A Modular Algorithm for Computing Cohomologies of Lie Algebras and Superalgebras

The paper describes an improved algorithm for computing cohomologies of Lie (super)algebras. The original algorithm developed earlier by the author of this paper is based on the decomposition of the entire cochain complex into minimal subcomplexes. The suggested improvement consists in the replacement of the arithmetic of rational or integer numbers by a more efficient arithmetic of modular fields and the use of the relationship dim H ^k( _p) dimH ^k() between the dimensions of cohomologies over an arbitrary modular field _p = /p and the filed of rational numbers . This inequality allows us to rapidly find subcomplexes for which dimH ^k( _p) > 0 (the number of such subcomplexes is usually not great) using computations over an arbitrary _p and, then, carry out all required computations over in these subcomplexes.     

Stability of a Class of Differential Inclusions

Stability of differential inclusions F(x(t)) is studied by using minorant and majorant mappings F ^– and F ⁺, F ^–(x)F(x)F ⁺(x). Properties of F ^–,F ⁺ are developed in terms of partial orderings, with the condition that F ^–, F ⁺ are either heterotone or pseudoconcave. The main results concern asymptotically stable absorbing sets, including the case of a single equilibrium point, and are illustrated by examples of control systems.     

Representation of Algebraic Functions As Power and Fractional Power Series of Special Type

A Maple procedure is described by means of which an algebraic function given by an equation f(x y) = 0 can be expanded into a fractional power series (Puiseux series)

A Representation of the Interval Hull of a Tolerance Polyhedron Describing Inclusions of Function Values and Slopes

Given a nonempty set of functions

Solution of Systems of Linear Equations by the p-Adic Method

A new algorithm for solving systems of linear equations Ax = b in an Euclidean domain is suggested. In the case of the ring of integers, the complexity of this algorithm is O (n ³ mlog² ||A||), where n)$$ " align="middle" border="0"> is a matrix of rank n and , if standard algorithms for the multiplication of integers and matrices are used. Under the same conditions, the best algorithm of this kind among those published earlier, which was suggested by Labahn and Storjohann in [1], has complexity O (n ⁴ mlog² ||A||). True, when using fast algorithms for the multiplication of numbers and matrices, the theoretical complexity estimate for the latter algorithm is O (n mlog² ||A||), which is better than the similar estimate O (n ³ mlog||A||) for the new algorithm.     

Analytical Expansion and Numerical Approximation of the Fermi–Dirac Integrals F j(x) of Order j=−1/2 and j=1/2

This paper uses properties of the Weyl semiintegral and semiderivative, along with Oldham's representation of the Randles–Sevcik function from electrochemistry, to derive infinite series expansions for the Fermi–Dirac integrals _j (x), –j=–1/2, 1/2. The practical use of these expansions for the numerical approximation of _–1/2(x) and _1/2(x) over finite intervals is investigated and an extension of these results to the higher order cases j=3/2, 5/2, 7/2 is outlined.     

An Estimation Problem for Quasilinear Stochastic Partial Differential Equations

A problem of estimating a functional parameter (x) and functionals () based on observation of a solution u (t, x) of the stochastic partial differential equation is considered. The asymptotic problem setting, as the noise intensity 0, is investigated.     

Definability of Polyadic Lifts of Generalized Quantifiers

We study generalized quantifiers on finite structures.With every function : we associate a quantifier Q by letting Q x say there are at least (n) elementsx satisfying , where n is the sizeof the universe. This is the general form ofwhat is known as a monotone quantifier of type < 1 >.We study so called polyadic liftsof such quantifiers. The particular lifts we considerare Ramseyfication, branching and resumption.In each case we get exact criteria fordefinability of the lift in terms of simpler quantifiers.     

Constructing sets of functions which have a givenF-cardinality

The termF-cardinality of (=F-card()) is introduced whereF: ^{n n} is a partial function and is a set of partial functionsf: ^{n n} . TheF-cardinality yields a lower bound for the worst-case complexity of computingF if only functionsf can be evaluated by the underlying abstract automaton without conditional jumps. This complexity bound isindependent from the oracles available for the abstract machine. Thus it is shown that any automaton which can only apply the four basic arithmetic operations needs (n logn) worst-case time to sortn numbers; this result is even true if conditional jumps witharbitrary conditions are possible. The main result of this paper is the following: Given a total functionF: ^{n n} and a natural numberk, it is almost always possible to construct a set such that itsF-cardinality has the valuek; in addition, can be required to be closed under composition of functionsf,g . Moreover, ifF is continuous, then consists of continuous functions.     

