Stabilizability of linear systems defined overC *-algebras

LetF andG be elements of aC ^*-algebraA. Assume that, for each irreducible^*-representation ofA on a Hilbert space210B; , there is a bounded linear operatorL B( ) such that the spectrum of(F) –(G)L is contained in the open left half plane. We prove that there is then an elementL A such that the spectrum ofF — GL is contained in the open left half plane. That is, if the system (F, G) is locally stabilizable, then it is stabilizable. We also consider the analogous problem with the open left half plane replaced by the open unit disk.This paper was supported in part by the National Science Foundation under Grant NSF-MCS-8002138.     

Splitting a configuration in a simplex

This paper presents a new method of partition, named-splitting, of a point set ind-dimensional space. Given a pointG in ad-dimensional simplexT, T(G;i) is the subsimplex spanned by G and the ith facet ofT. LetS be a set ofn points inT, and let be a sequence of nonnegative integers ₁, ..., n_d+1 satisfying _i=1 ^d+1 ₁=n The-splitter of (T, S) is a pointG inT such thatT(G;i) contains at least _i points ofS in its closure for everyi=1, 2, ...,d + 1. The associated dissection is the re-splitting.The existence of a-splitting is shown for any (T, S) and, and two efficient algorithms for finding such a splitting are given. One runs inO(d²n logn + d³n) time, and the other runs inO(n) time if the dimensiond can be considered as a constant. Applications of re-splitting to mesh generation, polygonal-tour generation, and a combinatorial assignment problem are given.     

Using Metric Spaces in Optimum Scheduling

The solution of an optimum problem of scheduling with n workpieces and m machine tools represents an optimum schedule of putting pieces on machines. In turn, the schedule is defined by an optimum collection of m permutations out of n objects, i.e., the vector permutation = (₁, ..., _m ), where each permutation _i (1 i m) points up the sequence of working of all pieces on the ith machine. In this case, to each admissible schedule there must correspond an integral point from the m-dimensional Euclidean space of permutations (or, which is practically the same, the permutation out of numbers {1, 2, ..., mn}. In an effort to seek an optimum schedule, use is made of the notion of a metric space in the set of admissible schedules and the justified methodology of the search for an optimum schedule. A few metric spaces are described and analyzed and their comparative effectiveness is investigated for the solution of a different-route problem of scheduling.     

The growth of configurations

The transition ruleF of a cellular automaton may sometimes be regarded as a rule of growth of a crystal from a seed. A study is made of the iterates,F,F ² .For certain one-dimensional growth rules, the limiting shapes of the crystals are computed, and an asymptotic formula for the size of the crystal as a function of time is obtained.     

A logical encoding of the π-calculus: model checking mobile processes using tabled resolution

We present MMC, a model checker for mobile systems specified in the style of the -calculus. MMCs development builds on that of XMC, a model checker for an expressive extension of Milners value-passing calculus implemented using the XSB tabled logic-programming engine. MMC addresses the salient issues that arise in the -calculus, including scope extrusion and intrusion and dynamic generation of new names to avoid name capture. We show that logic programming provides an efficient implementation platform for model checking -calculus specifications and can be used to obtain an exact encoding of the -calculuss transitional semantics. Moreover, MMC is easily extended to handle process expressions in the spi-calculus of Abadi and Gordon. Our experimental data show that MMC outperforms other known tools for model checking the -calculus.     

