L 2-lower bounds for a special class of random walks

Summary We investigate theL ₂-speed of convergence to stationarity for a certain class of random walks on a compact connected Lie group. We give a lower bound on the number of stepsk necessary such that thek-fold convolution power of the original step distribution has anL ₂-density. Our method uses work by Heckman on the asymptotics of multiplicities along a ray of representations. Several examples are presented.This paper is based on parts of the author's doctoral dissertation written at The Johns Hopkins University     

A counterexample on extreme operators

We give examples of extreme operators in the unit ball ofL(C(X),C(Y)) which are not nice operators (i.e., their adjoints do not carry extreme points into extreme points in the corresponding unit balls.) Such counterexamples exist in fact for every non-dispersed compact Hausdorff spaceX when the scalars are complex. This is part of the author’s Ph.D. thesis, prepared at Tel Aviv University under the supervision of Prof. A. J. Lazar.     

Representations and properties of the W-Weighted Drazin inverse

Several new representations of the W-weighted Drazin inverse are introduced. These representations are expressed in terms of various matrix powers as well as in terms of matrix products involving the Moore–Penrose inverse and the usual matrix inverse. Also, the properties of various generalized inverses which arise from derived representations are investigated. The computational complexity and efficiency of the proposed representations are considered. Representations are tested and compared among themselves in a substantial number of randomly generated test examples.     

Representations with scalarK-types and applications

We discuss some results of Shimura on invariant differential operators and extend a folklore theorem about spherical representationas to representations with scalarK-types. We then apply the result to obtain non-trivial isomorphisms of certain representations arising from local theta correspondence, many of which are unipotent in the sense of Vogan. The author was supported in part by the NUS-MOE grant R-146-000-017-112. The article is based on a talk at the National University of Singapore in February 2000.     

Analysis of a linearization heuristic for single-machine scheduling to maximize profit

We consider the problem of schedulingn jobs without preemption on a single machine to maximize total profit, where profit is given by a nonincreasing, concave separable function of job starting times. A heuristic is given in which jobs are sequenced optimally relative to a specific linear approximation of the profit, function. This heuristic always obtains at least 2/3 of the optimal profit, and examples exist where the heuristic obtains only 2/3 of the optimal profit. A large class of alternative linearizations is considrred and shown to give arbitrarily bad results. Work supported in part by NSF Grant ECS 82-05438 to the University of Pennsylvania and ONR Contract N00014-81-C-0302.     

The 2-characters of a group and the group determinant

A new look at Frobenius' original papers on character theory has produced the following: (a) the group determinant determines the group (Formanek-Sibley); (b) the group is determined by the 1-, 2- and 3-characters of the irreducible representations (Hoehnke-Johnson); and (c) pairs of non-isomorphic groups exist with the same irreducible 1- and 2-characters (Johnson-Sehgal). The examples produced in Johnson and Sehgal have large orders but, recently, McKay and Sibley have proved that the ten Brauer pairs of order 256 have the same irreducible 2-characters. It is shown here that the pairs of non-isomorphic groups of order p³, p and odd prime, have the same irreducible 2-characters. Further results are given on the k-characters of the regular representation/rod a shorter proof of the result mentioned in (c) is indicated. A criterion is given which is sufficient for the 3-character of an arbitrary representation to determine the group.     

Finite-dimensional algebraic representations of the infinite phylon group

The concept of phylon is introduced as a generalisation of derivative strings, differential strings and new tensors. The behaviour of phyla under change of coordinates is given by finite-dimensional algebraic representations of a very large group, the infinite phylon group. These representations are studied from both the general and the matrix points of view. Various examples of phyla are given, mainly from a statistical context. The basic structure of these representations is given.     

Invertibility symbol for a Banach algebra generated by two idempotents and a shift

A Banach algebra generated by two idempotentsp, r, identitye and a shiftv which satisfy the conditionspv=vp andrv=v(e–r) is investigated. It is proved that all irreducible representations of algebraA are two- and four-dimensional. The explicit form of these representations is obtained. An Invertibility Symbol is constructed. Some examples are considered.     

Irreducibility of unitary group representations and reproducing kernels Hilbert spaces

We discuss unitary representations of groups in Hilbert spaces of functions given together with reproducing kernels, and in particular irreducibility. Our main focus is on examples, including spherical representations of orthogonal groups, distance-transitive finite graphs, irreducibility of induced representations, the discrete series of SL(2, ), and some representations of PGL(2, _p).The appendix contains examples involving groups acting on rooted trees.     

Nonanticipative transformations of the two-parameter Wiener process and a Girsanov theorem

Nonanticipative linear transformations of the two-parameter Wiener process W are studied. It is shown that they induce measures equivalent to two-parameter Wiener measure and the corresponding Radon-Nikodym derivatives are calculated. A two-parameter extension of Girsanov's theorem is established for a class of nonanticipative, possibly nonlinear transformations of W.     

