Unextendible Product Bases and Locally Unconvertible Bound Entangled States

Mutual convertibility of bound entangled states under local quantum operations and classical communication (LOCC) is studied. We focus on states associated with unextendible product bases (UPB) in a system of three qubits. A complete classification of such UPBs is suggested. We prove that for any pair of UPBs S and T the associated bound entangled states _S and _T cannot be converted to each other by LOCC, unless S and T coincide up to local unitaries. More specifically, there exists a finite precision (S,T) > 0 such that for any LOCC protocol mapping _S into a probabilistic ensemble (p, ), the fidelity between _T and any possible final state satisfies F(_T, ) = 1 - (S,T).PACS: 03.65.Bz; 03.67.-a; 89.70+c.     

On characterizations of recursively enumerable languages

Summary Geffert has shown that earch recursively enumerable languageL over can be expressed in the formL{h(x) ^–1 g(x)x in ⁺} ^* where is an alphabet andg, h is a pair of morphisms. Our purpose is to give a simple proof for Geffert's result and then sharpen it into the form where both of the morphisms are nonerasing. In our method we modify constructions used in a representation of recursively enumerable languages in terms of equality sets and in a characterization of simple transducers in terms of morphisms. As direct consequences, we get the undecidability of the Post correspondence problem and various representations ofL. For instance,L =(L ₀) ^* whereL ₀ is a minimal linear language and is the Dyck reductiona, A.     

Logic Based Abstractions of Real-Time Systems

When verifying concurrent systems described by transition systems, state explosion is one of the most serious problems. If quantitative temporal information (expressed by clock ticks) is considered, state explosion is even more serious. We present a notion of abstraction of transition systems, where the abstraction is driven by the formulae of a quantitative temporal logic, called qu-mu-calculus, defined in the paper. The abstraction is based on a notion of bisimulation equivalence, called , n-equivalence, where is a set of actions and n is a natural number. It is proved that two transition systems are , n-equivalent iff they give the same truth value to all qu-mu-calculus formulae such that the actions occurring in the modal operators are contained in , and with time constraints whose values are less than or equal to n. We present a non-standard (abstract) semantics for a timed process algebra able to produce reduced transition systems for checking formulae. The abstract semantics, parametric with respect to a set of actions and a natural number n, produces a reduced transition system , n-equivalent to the standard one. A transformational method is also defined, by means of which it is possible to syntactically transform a program into a smaller one, still preserving , n-equivalence.     

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.     

On States,Channels, and Purification

In this note we introduce purification for a pair (, ), where is a quantum state and is a channel, which allows in particular a natural extension of the properties of related information quantities (mutual and coherent informations) to the channels with arbitrary input and output spaces. PACS: 03.67.Hk     

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.     

Conditions for the Light-Traffic Insensitivity to Loss Probability in a GI/G/m/0 Queueing System

A loss queueing system GI/G/m/0 is considered. Let a(x) be a p.d.f. of interarrival intervals. Assume that this function behaves like cx^-1 for small x. Further let B(x) be a d.f. of service time; (1/) be the mean service time. Conditions are derived for the light-traffic insensitivity of the loss probability to the form of B(x) as (/ ) 0. In particular, the condition = 1 is necessary. Estimates for the loss probability are obtained.     

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.     

Minimization of norms and logarithmic norms by diagonal similarities

It is a commonly occurring problem to find good norms · or logarithmic norms (·) for a given matrix in the sense that they should be close to respectively the spectral radius (A) and the spectral abscissa (A). Examples may be the certification thatA is convergent, i.e. (A)A<1 or stable, i.e. (A)(A)<0. Often the ordinary norms do not suffice and one would like to try simple modifications of them such as using an ordinary norm for a diagonally transformed matrix. This paper treats this problem for some of the ordinary norms.

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Minimisierung von Normen und Logarithmischen Normen durch Diagonale Transformationen

Zusammenfassung Ein oft vorkommendes praktisches Problem ist die Konstruktion von guten Normen · und logarithmischen Normen (·) für eine gegebene MatrixA. Mit gut wird dann verstanden, daß A den Spektralradius (A)=max |₁| und (A) die Spektralabszisse (A)=max Re _i gut approximieren. Beispiele findet man für konvergente Matrizen wo (A)A<1 gewünscht ist, und für stabile Matrizen wo (A)(A)<0 zu zeigen ist. Wir untersuchen hier, wie weit man mit Diagonaltransformationen und dengewöhnlichsten Normen kommen kann.

