Showing papers on "Bounded function published in 1985"

The Glory of the Past

Abstract: An extension of propositional temporal logic that includes operators referring to a bounded past is considered. An exponential time decision procedure and a complete axiomatic system are presented. A suggested normal form leads to a syntactic classification of safety and liveness formulae. The adequacy of temporal logic to modular verification is examined. Finally we present the notion of α-fairness which is proved to fully capture the behavior of probabilistic finite state programs.

A Local-Ratio Theorem for Approximating the Weighted Vertex Cover Problem

Abstract: A local-ratio theorem for approximating the weighted vertex cover problem is presented. It consists of reducing the weights of vertices in certain subgraphs and has the effect of local-approximation. Putting together the Nemhauser-Trotter local optimization algorithm and the local-ratio theorem yields several new approximation techniques which improve known results from time complexity, simplicity and performance-ratio point of view. The main approximation algorithm guarantees a ratio of where K is the smallest integer s.t. † This is an improvement over the currently known ratios, especially for a “practical” number of vertices (e.g. for graphs which have less than 2400, 60000, 10 12 vertices the ratio is bounded by 1.75, 1.8, 1.9 respectively).

Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups

Abstract: For any amenable locally compact group G, the space of multipliers MA(G) of the Fourier algebra A(G) coincides with the space B(G) of functions on G that are linear combinations of continuous positive definite functions. We prove that MA(G)\B(G) * 0 for many non-amenable connected groups. More specifically we prove that MOA(G)\B(G) * 0 for the classical complex Lie groups, and the general Lorentz groups SOO(n, 1), n > 2. MOA(G) is a certain subspace of MA(G), which we call the space of completely bounded multipliers of A(G). Unlike MA(G), the space MOA(G) has nice stability properties with respect to direct products of groups. It is known that the Fourier algebra of the free group on N generators (N 2 2) admits an unbounded approximate unit ((Pn), which is bounded in the multiplier norm. We extend this result to any closed subgroup of the general Lorentz group SOO(n, 1). Moreover we show that for these groups ((Pn) can be chosen to be bounded with respect to the MOA(G)-norm. By a duality argument we obtain that the reduced C*-algebra of every discrete subgroup of SOO(n, 1) has "the completely bounded approximation property. " In particular this property holds for C* (F2), the reduced C*-algebra of the free group on two generators. We also prove

Motion planning in the presence of moving obstacles

Abstract: This paper investigates the computational complexity of planning the motion of a body B in 2-D or 3-D space, so as to avoid collision with moving obstacles of known, easily computed, trajectories. Dynamic movement problems are of fundamental importance to robotics, but their computational complexity has not previously been investigated. We provide evidence that the 3-D dynamic movement problem is intractable even if B has only a constant number of degrees of freedom of movement. In particular, we prove the problem is PSPACE-hard if B is given a velocity modulus bound on its movements and is NP hard even if B has no velocity modulus bound, where in both cases B has 6 degrees of freedom. To prove these results we use a unique method of simulation of a Turing machine which uses time to encode configurations (whereas previous lower bound proofs in robotics used the system position to encode configurations and so required unbounded number of degrees of freedom). We also investigate a natural class of dynamic problems which we call asteroid avoidance problems: B, the object we wish to move, is a convex polyhedron which is free to move by translation with bounded velocity modulus, and the polyhedral obstacles have known translational trajectories but cannot rotate. This problem has many applications to robot, automobile, and aircraft collision avoidance. Our main positive results are polynomial time algorithms for the 2-D asteroid avoidance problem with bounded number of obstacles as well as single exponential time and nO(log n) space algorithms for the 3-D asteroid avoidance problem with an unbounded number of obstacles. Our techniques for solving these asteroid avoidance problems are novel in the sense that they are completely unrelated to previous algorithms for planning movement in the case of static obstacles. We also give some additional positive results for various other dynamic movers problems, and in particular give polynomial time algorithms for the case in which B has no velocity bounds and the movements of obstacles are algebraic in space-time.