A priori Probability That Two Qubits Are Unentangled

In a previous study (P. B. Slater, Eur. Phys. J. B. 17, 471 (2000)), several remarkably simple exact results were found, in certain specialized m-dimensional scenarios (m 4), for the a priori probability that a pair of qubits is unentangled/separable. The measure used was the volume element of the Bures metric (identically one-fourth the statistical distinguishability [SD] metric). Here, making use of a newly-developed (Euler angle) parameterization of the 4 × 4 density matrices of Tilma, Byrd and Sudarshan, we extend the analysis to the complete 15-dimensional convex set (C) of arbitrarily paired qubits—the total SD volume of which is known to be ⁸ / 1680 = ⁸/2⁴ 3 5 7 5.64794. Using advanced quasi-Monte Carlo procedures (scrambled Halton sequences) for numerical integration in this high-dimensional space, we approximately (5.64851) reproduce that value, while obtaining an estimate of 0.416302 for the SD volume of separable states. We conjecture that this is but an approximation to ⁶/2310 = ⁶ / (2 3 5 7 11) 0.416186. The ratio of the two volumes, 8/11²2 .0736881, would then constitute the exact Bures/SD probability of separability. The SD area of the 14-dimensional boundary of C is 142⁷/12285 = 2 71⁷/3³ 5 7 13 34.911, while we obtain a numerical estimate of 1.75414 for the SD area of the boundary of separable states. PACS: 03.67.-; 03.65.Ud; 02.60.Jh; 02.40.Ky     

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].     

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.     

Maintaining the visibility map of spheres while moving the viewpoint on a circle at infinity

We investigate three-dimensional visibility problems for scenes that consist ofn non-intersecting spheres. The viewing point moves on a flightpath that is part of a circle at infinity given by a planeP and a range of angles {(t)¦t[01]} [02]. At timet, the lines of sight are parallel to the ray inP, which starts in the origin ofP and represents the angle(t) (orthographic views of the scene). We give an algorithm that computes the visibility graph at the start of the flight, all time parameters at which the topology of the scene changes, and the corresponding topology changes. The algorithm has running time0(n + k + p) logn), wheren is the number of spheres in the scene;p is the number of transparent topology changes (the number of different scene topologies visible along the flight path, assuming that all spheres are transparent); andk denotes the number of vertices (conflicts) which are in the (transparent) visibility graph at the start and do not disappear during the flight.The second author was supported by the ESPRIT II Basic Research Actions Program, under Contract No. 3075 (project ALCOM).     

Maintaining Longest Paths Incrementally

Modeling and programming tools for neighborhood search often support invariants, i.e., data structures specified declaratively and automatically maintained incrementally under changes. This paper considers invariants for longest paths in directed acyclic graphs, a fundamental abstraction for many applications. It presents bounded incremental algorithms for arc insertion and deletion which run in O( + || log||) time and O() time respectively, where || and are measures of the change in the input and output. The paper also shows how to generalize the algorithm to various classes of multiple insertions/deletions encountered in scheduling applications. Preliminary experimental results show that the algorithms behave well in practice.     

A universal topological transformation group inL 2 (G × G)

LetG denote an infinite,-compact, locally compact topological group. In this paper a construction is given for a topological transformation groupH _G with the Hilbert spaceL ² (G × G) as a phase space such that any topological transformation group (G, X, ) can be embedded inH _G , providedX is a separable metrizable space and is a bounded action. The class of such topological transformation groups contains all actions ofG on separable, metrizable, locally compact spaces.     

A Delay-Averaged Logistic Model

For the equation x(t) = x(t) (1-(1/) _t-- ^t- x(u)du), > 0, > 0, > 0, conditions for the stability of a nonzero stationary solution under small perturbations are determined.     

A geometric view of parametric linear programming

We present a new definition of optimality intervals for the parametric right-hand side linear programming (parametric RHS LP) Problem () = min{c ^t x¦Ax =b + ¯b,x 0}. We then show that an optimality interval consists either of a breakpoint or the open interval between two consecutive breakpoints of the continuous piecewise linear convex function (). As a consequence, the optimality intervals form a partition of the closed interval {; ¦()¦ < }. Based on these optimality intervals, we also introduce an algorithm for solving the parametric RHS LP problem which requires an LP solver as a subroutine. If a polynomial-time LP solver is used to implement this subroutine, we obtain a substantial improvement on the complexity of those parametric RHS LP instances which exhibit degeneracy. When the number of breakpoints of () is polynomial in terms of the size of the parametric problem, we show that the latter can be solved in polynomial time.This research was partially funded by the United States Navy-Office of Naval Research under Contract N00014-87-K-0202. Its financial support is gratefully acknowledged.     

