Showing papers in "Journal of Mathematical Sciences in 2006"

On the Translocation of Masses

Abstract: ON THE TRANSLOCATION OF MASSES L. V. Kantorovich∗ The original paper was published in Dokl. Akad. Nauk SSSR, 37, No. 7–8, 227–229 (1942). We assume that R is a compact metric space, though some of the definitions and results given below can be formulated for more general spaces. Let Φ(e) be a mass distribution, i.e., a set function such that: (1) it is defined for Borel sets, (2) it is nonnegative: Φ(e) ≥ 0, (3) it is absolutely additive: if e = e1 + e2+ · · · ; ei∩ ek = 0 (i = k), then Φ(e) = Φ(e1)+ Φ(e2) + · · · . Let Φ′(e′) be another mass distribution such that Φ(R) = Φ′(R). By definition, a translocation of masses is a function Ψ(e, e′) defined for pairs of (B)-sets e, e′ ∈ R such that: (1) it is nonnegative and absolutely additive with respect to each of its arguments, (2) Ψ(e, R) = Φ(e), Ψ(R, e′) = Φ′(e′). Let r(x, y) be a known continuous nonnegative function representing the work required to move a unit mass from x to y. We define the work required for the translocation of two given mass distributions as W (Ψ,Φ,Φ′) = ∫

On a problem of Monge.

Abstract: In 1942, I considered a general problem on the most profitable translocation of masses in a compact metric space. The problem is as follows: Assume that we are given two mass distributions determined by additive set functions Φ(e) and Φ′(e) with Φ(R) = Φ′(R) = 1. A translocation of masses is a function Ψ(e, e′) that determines the mass translocated from a set e to a set e′ with [Ψ(e, R) = Φ(e); Ψ(R, e) = Φ′(e)]. The translocation work is defined by the integral

Exchangeable Gibbs partitions and Stirling triangles

Abstract: For two collections of nonnegative and suitably normalized weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to a generic partition {Aj, j ≤ k} the probability Vn,k % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgatC% vAUfeBSjuyZL2yd9gzLbvyNv2CaeHbd9wDYLwzYbItLDharyavP1wz% ZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0xbb% L8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpe% pae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaam% aaeaqbaaGcbaGaemOvay1aaSbaaSqaaiabd6gaUjabcYcaSiabdUga% RbqabaacbaGccqWFGaaicqWGxbWvdaWgaaWcbaWaaqWaaeaacqWGbb% qqdaWgaaadbaGaeGymaedabeaaaSGaay5bSlaawIa7aaqabaGccqWI% VlctcqWGxbWvdaWgaaWcbaWaaqWaaeaacqWGbbqqdaWgaaadbaGaem% 4AaSgabeaaaSGaay5bSlaawIa7aaqabaaaaa!507F! $$V_{n,k} W_{\left| {A_1 } \right|} \cdots W_{\left| {A_k } \right|} $$ , where |Aj| is the number of elements of Aj. We impose constraints on the weights by assuming that the resulting random partitions Π n of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [− ∞, 1]. The case α = 1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α,θ)-partition on the asymptotics of the number of blocks of Πn as n tends to infinity. Bibliography: 29 titles.

Optimal Regularity of Lower-Dimensional Obstacle Problems

Abstract: In this paper, we prove that solutions to the “boundary obstacle problem” have the optimal regularity, C1,1/2, in any space dimension. This bound depends only on the local L2-norm of the solution. Main ingredients in the proof are the quasiconvexity of the solution and a monotonicity formula for an appropriate weighted average of the local energy of the normal derivative of the solution. Bibliography: 8 titles.

Density of Smooth Functions in Sobolev-Orlicz Spaces

Abstract: We study the problem on density of smooth functions in Sobolev spaces of variable orders, also called Sobolev-Orlicz spaces. In particular, we generalize the earlier obtained logarithmic condition and give a number of examples. Bibliography: 9 titles.

Kantorovich metric: initial history and little-known applications

Abstract: We remind of the history of the transportation (Kantorovich) metric and the Monge-Kantorovich problem. We also describe several little-known applications: the first one concerns the theory of decreasing sequences of partitions (tower of measures and iterated metric), the second one relates to Ornstein's theory of Bernoulli automorphisms (d¯-metric), and the third one is the formulation of the strong Monge-Kantorovich problem in terms of matrix distributions. Bibliography: 29 titles

A Chi-Squared Test for the Generalized Power Weibull Family for the Head-and-Neck Cancer Censored Data

Abstract: We propose a chi-squared type statistic to test the validity of the generalized power Weibull family based on the Head-and-Neck cancer censored data. Bibliography: 18 titles.

