Matroid Shellability, β-Systems,and Affine Hyperplane Arrangements

The broken-circuit complex is fundamental to the shellability and homology of matroids, geometric lattices, and linear hyperplane arrangements. This paper introduces and studies the -system of a matroid, nbc(M), whose cardinality is Crapo's -invariant. In studying the shellability and homology of base-pointed matroids, geometric semilattices, and afflne hyperplane arrangements, it is found that the -system acts as the afflne counterpart to the broken-circuit complex. In particular, it is shown that the -system indexes the homology facets for the lexicographic shelling of the reduced broken-circuit complex , and the basic cycles are explicitly constructed. Similarly, an EL-shelling for the geometric semilattice associated with M is produced,_and it is shown that the -system labels its decreasing chains.Basic cycles can be carried over from The intersection poset of any (real or complex) afflnehyperplane arrangement is a geometric semilattice. Thus the construction yields a set of basic cycles, indexed by nbc(M), for the union of such an arrangement.     

Bounds for optimalα-β binary trees

In this paper we consider a special kind of binary trees where each right edge is associated with a positive number and each left edge with a positive number( ). Given, and the number of nodesn, an optimal tree is one which minimizes the total weighted path length. An algorithm for constructing an optimal tree for given, , n is presented, based on which bounds for balances and total weighted path lengths of optimal trees are derived.     

Strictly Cyclic Algebra of Operators Acting on Banach Spaces H p(β)

Let be a sequence of positive numbers and 1 p< . We consider the space H ^p() of all power series such that . We investigate strict cyclicity of the weakly closed algebra generated by the operator of multiplication by zacting on H ^p(), and determine the maximal ideal space, the dual space and the reflexivity of the algebra . We also give a necessary condition for a composition operator to be bounded on H ^p() when is strictly cyclic.     

Exchange Rings with Artinian Primitive Factors

We prove that every exchange ring with primitive factors Artinian is clean. Also, it is shown that for exchange rings with Artinian primitive factors, the following are equivalent: (1) Every element in R is a sum of two units. (2) There exist , U(R) such that + = 1. (3) R does not have Z / 2 Z as a homomorphic image. Finally, we prove that exchange ring R is strongly -regular if the Jacobson radical of any homomorphic image of R is T-nilpotent or locally nilpotent. These are generalizations of the corresponding results of A. Badawi, W. D. Burgess and P. Menal, Fisher and Snider, and J. Stock.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

On θ -Preirresolute Functions

We introduce a new class of functions called -preirresolute functions which is contained in the class of quasi preirresolute functions and contains the class of preirresolute functions. It is shown that p-closedness and -connectedness are preserved by -preirresolute surjections.     

Explicit Formulas for Green’s Functions on the Annulus and on the Möbius Strip

Given a fixed point free antianalytic involution k of a domain G in thecomplex plane, bounded by a finite number of analytic curves, k-invariant Greensfunctions are defined on G. The Lindelöfs principle is extended to k-invariantGreens functions. When G is the annulus, k-invariant Greens functions areobtained in the explicit form. Since the factorization of the annulus by the group kgenerated by k produces a Möbius strip, the respective result helped us to obtain explicitforms for Greens functions on the Möbius strip.     

Latteninterpolation bei einer Funktion von zwei Veränderlichen

Summary In each lattice point , of a rectangular net a numerical valueu is given. A bicubical and twice continuously differentiable function is constructed interpolating the valuesu . The method is known as «spline interpolation».     

Some Asymptotic Results for the M/M/∞ Queue with Ranked Servers

We consider an M/M/ model with m primary servers and infinitely many secondary ones. An arriving customer takes a primary server, if one is available. We derive integral representations for the joint steady state distribution of the number of occupied primary and secondary servers. Letting =/ be the ratio of arrival and service rates (all servers work at rate ), we study the joint distribution asymptotically for . We consider both m=O(1) and m scaled to be of the same order as . We also give results for the marginal distribution of the number of secondary servers that are occupied.     

(φ,ξ,η,λ)‐Structures of Parabolic Type

We consider (,,,)structures of parabolic type on hypersurfaces of dual spaces and study the rank of the affinor . We consider almost contact metric structures of parabolic type of the first kind on hypersurfaces of 4dimensional dual metric space. We study the properties of these structures and give examples of normal, integrable, and Sasakian parabolic structures.     

Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Let (0,1) be an irrational, and [a ₁,a ₂,...] be the continued fraction expansion of . Let H be the one-dimensional Schrödinger operator with Sturmian potentials. We show that if the potential strength V>20, then the Hausdorff dimension of the spectrum (H ) is strictly great than zero for any irrational , and is strictly less than 1 if and only if liminf _k(a ₁ a ₂a _k ))^1/k <.     

