Regularity and Unit-regularity of Generalized Semigroups of Linear Transformations

Let V and W be vector spaces over a division ring D and L_D (V, W) the set of all linear transformations from V into W. For L_D(W, V), let (L_D (V, W), ) denote the semigroup L_D (V, W) with the operation * defined by * = for all , L_D(V, W). By a unit-regular semigroup we mean a semigroup S with identity having the property that for each a S, a = aua for some unit u S. The main purpose of this paper is to prove the following statements. The semigroup (L_D(V, W), ) is regular if and only if V = {0}, W = {0} or is an isomorphism from W onto V. The semigroup (L_D (V, W), ) is unit-regular if and only if (i) V = {0}, (ii) W = {0} or (iii) is an isomorphism from W onto V and dim_D V .     

Optimal selection of the four best of a sequence

We consider the situation in which the decision-maker is allowed to have four choices with purpose to choose exactly the four absolute best candidates fromN applicants. The optimal stopping rule and the maximum probability of making the right choice are given for largeNN, the maximum asymptotic value of the best choice being lim _N P(win)0.12706.     

Explicitly solvable model of Mössbauer scattering

An explicitly solvable model of Mössbauer scattering of rays by a nucleus bound in a harmonic-oscillator potential is constructed. The probability of elastic scattering, which is proportional to the Debye—Waller factor, is calculated in the framework of the explicitly solvable scattering problem. It is assumed that the rms deviation x of the nucleus and the photon wave numberk satisfykxE /E , whereE andE are typical energy levels of the photon and the oscillator states.St Petersburg State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 3, pp. 439–430, June, 1993.     

Multivalued Stochastic Differential Equations: Convergence of a Numerical Scheme

In this paper we show the strong mean square convergence of a numerical scheme for a R ^d -multivalued stochastic differential equation: dX _t +A(X _t )dtb(t,X _t )dt+(t,X _t )dW _t and obtain the rate of convergence O(( log(1/)^1/2) when the diffusion coefficient is bounded. By introducing a discrete Skorokhod problem, we establish L ^p -estimates (p2) for the solutions and prove the convergence by using a deterministic result. Numerical experiments for the rate of convergence are presented.     

Discrete Series Characters for GL(n, q)

The discrete series characters of the finite general linear group GL(n, q) are expressed as uniquely defined integral linear combinations of characters induced from linear characters on certain subgroups Hd, n of GL(n, q). The coefficients in these linear combinations are determined (for all n, q) by a family of polynomials r(T) Z[T] indexed by the set of all partitions .     

Eine Charakterisierung der (Rk,Sk+1)-Ringe

Ohne ZusammenfassungDie Ergebnisse dieser Note sind enthalten in der vom Verfasser als Habilitationsschrift eingereichten Arbeit Basische Elemente in Moduln über noetherschen Ringen.     

Level set representation for the Gibbs states of the ferromagnetic ising model

Summary We prove the existence of a real valued random field with parameters in thed-dimensional cubic lattice, such that the distribution of the level set of this random field is a Gibbs state for the nearest neighbour ferromagnetic Ising model. Using this, we prove the continuity of the percolation probability with respect to the parameter (,h) in the uniqueness region except on the critical curve _c ={(,h _c ())}, whereh _c() is the critical level of the external field above which percolation takes place.Supported in part by JSPS, BiBoS, Grant in Aid for Cooperative research no. 62303006 and Grant in Aid for Scientific Research no. 63540168     

On translations of double rays in graphs

Let be an infinite graph, let be a double ray in , and letd andd denote the distance functions in and in , respectively. One calls anaxis ifd(x,y)=d (x,y) and aquasi-axis if lim infd(x,y)/d (x,y)>0 asx, y range over the vertex set of andd (x,y). The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if is a translation whose directionD() is a thin end, then there exists an axis inD() andD(^–1) invariant under ^r for somer not exceeding the maximum number of disjoint rays inD().The thinness ofD() is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.     