Homogenization of a multidimensional periodic elliptic operator in a neighborhood of the edge of an internal gap

Abstract: The homogenization procedure for a multidimensional periodic Schrodinger operator near the edge of an internal gap is discussed. We obtain an approximation for the resolvent in the small period limit with respect to the operator norm in L2(ℝd). This approximation contains oscillations but in a simpler form than the resolvent of the initial operator. Bibliography: 8 titles.

Sequential Importance Sampling Algorithms for Dynamic Stochastic Programming

Abstract: This paper gives a comprehensive treatment of EVPI-based sequential importance sampling algorithms for dynamic (multistage) stochastic programming problems. Both theory and computational algorithms are discussed. Under general assumptions it is shown that both an expected value of perfect information (EVPI) process and the corresponding marginal EVPI process (the supremum norm of the conditional expectation of its generalized derivative) are nonanticipative nonnegative supermartingales. These processes are used as importance criteria in the class of sampling algorithms treated in the paper. When their values are negligible at a node of the current sample problem scenario tree, scenarios descending from the node are replaced by a single scenario at the next iteration. On the other hand, high values lead to increasing the number of scenarios descending from the node. Both the small sample and asymptotic properties of the sample problem estimates arising from the algorithms are established, and the former are evaluated numerically in the context of a financial planning problem. Finally, current and future research is described. Bibliography: 49 titles.

Automorphic L-Functions in the Weight Aspect

Abstract: Let Sk(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let Sk(Γ)+ be the set of all Hecke eigenforms from this space with the first Fourier coefficient af(1) = 1. For f ∈ Sk(Γ)+, consider the Hecke L-function L(s, f). Let $$S\left( {k \leqslant K} \right) = \bigcup\limits_{\mathop {12 \leqslant k \leqslant K}\limits_{k\quad \operatorname{even} } } {Sk\left( \Gamma \right)^ + .}$$ It is proved that for large K, $$\sum\limits_{f \in S\left( {k \leqslant K} \right)} {L(\frac{1}{2},f)^4 \ll K^{2 + \varepsilon } } ,$$ where e > 0 is arbitrary. For f ∈ Sk(Γ)+, let L(s, sym2 f) denote the symmetric square L-function. It is proved that as k → ∞ the frequence $$\frac{{\# \left\{ {f|f \in S_k (\Gamma )^ + ,L(1,\operatorname{sym} ^2 f) \leqslant x} \right\}}}{{\# \left\{ {f|f \in S_k (\Gamma )^ + } \right\}}}$$ converges to a distribution function G(x) at every point of continuity of the latter, and for the corresponding characteristic function an explicit expression is obtained. Bibliography: 17 titles.

Student’s t-test for Gaussian scale mixtures

Abstract: A Student-type test is constructed under a condition weaker than normal. We assume that the errors are scale mixtures of normal random variables and compute the critical values of the suggested s-test. Our s-test is optimal in the sense that if the level is at most α, then the s-test provides the minimum critical values. (The most important critical values are tabulated at the end of the paper.) For α ≤.05, the two-sided s-test is identical with Student’s classical t-test. In general, the s-test is a t-type test, but its degree of freedom should be reduced depending on α. The s-test is applicable for many heavy-tailed errors, including symmetric stable, Laplace, logistic, or exponential power. Our results explain when and why the P-value corresponding to the t-statistic is robust if the underlying distribution is a scale mixture of normal distributions. Bibliography: 24 titles.

Newton-Kantorovich Method and Its Global Convergence

Abstract: In 1948, L. V. Kantorovich extended the Newton method for solving nonlinear equations to functional spaces. This event cannot be overestimated: the Newton-Kantorovich method became a powerful tool in numerical analysis as well as in pure mathematics. We address basic ideas of the method in historical perspective and focus on some recent applications and extensions of the method and some approaches to overcoming its local ture. Bibliography: 56 titles.

Hardy-Sobolev Inequalities in a Cone

Abstract: The attainability of the exact constant in the Hardy-Sobolev inequality is established in an arbitrary cone in ℝ n . Bibliography: 17 titles.

