Khachiyan's linear programming algorithm

L. G. Khachiyan's polynomial time algorithm for determining whether a system of linear inequalities is satisfiable is presented together with a proof of its validity. The algorithm can be used to solve linear programs in polynomial time.     

The Dynamics of Cultural Influence Networks

This article investigates the behavior of cultural influence networks over time, using a computer simulation based on a formal model of cultural transmission in organizations. In the formal model, every organizational member exerts some cultural influence on, and is influenced by, every other member; these influence paths constitute a dense social network and the weights of paths (ties) vary throughout the network. Over time, each organizational member's enculturation level changes in response to influence from other members, and the influence weight of each path changes in relationship to the cultural similarity of the individuals connected by the path. Virtual experiments explore the configuration and evolution of the cultural influence network under varying demographic conditions and influence principles. Demographic effects are studied by varying organizational size, hiring selectivity and turnover rates. Two principles for determining initial influence path weights are examined, cohort-based influence and random influence. The simulations show that the cultural influence network evolves over time to a robust configuration, fluctuating around a stable dynamic equilibrium as individuals enter and leave the organization. As turnover rates rise, cohort-based influence strengthens the influence network and reduces network inequality. In this model, cohort-based influence processes promote cultural stability in organizations.     

Morita equivalences of Ariki–Koike algebras

We prove that every Ariki–Koike algebra is Morita equivalent to a direct sum of tensor products of smaller Ariki–Koike algebras which have q–connected parameter sets. A similar result is proved for the cyclotomic q–Schur algebras. Combining our results with work of Ariki and Uglov, the decomposition numbers for the Ariki–Koike algebras defined over fields of characteristic zero are now known in principle. Received: 22 March 2000; in final form: 19 September 2001 / Published online: 29 April 2002     

The Nielsen Method For Groups Acting on Trees

We use Nielsen methods to study generating sets of subgroupsof groups that act on simplicial trees and give several applications.In particular, we exhibit an explicit bound for the complexityof acylindrical splittings of a finitely generated group interms of its rank. This is applied to JSJ-splittings of word-hyperbolicgroups and 3-manifolds. As a last application we construct examplesof amalgamated products that show that there exists no non-trivialrank formula for amalgamated products. 2000 Mathematical Subject Classification: 20E06, 20E08, 57M27.     

Exact Misclassification Probabilities for Plug-In Normal Quadratic Discriminant Functions: II. The Heterogeneous Case

We consider the problem of discriminating between two independent multivariate normal populations, N_p(μ₁, Σ₁) and N_p(μ₂, Σ₂), having distinct mean vectors μ₁ and μ₂ and distinct covariance matrices Σ₁ and Σ₂. The parameters μ₁, μ₂, Σ₁, and Σ₂ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the “plug-in” quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and F distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher (Ann. Eugen.7 (1936), 179–188); a data set on corporate financial ratios provided by Johnson and Wichern (Applied Multivariate Statistical Analysis, 4th ed., Prentice–Hall, Englewood Cliffs, NJ, 1998); and a data set analyzed by Reaven and Miller (Diabetologia16 (1979), 17–24) in a classification of diabetic status.     

The Fischer decomposition for Hodge‐de Rham systems in Euclidean spaces

The classical Fischer decomposition of spinor‐valued polynomials is a key result on solutions of the Dirac equation in the Euclidean space . As is well‐known, it can be understood as an irreducible decomposition with respect to the so‐called L‐action of the Pin group Pin(m). But, in Clifford algebra valued polynomials, we can consider also the H‐action of Pin(m). In this paper, the corresponding Fischer decomposition for the H‐action is obtained. It turns out that, in this case, basic building blocks are the spaces of homogeneous solutions to the Hodge‐de Rham system. Moreover, it is shown that the Fischer decomposition for the H‐action can be viewed even as a refinement of the classical one. Copyright © 2011 John Wiley & Sons, Ltd.     

The maximum of a Lévy process reflected at a general barrier

We investigate the reflection of a Lévy process at a deterministic, time-dependent barrier and in particular properties of the global maximum of the reflected Lévy process. Under the assumption of a finite Laplace exponent, ψ(θ)$\psi \left(\theta \right)$, and the existence of a solution θ^∗>0${\theta }^{\ast }>0$ to ψ(θ)=0$\psi \left(\theta \right)=0$ we derive conditions in terms of the barrier for almost sure finiteness of the maximum. If the maximum is finite almost surely, we show that the tail of its distribution decays like Kexp(−θ^∗x)$Kexp(-{\theta }^{\ast }x)$. The constant K$K$ can be completely characterized, and we present several possible representations. Some special cases where the constant can be computed explicitly are treated in greater detail, for instance Brownian motion with a linear or a piecewise linear barrier. In the context of queuing and storage models the barrier has an interpretation as a time-dependent maximal capacity. In risk theory the barrier can be interpreted as a time-dependent strategy for (continuous) dividend pay out.     

Steinberg symbols modulo the trace class, holonomy, and limit theorems for Toeplitz determinants

Suppose that where and , and the Toeplitz operator is invertible. Let be the determinant of the Toeplitz matrix where . Let be the orthogonal projection onto where ; set , let denote the Hankel operator associated to , and set for . For the Wiener-Hopf factorization where and , put , Theorem A.    

Let be a decomposition into invariant subspaces, and , so that restricted to is invertible, is finite dimensional, and restricted to is nilpotent. Let be the basis for the null space of , and let be the top vector in a Jordan root vector chain of length lying over , i.e., where . Theorem B.     , the holonomy of a Deligne bundle with connection defined by the factorization . Note that the generalizations of the Szegö limit theorem for which have appeared in the literature with instead of have the defect that the limit of does not exist in general. An example is given with yet for infinitely many .