Linear Interval Tolerance Problem and Linear Programming Techniques

In this paper, we consider the linear interval tolerance problem, which consists of finding the largest interval vector included in ([A], [b]) = {x R ⁿ | A [A], b [b], Ax = b}. We describe two different polyhedrons that represent subsets of all possible interval vectors in ([A], [b]), and we provide a new definition of the optimality of an interval vector included in ([A], [b]). Finally, we show how the Simplex algorithm can be applied to find an optimal interval vector in ([A], [b]).     

Records of turing machines

Suppose a Turing machine is equipped with an extra tape. At each step of a computation being performed, it prints symbol read move symbol symbol printed on a square of the extra tape. It then moves the extra tape one square to the left. This procedure yields arecord of the computation.If is a finite alphabet, let be the set of triples (a, b, c) wherea ,c , andb {–1, 0, 1}. We will characterize those sequences of symbols of that are records of computations of Turing machines.This research was supported in part by AFOSR Grant GP-68-1402.     

Polynomial-Time Decomposition Algorithms for Support Vector Machines

This paper studies the convergence properties of a general class of decomposition algorithms for support vector machines (SVMs). We provide a model algorithm for decomposition, and prove necessary and sufficient conditions for stepwise improvement of this algorithm. We introduce a simple rate certifying condition and prove a polynomial-time bound on the rate of convergence of the model algorithm when it satisfies this condition. Although it is not clear that existing SVM algorithms satisfy this condition, we provide a version of the model algorithm that does. For this algorithm we show that when the slack multiplier C satisfies 1/2 C mL, where m is the number of samples and L is a matrix norm, then it takes no more than 4LC ² m ⁴/ iterations to drive the criterion to within of its optimum.     

Almost surely continuous solutions of a nonlinear stochastic integral equation

A nonlinear stochastic integral equation of the Hammerstein type in the formx(t; ) = h(t, x(t; )) + _s k(t, s; )f(s, x(s; ); )d(s) is studied wheret S, a measure space with certain properties, , the supporting set of a probability measure space (,A, P), and the integral is a Bochner integral. A random solution of the equation is defined to be an almost surely continuousm-dimensional vector-valued stochastic process onS which is bounded with probability one for eacht S and which satisfies the equation almost surely. Several theorems are proved which give conditions such that a unique random solution exists. AMS (MOS) subject classifications (1970): Primary; 60H20, 45G99. Secondary: 60G99.     

Duality in Quantum Information Geometry

Let H be a separable Hilbert space. We consider the manifold M consisting of density operators on H such that ^p is of trace class for some p (0, 1). We say M is nearby if there exists C > 1 such that C ^–1C. We show that the space of nearby points to can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish M with a norm making it into a Banach manifold.     

Directional derivatives of the maximum function

Directional differentiability of the function (x) = sup{f(x, u), u U} is proved for a class of smooth functions f. The result is applied to study the directional differentiability of the function (x) = sup{f(x, y), y F(x)}, where F is a multivalued mapping.Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 171–173, March–April, 1992.     

Towards optimal simulations of formulas by bounded-width programs

We show that, over an arbitrary ring, for any fixed >0, all balanced algebraic formulas of sizes are computed by algebraic straight-line programs that employ a constant number of registers and have lengthO (s ¹⁺). In particular, in the special case where the ring isGF(2), we obtain a technique for simulating balanced Boolean formulas of sizes by bounded-width branching programs of lengthO(s ¹⁺), for any fixed >0. This is an asymptotic improvement in efficiency over previous simulations in both the Boolean and algebraic settings.     

On the Axioms of Scale Space Theory

We consider alternative scale space representations beyond the well-established Gaussian case that satisfy all reasonable axioms. One of these turns out to be subject to a first order pseudo partial differential equation equivalent to the Laplace equation on the upper half plane {(x, s) ^d × | s > 0}. We investigate this so-called Poisson scale space and show that it is indeed a viable alternative to Gaussian scale space. Poisson and Gaussian scale space are related via a one-parameter class of operationally well-defined intermediate representations generated by a fractional power of (minus) the spatial Laplace operator.     

