On the inverse braid and reflection monoids of type <Emphasis Type="Italic">B</Emphasis>

There are well-known relations between braid and symmetric groups as well as Artin-Brieskorn braid groups and Coxeter groups: the latter are the factor-groups of the Artin-Brieskorn braid groups. The inverse braid monoid is related to the inverse symmetric monoid in the same way. We show that similar relations exist between the inverse braid monoid of type B and the inverse reflection monoid of type B. This gives a presentation of the latter monoid.     

SOME RAMSEY THEORY IN BOOLEAN ALGEBRA FOR COMPLEXITY CLASSES

It is known that for two given countable sets of unary relations A and B on ω there exists an infinite set H ? ω on which A and B are the same. This result can be used to generate counterexamples in expressibility theory. We examine the sharpness of this result.     

Quasi-Homeomorphisms and Measures of Finite Energy Integrals of Generalized Dirichlet Forms

We introduce the quasi-homeomorphisms of generalized Dirichlet forms and prove that any quasi-regular generalized Dirichlet form is quasi-homeomorphic to a semi-regular generalized Dirichlet form. Moreover, we apply this quasi-homeomorphism method to study the measures of finite energy integrals of generalized Dirichlet forms. We show that any 1-coexcessive function which is dominated by a function in is associated with a measure of finite energy integral. Consequently, we prove that a Borel set B is ɛ-exceptional if and only if μ (B) = 0 for any measure μ of finite energy integral. Received May 28, 1999, Revised September 8, 1999, Accepted December 10, 1999     

Endomorphism Seminear-Rings of Brandt Semigroups

We consider the endomorphisms of a Brandt semigroup B _n and the semigroup of mappings E(B _n ) that they generate under pointwise composition. We describe all the elements of this semigroup, determine Green's relations, consider certain special types of mapping, which we can enumerate for each n, and give complete calculations for the size of E(B _n ) for small n.     

Rational obstruction theory and rational homotopy sets

We develop an obstruction theory for homotopy of homomorphisms between minimal differential graded algebras. We assume that has an obstruction decomposition given by and that f and g are homotopic on . An obstruction is then obtained as a vector space homomorphism . We investigate the relationship between the condition that f and g are homotopic and the condition that the obstruction is zero. The obstruction theory is then applied to study the set of homotopy classes . This enables us to give a fairly complete answer to a conjecture of Copeland-Shar on the size of the homotopy set [A,B] whenA and B are rational spaces. In addition, we give examples of minimal algebras (and hence of rational spaces) that have few homotopy classes of self-maps. Received February 22, 1999; in final form July 7, 1999 / Published online September 14, 2000     

On wavelet decomposition of spaces of first order splines

We consider approximate relations in the form of a system of linear algebraic equations that yield B _φ -splines. We construct Lagrange type splines of the first order and give examples of polynomial, trigonometric, hyperbolic, and exponential B _φ -splines. We also construct a system of linear functionals biorthogonal to the B _φ -splines and resolve an interpolation problem generated by this system. For refined nonuniform grids we establish an embedding of spaces of B _φ -splines. The decomposition and reconstruction formulas are obtained. Bibliography: 20 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 47–60.     

Hamilton cycles in block-intersection graphs of triple systems

Given a BIBD S = (V, B), its 1-block-intersection graph G_s has as vertices the elements of B; two vertices B₁, B₂ ∈ B are adjacent in G_s if |B₁ ∩ B₂| = 1. If S is a triple system of arbitrary index λ, it is shown that G_S is hamiltonian. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 243-246, 1999     

Vandermonde matrices on integer nodes: the rectangular case

Summary. We consider a rectangular Vandermonde matrix V on integer nodes. Using combinatorial identities, we give an explicit Cholesky factorization of the matrix , a factorization of the pseudo-inverse of V, and asymptotic estimates of the extremal eigenvalues of B. We also discuss the numerical properties of the proposed formulas. Received March 9, 1999 / Revised version received July 12, 1999 / Published online August 24, 2000     

Operator space projective tensor product of $C^*$-algebras

For a finite dimensional -algebra A and any -algebra B, we determine a constant of equivalence of operator space projective norm and the Banach space projective norm on . We also discuss the *-Banach algebra . Received May 12, 1999; in final form September 8, 1999 / Published online April 12, 2001     

On Compatibility of Interval Fuzzy Preference Relations

This paper defines the concept of compatibility degree of two interval fuzzy preference relations, and gives a compatibility index of two interval fuzzy preference relations. It is proven that an interval fuzzy preference relation B and the synthetic interval fuzzy preference relation of interval fuzzy preference relations A ₁,A ₂,...,A _s are of acceptable compatibility under the condition that the interval fuzzy preference relation B and each of the interval fuzzy preference relations A _l,A ₂,...,A _s are of acceptable compatibility, and thus a theoretic basis has been developed for the application of the interval fuzzy preference relations in group decision making.     

