¹ which guarantee that a certain determinantal formula holds.     

Characterization of affine Steiner triple systems and Hall triple systems

It is known that a Steiner triple system is projective if and only if it does not contain the four-triple configuration C₁₄. We find three configurations such that a Steiner triple system is affine if and only if it does not contain any of these configurations. Similarly, we characterize Hall triple systems, a superclass of affine Steiner triple systems, using two forbidden configurations.     

Rank-preserving diagonal completions of a matrix

Suppose A is an n-by-n matrix over a field F. We prove that it is possible to complete the diagonal entries of A so that the resulting rank of A is as small as possible when n⩾3r, where r is the “off-diagonal rank” of A and (n,r)≠(3,1).     

Reduced convex bodies in the plane

A convex bodyR of Euclideand-spaceE ^d is called reduced if there is no convex body properly contained inR of thickness equal to the thickness Δ(R) ofR. The paper presents basic properties of reduced bodies inE ². Particularly, it is shown that the diameter of a reduced bodyR?E ² is not greater than √2Δ(R), and that the perimeter is at most (2+½π)Δ(R). Both the estimates are the best possible.     

Abelian groups as UA-modules over their endomorphism ring

Suppose that V is a module over a ring R. Themodule V is called a unique addition module (UA-module) if it is not possible to change the addition on the set V without changing the action of R on V. In this paper, we find Abelian groups that are UA-modules over their endomorphism ring.     

Wagner's theorem for Torus graphs

Wagner's theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph having p vertices may be diagonally transformed into any other maximal torus graph having p vertices. As with Wagner's theorem, a normal form representing an intermediate stage in the above transformation is displayed. This result, along with Wagner's theorem, may make possible constructive characterizations of planar and toroidal graphs, through a wholly combinatorial definition of diagonal transformation.     

Strongly rational comodules and semiperfect Hopf algebras over QF rings

Let C be a coalgebra over a QF ring R. A left C-comodule is called strongly rational if its injective hull embeds in the dual of a right C-comodule. Using this notion a number of characterizations of right semiperfect coalgebras over QF rings are given, e.g., C is right semiperfect if and only if C is strongly rational as left C-comodule. Applying these results we show that a Hopf algebra H over a QF ring R is right semiperfect if and only if it is left semiperfect or — equivalently — the (left) integrals form a free R-module of rank 1.     

Isometry theorem for the Segal-Bargmann transform on a noncompact symmetric space of the complex type

We consider the Segal-Bargmann transform on a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the dual compact case. The isometry theorem involves integration over a tube of radius R in the complexification, followed by analytic continuation with respect to R. A cancellation of singularities allows the relevant integral to have a nonsingular extension to large R, even though the function being integrated has singularities.