Multiplicative B-Product and Its Properties

The author considers a special kind of B-product which is called the multiplicative B-product. The most important property of multiplicative B-product is that it preserves multiplicativity of functions.     

B-Theory of Runge-Kutta methods for stiff volterra functional differential equations

B-stability andB-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra functional differential equations (VFDEs) are established which provide unified theoretical foundation for the study of Runge-Kutta methods when applied to nonlinear stiff initial value problems (IVPs) in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs) and VFDEs of other type which appear in practice.     

On the rearrangement estimates and the boundedness of the generalized fractional integrals associated with the Laplace-Bessel differential operator

We introduce the generalized fractional integrals (generalized B-fractional integrals) generated by the Δ _B Laplace-Bessel differential operator and give some results for them. We obtain O’Neil type inequalities for the B-convolutions and give pointwise rearrangement estimates of the generalized B-fractional integrals. Then we get the L _p,γ -boundedness of the generalized B-convolution operator, the generalized B-Riesz potential and the generalized fractional B-maximal function. Finally, we prove a sharp pointwise estimate of the nonincreasing rearrangement of the generalized fractional B-maximal function. V. S. Guliyev was partially supported by the grant of INTAS (Nr. 05-1000008-8157). Z. V. Safarov was partially supported by INTAS YS Post Doctoral Fellowship (Nr. 05-113-4671).     

Pointwise and integral estimates for the B-Riesz potential in terms of B-maximal and B-fractional maximal functions

We study the maximal and fractional maximal functions and Riesz potentials that are generated by the generalized shift operator associated with the Laplace-Bessel operator. We obtain some pointwise and integral estimates that give a relation between the B-maximal and B-fractional maximal functions and B-Riesz potentials and extend the available results to the objects of a more general nature. Basing on these results, we prove interpolation theorems for the B-fractional maximal functions and B-Riesz potentials.     

The Role of the Inner Product in Stopping Criteria for Conjugate Gradient Iterations

Two natural and efficient stopping criteria are derived for conjugate gradient (CG) methods, based on iteration parameters. The derivation makes use of the inner product matrix B-defining the CG method. In particular, the relationship between the eigenvalues and B-norm of a matrix is investigated, and it is shown that the ratio of largest to smallest eigenvalues defines the B-condition number of the matrix. Upper and lower bounds on various measures of the error are also given. The compound stopping criterion presented here is an obvious default in software packages because it does not require any additional norm computations.     

Game-perfect graphs

A graph coloring game introduced by Bodlaender (Int J Found Comput Sci 2:133–147, 1991) as coloring construction game is the following. Two players, Alice and Bob, alternately color vertices of a given graph G with a color from a given color set C, so that adjacent vertices receive distinct colors. Alice has the first move. The game ends if no move is possible any more. Alice wins if every vertex of G is colored at the end, otherwise Bob wins. We consider two variants of Bodlaender’s graph coloring game: one (A) in which Alice has the right to have the first move and to miss a turn, the other (B) in which Bob has these rights. These games define the A-game chromatic number resp. the B-game chromatic number of a graph. For such a variant g, a graph G is g-perfect if, for every induced subgraph H of G, the clique number of H equals the g-game chromatic number of H. We determine those graphs for which the game chromatic numbers are 2 and prove that the triangle-free B-perfect graphs are exactly the forests of stars, and the triangle-free A-perfect graphs are exactly the graphs each component of which is a complete bipartite graph or a complete bipartite graph minus one edge or a singleton. From these results we may easily derive the set of triangle-free game-perfect graphs with respect to Bodlaender’s original game. We also determine the B-perfect graphs with clique number 3. As a general result we prove that complements of bipartite graphs are A-perfect.      

On Some Rings of Arithmetical Functions

In this paper, we consider several constructions which from a given B-product * _B lead to another one We shall be interested in finding what algebraic properties of the ring are shared also by the ring . In particular, for some constructions the rings R _B and will be isomorphic and therefore have the same algebraic properties.     

An SQP algorithm for extended linear-quadratic problems in stochastic programming

Extended Linear-Quadratic Programming (ELQP) problems were introduced by Rockafellar and Wets for various models in stochastic programming and multistage optimization. Several numerical methods with linear convergence rates have been developed for solving fully quadratic ELQP problems, where the primal and dual coefficient matrices are positive definite. We present a two-stage sequential quadratic programming (SQP) method for solving ELQP problems arising in stochastic programming. The first stage algorithm realizes global convergence and the second stage algorithm realizes superlinear local convergence under a condition calledB-regularity.B-regularity is milder than the fully quadratic condition; the primal coefficient matrix need not be positive definite. Numerical tests are given to demonstrate the efficiency of the algorithm. Solution properties of the ELQP problem underB-regularity are also discussed.Supported by the Australian Research Council.     

Polynomials of Least Deviation from Zero

A new family of polynomials of least deviation from zero is defined on the unit disk B. Lower bounds for best approximations in the space L _p (B), p ≥ 1, are given.__________Translated from Matematicheskie Zametki, vol. 78, no. 2, 2005, pp. 308–313.Original Russian Text Copyright © 2005 by V. A. Yudin.     

Continuity Properties of Distributions with Some Decomposability

Absolute continuity and smoothness of distributions in the nested subclasses ~L _m (B), m = 0, 1, 2,..., of the class of all B-decomposable distributions are studied. All invertible matrices are classified into two types in terms of P.V. numbers. The minimum integer m for which all full distributions in ~L _m (B) are absolutely continuous and the minimum integer m for which all absolutely continuous distributions in ~L _m (B) have the densities of class C ^r for 0 r are discussed according to the type of the matrix B related to P.V. numbers.     

