Finite quantifier hierarchies in relational algebras

For a given structure of finite signature, one can construct a hierarchy of classes of relations definable in this structure according to the number of quantifier alternations in the formulas expressing the relations. In ordinary examples, this hierarchy is either infinite (as in the arithmetic of addition and multiplication of natural numbers) or stabilizes very rapidly (in structures with decidable theories, such as the field of real numbers). In the present paper, we construct a series of examples showing that the above-mentioned hierarchy may have an arbitrary finite length. The proof employs a modification of the Ehrenfeucht game.     

A new approach to the representation theory of semisimple Lie algebras and quantum algebras

We propose a method for explicitly constructing the simple-root generators in an arbitrary finite-dimensional representation of a semisimple quantum algebra or Lie algebra. The method is based on general results from the global theory of representations of semisimple groups. The rank-two algebras A₂, B₂=C₂, D₂, and G₂ are considered as examples. The simple-root generators are represented as solutions of a system of finite-difference equations and are given in the form of N_l×N_l matrices, where N_l is the dimension of the representation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 264–284, May, 2000.     

Changing algebras in low energy biologic relational processes

Finite distributive lattices belonging to different varieties of pseudo-Boolean algebras have identified normal biological processes in terms of their qualitative relationships. When the normal processes evolve or deviate, the H₅ equational variety is produced as an algebra satisfying an intermediate step. Propositions concern about the study of the H₅ equational variety in the sense of getting the conditions of how to arrive to it and also which are the lattices belonging to the H₅ equational variety which evolve to a nonmodular algebra when the dual Heyting arrow operation is applied.     

A new approach to the Tomita-Takesaki theory of generalized Hilbert algebras

Let M be a von Neumann algebra with separating and cyclic vector ξ₀. The map $\text{xξ}{}_0\text{→ x}{}^1\text{ξ}{}_0$ with x?M has a least closed extension S. Tomita proved that the isometric involution J and the positive self-adjoint operator Δ obtained from the polar decomposition $\text{S = JΔ}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ of S satisfy JMJ = M′ and Δ^itMΔ^?it = M for any real t. More generally, he obtained similar results for the left von Neumann algebra of any generalized Hilbert algebra. In this paper a shorter proof of his results is given.     

Flexibility in manufacturing systems: A relational and a dynamic approach

In this paper, we propose that flexibility in a manufacturing system is not independent of the environment with which it interacts. Furthermore, we propose that flexibility is a multidimensional concept relating the degree, effort and time of adaptation. In order to arrive at this approach, a dynamic perspective is adopted in which the flexibility dimensions are related. A number of examples are presented to point out some problems where this approach may be used.     

Heuristics—the relational approach

A review of the literature on heuristics would suggest two approaches to their use in problem solving: mathematical and engineering. It is suggested however that there is a third approach for real world application, which the authors have called relational. Instead of investigating a problem through the medium of mathematical models and then deriving heuristics because direct optimising techniques are not available, it is advocated that a close relationship between problem owner and problem solver can be achieved by setting down together the decision rules that the owner employs. In this way, a heuristic model is developed directly, with the opportunity to introduce additional procedures as the situation allows. The result is a consistent control mechanism which is invaluable for both strategic and operational decision making. The model can be a predictor of the effects of policy decisions as well as a means by which those decisions can be implemented and monitored, dependent as much upon the balancing of sometimes conflicting objectives by management as upon the setting of bounds to achieve a guaranteed performance.     

Central simple algebras with involution: a geometric approach

Let k be an algebraically closed base field of characteristic zero. The category equivalence between central simple algebras and irreducible, generically free PGL _n -varieties is extended to the context of central simple algebras with involution. The associated variety of a central simple algebra with involution comes with an action of , where τ is the automorphism of PGL _n given by τ (h) = (h ⁻¹)^transpose. Basic properties of an involution are described in terms of the action of on the associated variety, and in particular in terms of the stabilizer in general position for this action.     

A homological approach to representations of algebras II: tame hereditary algebras

We determine the pure global dimension of finite dimensional hereditary or radical-squared zero algebras over algebraically closed fields. The results are applied to algebras of dimension four and to the incidence algebras of critical ordered sets studied by Loupias. We further prove that the path algebra of an oriented cycle shares with Dedekind domains the Kulikov property (submodules of pure-projective modules are pure-projective).     

On lattices of congruences of relational systems and universal algebras

Let \(\mathfrak{X}\) =〈X;R〉 be a relational system.X is a non-empty set andR is a collection of subsets ofX ^α, α an ordinal. The system of equivalence relations onX having the substitution property with respect to members ofR form a complete latticeC( \(\mathfrak{X}\) ) containing the identity but not necessarilyX×X. It is shown that for any relational system (X;R) there is a groupoid definable onX whose congruence lattice isC( \(\mathfrak{X}\) )U{X×X} . Theorem 2 and Corollary 2 contain some interesting combinatorial pecularities associated with oriented complete graphs and simple groupoids.     

