One- and two-sample match statistics

A mathc statistic considered by Khidr (1981) is interpreted in terms of crossings of the empirical and true distribution functions and a simpler alternate derivation of its distribution provided. This approach can also be used to obtain the distribution of a two-sample match statistic, considered earlier by Takács (1971).     

Separable statistics derived from a given separable statistic by “restricting” the probability space

The problem of derivation of a separable statistic from another separable statistic is considered for the case of a distribution with atoms. This work was performed with the financial support of the International Science Foundation's “Cultural Initiative”. Translated fromStatisticheskie Metody Proverki i Otsenivaniya Gipotez, pp. 109–117, Perm, 1993.     

Smirnov's two-sample problem

An exact distribution and complete asymptotic expansions are obtained for a statistic studied by Smirnov in connection with the calculation of limiting distributions for the statistics D⁺(n₁, n₂), D^–(n₁, n₂, and Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 53, pp. 54–63, 1975.     

Separable Bimodules and Approximation

Using approximations, we give several characterizations of separability of bimodules. We also discuss how separability properties can be used to transfer some representation theoretic properties from one ring to another: contravariant finiteness of the subcategory of (finitely generated) left modules with finite projective dimension, finitistic dimension, finite representation type, Auslander algebra, tame or wild representation type. Presented by A. VerschorenMathematics Subjects Classifications (2000) 16L60, 16H05, 16G10.Research supported by the bilateral project BIL99/43 “New computational, geometric and algebraic methods applied to quantum groups and diffferential operators” of the Flemish and Chinese governments.     

Separable d-Permutations and Guillotine Partitions

We characterize separable multidimensional permutations in terms of forbidden patterns and enumerate them by means of generating function, recursive formula, and explicit formula. We find a connection between multidimensional permutations and guillotine partitions of a box. In particular, a bijection between separable d-dimensional permutations and guillotine partitions of a 2 ^d-1-dimensional box is constructed. We also study enumerating problems related to guillotine partitions under certain restrictions revealing connections to other combinatorial structures. This allows us to obtain several results on patterns in permutations.     

Separable and low-rank continuous games

In this paper, we study nonzero-sum separable games, which are continuous games whose payoffs take a sum-of-products form. Included in this subclass are all finite games and polynomial games. We investigate the structure of equilibria in separable games. We show that these games admit finitely supported Nash equilibria. Motivated by the bounds on the supports of mixed equilibria in two-player finite games in terms of the ranks of the payoff matrices, we define the notion of the rank of an n-player continuous game and use this to provide bounds on the cardinality of the support of equilibrium strategies. We present a general characterization theorem that states that a continuous game has finite rank if and only if it is separable. Using our rank results, we present an efficient algorithm for computing approximate equilibria of two-player separable games with fixed strategy spaces in time polynomial in the rank of the game. This research was funded in part by National Science Foundation grants DMI-0545910 and ECCS-0621922 and AFOSR MURI subaward 2003-07688-1.     

Separable states and positive maps

Using the natural duality between linear functionals on tensor products of C^∗-algebras with the trace class operators on a Hilbert space H and linear maps of the C^∗-algebra into B(H), we study the relationship between separability, entanglement and the Peres condition of states and positivity properties of the linear maps.     

Separable Groupoid Rings

We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity.     

Separable anisotropic elliptic operators and applications

This paper focuses on boundary value problems for anisotropic differential-operator equations of high order with variable coefficients in the half plane. Several conditions are obtained which guarantee the maximal regularity of anisotropic elliptic and parabolic problems in Banach-valued L _p -spaces. Especially, it is shown that this differential operator is R-positive and is a generator of an analytic semigroup. These results are also applied to infinite systems of anisotropic type partial differential equations in the half plane.     

Separable integral extensions and plus closure

We show that an excellent local domain of characteristic p has a separable big Cohen–Macaulay algebra. In the course of our work we prove that an element which is in the Frobenius closure of an ideal can be forced into the expansion of the ideal to a module-finite separable extension ring. Received: 28 May 1998 / Revised version: 30 November 1998     

Bartlett correctable two-sample adjusted empirical likelihood

We propose a two-sample adjusted empirical likelihood (AEL) to construct confidence regions for the difference of two d-dimensional population means. This method eliminates the non-definition of the usual two-sample empirical likelihood (EL) and is shown to be Bartlett correctable. We further show that when the adjustment level is half the Bartlett correction factor for the usual two-sample EL, the two-sample AEL has the same high-order precision as the EL with Bartlett correction. To enhance the performance of the two-sample AEL with adjustment level being half the Bartlett correction factor, we propose a less biased estimate of the Bartlett correction factor. The efficiency of the proposed method is illustrated by simulations and a real data example.     

Nilpotent schemes and group-like schemes

We give a definition of nilpotent association schemes as a generalization of nilpotent groups and investigate their basic properties. Moreover, for a group-like scheme, we characterize the nilpotency by its character products.     

Separable anisotropic differential operators and applications

The nonlocal boundary value problems for anisotropic partial differential-operator equations with a dependent coefficients are studied. The principal parts of the appropriate generated differential operators are nonself-adjoint. Several conditions for the maximal regularity and the fredholmness in Banach-valued Lp-spaces of these problems are given. These results permit us to establish that the inverse of corresponding differential operators belongs to Schatten q-class. Some spectral properties of the operators are investigated. In applications, the nonlocal BVP's for quasielliptic partial differential equations and for systems of quasielliptic equations on cylindrical domain are studied.     

Complete asymptotic expansions for the two-sample problem

Let x₁, x₂,...,x_n and y₁, y₂,...,y_n be the results of two series of independent observations. Let us denote by F_{R 1} (x) and G_{R 2} (y) the empirical distribution functions constructed on the basis of the first and the second sample, respectively. Let us write This paper deals with a complete asymptotic expansion, for the case n₁=n, n₂=np, n of the probability in a power series 1/n, where p1 is a fixed integer, and ₁>0 and ₂> 0 are fixed positive numbers.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 53, pp. 4–53, 1975.