Correlation structure of training data and the fitting ability of back propagation networks: Some experimental results

White [6–8] has theoretically shown that learning procedures used in network training are inherently statistical in nature. This paper takes a small but pioneering experimental step towards learning about this statistical behaviour by showing that the results obtained are completely in line with White's theory. We show that, given two random vectorsX (input) andY (target), which follow a two-dimensional standard normal distribution, and fixed network complexity, the network's fitting ability definitely improves with increasing correlation coefficient r_XY (0r_XY1) betweenX andY. We also provide numerical examples which support that both increasing the network complexity and training for much longer do improve the network's performance. However, as we clearly demonstrate, these improvements are far from dramatic, except in the case r_XY=+ 1. This is mainly due to the existence of a theoretical lower bound to the inherent conditional variance, as we both analytically and numerically show. Finally, the fitting ability of the network for a test set is illustrated with an example.Nomenclature X Generalr-dimensional random vector. In this work it is a one-dimensional normal vector and represents the input vector - Y Generalp-dimensional random vector. In this work it is a one-dimensional normal vector and represents the target vector - Z Generalr+p dimensional random vector. In this work it is a two-dimensional normal vector - E Expectation operator (Lebesgue integral) - g(X)=E(Y¦X) Conditional expectation - Experimental random error (defined by Eq. (2.1)) - y Realized target value - o Output value - f Network's output function. It is formally expressed asf: R ^r×WR ^p, whereW is the appropriate weight space - Average (or expected) performance function. It is defined by Eq. (2.2) as (w)=E[(Yf(X,w)],w W - Network's performance - w ^* Weight vector for optimal solution. That is, the objective of network is such that (w ^*) is minimum - C ₁ Component one - C ₂ Component two - Z Matrix of realised values of the random vectorZ overn observations - Z _t Transformed matrix version ofZ in such a way thatX andY have values in [0,1] - X _t ,Y _t Transformed versions ofX andY and both are standard one-dimensional normal vectors - n _h Number of hidden nodes (neurons) - r _XY Correlation coefficient between eitherX andY orX _t andY _t - _s and _k in Eq. (3.1) and afterwards _s is average value of 100 differentZ _t matrices. _k is the error function ofkthZ _t , matrix. In Eq. (3.1), the summation is fromk=1 to 100, and in Eq. (3.2) fromi=1 ton. In Eq. (3.2)o _ki andy _ki are the output and target values for the kthZ _t matrix and ith observation, respectively - ^1/2(w ^*) and _k (w_n) k(w_n) is the sample analogue of ^1/2(w ^*) - _Y ² In Eq. (4.1) and afterwards, _Y ² is the variance ofY - _Y ² variance ofY _t . In Sect. 4.3 the transformation isY _t=a Y+b - Y _max,Y _min the maximum and minimum values ofY - R Correlation matrix ofX andY - Covariance matrix ofX andY - Membership symbol in set theory     

Priority as extremal probability

We extend the stratified model of probabilistic processes to obtain a very general notion ofprocess priority. The main idea is to allow probability guards of value 0 to be associated with alternatives of a probabilistic summation expression. Such alternatives can be chosen only if the non-zero alternatives are precluded by contextual constraints. We refer to this model as one of extremal probability and to its signature asPCCS . We providePCCS with a structural operational semantics and a notionof probabilistic bisimulation, which is shown to be a congruence. Of particular interest is the abstractionPCCS ofPCCS in which all non-zero probability guards are identified.PCCS represents a customized framework for reasoning about priority, and covers all features of process algebras proposed for reasoning about priority that we know of.A preliminary version of this paper appeared inProceedings of CONCUR '90 — First International Conference on Concurrency Theory, Vol. 458 of the Springer-Verlag seriesLecture Notes in Computer Science, pp. 456–466, Aug. 1990. The research of Scott Smolka was supported in part by NSF Grants CCR-9120995, CCR-9208585 and CCR-9505562; and AFOSR Grants F49620-93-1-0250, F49620-95-1-0508 and F49620-96-1-0087.     