Least Squares Policy Evaluation Algorithms with Linear Function Approximation

We consider policy evaluation algorithms within the context of infinite-horizon dynamic programming problems with discounted cost. We focus on discrete-time dynamic systems with a large number of states, and we discuss two methods, which use simulation, temporal differences, and linear cost function approximation. The first method is a new gradient-like algorithm involving least-squares subproblems and a diminishing stepsize, which is based on the -policy iteration method of Bertsekas and Ioffe. The second method is the LSTD() algorithm recently proposed by Boyan, which for =0 coincides with the linear least-squares temporal-difference algorithm of Bradtke and Barto. At present, there is only a convergence result by Bradtke and Barto for the LSTD(0) algorithm. Here, we strengthen this result by showing the convergence of LSTD(), with probability 1, for every [0, 1].     

Modelling default and likelihood reasoning as probabilistic reasoning

This paper presents a probabilistic analysis of plausible reasoning about defaults and about likelihood. Likely and by default are in fact treated as duals in the same sense as possibility and necessity. To model these four forms probabilistically, a logicQDP and its quantitative counterpartDP are derived that allow qualitative and corresponding quantitative reasoning. Consistency and consequence results for subsets of the logics are given that require at most a quadratic number of satisfiability tests in the underlying prepositional logic. The quantitative logic shows how to track the propagation error inherent in these reasoning forms. The methodology and sound framework of the system highlights their approximate nature, the dualities, and the need for complementary reasoning about relevance.Much of this research was done while at the University of Technology, Sydney, Broadway, NSW, Australia, and some at the Turing Institute, 36 Nth. Hanover Str., Glasgow, Scotland.     

An Interval Linear Tolerance Problem

This paper is concerned with a tolerance problem for an interval linear system A x = b requiring inner estimation of the admissible solution set {x ⁿ | (A A)(Ax b)} formed by vectors x for which the product Ax remains within b for any possible A A. Methods for verifying the emptiness and nonemptiness of admissible solution sets are developed. Formulas for the dimensions of the interval solution of a tolerance problem with known center are derived.     

On a random Volterra integral equation

Summary Tsokos [12] showed the existence of a unique random solution of the random Volterra integral equation (*)x(t; ) = h(t; ) + _o ^t k(t, ; )f(, x(; )) d, where , the supporting set of a probability measure space (,A, P). It was required thatf must satisfy a Lipschitz condition in a certain subset of a Banach space. By using an extension of Banach's contraction-mapping principle, it is shown here that a unique random solution of (*) exists whenf is (, )-uniformly locally Lipschitz in the same subset of the Banach space considered in [12].     

Multi-time scale zero-sum differential games with perfect state measurements

We obtain necessary and sufficient conditions for the existence of approximate saddle-point solutions in linear-quadratic zero-sum differential games when the state dynamics are defined on multiple (three) time scales. These different time scales are characterized in terms of two small positive parameters ₁, and ₂, and the terminology approximate saddle-point solution is used to refer to saddle-point policies that do not depend on ₁ and ₂, while providing cost levels withinO(₁) of the full-order game. It is shown in the paper that, under perfect state measurements, the original game can be decomposed into three subgames-slow, fast and fastest, the composite saddle-point solution of which make up the approximate saddle-point solution of the original game. Specifically, for the minimizing player, it is necessary to use a composite policy that uses the solutions of all three subgames, whereas for the maximizing player, it is sufficient to use a slow policy. In the finite-horizon case this slow policy could be a feedback policy, whereas in the infinite-horizon case it has to be chosen as an open-loop policy that is generated from the solution and dynamics of the slow subgame. These results have direct applications in theH -optimal control of three-time scale singularly perturbed linear systems under perfect state measurements.Research supported in part by the National Science Foundation under Grant ECS 91-13153, and in part by the U.S. Department of Energy under Grant DE-FG-02-88-ER-13939. This paper is dedicated to the memory of John Breakwell.     