An existence result for nonlinear elliptic problems involving critical Sobolev exponent

Abstract: In this paper we consider the following problem: (1) { − Δ u − λ u = | u | 2 ⁎ − 2 ⋅ u u = 0 on ∂ Ω 2 ⁎ = 2 n / ( n − 2 ) where Ω ⊂ Rn is a bounded domain and λ ∈ R. We prove the existence of a nontrivial solution of (1) for any λ > 0, if n ⩾ 4.

Linear turning point theory

Abstract: I Historical Introduction.- 1.1. Early Asymptotic Theory Without Turning Points.- 1.2. Total Reflection and Turning Points.- 1.3. Hydrodynamic Stability and Turning Points.- 1.4. The So-Called WKB Method.- 1.5. The Contribution of R. E. Langer.- 1.6. Remarks on Recent Trends.- II Formal Solutions.- 2.1. Introduction.- 2.2. The Jordan Form of Holomorphic Functions.- 2.3. A Formal Block Diagonalization.- 2.4. Parameter Shearing: Its Nature and Purpose.- 2.5. Simplification by a Theorem of Arnold.- 2.6. Parameter Shearing: Its Application.- 2.7. Parameter Shearing: The Exceptional Case.- 2.8. Formal Solution of the Differential Equation.- 2.9. Some Comments and Warnings.- III Solutions Away From Turning Points.- 3.1. Asymptotic Power Series: Definition of Turning Points.- 3.2. A Method for Proving the Analytic Validity of Formal.- Solutions: Preliminaries.- 3.3. A General Theorem on the Analytic Validity of Formal.- Solutions.- 3.4. A Local Asymptotic Validity Theorem.- 3.5. Remarks on Points That Are Not Asymptotically Simple.- IV Asymptotic Transformations of Differential Equations.- 4.1. Asymptotic Equivalence.- 4.2. Formal Invariants.- 4.3. Formal Circuit Relations with Respect to the Parameter.- V Uniform Transformations at Turning Points: Formal Theory.- 5.1. Preparatory Simplifications.- 5.2. A Method for Formal Simplification in Neighborhoods of a Turning Point.- 5.3. The Case h > 1.- 5.4. The General Theory for n = 2.- VI Uniform Transformations at Turning Points: Analytic Theory.- 6.1. Preliminary General Results.- 6.2. Differential Equations Reducible to Airy's Equation.- 6.3. Differential Equations Reducible to Weber's Equation.- 6.4. Uniform Transformations in a Full Neighborhood of.- a Turning Point.- 6.5. Complete Reduction to Airy's Equation.- 6.6. Reduction to Weber's Equation in Wider Sectors.- 6.7. Reduction to Weber's Equation in a Full Disk.- VII Extensions of the Regions of Validity of the Asymptotic Solutions.- 7.1. Introduction.- 7.2. Regions of Asymptotic Validity Bounded by Separation Curves: The Problem.- 7.3. Solutions Asymptotically Known in Sectors Bounded by.- Separation Curves.- 7.4. Singularities of Formal Solutions at a Turning Point.- 7.5. Asymptotic Expansions in Growing Domains.- 7.6. Asymptotic Solutions in Expanding Regions: A General Theorem.- 7.7. Asymptotic Solutions in Expanding Regions: A Local Theorem.- VIII Connection Problems.- 8.1. Introduction.- 8.2. Stretching and Parameter Shearing.- 8.3. Calculation of the Restraint Index.- 8.4. Inner and Outer Solutions for a Particular nth-Order System.- 8.5. Calculation of a Central Connection Matrix.- 8.6. Connection Formulas Calculated Through Uniform Simplification.- IX Fedoryuk's Global Theory of Second-Order Equations.- 9.1. Global Formal Solutions of ?2u"=a(x)u2u" = a(x)u.- 9.2. Separation Curves for ?2u"=a(x)u2u" = a(x)u.- 9.3. A Global Asymptotic Existence Theorem for ?2u"=a(x)u2u" = a(x)u.- X Doubly Asymptotic Expansions.- 10.1. Introduction.- 10.2. Formal Solutions for Large Values of the.- Independent Variable.- 10.3. Asymptotic Solutions for Large Values of the.- Independent Variable.- 10.4. Some Properties of Doubly Asymptotic Solutions.- 10.5. Central Connection Problems in Unbounded Regions.- XI A Singularly Perturbed Turning Point Problem.- 11.1. The Problem.- 11.2. A Simple Example.- 11.3. The General Case: Formal Part.- 11.4. The General Case: Analytic Part.- XII Appendix: Some Linear Algebra for Holomorphic Matrices.- 12.1. Vectors and Matrices of Holomorphic Functions.- 12.2. Reduction to Jordan Form.- 12.3. General Holomorphic Block Diagonalization.- 12.4. Holomorphic Transformation of Matrices into Arnold's Form.- References.

