Averaging of random walks and shift-invariant measures on a Hilbert space

We study random walks in a Hilbert space H and representations using them of solutions of the Cauchy problem for differential equations whose initial conditions are numerical functions on H. We construct a finitely additive analogue of the Lebesgue measure: a nonnegative finitely additive measure λ that is defined on a minimal subset ring of an infinite-dimensional Hilbert space H containing all infinite-dimensional rectangles with absolutely converging products of the side lengths and is invariant under shifts and rotations in H. We define the Hilbert space H of equivalence classes of complex-valued functions on H that are square integrable with respect to a shift-invariant measure λ. Using averaging of the shift operator in H over random vectors in H with a distribution given by a one-parameter semigroup (with respect to convolution) of Gaussian measures on H, we define a one-parameter semigroup of contracting self-adjoint transformations on H, whose generator is called the diffusion operator. We obtain a representation of solutions of the Cauchy problem for the Schrödinger equation whose Hamiltonian is the diffusion operator.     

A Sufficient Condition for a Graph to Have a k-tree

 A k-tree of a connected graph is a spanning tree with maximum degree at most k. We obtain a sufficient condition for a graph to have a k-tree, as a generalization of the condition of E. Flandrin, H. A. Jung and H. Li [3] for traceability. We also extend early results of Y. Caro, I. Krasikov and Y. Roditty [2] and Min Aung and Aung Kyaw [4] for the maximal order of a tree with bounded maximum degree in a graph. Received: July 28, 1997 Final version received: April 13, 1998     

Essential and discrete spectra of the three-particle Schrödinger operator on a lattice

We study the position of the essential spectrum of a three-body Schrödinger operator H. We evaluate the lower boundary of the essential spectrum of H and prove that the number of eigenvalues located below the lower edge of the essential spectrum in the H model is finite.     

The Weierstrass Subgroup of a Curve has Maximal Rank

We show that the Weierstrass points of the generic curve ofgenus g over an algebraically closed field of characteristic0 generate a group of maximal rank in the Jacobian. 2000 MathematicsSubject Classification 11G30, 14H40, 14H10, 14H55, 14Q05.     

Electron scattering by a crystal layer

We consider the one-particle discrete Schrödinger operator H with a periodic potential perturbed by a function ?W that is periodic in two variables and exponentially decreasing in the third variable. Here, ? is a small parameter. We study the scattering problem for H near the point of extremum with respect to the third quasimomentum coordinate for a certain eigenvalue of the Schrödinger operator with a periodic potential in the cell, in other words, for the small perpendicular component of the angle of particle incidence on the potential barrier ?W. We obtain simple formulas for the transmission and reflection probabilities.     

Investigation of the spectrum of a model operator in a Fock space

We consider a model operator H corresponding to a quantum system with a nonconserved finite number of particles on a lattice. Based on an analysis of the spectrum of the channel operators, we describe the position of the essential spectrum of H. We obtain a Faddeev-type equation for the eigenvectors of H.     

Optimal versus satisfactory decision making: a case study of sales with a target

We consider a common managerial problem where an agent strives to achieve a prescribed target. We use this problem as a vehicle for comparing and discussing two approaches to decision making: optimising and “satisficing” (where the latter is a neologism due to H. Simon, 1976 Economics Nobel Prize Winner). We compute the optimising and satisfactory strategies for the problem at hand and highlight the differences in their outcomes.     

On a theorem of corson and lindenstrauss on Lindelöf function spaces

We give two generalizations of a theorem of H. H. Corson and J. Lindenstrauss concerning Lindelöf property in spaces of functions vanishing at infinity.     

Perfect equilibria of a model of n-person noncooperative bargaining

We study the set of subgame perfect equilibria associated with then-person noncooperative bargaining mechanism proposed by Hart and Mas-Colell (1992). Our results pertain to transferable utility games. The set of perfect equilibria depends on the parameter representing the continuation probability, . For general TU games, we characterize the set of payoffs from perfect equilibria for (a) small values of and (b) large values of . For symmetric games a complete characterization for all values of is provided.We are grateful to Andreu Mas-Colell for help and encouragement and to two referees for every helpful comments.     

