r-Numbers completion problems of partial upper canonical form I

A pair (A, B), where A is an n × n matrix and B is an n × m matrix, is said to have the nonnegative integers sequence {r_j}_j=1^p as the r-numbers sequence if r₁ = rank(B) and r_j = rank[B AB … A^j−1 B] − rank[B AB … A^j−2B], 2 ≤ j ≤ p. Given a partial upper triangular matrix A of size n × n in upper canonical form and an n × m matrix B, we develop an algorithm that obtains a completion A_c of A, such that the pair (A^c, B) has an r-numbers sequence prescribed under some restrictions.     

Imbedding conditions for λ-matrices

We say that A(λ) is λ-imbeddable in B(λ) whenever B(λ) is equivalent to a λ-matrix having A(λ) as a submatrix. In this paper we solve the problem of finding a necessary and sufficient condition for A(λ) to be λ-imbeddable in B(λ). The solution is given in terms of the invariant polynomials of A(λ) and B(λ). We also solve an analogous problem when A(λ) and B(λ) are required to be equivalent to regular λ-matrices. As a consequence we give a necessary and sufficient condition for the existence of a matrix B, over a field F, with prescribed similarity invariant polynomials and a prescribed principal submatrix A.     

Representing matrices

Let A be a nonnegative real matrix whose column set is countable. We give a necessary and sufficient condition on A for the existence of a nonnegative matrix B, B ? A, with column sums equal to prescribed numbers, and row sums not greater than prescribed numbers. This is a generalization of a result of Damerell and Milner, who solved the problem for (0, 1) matrices.     

Hopf extensions of algebras and Maschke type theorems

LetB/C be anA-extension for a Hopf algebraA. We consider two Maschketype questions: first, for an exact sequence of (A, B)-Hopf modules which splitsB-linearly, when does it split (A, B)-linearly? Second, for an exact sequence ofB-modules which splitsC-linearly, when does it splitB-linearly? Finally, we consider when the pair of the restriction functor from mod-B to mod-C and the induction functor ( ) ? _C B is an adjoint pair of functors.     

A Mayer–Vietoris Formula for Persistent Homology with an Application to Shape Recognition in the Presence of Occlusions

In algebraic topology it is well known that, using the Mayer–Vietoris sequence, the homology of a space X can be studied by splitting X into subspaces A and B and computing the homology of A, B, and A∩B. A natural question is: To what extent does persistent homology benefit from a similar property? In this paper we show that persistent homology has a Mayer–Vietoris sequence that is generally not exact but only of order 2. However, we obtain a Mayer–Vietoris formula involving the ranks of the persistent homology groups of X, A, B, and A∩B plus three extra terms. This implies that persistent homological features of A and B can be found either as persistent homological features of X or of A∩B. As an application of this result, we show that persistence diagrams are able to recognize an occluded shape by showing a common subset of points.     

A note on “Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials”

Let F be any field and let B a matrix of F^q×p. Zaballa found necessary and sufficient conditions for the existence of a matrix A=[A_ij]_i,j∈{1,2}∈F^(p+q)×(p+q) with prescribed similarity class and such that A₂₁=B. In an earlier paper [A. Borobia, R. Canogar, Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials, Linear Algebra Appl. 424 (2007) 615-633] we obtained, for fields of characteristic different from 2, a finite step algorithm to construct A when it exists. In this short note we extend the algorithm to any field.     

Some isomorphisms between pairs of latin squares

Let A, B, C, D be latin squares with A orthogonal to B and C orthogonal to D. The pair A, B is isomorphic with the pair C, D if the graph of A, B is graph-isomorphic with the graph of C, D. A characterization is given for determining when a pair A, B of latin squares is isomorphic with a self-orthogonal square C and its transpose. Self-orthogonal squares are important because they are both abundant and easy to store. An algorithm either displays a self-orthogonal square C and an isomorphism from A, B to C, C^T or, if none exists, gives a small set of blocks to the existence of such a square isomorphism.     