Circuits constructed with MOD q gates cannot compute “AND” in sublinear size

Algebraic techniques are used to prove that any circuit constructed with MOD _q gates that computes the AND function must use (n) gates at the first level. The best bound previously known to be valid for arbitraryq was (logn).     

On the group of automorphisms of strongly connected automata

LetA = (S, I, M) be a strongly connected finite automaton withn states. Weeg has shown that ifA has a group of automorphisms of orderm, then there is a partition of the setS inton/m blocks each withm states. Furthermore ifs _i ands _j are in the same block of, thenT _ii =T _jj , whereT _ii = {x|x ^* and thenM(s _i , x) =s _i }. It will be shown that the partition also must have the substitution property and that these two conditions are sufficient for ann state strongly connected automaton to have a group of automorphisms of orderm.Necessary and sufficient conditions for twon-state strongly connected automata to have isomorphic automorphism groups are given. Also, it is demonstrated that forT _ii to equalT _jj it is necessary to check only a finite number of tapes and consequently provide an algorithm for determining whether or notA has a group of automorphisms of orderm. This research was partially supported by the National Science Foundation under Grant No. G.P. 7077.     

A note on dense and nondense families of complexity classes

In a model for a measure of computational complexity, , for a partial recursive functiont, letR _t denote all partial recursive functions having the same domain ast and computable within timet. Let = {R _t |t is recursive} and let = { |_i is actually the running time function of a computation}. and are partially ordered under set-theoretic inclusion. These partial orderings have been extensively investigated by Borodin, Constable and Hopcroft in [3]. In this paper we present a simple uniform proof of some of their results. For example, we give a procedure for easily calculating a model of computational complexity for which is not dense while is dense. In our opinion, our technique is so transparent that it indicates that certain questions of density are not intrinsically interesting for general abstract measures of computational complexity, . (This is not to say that similar questions are necessarily uninteresting for specific models.)Supported by NSF Research Grants GP6120 and GJ27127.     

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.     

Probabilistic analysis of the application of the cross ratio to model based vision: Misclassification

The cross ratio of four colinear points is of fundamental importance in model based vision, because it is the simplest numerical property of an object that is invariant under projection to an image. It provides a basis for algorithms to recognise objects from images without first estimating the position and orientation of the camera.A quantitative analysis of the effectiveness of the cross ratio in model based vision is made. A given imageI of four colinear points is classified by making comparisons between the measured cross ratio of the four image points and the cross ratios stored in the model database. The imageI is accepted as a projection of an objectO with cross ratio if |–|ntu, wheren is the standard deviation of the image noise,t is a threshold andu=. The performance of the cross ratio is described quantitatively by the probability of rejectionR, the probability of false alarmF and the probability of misclassificationp (), defined for two model cross ratios , . The trade off between these different probabilities is determined byt.It is assumed that in the absence of an object the image points have identical Gaussian distributions, and that in the presence of an object the image points have the appropriate conditional densities. The measurements of the image points are subject to small random Gaussian perturbations. Under these assumptions the trade offs betweenR,F andp () are given to a good approximation byR=2(1–(t)),F=r _F t, t|–|^–1, where is the relative noise level, is cumulative distribution function for the normal distribution,r _F is constant, ande is a function of only. The trade off betweenR andF is obtained in Maybank (1994). In this paper the trade off betweenR andp () is obtained.It is conjectured that the general form of the above trade offs betweenR,F andp () is the same for a range of invariants useful in model based vision. The conjecture prompts the following definition: an invariant which has trade offs betweenR,F,p () of the above form is said to benon-degenerate for model based vision.The consequences of the trade off betweenR andp () are examined. In particular, it is shown that for a fixed overall probability of misclassification there is a maximum possible model cross ratio _m , and there is a maximum possible numberN of models. Approximate expressions for _m andN are obtained. They indicate that in practice a model database containing only cross ratio values can have a size of order at most ten, for a physically plausible level of image noise, and for a probability of misclassification of the order 0.1.