XX0 Heisenberg chain and random walks

Abstract: The time-dependent correlation functions of the XX0 Heisenberg chain over the ferromagnetic vacuum are considered as the generating functions of random walks on a one-dimensional lattice with different boundary conditions. The long-time asymptotic behaviour of these functions is studied. Bibliography: 18 titles.

Necessary conditions of the minimum in an impulse optimal control problem

Abstract: The paper is devoted to studying the impulse optimal control problem with inequality-type state constraints and geometric control constraints defined by a measurable multivalued mapping. The author obtains necessary optimality conditions in the form of the Pontryagin maximum principle and nondegeneracy conditions for the latter.

Contact Equivalence Problem for Linear Hyperbolic Equations

Abstract: We consider the local equivalence problem for the class of linear second-order hyperbolic equations in two independent variables under an action of the pseudo-group of contact transformations. E. Cartan’s method is used for finding the Maurer-Cartan forms for symmetry groups of equations from the class and computing structure equations and complete sets of differential invariants for these groups. The solution of the equivalence problem is formulated in terms of these differential invariants.

Coherent random allocations, and the Ewens-Pitman formula

Abstract: Assume that there is a random number K of positive integer random variables S1, …, SK that are conditionally independent given K and all have identical distributions. A random integer partition N = S1 + S2 + … + SK arises, and we denote by PN the conditional distribution of this partition for a fixed value of N. We prove that the distributions {PN}N=1∞ form a partition structure in the sense of Kingman if and only if they are governed by the Ewens-Pitman Formula. The latter generalizes the celebrated Ewens sampling formula, which has numerous applications in pure and applied mathematics. The distributions of the random variables K and Sj belong to a family of integer distributions with two real parameters, which we call quasi-binomial. Hence every Ewens-Pitman distribution arises as a result of a two-stage random procedure based on this simple class of integer distributions. Bibliography: 25 titles.

Heterogeneous Finite-Source Retrial Queues with Server Subject to Breakdowns and Repairs

Abstract: This paper deals with a retrial queueing system with a finite number of heterogeneous sources of calls and a single non-reliable server, which means that the server is subject to random breakdowns depending on whether it is busy or idle. The failure of the server may block or unblock the system’s operations and the service of the interrupted request may be resumed or the call can be transmitted to the orbit. All random variables involved in the model constructions are supposed to be exponentially distributed and independent of each other. The novelty of the investigation is the heterogeneous sources and the variability of this non-reliablility of the server which makes the system rather complicated. The MOSEL tool was used to formulate and solve the problem and the main performance and reliability measures were derived and graphically displayed. Several numerical calculations were performed to show the effect of the non-reliability of the server on the mean response times of the calls and the mean number of requests staying at the service facility.

Absence of the singular continuous component in spectra of analytic direct integrals

Abstract: We give a simple proof of the absence of the singular continuous component in spectra of self-adjoint operators that are representable as analytic direct integrals. Bibliography: 7 titles.

Fully closed mappings and their applications

Abstract: This survey is devoted to the theory of fully closed mappings and their applications. The theoretical part includes a systematic study of relations between fully closed mappings and fiber products, inverse systems, and resolutions. The projective properties of fully closed mappings are studied. The applications are largely related to dimension and cardinal functions. The results are mainly due to the author and his students.

Towards Applying Computational Complexity to Foundations of Physics

Abstract: In one of his early papers, D. Grigoriev analyzed the decidability and computational complexity of different physical theories. This analysis was motivated by the hope that it would help physicists. In this paper, we survey several similar ideas that may be of help to physicists. We hope that further research may lead to useful physical applications. Bibliography: 41 titles.

On the Littlewood-Paley theorem for arbitrary intervals

Abstract: We extend the results of Rubio de Francia and Bourgain by showing that, for arbitrary mutually disjoint intervals Δk ⊂ ℤ+, arbitrary p ∈, (0, 2], and arbitrary trigonometric polynomials fk with supp \(\hat f_k \subset \Delta _k \), we have $$\left\| {\sum\limits_k {f_k } } \right\|_{H^p (\mathbb{T})} \leqslant a_p \left\| {\left( {\sum\limits_k {\left| {f_k } \right|} ^2 } \right)^{1/2} } \right\|_{L^p (\mathbb{T})} $$ . The method is a development of that by Rubio de Francia. Bibliography: 9 titles.