Hyperbolic reflection groups

Abstract: CONTENTSIntroductionChapter I. Acute-angled polytopes in Lobachevskii spaces ??1. The Gram matrix of a convex polytope ??2. The existence theorem for an acute-angled polytope with given Gram matrix ??3. Determination of the combinatorial structure of an acute-angled polytope from its Gram matrix ??4. Criteria for an acute-angled polytope to be bounded and to have finite volumeChapter II. Crystallographic reflection groups in Lobachevskii spaces ??5. The language of Coxeter schemes. Construction of crystallographic reflection groups ??6. The non-existence of discrete reflection groups with bounded fundamental polytope in higher-dimensional Lobachevskii spacesReferences

On systems of equations in a free group

Abstract: A description of the general solution of given bounded periodicity exponent is obtained for an arbitrary system of equations in a free group. On the basis of this result an algorithm is constructed that computes the rank of coefficient-free systems of equations. Bibliography: 7 titles.

Regularity properties of solutions to the basic problem in the calculus of variations

Abstract: This paper concerns the basic problem in the calculus of variations: minimize a functional J defined by J(x) =j L(t, x(t), x(t)) dt a over a class of arcs x whose values at a and b have been specified. Existence theory provides rather weak conditions under which the problem has a solution in the class of absolutely continuous arcs, conditions which must be strengthened in order that the standard necessary conditions apply. The question arises: What necessary conditions hold merely under hypotheses of existence theory, say the classical Tonelli conditions? It is shown that, given a solution x, there exists a relatively open subset Q of [a, b], of full measure, on which x is locally Lipschitz and satisfies a form of the Euler-Lagrange equation. The main theorem, of which this is a corollary, can also be used in conjunction with various classes of additional hypotheses to deduce the global smoothness of solutions. Three such classes are identified, and results of Bernstein, Tonelli, and Morrey are extended. One of these classes is of a novel nature, and its study implies the new result that when L is independent of t, the solution has essentially bounded derivative.

Generalized bounded variation and applications to piecewise monotonic transformations

Abstract: We prove the quasi-compactness of the Perron-Frobenius operator of piecewise monotonic transformations when the inverse of the derivative is Holder-continuous or, more generally, of bounded p-variation.

Spline Smoothing in Regression Models and Asymptotic Efficiency in $L_2$

Abstract: For nonparametric regression estimation on a bounded interval, optimal rates of decrease for integrated mean square error are known but not the best possible constants. A sharp result on such a constant, i.e., an analog of Fisher's bound for asymptotic variances is obtained for minimax risk over a Sobolev smoothness class. Normality of errors is assumed. The method is based on applying a recent result on minimax filtering in Hilbert space. A variant of spline smoothing is developed to deal with noncircular models.

A globally convergent adaptive pole placement algorithm without a persistency of excitation requirement

Abstract: This paper presents an indirect adaptive control scheme for deterministic plants which are not necessarily minimum phase. Global convergence is established for the scheme in the sense that the closed-loop poles are asymptotically assigned for the given data sequence and the system input and output remain bounded for all time. A key feature of the scheme is that no persistency of excitation condition is required. The algorithm uses recursive least squares with variable forgetting factor, normalized regression vectors, and a matrix gain with constant trace.

