Seminorm Related to Banach-Saks Property and Real Interpolation of Operators

We introduce the arithmetic separation of a sequence—a geometric characteristic for bounded sequences in a Banach space which describes the Banach-Saks property. We define an operator seminorm vanishing for operators with the Banach-Saks property. We prove quantitative stability of the seminorm for a class of operators acting between l _p -sums of Banach spaces. We show logarithmically convex-type estimates of the seminorm for operators interpolated by the real method of Lions and Peetre.      

Aluthge transforms of unbounded weighted composition operators in L2-spaces

We describe the Aluthge transform of an unbounded weighted composition operator acting in an L²-space. We show that its closure is also a weighted composition operator with the same symbol and a modified weight function. We investigate its dense definiteness. We characterize p-hyponormality of unbounded weighted composition operators and provide results on how it is affected by the Aluthge transformation. We show that the only fixed points of the Aluthge transformation on weighted composition operators are quasinormal ones.     

Fuzzy capital rationing model

In this paper, we study the fuzzification of Weingartner’s pure capital rationing model and its analysis. We develop a primal–dual pair based on t-norm/t-conorm relation for the constraints and objective function for a fully fuzzified pure capital rationing problem except project selection variables. We define the α$\alpha$-interval under which the weak duality is proved. We perform sensitivity analysis for a change in a budget level or in a cash flow level of a non-basic as well as a basic variable. We analyze the problem based on duality and complementary slackness results. We illustrate the proposed model by computational analysis, and interpret the results.     

Capturing incomplete information in resource allocation problems through numerical patterns

We look at the problem of optimizing complex operations with incomplete information where the missing information is revealed indirectly and imperfectly through historical decisions. Incomplete information is characterized by missing data elements governing operational behavior and unknown cost parameters. We assume some of this information may be indirectly captured in historical databases through flows characterizing resource movements. We can use these flows or other quantities derived from these flows as “numerical patterns” in our optimization model to reflect some of the incomplete information. We develop our methodology for representing information in resource allocation models using the concept of pattern regression. We use a popular goodness-of-fit measure known as the Cramer–Von Mises metric as the foundation of our approach. We then use a hybrid approach of solving a cost model with a term known as the “pattern metric” that minimizes the deviations of model decisions from observed quantities in a historical database. We present a novel iterative method to solve this problem. Results with real-world data from a large freight railroad are presented.     

A cohomological construction of quantization functors of Lie bialgebras

We propose a variant to the Etingof-Kazhdan construction of quantization functors. We construct the twistor J_Φ associated to an associator Φ using cohomological techniques. We then introduce a criterion ensuring that the “left Hopf algebra” of a quasitriangular QUE algebra is flat. We prove that this criterion is satisfied at the universal level. This gives a construction of quantization functors, equivalent to the Etingof-Kazhdan construction.     

On quotient spaces of compact lie groups by Tori centralizers

We consider the graph of the homogeneous space K/L, where K is a compact Lie group and L is the centralizer of a torus in K. We obtain a characterization of those spaces whose graphs admit embeddings in a certain standard graph. We compute the number of arcs in such graphs. We also give a simple expression for the Euler class of the homogeneous space K/L.     

A generalized Harish-Chandra method of descent for reductive Lie algebras

Let G be a connected real reductive group and M a connected reductive subgroup of G with Lie algebras g and m, respectively. We assume that g and m have the same rank. We define a map from the space of orbital integrals of m into the space of orbital integrals of g which we call a transfer. We then consider the transpose of the transfer. This can be viewed as a map from the space of G-invariant distributions of g to the space of M-invariant distributions of m and can be considered as a restriction map from g to m. We prove that this map extends Harish-Chandra method of descent and we obtain a generalization of the radial component theorem. We give an application.     

Inverse problems for multi-valued quasi variational inequalities and noncoercive variational inequalities with noisy data

ABSTRACT

We study the inverse problem of identifying a variable parameter in variational and quasi-variational inequalities. We consider a quasi-variational inequality involving a multi-valued monotone map and give a new existence result. We then formulate the inverse problem as an optimization problem and prove its solvability. We also conduct a thorough study of the inverse problem of parameter identification in noncoercive variational inequalities which appear commonly in applied models. We study the inverse problem by posing optimization problems using the output least-squares and the modified output least-squares. Using regularization, penalization, and smoothing, we obtain a single-valued parameter-to-selection map and study its differentiability. We consider optimization problems using the output least-squares and the modified output least-squares for the regularized, penalized and smoothened variational inequality. We give existence results, convergence analysis, and optimality conditions. We provide applications and numerical examples to justify the proposed framework.     

On Lebesgue-type inequalities for greedy approximation

We study the efficiency of greedy algorithms with regard to redundant dictionaries in Hilbert spaces. We obtain upper estimates for the errors of the Pure Greedy Algorithm and the Orthogonal Greedy Algorithm in terms of the best m-term approximations. We call such estimates the Lebesgue-type inequalities. We prove the Lebesgue-type inequalities for dictionaries with special structure. We assume that the dictionary has a property of mutual incoherence (the coherence parameter of the dictionary is small). We develop a new technique that, in particular, allowed us to get rid of an extra factor m^1/2 in the Lebesgue-type inequality for the Orthogonal Greedy Algorithm.     

