Stability of Linear Difference and Differential Inclusions

Any given n×n matrix A is shown to be a restriction, to the A-invariant subspace, of a nonnegative N×N matrix B of spectral radius (B) arbitrarily close to (A). A difference inclusion , where is a compact set of matrices, is asymptotically stable if and only if can be extended to a set of nonnegative matrices B with or . Similar results are derived for differential inclusions.     

The Complexity of Learning According to Two Models of a Drifting Environment

We show that a bound on the rate of drift of the distribution generating the examples is sufficient for agnostic learning to relative accuracy , where c > 0 is a constant; this matches a known necessary condition to within a constant factor. We establish a sufficient condition for the realizable case, also matching a known necessary condition to within a constant factor. We provide a relatively simple proof of a bound of + on the sample complexity of agnostic learning in a fixed environment.     

Lower Bound Methods and Separation Results for On-Line Learning Models

We consider the complexity of concept learning in various common models for on-line learning, focusing on methods for proving lower bounds to the learning complexity of a concept class. Among others, we consider the model for learning with equivalence and membership queries. For this model we give lower bounds on the number of queries that are needed to learn a concept class in terms of the Vapnik-Chervonenkis dimension of , and in terms of the complexity of learning with arbitrary equivalence queries. Furthermore, we survey other known lower bound methods and we exhibit all known relationships between learning complexities in the models considered and some relevant combinatorial parameters. As it turns out, the picture is almost complete. This paper has been written so that it can be read without previous knowledge of Computational Learning Theory.     

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.     

Stochastic Finite Learning of the Pattern Languages

The present paper proposes a new learning model—called stochastic finite learning—and shows the whole class of pattern languages to be learnable within this model.This main result is achieved by providing a new and improved average-case analysis of the Lange–Wiehagen (New Generation Computing, 8, 361–370) algorithm learning the class of all pattern languages in the limit from positive data. The complexity measure chosen is the total learning time, i.e., the overall time taken by the algorithm until convergence. The expectation of the total learning time is carefully analyzed and exponentially shrinking tail bounds for it are established for a large class of probability distributions. For every pattern containing k different variables it is shown that Lange and Wiehagen's algorithm possesses an expected total learning time of , where and are two easily computable parameters arising naturally from the underlying probability distributions, and E[] is the expected example string length.Finally, assuming a bit of domain knowledge concerning the underlying class of probability distributions, it is shown how to convert learning in the limit into stochastic finite learning.     

On the Complexity of Function Learning

The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger range such as or . In particular relatively few results exist about the general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models.We introduce in this paper the notion of a binary branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization of the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not involve any learning model).Another general structural result of this paper relates the cost for learning a union of function classes to the learning costs for the individual function classes.Furthermore, we exhibit an efficient learning algorithm for learning convex piecewise linear functions from ^d into . Previously, the class of linear functions from ^d into was the only class of functions with multidimensional domain that was known to be learnable within the rigorous framework of a formal model for online learning.Finally we give a sufficient condition for an arbitrary class of functions from into that allows us to learn the class of all functions that can be written as the pointwise maximum ofk functions from . This allows us to exhibit a number of further nontrivial classes of functions from into for which there exist efficient learning algorithms.     

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

Given a nonempty set of functions

The Power of Self-Directed Learning

This article studies self-directed learning, a variant of the on-line (or incremental) learning model in which the learner selects the presentation order for the instances. Alternatively, one can view this model as a variation of learning with membership queries in which the learner is only charged for membership queries for which it could not predict the outcome. We give tight bounds on the complexity of self-directed learning for the concept classes of monomials, monotone DNF formulas, and axis-parallel rectangles in {0, 1, , n – 1} ^d . These results demonstrate that the number of mistakes under self-directed learning can be surprisingly small. We then show that learning complexity in the model of self-directed learning is less than that of all other commonly studied on-line and query learning models. Next we explore the relationship between the complexity of self-directed learning and the Vapnik-Chervonenkis (VC-)dimension. We show that, in general, the VC-dimension and the self-directed learning complexity are incomparable. However, for some special cases, we show that the VC-dimension gives a lower bound for the self-directed learning complexity. Finally, we explore a relationship between Mitchell's version space algorithm and the existence of self-directed learning algorithms that make few mistakes.     

Factorisation of finite state machines under strong and observational equivalences

The factorisation problem is to construct the specification of a submoduleX when the specifications of the system and all submodules butX are given. It is usually described by the equation where P and X are submodules of system Q, ¦ is a composition operator, and is the equivalence criterion. In this paper we use a finite state machine (FSM) model consistent with CCS and study two factorisation problems:P |||P Q andP |||P Q, where ||| is a derived CCS composition operator, and represent strong and observational equivalences. Algorithms are presented and proved correct to find the most general specification of submoduleX forP |||P Q withQ -deterministic and forP |||P Q withQ deterministic. Conditions on the submachines of the most general solutions that remain solutions toP |||P Q(P |||P Q) are given. This paper extends and is based on the work of M. W. Shields.     

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.     