     

On improving the accuracy of the Hough transform

The subject of this paper is very high precision parameter estimation using the Hough transform. We identify various problems that adversely affect the accuracy of the Hough transform and propose a new, high accuracy method that consists of smoothing the Hough arrayH(, ) prior to finding its peak location and interpolating about this peak to find a final sub-bucket peak. We also investigate the effect of the quantizations and ofH(, ) on the final accuracy. We consider in detail the case of finding the parameters of a straight line. Using extensive simulation and a number of experiments on calibrated targets, we compare the accuracy of the method with results from the standard Hough transform method of taking the quantized peak coordinates, with results from taking the centroid about the peak, and with results from least squares fitting. The largest set of simulations cover a range of line lengths and Gaussian zero-mean noise distributions. This noise model is ideally suited to the least squares method, and yet the results from the method compare favorably. Compared to the centroid or to standard Hough estimates, the results are significantly better—for the standard Hough estimates by a factor of 3 to 10. In addition, the simulations show that as and are increased (i.e., made coarser), the sub-bucket interpolation maintains a high level of accuracy. Experiments using real images are also described, and in these the new method has errors smaller by a factor of 3 or more compared to the standard Hough estimates.     

Simultaneous inner and outer approximation of shapes

For compact Euclidean bodiesP, Q, we define (P, Q) to be the smallest ratior/s wherer > 0,s > 0 satisfy . HeresQ denotes a scaling ofQ by the factors, andQ,Q are some translates ofQ. This function gives us a new distance function between bodies which, unlike previously studied measures, is invariant under affine transformations. If homothetic bodies are identified, the logarithm of this function is a metric. (Two bodies arehomothetic if one can be obtained from the other by scaling and translation.)For integerk 3, define (k) to be the minimum value such that for each convex polygonP there exists a convexk-gonQ with (P, Q) (k). Among other results, we prove that 2.118 ... <-(3) 2.25 and (k) = 1 + (k ^–2). We give anO(n ² log² n)-time algorithm which, for any input convexn-gonP, finds a triangleT that minimizes (T, P) among triangles. However, in linear time we can find a trianglet with (t, P)<-2.25.Our study is motivated by the attempt to reduce the complexity of the polygon containment problem, and also the motion-planning problem. In each case we describe algorithms which run faster when certain implicitslackness parameters of the input are bounded away from 1. These algorithms illustrate a new algorithmic paradigm in computational geometry for coping with complexity.Work of all authors was partially supported by the ESPRIT II Basic Research Actions Program of the EC under Contract No. 3075 (project ALCOM). Rudolf Fleischer and Kurt Mehlhorn acknowledge also DFG (Grant SPP Me 620/6). Chee Yap acknowledges also DFG (Grant Be 142/46-1) and NSF (Grants DCR-84-01898 and CCR-87-03458). This research was performed when Günter Rote and Chee Yap were at the Freie Universität Berlin.     

Utilizing deontic operators in information systems specification

One major task in requirements specification is to capture the rules relevant to the problem at hand. Declarative, rule-based approaches have been suggested by many researchers in the field. However, when it comes to modeling large systems of rules, not only for the behavior of the computer system but also for the organizational environment surrounding it, current approaches have problems with limited expressiveness, flexibility, and poor comprehensibility. Hence, rule-based approaches may benefit from improvements in two directions: (1) improvement of the rule languages themselves and (2) better integration with other, complementary modeling approaches.In this article, both issues are addressed in an integrated manner. The proposal is presented in the context of the Tempora project on rule-based information systems development, but has also been integrated with PPP. Tempora has provided a rule language based on an executable temporal logic working on top of a temporal database. The rule language is integrated with static (ER-like) and dynamic (SA/RT-like) modeling approaches. In the current proposal, the integration with complementary modeling approaches is extended by including organization modeling (actors, roles), and the expressiveness of the rule language is increased by introducing deontic operators and rule hierarchies. The main contribution of the article is not seen as any one of the above-mentioned extensions, but as the resulting comprehensive modeling support. The approach is illustrated by examples taken from an industrial case study done in connection with Tempora.C. List of Symbols Subset of set - Not subset of set - Element of set - Not element of set - Equivalent to - Not equivalent to - ¬ Negation - Logical and - Logical or - Implication - Sometime in past - Sometime in future - Always in past - Always in future - Just before - Just after - u Until - s Since - Trigger - Condition - _s State condition - Consequence - _a Action - _s State - Role - Actor - ¬ - General deontic operator - O Obligatory - R Recommended - P Permitted - D Discouraged - F Forbidden - (/–) General rule - t _R Real time - t _M Model time     