On BF-algebras

In this paper we introduce the notion of BF-algebras, which is a generalization of B-algebras. We also introduce the notions of an ideal and a normal ideal in BF-algebras. We investigate the properties and characterizations of them.      

Representation of finite groups and the first Betti number of branched coverings of a universal Borromean orbifold

The paper studies the first homology of finite regular branched coverings of a universal Borromean orbifold called B _4,4,4ℍ³. We investigate the irreducible components of the first homology as a representation space of the finite covering transformation group G. This gives information on the first betti number of finite coverings of general 3-manifolds by the universality of B _4,4,4. The main result of the paper is a criterion in terms of the irreducible character whether a given irreducible representation of G is an irreducible component of the first homology when G admits certain symmetries. As a special case of the motivating argument the criterion is applied to principal congruence subgroups of B _4,4,4. The group theoretic computation shows that most of the, possibly nonprincipal, congruence subgroups are of positive first Betti number. This work is partially supported by the Sonderforschungsbereich 288.     

Properties A and B of nth Order Linear Differential Equations with Deviating Argument

Sufficient conditions for the nth order linear differential equation

Solutions to the XXX type Bethe ansatz equations and flag varieties

We consider a version of the A _N Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from a given one is called a population. We show that a population is isomorphic to the sl _N +1 flag variety and that the populations are in one-to-one correspondence with intersection points of suitable Schubert cycles in a Grassmanian variety. We also obtain similar results for the root systems B _N and C _N . Populations of B _N and C _N type are isomorphic to the flag varieties of C _N and B _N types respectively.     

Equivalence of the C*-Algebras qℂ and C 0(ℝ2) in the Asymptotic Category

The results of Kasparov, Connes, Higson, and Loring imply the coincidence of the functors [[qℂ ⊗ K, B ⊗ K]] = [[C ₀(ℝ²) ⊗ K, B ⊗ K]] for any C*-algebra B; here[[A, B]] denotes the set of homotopy classes of asymptotic homomorphisms from A to B. Inthe paper, this assertion is strengthened; namely, it is shown that the algebras qℂ ⊗ K and C ₀(ℝ²) ⊗ K are equivalent in the category whose objects are C*-algebras and morphisms are classes of homotopic asymptotic homomorphisms. Some geometric properties of the obtained equivalence are studied. Namely, the algebras qℂ ⊗ K and C ₀(ℝ²) ⊗ K are represented as fields of C*-algebras; it is proved that the equivalence is not fiber-preserving, i.e., is does not take fibers to fibers. It is also proved that the algebras under consideration are not homotopy equivalent.__________Translated from Matematicheskie Zametki, vol. 77, no. 5, 2005, pp. 788–796.Original Russian Text Copyright ©2005 by T. V. Shul’man.     

Nonadaptive search problem with sets of equal sum

Consider the set A={1,2,3,…,2 ⁿ }, n≥3 and let x∈ A be unknown element. For given natural number S we are allowed to ask whether x belongs to a subset B of A such that the sum of the elements of B equals S. We investigate for which S it is possible to find x using a nonadaptive search.     

Controlling the space with preassigned responses

It is shown that large classes of control systems, which include certain systems of the typex+A(t)x=B(t)u, can be handled in such a way that the control functionsu(t) are actually associated with responsesx(t) that belong to known families of functions. In particular, it is possible, for a variety of perturbationsB(t)u and operatorsA(t) with convex domains, to have responses that are line segments joining the origin to the reachable states.The present approach establishes the fact that a vast number of results from functional analysis concerning ranges of operators can be effectively applied to the general theory of control. It is also rather significant that the present theory does not necessarily require the solvability of the associated Cauchy problem.The operatorsB(t)u do not have to be invertible inu. However, it is shown that continuous controlsu(t) can be obtained for a variety of problems whenB ^–1(t)u exists and is continuous int.     

A new characterization of Anderson’s inequality in <Emphasis Type="Italic">C</Emphasis><Subscript>1</Subscript>-classes

Let ℋ be a separable infinite dimensional complex Hilbert space, and let ℒ(H) denote the algebra of all bounded linear operators on ℋ into itself. Let A = (A ₁, A ₂,..., A _n), B = (B ₁, B ₂,..., B _n) be n-tuples of operators in ℒ(H); we define the elementary operators Δ _A,B : ℒ(H) ↦ ℒ(H) by

On almost-divided domains

Let B be a domain, Q a maximal ideal of B, π: B → B/Q the canonical surjection, D a subring of B/Q, and A:=π ⁻¹(D). If both B and D are almost-divided domains (resp., n-divided domains), then A = B × _B/Q D is an almost-divided domain (resp., an n-divided domain); the converse holds if B is quasilocal. If 2 ≤ d ≤ ∞, an example is given of an almost-divided domain of Krull dimension d which is not a divided domain.      

Simulation input data modeling

Input data modeling is a critical component of a successful simulation application. A perspective of the area is given with an emphasis on available probability distributions as models, estimation methods, model selection and discrimination, and goodness of fit. Three specific distribution classes (lambda,S _B , TES processes) are discussed in some detail to illustrate characteristics that favor input models. Regarding estimation, we argue for maximum likelihood estimation over method of moments and other matching schemes due to intrinsic superior properties (presuming a specific model) and the capability of accommodating messy data types. We conclude with a list of specific research problems and areas warranting additional attention.