Uncertainty principle and uncertainty relations

Abstract: It is generally believed that the uncertainty relation Δq Δp≥1/2ħ, where Δq and Δp are standard deviations, is the precise mathematical expression of the uncertainty principle for position and momentum in quantum mechanics. We show that actually it is not possible to derive from this relation two central claims of the uncertainty principle, namely, the impossibility of an arbitrarily sharp specification of both position and momentum (as in the single-slit diffraction experiment), and the impossibility of the determination of the path of a particle in an interference experiment (such as the double-slit experiment). The failure of the uncertainty relation to produce these results is not a question of the interpretation of the formalism; it is a mathematical fact which follows from general considerations about the widths of wave functions. To express the uncertainty principle, one must distinguish two aspects of the spread of a wave function: its extent and its fine structure. We define the overall widthW Ψ and the mean peak width wψ of a general wave function ψ and show that the productW Ψ w φ is bounded from below if φ is the Fourier transform of ψ. It is shown that this relation expresses the uncertainty principle as it is used in the single- and double-slit experiments.

A general approach to d-dimensional geometric queries

Abstract: It is shown that any bounded region in Ed can be divided into 2d subregions of equal volume in such a way that no hyperplane in Ed can intersect all 2d of the subregions. This theorem provides the basis of a data structure scheme for organizing n points in d dimensions. Under this scheme, a broad class of geometric queries in d dimensions, including many common problems in range search and optimization, can be solved in linear storage space and sublinear time.

Bounded cohomology of certain groups of homeomorphisms

Abstract: We consider the condition when bounded cohomology injects into ordinary cohomology and prove the vanishing of bounded cohomology of the group of all compactly supported homeomorphisms of RW. Introduction. In this note we consider relations among bounded cohomology, ordinary real cohomology and 1 homology of spaces or groups. In particular we present a necessary and sufficient condition under which bounded cohomology injects into ordinary cohomology and by using it prove the vanishing of bounded cohomology and 1 homology of HomeoKR', the group of all homeomorphisms of R'7 with compact support. We also determine the second bounded cohomology of SL2R. 1. Bounded cohomology. Let us quickly review the theory of bounded cohomology developed by Gromov [2] (see also Brooks [1] and Mitsumatsu [5]). Let X be a topological space and let W'*(X) = {CQ(X), aq} be the singular chain complex of X with real coefficients. Define a norm on Cq(X) by lIIn=1aaiill = ,-=jail. The differentials a are then bounded linear operators. Let W'((X) = {(C'(X), aq} be the norm completion of W',(X). Thus Cql(X) (Y?=1laiailD%1jail < 00) is a Banach space. Passing to the dual Banach spaces, we obtain a cochain complex if b(X) = (Cq( X), b q It is a subcomplex of the ordinary singular cochain complex consisting of bounded cochains. The homology of W*' (X), denoted by Hl (X), is called 11 homology of X and the cohomology of Wb (X), denoted by Hb*(X), is called bounded cohomology of X. The inclusions induce homomorphisms H*(X) -* H* (X) and Hb*( X) -* H *(X). Since the image of a bounded operator is not necessarily a closed subspace, it may happen that the pseudonorms induced on H* (X) or Hb*( X) are not norms. Following Mitsumatsu [5], we define Hl*(X) (resp. H,b*(X)) to be the quotient of Hl (X) (resp. Hb*(X)) by the subspace of pseudonorm zero. In other words, H"'(X) Z('( X)/Bl$ ( X) and Hbq( X) = Zb( X)/Bbf( X), where Z or B denotes the spaces of (co)cycles or (co)boundaries of the corresponding complex and B denotes the closure of B. Notice that H' (X) and Hbq(X) are Banach spaces. There is a Received by the editors July 19, 1984. 1980 Mathematics Subject Classification. Primary 55N99; Secondary 57T99.