NOTE – A Min-Max Relation for Monotone Path Systems in Simple Regions

xy -plane which bounds the simple polygonal (closed) region D. Let T and B be two finite, disjoint, equicardinal sets of points of D. We give a min-max relation for the maximum number of points of T and B which can be joined by a MPS in D, and a polytime algorithm for finding such a MPS. Received July 15, 1996/Revised October 7, 1999     

G-frames and g-frame sequences in Hilbert spaces

In this paper, we first determine the relations among the best bounds A and B of the g-frame, the g-frame operator S and the pre-frame operator Q and give a necessary and sufficient condition for a g-frame with bounds A and B in a complex Hilbert space. We also introduce the definition of a g-frame sequence and obtain a necessary and sufficient condition for a g-frame sequence with bounds A and B in a complex Hilbert space. Lastly, we consider the stability of a g-frame sequence for a complex Hilbert space under perturbation.     

Binary relations as lattice isomorphisms

Summary In 1963, Zaretskiį established a one-to-one correspondence between the setB _X of binary relations on a set X and the set of triples of the form (W, ϕ, V) where W and V are certain lattices and ϕ: W→V is an isomorphism. We provide a multiplication for these triples making the Zaretskiį correspondence a semigroup isomorphism. In addition, we consider faithful representations ofB _X by pairs of partial transformations and also as the translational hull of its rectangular relations. Using these triples, we study idempotents, regular and completely regular elements and relationsH-equivalent to some relations with familiar properties such as reflexivity, transitivity, etc. Entrata in Redazione il 14 aprile 1998.     

Approximating the complexity measure of Vavasis-Ye algorithm is NP-hard

Given an m×n integer matrix A of full row rank, we consider the problem of computing the maximum of ∥B ^-1 A∥₂ where B varies over all bases of A. This quantity appears in various places in the mathematical programming literature. More recently, logarithm of this number was the determining factor in the complexity bound of Vavasis and Ye’s primal-dual interior-point algorithm. We prove that the problem of approximating this maximum norm, even within an exponential (in the dimension of A) factor, is NP-hard. Our proof is based on a closely related result of L. Khachiyan [1]. Received November 13, 1998 / Revised version received January 20, 1999? Published online May 12, 1999     

Notes on R1-ideals in partial abelian monoids

In the present paper, we deal with a class of R ₁-ideals of cancellative positive partial abelian monoids (CPAMs). We prove that, for I being an R ₁-ideal of a CPAM P, P/I is a CPAM. The lattice of congruence relations associated with R ₁-ideals is a sublattice of the lattice of all equivalence relations. Finally, we prove that an intersection of two Riesz ideals is a Riesz ideal and that the lattice of Riesz ideals is a sublattice of the lattice of all ideals. Received March 19, 1999; accepted in final form December 16, 1999.     

Imaginaries in Boolean algebras

Given an infinite Boolean algebra B, we find a natural class of $\varnothing$‐definable equivalence relations $\mathcal {E}_{B}$ such that every imaginary element from B^eq is interdefinable with an element from a sort determined by some equivalence relation from $\mathcal {E}_{B}$. It follows that B together with the family of sorts determined by $\mathcal {E}_{B}$ admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of imaginaries for Boolean algebras, obtained in 10 .     

(A,B)-Invariant Polyhedral Sets of Linear Discrete-Time Systems

The problem of confining the trajectory of a linear discrete-time system in a given polyhedral domain is addressed through the concept of (A, B)-invariance. First, an explicit characterization of (A, B)-invariance of convex polyhedra is proposed. Such characterization amounts to necessary and sufficient conditions in the form of linear matrix relations and presents two major advantages compared to the ones found in the literature: it applies to any convex polyhedron and does not require the computation of vertices. Such advantages are felt particularly in the computation of the supremal (A, B)-invariant set included in a given polyhedron, for which a numerical method is proposed. The problem of computing a control law which forces the system trajectories to evolve inside an (A, B)-invariant polyhedron is treated as well. Finally, the (A, B)-invariance relations are generalized to persistently disturbed systems.     

On equivalent reformulations for absolute value equations

In this note we consider absolute value equations (AVE) of the type Ax+B|x|=c. We discuss unique solvability of AVE, and its relations with linear complementarity problem (LCP) and mixed integer programming.     

Continuous versus Borel reductions

We present some natural examples of countable Borel equivalence relations E, F with E ≤ _B  F such that there does not exist a continuous reduction from E to F.     

Majorant estimates for partial sums of multiple fourier series of functions from Orlicz spaces vanishing on some set

We prove relations in the braid group that generalize the relations of the formaba=bab. These relations are obtained by means of the generalization to the higher Bruhat orders of the well-known relation between the decompositions of the element of maximal length in the Weyl groupA _n−1 and some specific linear orders on the system of positive roots ofA _n−1 . Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 840–848, December, 1999.