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     

Reasoning about programs in continuation-passing style

Plotkin's _v -calculus for call-by-value programs is weaker than the -calculus for the same programs in continuation-passing style (CPS). To identify the call-by-value axioms that correspond to on CPS terms, we define a new CPS transformation and an inverse mapping, both of which are interesting in their own right. Using the new CPS transformation, we determine the precise language of CPS terms closed under -transformations, as well as the call-by-value axioms that correspond to the so-called administrative -reductions on CPS terms. Using the inverse mapping, we map the remaining and equalities on CPS terms to axioms on call-by-value terms. On the pure (constant free) set of -terms, the resulting set of axioms is equivalent to Moggi's computational -calculus. If the call-by-value language includes the control operatorsabort andcall-with-current-continuation, the axioms are equivalent to an extension of Felleisenet al.'s _v -C-calculus and to the equational subtheory of Talcott's logic IOCC.This article is a revised and extended version of the conference paper with the same title [42]. The technical report of the same title contains additional material.The authors were supported in part by NSF grant CCR 89-17022 and by Texas ATP grant 91-003604014.     

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.     

Symbolic computation of robot models for geometric parameters identification with singularity analysis

The computation of the geometrical parameters identification model of a robot arm leads to an intrinsic extended Jacobian matrix. This matrix is often singular and its manual computation is tedious and may lead to error. A computer assisted procedure is developed to compute this matrix symbolically and to check its singularity. The program GPIM is developed in Pascal and works on PC's for this purpose. Jacobian singularity is checked using its symbolic expressions with the aid of a proposed table for robot arm parameters to obtain a minimum set of identified parameters. This program can be used for simple chain robot arms having revolute (R) and/or prismatic (P) joints including the case of consecutive near parallel axes. TH8 robot arm of type RPPRRR having two pairs of consecutive near parallel axes is studied to show the program efficiency and how singularity is eliminated.Nomenclature a _i Denavit-Hartenberg parameter. - C _i ,C _i andC ₁₂ cos _i , cos _i and cos ( ₁ + ₂). - J ₀, J Basic Jacobian and Jacobian matrices. - J _a , J _r , J , J and J Sub-Jacobian matrices. - J _0a , J _0r , J ₀, J ₀ and J ₀ Intrinsic sub-Jacobian matrices. - m Work space dimensions. - n Number of moving links of the robot arm. - p Number of pairs of consecutive near parallel axes. - P _i,i+1 and P _li [3×1] position vectors. - Q _pi , Q _ai , Q _ri , Q _i , Q _i and Q _i [4×4] differential operator matrices. - r _i Denavit-Hartenberg parameter. - R _l , R _i , R _i+1 and R _n+1 Base, links i–1, i and n coordinate frames. - R _i,i+1 and R _1i [3×3] orientation matrices. - S _i , S _i and S ₁₂ sin _i , sin _i and sin ₁ + ₂. - T _i,i+1 and T _1i [4×4] homogeneous transformation matrices. - X [m×1] robot arm operational coordinates. - 1st columns of the orientation matrices, and associated tensors. - 2nd columns of the orientation matrices, and associated tensors. - 3rd columns of the orientation matrices, and associated tensors. - _i Denavit-Hartenberg parameter. - _i Twist angle parameter. - X Changes in robot arm operational coordinates. - , ls and rr Changes in robot arm geometrical parameters, and their least square and ridge regression estimated changes. - _i Denavit-Hartenberg parameter. - , ^c and ⁿ Robot geometrical parameters and their correct and nominal values. - pi, ai, ri, i, i and i [4×4] differential operator matrices. - and ^–1 [3×3] conversion matrix and its inverse.     

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)     

To the Metrical Rigidity of Binary Codes

A code C in the n-dimensional metric space E ⁿ over GF(2) is called metrically rigid if each isometry I : C E ⁿ can be extended to an isometry of the whole space E ⁿ . For n large enough, metrical rigidity of any length-n binary code that contains a 2-(n, k, )-design is proved. The class of such codes includes, for instance, all families of uniformly packed codes of large enough lengths that satisfy the condition d – 2, where d is the code distance and is the covering radius.     

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].     

On restricted one-counter machines

Let be the class of real-time nondeterministic one-counter machines whose counters make at mostone reversal. Let ₁ (respectively, ₂) be the subclass consisting of machines whose only nondeterministic move is in the choice of when to reverse the counter (respectively, when to start using the counter). ₁ and ₂ are among the simplest known classes of machines for which the universe problem has been shown undecidable. (The universe problem for a class of machines is the problem of deciding if an arbitrary machine in the class accepts all its inputs.) Here, we show that the classes of languages accepted by machines in ₁ and ₂ are incomparable. Moreover, the union of the language classes is properly contained in the class defined by . We also, briefly, look at the closure properties of these machines.This research was supported in part by NSF Grant MCS78-01736.