SUBEXPONENTIAL DISTRIBUTION FUNCTIONS IN R d

Abstract: ∞n=0 ∞ pnF ∗n (x), where F ∗n (x) denotes the n-fold convolution of F (x) and where F ∗0 (x) denotes the unit mass at 0. The d.f. W (x )i s called subordinate to F (x) with subordinator {pn}. As in the univariate case in the paper, we shall assume that N satisfies condition (A): N has a generating function P (z )= E(z N ) that is analytic at z =1 . In the present paper, we discuss the relation between the asymptotic behavior of 1 − F (x) and that of 1 − F ∗n (x) and 1 − W (x). It turns out that, as in the univariate case, there are many cases in which 1 − F ∗n (x) asymptotically behaves as n(1 − F (x)) and 1 − W (x) behaves as E(N )(1 − F (x)). To specify the precise kind of asymptotic behavior, we present a form of multivariate subexponentiality. The paper is organized as follows. In Sec. 2, we briefly recall some basic properties and definitions concerning univariate subexponential d.f. In Sec. 3, we introduce and study multivariate subexponential d.f.’s. In Sec. 4, we discuss the relation with regular variation and, in Sec. 5, we provide some extensions. In our main results, we obtain first-order estimates for 1 − F ∗n (x) and 1 − W (x). In a forthcoming paper, we discuss second-order estimates. Without further comment, in the paper, we shall assume that all random vectors X, Y, Z, etc. are positive and have infinite support, i.e., the d.f. satisfies F (0+) = 0 and F (x) < 1, ∀x ∈ R d . We also use the notation F (x )=1 − F (x), and for vectors x and a ,w e setx ◦ = min(xi )a nda ∗ x =( a1x1 ,a 2x2 ,...,a dxd). 2. Univariate Subexponential Distributions In the one-dimensional case, many papers have been devoted to the tail behavior of subordinated d.f.’s. In doing so, the class of subexponential d.f.’s (notation: S) plays an important role. Extending the class S, Chover et al. [6, 7], introduced the class S(γ), where γ ≥ 0. To define these classes, let F (x) denote a d.f. in R such that F (0+) = 0 and F (x) < 1, ∀x ∈ R. Also, let f (s )= E(e −sX ) denote the generating function of X or F (x). The d.f. F (x) belongs to the subexponential class S (notation: F ∈ S) if it satisfies lim

Discrete and Continuous Random Walk Models for Space-Time Fractional Diffusion

Abstract: A mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. A more general approach is however provided by the integral equation for the so-called continuous time random walk (CTRW), which can be understood as a random walk subordinated to a renewal process. We show how this integral equation reduces to our fractional diffusion equations by a properly scaled passage to the limit of compressed waiting times and jumps. The essential assumption is that the probabilities for waiting times and jumps behave asymptotically like powers with negative exponents related to the orders of the fractional derivatives. Illustrating examples are given, numerical results and plots of simulations are displayed.

A Discrete-Time Retrial Queue with Unreliable Server and General Server Lifetime

Abstract: Proceedings of the Seminar on Stability Problems for Stochastic Models, Pamplona, Spain, 2003, Part II.

Identities Involving Coefficients of Automorphic L-Functions

Abstract: Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2, ℤ) and let $$L(s,\operatorname{sym} ^2 f) = \sum\limits_{n = 1}^\infty {c_n n^{ - s} } ,\quad \operatorname{Re} s > 1,$$ denote the symmetric square L-function of f. A Voronoi type formula for $$C(x) = \sum\limits_{n \leqslant x} {c_n }$$ and the relation $$C(x) = \Omega _ \pm (x^{{1 \mathord{\left/ {\vphantom {1 3}} \right. \kern- ulldelimiterspace} 3}} )$$ are proved. Heuristic approaches to estimation of exponential sums arising in this connection are considered. Bibliography: 9 titles.

Automated Proofs of Upper Bounds on the Running Time of Splitting Algorithms

Abstract: The splitting method is one of the most powerful and well-studied approaches for solving various NP-hard problems. The main idea of this method is to split an input instance of a problem into several simpler instances (further simplified by certain simplification rules), such that when the solution for each of them is found, one can construct the solution for the initial instance in polynomial time. There exists a huge number of articles describing algorithms of this type and usually a considerable part of such an article is devoted to case analysis. In this paper we show how it is possible to write a program that given simplification rules would automatically generate a proof of an upper bound on the running time of a splitting algorithm that uses these rules. As an example we report the results of experiments with such a program for the SAT, MAXSAT, and (n,3)-MAXSAT (the MAXSAT problem for the case where every variable in the formula appears at most three times) problems.