Quasi-P*-Maps,P(τ, α, β)-Maps,Exceptional Family of Elements,and Complementarity Problems

Quasi-P_*-maps and P(, , )-maps defined in this paper are two large classes of nonlinear mappings which are broad enough to include P_*-maps as special cases. It is of interest that the class of quasi-P_*-maps also encompasses quasimonotone maps (in particular, pseudomonotone maps) as special cases. Under a strict feasibility condition, it is shown that the nonlinear complementarity problem has a solution if the function is a nonlinear quasi-P_*-map or P(, , )-map. This result generalizes a classical Karamardian existence theorem and a recent result concerning quasimonotone maps established by Hadjisawas and Schaible, but restricted to complementarity problems. A new existence result under an exceptional regularity condition is also established. Our method is based on the concept of exceptional family of elements for a continuous function, which is a powerful tool for investigating the solvability of complementarity problems.     

Tanaka Formulae for (α, d, β)-Superprocesses

We establish Tanaka like formulae for the local time of (, d, )-superprocess in the dimensions where the local time exists. The result generalizes the result of Adler, Lewin who proved existence of Tanaka formulae for a class of super-processes with finite variance. The fact that we abandon the finite variance assumption, requires using an L ¹⁺ convergence argument (with 0<<1) rather than L ² convergence, for the derivation of the Tanaka formulae.     

Boolean Term Orders and the Root System Bn

A Boolean term order is a total order on subsets of [n] ={1,..., n} such that for all [n], , and for all with ( ) = . Boolean term orders arise in several different areas of mathematics, including Gröbner basis theory for the exterior algebra, and comparative probability.The main result of this paper is that Boolean term orders correspond to one-element extensions of the oriented matroid M(B_n), where B_n is the root system {e_i : 1 i n} {e_i ± e_j : 1 i < j n}. This establishes Boolean term orders in the framework of the Baues problem, in the sense of (Reiner, 1998). We also define a notion of coherence for a Boolean term order, and a flip relation between different term orders. Other results include examples of noncoherent term orders, including an example exhibiting flip deficiency, and enumeration of Boolean term orders for small values of n.     

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings and on the power set of a quasi-ordering (Q,). We set X Y if every xX is majorized in by some yY, and X Y if every yY is minorized in by some xX. We show that both these quasi-orderings are -wqo if and only if the original quasi-ordering is ( )-wqo. For this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.     

On the question of the algebraic independence of algebraic powers of algebraic numbers

We obtain results showing that transcendental numbers of the form , wherea0, 1, is irrational, anda and are algebraic numbers, cannot be expressed algebraically in terms of two of the numbers. The proof is carried out by A. O. Gel'fond's method.Translated from Matematicheskie Zametki, Vol. 11, No. 6, pp. 635–644, June, 1972.     

The self-similar solutions to a fast diffusion equation

A quasilinear equation u -x·u/2+f(u)=0 is studied, wheref(u)=–u+u , > 0, 0<. <1, >1 andx R ⁿ. The equation arises from the study of blow-up self-similar solutions of the heat equation _t =+. We prove the existence and non-existence of ground state for various combination of , and . In particular, we prove that when / < forn=1,2 or / < (n + 2) /(n – 2) forn 3 there exists no non-constant positive radial self-similar solution of the parabolic equation, but for many cases where / > (n + 2)/(n – 2) there exists an infinite number of non-constant positive radial self-similar solutions.     

On an extension of a lattice-continuous mapping with embeddability applications

It is well known that ifX andY are completely regularT ₂ spaces, then any continuous function,f, fromX toY, has a unique continuous extension,(f), fromX toY, whereX andY are the Stone—ech compactifications ofX andY, respectively. This function plays an important role in Stone—ech Theory, especially in questions pertaining to embeddability.In this paper, we first extend this construction to general Wallman spaces, and then apply the results to extend well-known embeddability theorems.     

On the Completeness and Decidability of the Horn‐Like Fragment of the First‐Order Linear Temporal Logic

We prove the completeness and decidability of the Hornlike sequents, specifically, the socalled D₂sequents (of the firstorder linear temporal logic) considered in the author's paper [Lith. Math. J., 41(3), 266–281 (2001)]. In this paper, with the help of the infinitary calculus G_L, grounded by the author in his earlier papers, for D₂sequents we construct a D₂Sat calculus of the socalled saturated type consisting of decidable deductive procedures replacing the omegarule for the always operator. In the present paper, in order to prove the completeness and decidability of the calculus D₂Sat, we construct the socalled invariant decidable calculus D₂IN. We prove the equivalence of the calculi D₂IN, D₂Sat, and G _L ^** for the socalled saturated D₂sequents. From this equivalence, by reducing an arbitrary D₂sequent to a saturated D₂sequent, and also from the completeness of the G _L ^** calculus and decidability of the invariant calculus D₂IN, we deduce the completeness and decidability of the calculus D₂Sat in the class of D₂sequents.     

Stability Tests for Two Common Classes of Linear Time‐Delay and Hybrid Systems

The stability of linear timedelay systems with point internal delays is difficult to deal with in practice because of the fact that their characteristic equation is usually of transcendent type rather than of polynomial type. This feature usually causes the system to possess an infinite number of poles. In this paper, stability tests for this class of systems are obtained based either on extensions of classical tests applicable to delayfree systems or on approaches within the framework of twodimensional digital filters. Some of those twodimensional stability tests are also proved to be useful for stability testing of a common class of linear hybrid systems which involve coupled continuous and digital substates after a slightadhoc adaptation of the tests for that situation.