Opérateurs différentiels associés à certaines représentations unitaires d'un groupe de Lie résoluble exponentiel

Soient G = exp g un groupe de Lie résoluble exponentiel et H = exp h un sous-groupe connexe de G. Soient un caractère unitaire de H et = Ind_H ^G. Soit D(G/H) l'algèbre des opérateurs différentiels G-invariants sur G/H. Une question posée par Duflo et Corwin-Greenleaf consiste à voir si la finitude des multiplicités de est équivalente à la commutativité de D (G/H). Nous répondons positivement à cette question quand H est normal dans G. Lorsque H n'est pas normal, nous préparons le terrain pour d'espaces homogènes nilpotents et nous répondons à la question dans différents cas. Nous étudions finalement l'algèbre D (G) ^H , des opérateurs différentiels qui laissent l'espace des vecteurs C de invariant et qui commuttent avec l'action de H sur cet espace.     

On Some Structural Properties of Banach Function Spaces and Boundedness of Certain Integral Operators

In this paper the notions of uniformly upper and uniformly lower -estimates for Banach function spaces are introduced. Further, the pair (X, Y) of Banach function spaces is characterized, where X and Y satisfy uniformly a lower -estimate and uniformly an upper -estimate, respectively. The integral operator from X into Y of the form

Commutation Relations in an Indefinite-Metric Space

We describe the irreducible regular representations of the algebra of operators a and b defined by [a,b]=1 and ba=a ⁺ b ⁺ in an arbitrary nondegenerate closed indefinite-metric space. We find the relation of this algebra to the generalized Heisenberg algebra.     

Odd Systems of Vectors and Related Lattices

We consider uniform odd systems, i.e. sets of vectors of constant odd norm with odd inner product, and the lattice L(V) linearly generated by a uniform odd system V of odd norm 2t+1. If uu p (mod 4) for all u V, one has v² p (mod 4) if v² is odd and v² 0 (mod 4) if v² is even, for any vector v L(V). The vectors of even norm form a double even sublattice L₀(V) of L(V), i.e. is an even lattice. The closure of V, i.e. all vectors of L(V) of norm 2t+1, are minimal vectors of L(V) for t=1, and they are almost always minimal for t=2. For such t, the convex hull of vectors of the closure of V is an L-polytope of L₀V and the contact polytope of L(V). As an example, we consider closed uniform odd systems of norm 5 spanning equiangular lines.     

Chapter 1 Introduction: Extensions and new developments in DEA

The extensions, new developments and new interpretations for DEA covered in this paper include: (1) new measures of efficiency, (2) new models and (3) new ways of implementing established models with new results and interpretations presented that include treatments of congestion, returns-to-scale and mix and technical inefficiencies and measures of efficiency that can be used to reflect all pertinent properties. Previously used models, such as those used to identify allocative inefficiencies, are extended by means of assurance region approaches which are less demanding in their information requirements and underlying assumptions. New opportunities for research are identified in each section of this chapter. Sources of further developments and possible sources for further help are also suggested with references supplied to other papers that appear in this volume and which are summarily described in this introductory chapter.     

Topological degree of a family of convex functions and localization of nash equilibrium points

We define the degree of a family of convex functionsf _x, wherex , and prove that, if this degree is different from zero, one has at least one Nash equilibrium point in . This is a criterion of localization of such equilibrium points.     

Reduction of a multidimensional poincare-invariant nonlinear equation to two-dimensional equations

We study the structure of the invariants of the extended isochronous Galilean algebra which is a subalgebra of the Poincaré algebra AP(1, n). Using these results we classify maximal subalgebras of rank n–2 and n–1 of AP(1, n). With respect to subalgebras of rank n–1 of AP(1, n) ansatzes are constructed reducing the equation (u, (u)², u)=0 to differential equations in two invariant variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 10, pp. 1311–1323, October, 1991.     

On summability of subsequences of partial sums of trigonometric Fourier series

n- (n1) fL ^p ([–, ] ⁿ ),=1 = (L C) . , , f([–, ] ⁿ ).     

On Interpolation of the Fourier Maximal Operator in Orlicz Spaces

Let and be positive increasing convex functions defined on [0, ). Suppose satisfies the 2-condition, that is, (t)2 (C1t) for sufficiently large t, and has some nice properties. If -1(u)log(u+1) C2-1(u) for sufficiently large uthen we have S*(f) L CfL for all f L ([-, ])where S*(f) is the majorant function of partial sums of trigonometric Fourier series and fL is the Orlicz norm of f. This result is sharp.     

On strong summability of polynomial expansions

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Dedicated to Professor L. Leindler on his 50^th birthday     

Large transversals to small families of unit disks

Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).     

On certain imbedding and extension theorems for a weighted class of functions

n- , S n–1 m (0m<n), () S . () , ().