On boundaries on the predual of the Lorentz sequence space

Let X be the canonical predual of the Lorentz sequence space and let Au(BX) be the Banach algebra of all complex valued functions defined on the closed unit ball BX of X which are uniformly continuous on BX and holomorphic on the interior of BX, endowed with the sup norm. A characterization of the boundaries for Au(BX) is given in terms of the distance to the strong peak sets of this algebra.     

Dimensionality of biinfinite systems

The problem of determining the uniqueness of the coefficient of interpolation of M compactly supported real functions, with a biinfinite sequence of interpolation points, leads to the study of the kernel Z of the biinfinite block Toeplitz matrix

$\text{D=}\text{?}\text{?}\text{A}\text{B}\text{A}\text{B}\text{?}\text{?}$

. The dimension of Z is found by considering the “maximal solvable subspace” V (relative to A and B). Further results are obtained using the Kronecker canonical form of the matrix pencil A+λB and “restricted generalized inverses” of A (and B).     

MAPS PRESERVING STRONG SKEW LIE PRODUCT ON FACTOR VON NEUMANN ALGEBRAS

Let A be a factor von Neumann algebra and Φ be a nonlinear surjective map from A onto itself.We prove that,if Φ satisfies that Φ(A)Φ(B) - Φ(B)Φ(A)* =AB - BA* for all A,B ∈ A,then there exist a linear b...     

Spectral sequence and finitely presented dimension for weak Gopf–Galois extensions

Let H be a weak Hopf algebra, A a right weak H-comodule algebra, and B the subalgebra of the H-coinvariant elements of A. Let A/B be a right weak H-Galois extension. In this paper, a spectral sequence for Ext which yields an estimate for the global dimension of A in terms of the corresponding data for H and B is constructed. Next, the relationship between the finitely presented dimensions of A and its subalgebra B are given. Further, the case in which A is an n-Gorenstein algebra is studied.     

On the Cohomology Sequence in a Semiabelian Category

In a semiabelian category, a strictly exact sequence 0→A→B→C→0 of cochain complexes gives rise to the cohomology sequence ...→H ⁿ(A) →H ⁿ(B)→ H ⁿ(C)→ H ⁿ⁺¹ (A) →.... We study conditions for exactness of the homology sequence at a given term.     

Lattice tensor products. III - Congruences

G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B, of the lattices Aand B. In Part I of this paper, we showed that for any finite lattice A, we can "coordinatize" A?B, that is, represent A?,B as a subset A of B ^A, and provide an effective criteria to recognize the A-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we prove, for a finite lattice A and a bounded lattice B, the isomorphism Con A ≌ (Con A)B>, which is a special case of a recent result of G. Grätzer and F. Wehrung and a generalization of a 1981 result of G. Grätzer, H. Lakser, and R.W. Quackenbush.     

Sensitivity of convergent matrices and polynomials

A matrix A is said to be convergent if and only if all its characteristic roots have modulus less than unity. When A is real an explicit expression is given for real matrices B such that A + B is also convergent, this expression depending upon the solution of a quadratic matrix equation of Riccati type. If A and A + B are taken to be in companion form, then the result becomes one of convergent polynomials (i.e., polynomials whose roots have modulus less then unity), and is much easier to apply. A generalization is given for the case when A and A + B are complex and have the same number of roots inside and outside a general circle.     

On matrices having equal corresponding principal minors

Let A, B be n × n matrices with entries in a field F. We say A and B satisfy property $\text{D}$ if B or B^t is diagonally similar to A. It is clear that if A and B satisfy property $\text{D}$, then they have equal corresponding principal minors, of all orders. The question is to what extent the converse is true. There are examples which show the converse is not always true. We modify the problem slightly and give conditions on a matrix A which guarantee that if B is any matrix which has the same principal minors as A, then A and B will satisfy property $\text{D}$. These conditions on A are formulated in terms of ranks of certain submatrices of A and the concept of irreducibility.     