Rigorous Formulation of a 2D Conformal Model in the Fock–Krein Space: Construction of the Global Op J *-Algebra of Fields and Currents

We use the formalism of the 2D massless scalar field model in an indefinite space of the Fock–Krein type as a basis for constructing a rigorous formulation of 2D quantum conformal theories. We show that the sought construction is a several-stage procedure whose central block is the construction of a new type of representation of the Virasoro algebra. We develop the first stage of this procedure, which is to construct a special global algebra of fields and currents generated by exponential generators. We obtain a system of commutation relations for the Wick-squared currents used in the definition of the Virasoro generators. We prove the existence of Wick exponentials of the current given by operator-valued generalized functions; the sought global algebra is rigorously defined as the algebra of current and field, Wick and normal exponentials on a common dense invariant domain in a Fock–Krein space.     

A single-copy minimal-time simulation of a torus of automata by a ring of automata

We consider cellular automata on Cayley graphs and we simulate the behavior of a torus of n×m automata (nodes) by a ring of n·m automata (cells). Our simulation technique requires the neighborhood of the nodes to be preserved. We achieve this constraint by copying the contents of nodes on the cells. We consider the problem of minimizing the number of the copies. We prove that it is possible to simulate the behavior of a torus on a ring with a single copy on each cell if and only if n and m satisfy a given condition. In that case we propose a time-optimal algorithm. We thus improve a previous work done by Martin where two copies were requested. When the condition on n and m is not fulfilled one can use the previous algorithm.     

Busy period analysis for <Emphasis Type="Italic">M</Emphasis>/<Emphasis Type="Italic">PH</Emphasis>/1 queues with workload dependent balking

We consider an M/PH/1 queue with workload-dependent balking. An arriving customer joins the queue and stays until served if and only if the system workload is no more than a fixed level at the time of his arrival. We begin by considering a fluid model where the buffer content changes at a rate determined by an external stochastic process with finite state space. We derive systems of first-order linear differential equations for the mean and LST (Laplace-Stieltjes Transform) of the busy period in this model and solve them explicitly. We obtain the mean and LST of the busy period in the M/PH/1 queue with workload-dependent balking as a special limiting case of this fluid model. We illustrate the results with numerical examples.      

Complex Cauchy Problems with Intersecting Singularities

We consider linear Cauchy problems of order two in a complex domain. We assume that the initial values have singularities along a family of hypersurfaces, which cross pairwise transversally along a single intersection. We study the propagation of the singularities of the solution. We show that the solution may have anomalous singularities, and study the monodromy of the solution.     

Performance of the Gibbs,Hit-and-Run,and Metropolis Samplers

Abstract

We consider the performance of three Monte Carlo Markov-chain samplers—the Gibbs sampler, which cycles through coordinate directions; the Hit-and-Run (H&R) sampler, which randomly moves in any direction; and the Metropolis sampler, which moves with a probability that is a ratio of likelihoods. We obtain several analytical results. We provide a sufficient condition of the geometric convergence on a bounded region S for the H&R sampler. For a general region S, we review the Schervish and Carlin sufficient geometric convergence condition for the Gibbs sampler. We show that for a multivariate normal distribution this Gibbs sufficient condition holds and for a bivariate normal distribution the Gibbs marginal sample paths are each an AR(1) process, and we obtain the standard errors of sample means and sample variances, which we later use to verify empirical Monte Carlo results. We empirically compare the Gibbs and H&R samplers on bivariate normal examples. For zero correlation, the Gibbs sampler provides independent data, resulting in better performance than H&R. As the absolute value of the correlation increases, H&R performance improves, with H&R substantially better for correlations above .9. We also suggest and study methods for choosing the number of replications, for estimating the standard error of point estimators and for reducing point-estimator variance. We suggest using a single long run instead of using multiple iid separate runs. We suggest using overlapping batch statistics (obs) to get the standard errors of estimates; additional empirical results show that obs is accurate. Finally, we review the geometric convergence of the Metropolis algorithm and develop a Metropolisized H&R sampler. This sampler works well for high-dimensional and complicated integrands or Bayesian posterior densities.     

Self-similar p-adic fractal strings and their complex dimensions

We develop a geometric theory of self-similar p-adic fractal strings and their complex dimensions. We obtain a closed-form formula for the geometric zeta functions and show that these zeta functions are rational functions in an appropriate variable. We also prove that every self-similar p-adic fractal string is lattice. Finally, we define the notion of a nonarchimedean self-similar set and discuss its relationship with that of a self-similar p-adic fractal string. We illustrate the general theory by two simple examples, the nonarchimedean Cantor and Fibonacci strings. The text was submitted by the authors in English.     

Two-edge connected subgraphs with bounded rings: Polyhedral results and Branch-and-Cut

We consider the network design problem which consists in determining at minimum cost a 2-edge connected network such that the shortest cycle (a “ring”) to which each edge belongs, does not exceed a given length K. We identify a class of inequalities, called cycle inequalities, valid for the problem and show that these inequalities together with the so-called cut inequalities yield an integer programming formulation of the problem in the space of the natural design variables. We then study the polytope associated with that problem and describe further classes of valid inequalities. We give necessary and sufficient conditions for these inequalities to be facet defining. We study the separation problem associated with these inequalities. In particular, we show that the cycle inequalities can be separated in polynomial time when K≤4. We develop a Branch-and-Cut algorithm based on these results and present extensive computational results.     

On types of growth for graph-different permutations

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results in case of complementary graphs. We also explore the close relationship between our problem and the concept of Shannon capacity “within a given type.”