Partial algebras—survey of a unifying approach towards a two-valued model theory for partial algebras

In this survey, with no proofs included, we collect some material scattered through recent papers and a planned monograph, which shows that partial algebras do have a two-valued first order model theory which is simpler and nicer than one might have expected it to be. In section 1 we comment and present some basic definitions. In section 2 a correct and complete two-valued first order logic is developed. In section 3 the three main concepts of “varieties” are presented, while sections 4 and 5 contain some additional axiomatizability results and some applications, respectively. Section 6 contains some additional remarks. Presented by E. Fried.     

Estimating the covariance matrix: a new approach

In this paper, we consider the problem of estimating the covariance matrix and the generalized variance when the observations follow a nonsingular multivariate normal distribution with unknown mean. A new method is presented to obtain a truncated estimator that utilizes the information available in the sample mean matrix and dominates the James-Stein minimax estimator. Several scale equivariant minimax estimators are also given. This method is then applied to obtain new truncated and improved estimators of the generalized variance; it also provides a new proof to the results of Shorrock and Zidek (Ann. Statist. 4 (1976) 629) and Sinha (J. Multivariate Anal. 6 (1976) 617).     

A note on effective descent morphisms of topological spaces and relational algebras

We formulate two open problems related to and, in a sense, suggested by the Reiterman-Tholen characterization of effective descent morphisms of topological spaces.     

Repeated Red-Black ordering: a new approach

Hereafter, we describe and analyze, from both a theoretical and a numerical point of view, an iterative method for efficiently solving symmetric elliptic problems with possibly discontinuous coefficients. In the following, we use the Preconditioned Conjugate Gradient method to solve the symmetric positive definite linear systems which arise from the finite element discretization of the problems. We focus our interest on sparse and efficient preconditioners. In order to define the preconditioners, we perform two steps: first we reorder the unknowns and then we carry out a (modified) incomplete factorization of the original matrix. We study numerically and theoretically two preconditioners, the second preconditioner corresponding to the one investigated by Brand and Heinemann [2]. We prove convergence results about the Poisson equation with either Dirichlet or periodic boundary conditions. For a meshsizeh, Brand proved that the condition number of the preconditioned system is bounded byO(h ^–1/2) for Dirichlet boundary conditions. By slightly modifying the preconditioning process, we prove that the condition number is bounded byO(h ^–1/3).     

Necessary conditions in optimal control: a new approach

 We present necessary conditions of optimality for a general control problem formulated in terms of a differential inclusion. These conditions unify and substantially extend previous results in the literature, and also incorporate a `stratified' feature of a novel nature. When specialized to the calculus of variations, the results yield necessary conditions and regularity theorems that go significantly beyond the previous standard. They also give rise to new and stronger maximum principles of Pontryagin type. The principal focus here is the development of structural criteria on the problem which guarantee a priori that the necessary conditions apply to any local minimizer. Received: November 20, 2002 / Accepted: April 15, 2003 / Published online: May 28, 2003 Professeur à l'Institut universitaire de France.     

Equivariant mappings: a new approach in stochastic simulations

Stochastic simulations on manifolds usually are traced back to ⁿ via charts. If a group G is acting on a manifold M and if the respective distribution v is invariant under this group action then in many cases of practical interest there exists a more convenient approach which uses equivariant mappings. The concept of equivariant mappings will be discussed intensively at the instance of the Grassman manifold in which case G equals the orthogonal group. Further advantages of this concept will be demonstrated by applying it to a probabilistic problem from the field of combinatorial geometry.     

Demiarcs: An atomistic approach to relational systems and group dynamics

In the course of extensive conversations during the past decade with Dorwin Cartwright, the concept of demiarcs was conceived, developed, explored, and applied to a great variety of relational systems. The present communication is not intended as definitive, but rather as introductory and exploratory. There is still much room for clarification, empirical justification, and application to a great variety of relational systems including the study of group dynamics, innovation, and the dissemination of information.     

A new construction of vertex algebras and quasi-modules for vertex algebras

In this paper, a new construction of vertex algebras from more general vertex operators is given and a notion of quasimodule for vertex algebras is introduced and studied. More specifically, a notion of quasilocal subset(space) of for any vector space W is introduced and studied, generalizing the notion of usual locality in the most possible way, and it is proved that on any maximal quasilocal subspace there exists a natural vertex algebra structure and that any quasilocal subset of generates a vertex algebra. Furthermore, it is proved that W is a quasimodule for each of the vertex algebras generated by quasilocal subsets of . A notion of Γ-vertex algebra is also introduced and studied, where Γ is a subgroup of the multiplicative group C^× of nonzero complex numbers. It is proved that any maximal quasilocal subspace of is naturally a Γ-vertex algebra and that any quasilocal subset of generates a Γ-vertex algebra. It is also proved that a Γ-vertex algebra exactly amounts to a vertex algebra equipped with a Γ-module structure which satisfies a certain compatibility condition. Finally, two families of examples are given, involving twisted affine Lie algebras and certain quantum torus Lie algebras.