A hierarchy theorem for multihead stack-counter automata

Summary Let L _b = {w ₁ ^*...* w _2b ¦w _i is in {0,1}^* and w _i = w _2b+1–i for 1i2b for b1. We show that the language L _b is not recognizable by any nondeterministic one-way k-head stack-counter automata if \left( {\begin{array}{*{20}c} k \\ 2 \\ \end{array} } \right)$$ " align="middle" border="0"> . As a corollary, we show that k+1 heads are better than k for one-way multihead stack-counter automata.     

The π-Calculus in Direct Style

We introduce a calculus which is a direct extension of both the and the calculi. We give a simple type system for it, that encompasses both Curry's type inference for the -calculus, and Milner's sorting for the -calculus as particular cases of typing. We observe that the various continuation passing style transformations for -terms, written in our calculus, actually correspond to encodings already given by Milner and others for evaluation strategies of -terms into the -calculus. Furthermore, the associated sortings correspond to well-known double negation translations on types. Finally we provide an adequate CPS transform from our calculus to the -calculus. This shows that the latter may be regarded as an assembly language, while our calculus seems to provide a better programming notation for higher-order concurrency. We conclude by discussing some alternative design decisions.     

Sufficiency conditions for existence of an optimal feedback control in stochastic mechanics

In the framework of stochastic mechanics, the following problem is considered: in a set of admissible feedback controls v, with range inE ⁿ , find one minimizing the expectationE _sx { _s ^T L(t, (t), (t, (t)))dt + W _T ((T))} for all (s, x) [0,T) E ⁿ , whereL(t, x, ) = (/12)m ² – U(t, x) is the classical action integrand and is an-dimensional diffusion process in the weak sense, (see Bensoussan, 1982) with drift and diffusion coefficientD constant > 0.W _T andU are given real functions. Sufficiency conditions for the existence of such an optimal feedback control are given. Dedicated to George Leitmann Recommended by G.J. Olsder Presented at the Third Workshop on Control Mechanics in honor of George Leitmann, January 22–24, 1990, University of Southern California, Los Angeles, California (USA).     

Exact Linear Thermodynamic Evolution Equations from the Robertson Formalism

A linear evolution equation for a thermodynamic variable F, odd under time-reversal, is obtained from the exact equation derived by Robertson from the Liouville equation for the information-theoretic phase-space distribution. One obtains an exact expression for , the relaxation time for F. For very short , is time-independent for t > if C(t) F{exp(-i t)}Fo, the equilibrium time correlation, decays exponentially for t > . is the Liouville operator. So long as C(t) is such that decays rapidly to a steady-state value, the t limit of agrees with the fluctuation-dissipation theorem in applications to fluid transport.     

On the Quadratic Convergence of the Levenberg-Marquardt Method without Nonsingularity Assumption

Recently, Yamashita and Fukushima [11] established an interesting quadratic convergence result for the Levenberg-Marquardt method without the nonsingularity assumption. This paper extends the result of Yamashita and Fukushima by using _k=||F(x_k)||, where [1,2], instead of _k=||F(x_k)||² as the Levenberg-Marquardt parameter. If ||F(x)|| provides a local error bound for the system of nonlinear equations F(x)=0, it is shown that the sequence {x_k} generated by the new method converges to a solution quadratically, which is stronger than dist(x_k,X^*)0 given by Yamashita and Fukushima. Numerical results show that the method performs well for singular problems.     

Formal Solutions of Linear Ordinary Differential Equations Containing m-Hypergeometric Series

Let a linear homogeneous ordinary differential equation with polynomial coefficients over a field be given. For a singular point of the equation, the fundamental system of formal solutions that contain a finite number of power series with coefficients belonging to the algebraic extension of can be constructed by known algorithms. In this paper, an algorithm is suggested for construction of a space of formal solutions such that all series containing in these solutions have m-hypergeometric coefficients. The implementation of the algorithm in the computer algebra system Maple is discussed.     