Minimally supported error representations and approximation by the constants

Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions from the constants in the uniform norm is estimated using our representations for the error. Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000     

A series of unitary irreducible representations induced from a symmetric subgroup of a semisimple Lie group

LetG/H be a semisimple symmetric space. Generalizing results of Flensted-Jensen we give a sufficient condition for the existence of irreducible closed invariant subspaces of the unitary representations ofG induced from unitary finite dimensional representations ofH. This provides a method of constructing unitary irreducible representations ofG, and we show by examples that for some irreducible admissible representations ofG, this method exhibits not previously known unitarity.This work was supported by the Danish Natural Science Research Council.     

Discrete analogues of the Heisenberg-Weyl algebra

Theq-analog and finite difference analog of the canonical Heisenberg-Weyl algebra are studied. The basic representations are found. Analogs of the standard boson Fock space are constructed. Various examples and illustrations are presented.     

Compatible Poisson Brackets on Lie Algebras

We discuss the relationship between the representation of an integrable system as an L-A-pair with a spectral parameter and the existence of two compatible Hamiltonian representations of this system. We consider examples of compatible Poisson brackets on Lie algebras, as well as the corresponding integrable Hamiltonian systems and Lax representations.     

Explicit Universal Deformations of Even Galois Representations

We investigate the case of deformations of even Galois representations. Our methods are the group theoretic ones mainly developed by Nigel Boston to study odd representations. We present conditions for Borel and tame cases under which the universal deformation ring is isomorphic to ?_p[[T]] and where we compute the universal deformation explicitly. Furthermore we produce a family of examples of totally real S₃ extensions which satisfy the above conditions in the tame case and we give examples in the Borel case. Finally we study the change of the deformation space under enlarging the ramification and thus give an example of an even representation that is not twist-finite.     

Simultaneous dominance representation of multiple posets

We characterize the codominance pairs-pairs of posets that admit simultaneous dominance representations in the (x, y)-and (–x, y)-coordinate systems-and present a linear algorithm to recognize them and construct codominance representations. We define dominance polysemy as a generalization of codominance and describe several related problems and preliminary results.This author's research, supported in part by NSF grant CCR-9300079, also appears in his doctoral thesis [15], written, at the Johns Hopkins University under the supervision of Professors Edward R. Scheinerman and Michael T. Goodrich.This author's research, supported in part by an NSERC operating grant and an FCAR team grant, was performed in part during a visit at INRIA Sophia Antipolis.     

Diophantine representations of linear recurrences. I

Direct constructions of Diophantine representations of linear recurrent sequences are discussed. These constructions generalize already known results for second-order recurrences. Some connections of this problem with the theory of units in rings of algebraic integers are shown. It is proved that the required representations exist only for second-, third-, and fourth-order sequences. In the two last-mentioned cases certain additional restrictions on their coefficients must be imposed. Bibliography:14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 52–60.     

Products of B-patches

Products and tensor products of multivariate polynomials in B-patch form are viewed as linear combinations of higher degree B-patches. Univariate B-spline segments and certain regions of simplex splines are examples of B-patches. A recursive scheme for transforming tensor product B-patch representations into B-patch representations of more variables is presented. The scheme can also be applied for transforming ann-fold product of B-patch expansions into a B-patch expansion of higher degree. Degree raising formulas are obtained as special cases. The scheme calculates the blossom of the (tensor) product surface and generalizes the pyramidal recursive scheme for B-patches.     

Inactivity times of coherent systems with dependent components under periodical inspections

If we know that a coherent system has failed before a time t, the inactivity time is the period (from t) in which the system has been broken. If T is the system lifetime, the inactivity time at t is (t?T|T<t). Under periodical inspections, we may typically know that the system was working at a time t₁, but that is broken at another time t₂>t₁. Under this assumption, we obtain representations for the reliability function of the system inactivity time (t₂?T|t₁<T<t₂). We consider both the cases of independent and dependent components. Similar representations are obtained under other assumptions with partial information about component failures at times t₁ and t₂. These representations are used to compare stochastically the inactivity times under different assumptions. Some illustrative examples are provided.     

Tannaka-Krein duality for compact matrix pseudogroups. TwistedSU(N) groups

Summary The notion of concrete monoidalW ^*-category is introduced and investigated. A generalization of the Tannaka-Krein duality theorem is proved. It leads to new examples of compact matrix pseudogroups. Among them we have twistedSU(N) groups denoted byS U(N). It is shown that the representation theory forS U(N) is similar to that ofSU(N): irreducible representations are labeled by Young diagrams and formulae for dimensions and multiplicity are the same as in the classical case.On leave from Department of Mathematical Methods in Physics, Faculty of Physics, University of Warsaw, Hoza 7400-682, Warsaw, Poland.