Pinning class of the Wiener measure by a functional: related martingales and invariance properties

For a given functional Y on the path space, we define the pinning class of the Wiener measure as the class of probabilities which admit the same conditioning given Y as the Wiener measure. Using stochastic analysis and the theory of initial enlargement of filtration, we study the transformations (not necessarily adapted) which preserve this class. We prove, in this non Markov setting, a stochastic Newton equation and a stochastic Noether theorem. We conclude the paper with some non canonical representations of Brownian motion, closely related to our study.Mathematics Subject Classification (2000): 60G44, 60H07, 60H20, 60H30     

Solvability of a convolution equation in convex domains

For an exponential functiona(z) we consider the convolutiona* x in the function space H(G), consisting of functions analytic in convex domains G. We obtain conditions (close to necessary and sufficient) on G and G₁ subject to which the equationa* (H(G₁)) = H(G) is satisfied.     

Continuous regularization of linear operator equations in a Hilbert space

For the linear operator equation Au=f we offer a continuous regularization based on the stabilization of solutions of differential equations in a Hilbert space H. We assume that A is a positive operator and that the equation Au=f has a solution in H.Translated from Matematicheskie Zametki, Vol. 4, No. 5, pp. 503–510, November, 1968.     

Minimal Submanifolds in a Sphere II

We use the method of H. Gauchman [2] to show that if the sectionalcurvature of a compact minimal n-dimensional submanifold ina unit sphere is everywhere bigger than n/2(n+1) then it mustbe totally geodesic. We also discuss some related results andconjectures.     

The extension of a quantized Borcherds superalgebra by a Hopf algebra

A two-parameter quantum group is obtained from the usual enveloping algebra by adding two commutative grouplike elements. In this paper, we generalize this procession further by adding commutative grouplike elements b_(ik), c_(ik), g_(ik), h_(ik)(i ∈I, k = 1,..., mi) of a Hopf algebra H to the quantized enveloping algebra U_q(G) of a Borcherds superalgebra G defined by a symmetrizable integral Borcherds–Cartan matrix A =(aij)i,j∈I. Therefore, we define an extended Hopf superalgebra HU_q(G). We also discuss the basis and the grouplike elements of HU_q(G).     

Every frame is a sum of three (but not two) orthonormal bases—and other frame representations

We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. We next show that this result is best possible by including a result of Kalton: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis. We further show that every frame can be written as a (multiple of a) sum of two tight frames with frame bounds one or a sum of an orthonormal basis and a Riesz basis for H. Finally, every frame can be written as a (multiple of a) average of two orthonormal bases for a larger Hilbert space. Acknowledgements and Notes. This research was supported by NSF DMS 9701234. Part of this research was conducted while the author was a visitor at the “Workshop on Linear Analysis and Probability”, Texas A&M University.     

A free boundary problem for the Laplacian with a constant Bernoulli-type boundary condition

We study a free boundary problem for the Laplace operator, where we impose a Bernoulli-type boundary condition. We show that there exists a solution to this problem. We use A. Beurling’s technique, by defining two classes of sub- and super-solutions and a Perron argument. We try to generalize here a previous work of A. Henrot and H. Shahgholian. We extend these results in different directions.     

Study of the essential spectrum of a matrix operator

We consider a matrix operator H corresponding to a system with a nonconserved finite number of particles on a lattice. We describe the structure of the essential spectrum of the operator H and prove that the essential spectrum is a union of at most four intervals.     

On Extremal Problems on a Set of Interpolation Polynomials in a Hilbert Space

We give solutions of two extremal problems, a generalization of a result of W. Porter on an interpolation polynomial in L₂(a,b) with minimal norm, which we get here in an abstract Hilbert space with a measure, and an analogue of a theorem of M. Golomb and H. Weinberger for a bounded set of operator interpolants.     

A Metric for Unbounded Linear Operators in a Hilbert Space

We introduce a metric in the set S(H){{\mathcal S}(H)} of all semiclosed operators in a Hilbert space H, and its topological structures are studied.     

On the normal subgroup lattice of a hypercentral p-group

We describe all hypercentral p-groups G whose lattice of normal subgroups n(G) is isomorphic to n(H) for a group H with hypercenwal derived subgroup and H not a pgroup.