Hierarchy theorems for two-way finite state transducers

Summary For a family of languages , CAL() is defined as the family of images of under nondeterministic two-way finite state transducers, while FINITE · VISIT() is the closure of under deterministic two-way finite state transducers; CAL⁰()= and for n0, CAL ⁿ⁺¹()=CAL ⁿ (CAL()). For any semiAFL , if FINITE · VISIT() CAL(), then CAL ⁿ () forms a proper hierarchy and for every n0, FINITE · VISIT(CALⁿ()) CAL ⁿ⁺¹() FINITE · VISIT(CAL ⁿ⁺¹()). If is a SLIP semiAFL or a weakly k-iterative full semiAFL or a semiAFL contained in any full bounded AFL, then FINITE · VISIT() CAL() and in the last two cases, FINITE · VISIT(). If is a substitution closed full principal semiAFL and FINITE · VISIT(), then FINITE · VISIT() CAL(). If is a substitution closed full principal semiAFL generated by a language without an infinite regular set and ₁ is a full semiAFL, then is contained in CAL^m(₁) if and only if it is contained in ₁. Among the applications of these results are the following. For the following families , CAL ⁿ () forms a proper hierarchy: =INDEXED, =ETOL, and any semiAFL contained in CF. The family CF is incomparable with CAL ^m (NESA) where NESA is the family of one-way nonerasing stack languages and INDEXED is incomparable with CAL ^m (STACK) where STACK is the family of one-way stack languages.This work was supported in part by the National Science Foundation under Grants No. DCR74-15091 and MCS-78-04725     

On deterministic approximation of DNF

We develop several quasi-polynomial-time deterministic algorithms for approximating the fraction of truth assignments that satisfy a disjunctive normal form formula. The most efficient algorithm computes for a given DNF formulaF onn variables withm clauses and > 0 an estimateY such that ¦Pr[F] –Y¦ in time which is , for any constant. Although the algorithms themselves are deterministic, their analysis is probabilistic and uses the notion of limited independence between random variables.Research supported in part by National Science Foundation Operating Grant CCR-9016468, National Science Foundation Operating Grant CCR-9304722, United States-Israel Binational Science Foundation Grant No. 89-00312, United States-Israel Binational Science Foundation Grant No. 92-00226, and ESPRIT Basic Research Grant EC-US 030.Research partially done while visiting the International Computer Science Institute and while at Carnegie Mellon University.     

Langages saturés et cônes décroissants langages et cônes bifidèles

Résumé C() (resp.C ^d (–),C ^f (–)) est la plus petite famille de langages contenant et fermée par homomorphisme (resp. homomorphisme alphabétique, homomorphisme non effaçant), homomorphisme inverse (resp. homomorphisme strictement alphabétique inverse, homomorphisme inverse) et par intersection rationnelle.C ^d (–) etC ^f (–) sont donc des cônes restreints deC(–). Nous donnons des relations entre cônes restreints deC(–) etC(–). Ce qui nous permet de construire une hiérarchie infinie de langages algébriques non effaçables (L est effaçable C({L})=C ^f ({L})).

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Saturated languages and decreasing cones bifaithful languages and bifaithful cones

Summary C() (resp.C ^d (–),C ^f (–)) is the smallest family of languages containing and closed under morphism (resp. alphabetic morphism, non erasing morphism), inverse morphism (resp. inverse alphabetic non erasing morphism, inverse morphism) and intersection with regular sets.C ^d (–) andC ^f (–) are restricted cones ofC(). We give relations between restricted cones ofC() andC(), that allows us to construct an infinite hierarchy of no-erasable context-fee languages (L is erasableC({L})=C ^f ({L}))

     

A Theoretical Analysis of Query Selection for Collaborative Filtering

We consider the problem of determining which of a set of experts has tastes most similar to a given user by asking the user questions about his likes and dislikes. We describe a simple algorithm for generating queries for a theoretical model of this problem. We show that the algorithm requires at most opt(F)(ln(|F|/opt(F)) + 1) + 1 queries to find the correct expert, where opt(F) is the optimal worst-case bound on the number of queries for learning arbitrary elements of the set of experts F. The algorithm runs in time polynomial in |F| and |X| (where X is the domain) and we prove that no polynomial-time algorithm can have a significantly better bound on the number of queries unless all problems in NP have n ^{O(log log n)} time algorithms. We also study a more general case where the user ratings come from a finite set Y and there is an integer-valued loss function on Y that is used to measure the distance between the ratings. Assuming that the loss function is a metric and that there is an expert within a distance from the user, we give a polynomial-time algorithm that is guaranteed to find such an expert after at most 2opt(F, ) ln + 2( + 1)(1 + deg(F, )) queries, where deg(F, ) is the largest number of experts in F that are within a distance 2 of any f F.