Efficient Solution of Multi-Term Fractional Differential Equations Using P(EC)<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>E Methods

We investigate strategies for the numerical solution of the initial value problem with initial conditions where 0<₁<₂<<. Here y ⁽ _j ) denotes the derivative of order _j >0 (not necessarily _j ) in the sense of Caputo. The methods are based on numerical integration techniques applied to an equivalent nonlinear and weakly singular Volterra integral equation. The classical approach leads to an algorithm with very high arithmetic complexity. Therefore we derive an alternative that leads to lower complexity without sacrificing too much precision. Mathematics Subject Classification: Primary 65L05; Secondary 65L06, 65L20     

Specification and verification of database dynamics

Summary A framework is proposed for the structured specification and verification of database dynamics. In this framework, the conceptual model of a database is a many sorted first order linear tense theory whose proper axioms specify the update and the triggering behaviour of the database. The use of conceptual modelling approaches for structuring such a theory is analysed. Semantic primitives based on the notions of event and process are adopted for modelling the dynamic aspects. Events are used to model both atomic database operations and communication actions (input/output). Nonatomic operations to be performed on the database (transactions) are modelled by processes in terms of trigger/reaction patterns of behaviour. The correctness of the specification is verified by proving that the desired requirements on the evolution of the database are theorems of the conceptual model. Besides the traditional data integrity constraints, requirements of the form Under condition W, it is guaranteed that the database operation Z will be successfully performed are also considered. Such liveness requirements have been ignored in the database literature, although they are essential to a complete definition of the database dynamics.

Notation

Classical Logic Symbols (Appendix 1) for all (universal quantifier) - exists at least once (existential quantifier) - ¬ no (negation) - implies (implication) - is equivalent to (equivalence) - and (conjunction) - or (disjunction) Tense Logic Symbols (Appendix 1) G always in the future - G ₀ always in the future and now - F sometime in the future - F ₀ sometime in the future or now - H always in the past - H ₀ always in the past and now - P sometime in the past - P ₀ sometime in the past or now - X in the next moment - Y in the previous moment - L always - M sometime Event Specification Symbols (Sects. 3 and 4.1) (x) means immediately after the occurrence of x - (x) means immediately before the occurrence of x - (x) means x is enabled, i.e., x may occur next - { } ({w ₁} x{w ₂}) states that if w ₁ holds before the occurrence of x, then w ₂ will hold after the occurrence of x (change rule) - [ ] ([oa₁, ..., oa_n]x) states that only the object attributes oa₁, ..., oa _n are modifiable by x (scope rule) - {{ }} ({{w}}x) states that if x may occur next, then w holds (enabling rule) Process Specification Symbols (Sects. 5.3 and 5.4) :: for causal rules - for behavioural rules Transition-Pattern Composition Symbols (Sects. 5.2 and 5.3) ; sequential composition - ¦ choice composition - parallel composition - :| guarded alternative composition Location Predicates (Sect. 5.2) (z) means immediately after the occurrence of the last event of z (after) - (z) means immediately before the occurrence of the first event of z (before) - (z) means after the beginning of z and before the end of z (during) - ( z) means before the occurrence of an event of z (at)     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

PAC Learning Axis-aligned Rectangles with Respect to Product Distributions from Multiple-Instance Examples

We describe a polynomial-time algorithm for learning axis-aligned rectangles in Q ^d with respect to product distributions from multiple-instance examples in the PAC model. Here, each example consists of n elements of Q^d together with a label indicating whether any of the n points is in the rectangle to be learned. We assume that there is an unknown product distribution D over Q ^d such that all instances are independently drawn according to D. The accuracy of a hypothesis is measured by the probability that it would incorrectly predict whether one of n more points drawn from D was in the rectangle to be learned. Our algorithm achieves accuracy with probability 1- in O (d⁵ n¹²/²⁰ log² nd/ time.     

Bounding Perturbations in Zeros of Nonlinear Systems

Algorithms for determining computationally rigorous componentwise bounds on the solutions x R ⁿ of equations F(x, t) = 0 R ^m containing parameters t R ^l due to small perturbations in t when m n and when F is at most twice continuously differentiable in x and in t are described. Numerical results which illustrate the behaviour of the algorithms are presented.     

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.     

The existence of solutions,stability, and linearization of volterra systems

LetB be a Banach space ofR _n valued continuous functions on [0, ) withfB. Consider the nonlinear Volterra integral equation (^*)x(t)+ _o ^t K(t,s,x(s))ds. We use the implicit function theorem to give sufficient conditions onB andK (t,s,x) for the existence of a unique solutionxB to (^*) for eachf B with f ^B sufficiently small. Moreover, there is a constantM>0 independent off with Mf^B.Part of this work was done while the author was visiting at Wright State University.