Mesh Selection for a Nearly Singular Boundary Value Problem

In this paper, we investigate the numerical solution of a model equation u _xx = exp(– ) (and several slightly more general problems) when 1 using the standard central difference scheme on nonuniform grids. In particular, we are interested in the error behaviour in two limiting cases: (i) the total mesh point number N is fixed when the regularization parameter 0, and (ii) is fixed when N. Using a formal analysis, we show that a generalized version of a special piecewise uniform mesh 12 and an adaptive grid based on the equidistribution principle share some common features. And the optimal meshes give rates of convergence bounded by |log()| as 0 and N is given, which are shown to be sharp by numerical tests.     

Algorithmic properties of maximal orders in simple algebras over Q

This paper addresses an algorithmic problem related to associative algebras. We show that the problem of deciding if the index of a given central simple algebra over an algebraic number field isd, whered is a given natural number, belongs to the complexity classN P co N P. As consequences, we obtain that the problem of deciding if is isomorphic to a full matrix algebra over the ground field and the problem of deciding if is a skewfield both belong toN P co N P. These results answer two questions raised in [25]. The algorithms and proofs rely mostly on the theory of maximal orders over number fields, a noncommutative generalization of algebraic number theory. Our results include an extension to the noncommutative case of an algorithm given by Huang for computing the factorization of rational primes in number fields and of a method of Zassenhaus for testing local maximality of orders in number fields.     

Generating Boolean \mu-expressions

In this paper, we consider the class of Boolean -functions, which are the Boolean functions definable by -expressions (Boolean expressions in which no variable occurs more than once). We present an algorithm which transforms a Boolean formulaE into an equivalent -expression-if possible-in time linear in E times , where E is the size ofE andn _m is the number of variables that occur more than once inE. As an application, we obtain a polynomial time algorithm for Mundici's problem of recognizing -functions fromk-formulas [17]. Furthermore, we show that recognizing Boolean -functions is co-NP-complete for functions essentially dependent on all variables and we give a bound close to co-NP for the general case.     

The computational complexity of recognizing permutation functions

Let be a finite field withq elements and a rational function over . No polynomial-time deterministic algorithm is known for the problem of deciding whetherf induces a permutation on . The problem has been shown to be in co-R co-NP, and in this paper we prove that it is inR NP and hence inZPP, and it is deterministic polynomial-time reducible to the problem of factoring univariate polynomials over . Besides the problem of recognizing prime numbers, it seems to be the only natural decision problem inZPP unknown to be inP. A deterministic test and a simple probabilistic test for permutation functions are also presented.     

Extremal Relations between Additive Loss Functions and the Kolmogorov Complexity

Conditions are presented under which the maximum of the Kolmogorov complexity (algorithmic entropy) K(₁... _N ) is attained, given the cost f( _i ) of a message ₁... _N . Various extremal relations between the message cost and the Kolmogorov complexity are also considered; in particular, the minimization problem for the function f( _i ) – K(₁... _N ) is studied. Here, is a parameter, called the temperature by analogy with thermodynamics. We also study domains of small variation of this function.     

Directional differentiation of marginal functions in mathematical programming problems

We consider regular mathematical programming problems of the form f(x, y) inf, y F(x), x Rⁿ, where F(x) = {y R^m h_i| (x, y) 0, , h_i (x, y) = 0, . The directional derivatives offunctions (x) = inf{f(x, y)|y F(x)} are estimated.Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 70–77, November–December, 1991.     

On a Stroboscopic Approach to Quantum Tomography of Qudits Governed by Gaussian Semigroups

In this paper, we discuss the minimal number of observables Q ₁, ..., Q , where expectation values at some time instants t ₁, ..., t _r determine the trajectory of a d-level quantum system (qudit) governed by the Gaussian semigroup . We assume that the macroscopic information about the system in question is given by the mean values E _j(Q _i) = tr(Q _i(t _j)) of n selfadjoint operators Q ₁, ..., Q _n at some time instants t ₁ < t ₂ < ... < t _r, where n < d ²– 1 and r deg (, ). Here (, ) stands for the minimal polynomial of the generator of the Gaussian flow (t).     

Exact optimal grillage layouts — Part I: Combinations of free and simply supported edges

In the first instalment of this three-part study, a comprehensive treatment of analytically derived, exact optimal grillage layouts for combinations of simply supported and free edges is given. In part two, grillages with combinations of simply supported, clamped and free edges will be considered.Notation k constant in specific cost function - M beam bending moment - r radius of circular edge - R ⁺,R ^–,S ⁺,S ^–,T optimal regions - x, x _j coordinate along a beam (j) - slope of the adjoint deflection at pointD in directionDA - t, v coordinates along the free edge - adjoint deflection - angle between long beams and free edge - angle between free and simply supported edges - curvature of the adjoint deflection - , angles for layouts with circular edge - total weight (cost) of grillage - coordinate along a beam in anR ⁺ region - distance defined in Fig. 3     

Solving Underdetermined Systems with Interval Methods

In this paper, we use the concepts of {1} - and {2} -inverse of a matrix to construct such a Krawczyk-like operator as