Classifying simple closed curve pairs in the 2-sphere and a generalization of the Schoenflies theorem

A classification theory is developed for pairs of simple closed curves (A,B) in the sphere S², assuming that A∩B has finitely many components. Such a pair of simple closed curves is called an SCC-pair, and two SCC-pairs (A,B) and (A^′,B^′) are equivalent if there is a homeomorphism from S² to itself sending A to A^′ and B to B^′. The simple cases where A and B coincide or A and B are disjoint are easily handled. The component code is defined to provide a classification of all of the other possibilities. The component code is not uniquely determined for a given SCC-pair, but it is straightforward that it is an invariant; i.e., that if (A,B) and (A^′,B^′) are equivalent and C is a component code for (A,B), then C is a component code for (A^′,B^′) as well. It is proved that the component code is a classifying invariant in the sense that if two SCC-pairs have a component code in common, then the SCC-pairs are equivalent. Furthermore code transformations on component codes are defined so that if one component code is known for a particular SCC-pair, then all other component codes for the SCC-pair can be determined via code transformations. This provides a notion of equivalence for component codes; specifically, two component codes are equivalent if there is a code transformation mapping one to the other. The main result of the paper asserts that if C and C^′ are component codes for SCC-pairs (A,B) and (A^′,B^′), respectively, then (A,B) and (A^′,B^′) are equivalent if and only if C and C^′ are equivalent. Finally, a generalization of the Schoenflies theorem to SCC-pairs is presented.     

Automorphisms of C1-algebra bundles

Let B be a unital C¹-algebra and A = Γ(E) be the C¹-algebra of sections of a bundle over the separable compact space X with fibre B and structure group Inn B. If B is the quotient of an AW¹-algebra, then an exact sequence: 0 → Inn A → PInn A →^ηH²(X, G), where PInn A is the group of pointwise inner automorphisms of A and G=H^°(Z(B)^, $\text{Z}$) is obtained. The map η is onto whenever A = C(X, B) and B is the quotient of a purely infinite AW¹-algebra. An essential part of the analysis is the result that Ad: U(B) → Inn B is a fibre bundle if and only if the space of inner derivations of B is norm closed. These results extend and clarify previous joint work with I. Raeburn (Indiana Univ. Math. J.29 (1980), 799).     

A sparse counterpart of Reichel and Gragg’s package QRUP

We describe how to maintain the triangular factor of a sparse QR factorization when columns are added and deleted and Q cannot be stored for sparsity reasons. The updating procedures could be thought of as a sparse counterpart of Reichel and Gragg’s package QRUP. They allow us to solve a sequence of sparse linear least squares subproblems in which each matrix B_k is an independent subset of the columns of a fixed matrix A, and B_k+1 is obtained by adding or deleting one column. Like Coleman and Hulbert [T. Coleman, L. Hulbert, A direct active set algorithm for large sparse quadratic programs with simple bounds, Math. Program. 45 (1989) 373-406], we adapt the sparse direct methodology of Björck and Oreborn of the late 1980s, but without forming A^TA, which may be only positive semidefinite. Our Matlab 5 implementation works with a suitable row and column numbering within a static triangular sparsity pattern that is computed in advance by symbolic factorization of A^TA and preserved with placeholders.     

Norms of certain Jordan elementary operators

Let H be a complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. For A,B∈B(H), the Jordan elementary operator UA,B is defined by UA,B(X)=AXB+BXA, ∀X∈B(H). In this short note, we discuss the norm of UA,B. We show that if dimH=2 and ‖UA,B‖=‖A‖‖B‖, then either AB^∗ or B^∗A is 0. We give some examples of Jordan elementary operators UA,B such that ‖UA,B‖=‖A‖‖B‖ but AB^∗≠0 and B^∗A≠0, which answer negatively a question posed by M. Boumazgour in [M. Boumazgour, Norm inequalities for sums of two basic elementary operators, J. Math. Anal. Appl. 342 (2008) 386-393].     

Lattice tensor products. IV

In 1999, for lattices A and B, G. Grätzer and F. Wehrung introduced the lattice tensor product, A?B. In Part I of this paper, we showed that for a finite lattice A and a bounded lattice B, this construction can be "coordinatized,'' that is, represented in B ^A so that the representing elements are easy to recognize. In this note, we show how to extend our method to an arbitrary bounded lattice A to coordinatize A?B.