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.     

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)

Interpretations of open default theories in non-monotonic logics

It is shown that the translation of an open default into a modal formula x(L(x)LM ₁ (x)...LM _m (x)w(x)) gives rise to an embedding of open default systems into non-monotonic logics.     

A short note on simultaneous splitting

We show that for any alphabet there is a setL ^* such that ifC is any infinite co-infinite context-free language over , thenL splitsC (i.e., each ofL C,L , C, and is infinite).Preparation of this paper was supported in part by the National Science Foundation under Grant No. MCS77-11360.     

Finiteness in Infinite-Valued Łukasiewicz Logic

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic _m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in _m. We also reduce the notion of logical consequence in to the same notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic.     

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.     

Modelling Asynchrony with a Synchronous Model

The I/O automaton paradigm of Lynch and Tuttle models asynchrony through an interleaving parallel composition. The recognition that such interleaving models in fact can be viewed as special cases of synchronous parallel composition has been very limited. Let be any set of finite-state I/O automata drawing actions from a fixed finite set containing a subset . In this article we establish a translation T : to a class of -automata closed under a synchronous parallel composition, for which T is monotonic with respect to implementation relative to , and linear with respect to composition. Thus, for A¹, ..., A, B¹, ..., B and A = A¹ ... A, B = B¹ ... B, if is the set of actions common to both A and B, then A implements B (in the sense of I/O automata) if and only if the -automaton language containment (T(A¹) ... T(A)) (T(B¹) ... T(B)) obtains, where denotes the interleaving parallel composition on and denotes the synchronous parallel composition on . For the class , we use the L-process model of -automata. This result enables one to verify systems specified by I/O automata through model-checkers such as COSPAN or SMV, that operate on models with synchronous parallel composition. The translation technique generalizes to other interleaving models, although in each case, the translation map must match the specific model.     

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

Given a nonempty set of functions

The Relative Entropy of Entanglement of Schmidt Correlated States and Distillation

We show that a mixed state = _mna_mn|mn| can be realized by an ensemble of pure states {p_k, |_k} where . Employing this form, we discuss the relative entropy of entanglement of Schmidt correlated states. Also, we calculate the distillable entanglement of a class of mixed states. PACS: 03.67.-a; 03.65.Bz; 03.65.Ud     

Stability of the Logistic Population Model with Delayed Environmental Response

The population dynamics model , was considered. For this model with uniform distribution of delays and a _n = 0, nonnegativeness and convexity of the sequence a _k (0 k n) was shown to be the sufficient stability condition. Therefore, there is no need to constrain the reproduction rate and the mean delay .     

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.     

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.     

Ramsey numbers and an approximation algorithm for the vertex cover problem

Summary We show two results. First we derive an upper bound for the special Ramsey numbers r _k(q) where r _k(q) is the largest number of nodes a graph without odd cycles of length bounded by 2k+1 and without an independent set of size q+1 can have. We prove The proof is constructive and yields an algorithm for computing an independent set of that size. Using this algorithm we secondly describe an O(¦V¦·¦E¦) time bounded approximation algorithm for the vertex cover problem, whose worst case ratio is , for all graphs with at most (2k+3) ^k (2k+2) nodes (e.g. 1.8, if ¦V¦146000).     

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     

Spigot Algorithm and Reliable Computation of Natural Logarithm

The spigot approach used in the previous paper (Reliable Computing 7 (3) (2001), pp. 247–273) for root computation is now applied to natural logarithms. The logarithm ln Q with Q , Q > 1 is decomposed into a sum of two addends k ₁× ln Q ₁+k ₂× ln Q ₂ with k ₁, k ₂ , then each of them is computed by the spigot algorithm and summation is carried out using integer arithmetic. The whole procedure is not literally a spigot algorithm, but advantages are the same: only integer arithmetic is needed whereas arbitrary accuracy is achieved and absolute reliability is guaranteed. The concrete procedure based on the decomposition with p, q ( – {0}), p < q is simple and ready for implementation. In addition to the mentioned paper, means for determining an upper bound for the biggest integer occurring in the process of spigot computing are now provided, which is essential for the reliability of machine computation.     

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.     

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.     

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.     

Staircase visibility and computation of kernels

Let be some set of orientations, that is, . We consider the consequences of defining visibility based on curves that are monotone with respect to the orientations in . We call such curves -staircases. Two points p andq in a polygonP are said to -see each other if an -staircase fromp toq exists that is completely contained inP. The -kernel of a polygonP is then the set of all points which -see all other points. The -kernel of a simple polygon can be obtained as the intersection of all {}-kernels, with . With the help of this observation we are able to develop an algorithm to compute the -kernel of a simple polygon, for finite . We also show how to compute theexternal -kernel of a polygon in optimal time . The two algorithms are combined to compute the ( -kernel of a polygon with holes in time .This work was supported by the Deutsche Forschungsgemeinschaft under Grant No. Ot 64/5-4 and the Natural Sciences and Engineering Research Council of Canada and Information Technology Research Centre of Ontario.