Lower curvature bounds, Toponogov's theorem, and bounded topology

Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1985, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

On submodular function minimization

Abstract: Earlier work of Bixby, Cunningham, and Topkis is extended to give a combinatorial algorithm for the problem of minimizing a submodular function, for which the amount of work is bounded by a polynomial in the size of the underlying set and the largest function value (not its length).

A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension

Abstract: It has been a challenge for mathematicians to confirm theoretically the extremely good performance of simplex-type algorithms for linear programming. In this paper the average number of steps performed by a simplex algorithm, the so-called self-dual method, is analyzed. The algorithm is not started at the traditional point (1, … , l)T, but points of the form (1, ϵ, ϵ2, …)T, with ϵ sufficiently small, are used. The result is better, in two respects, than those of the previous analyses. First, it is shown that the expected number of steps is bounded between two quadratic functions c1(min(m, n))2 and c2(min(m, n))2 of the smaller dimension of the problem. This should be compared with the previous two major results in the field. Borgwardt proves an upper bound of O(n4m1/(n-1)) under a model that implies that the zero vector satisfies all the constraints, and also the algorithm under his consideration solves only problems from that particular subclass. Smale analyzes the self-dual algorithm starting at (1, … , 1)T. He shows that for any fixed m there is a constant c(m) such the expected number of steps is less than c(m)(ln n)m(m+1); Megiddo has shown that, under Smale's model, an upper bound C(m) exists. Thus, for the first time, a polynomial upper bound with no restrictions (except for nondegeneracy) on the problem is proved, and, for the first time, a nontrivial lower bound of precisely the same order of magnitude is established. Both Borgwardt and Smale require the input vectors to be drawn from spherically symmetric distributions. In the model in this paper, invariance is required only under certain

Robust model tracking control for a class of nonlinear plants

Abstract: We propose a new model-following control scheme for a class of nonlinear plants. The feedback control signal is a continuous function of all its arguments. It is shown that this scheme guarantees that tracking error remains bounded and tends to a neighborhood of the origin with a rate not inferior to an exponential one; furthermore, it allows the designer to arbitrarily prescribe the rate of convergence and the size of the set of ultimate boundedness.

Injectivity and decomposition of completely bounded maps

Abstract: In 1979 Wittstock proved the striking result that any completely bounded map from a C*-algebra A into an injective C*-algebra B is a linear combination of completely positive maps from A to B More specificly he proved that if T : A ~ B is a completely bounded selfadjoint map (ie T(x*) = T(x)*, x £ A), then there exist completely positive maps TI, T 2 from A to B , such that T = T I T 2 and lIT I + T211 ~ llTl~b

Conditions for certain Higgs potentials to be bounded below

Abstract: The author obtains sufficient and necessary conditions for Higgs potentials to be bounded below when they are constructed from: two doublets, and also two doublets and a singlet of SU (2), and the adjoint and vector representations of SO (n). For a potential constructed from the adjoint and fundamental multiplets of SU (n), the problem of necessary and sufficient conditions has been only partly solved.

Regularity and large time behaviour of solutions of a conservation law without convexity

Abstract: Using the method of generalized characteristics, we discuss the regularity and large time behaviour of admissible weak solutions of a single conservation law, in one space variable, with one inflection point.We show that when the initial data are C∞ then, generically, the solution is C∞ except: (a) on a finite set of C∞ arcs across which it experiences jump discontinuities (genuine shocks or left contact discontinuities); (b) on a finite set of straight line characteristic segments across which its derivatives of order m, m = 1, 2,…, experience jump discontinuities (weak waves of order m); and (c) on the finite set of points of interaction of shocks and weak waves. Weak waves of order 1 are triggered by rays grazing upon contact discontinuities. Weak waves of order m, m ≥ 2, are generated by the collision of a weak wave of order m − 1 with a left contact discontinuity.We establish sharp decay rates for solutions with initial data of the following types: (a) with bounded primitive; (b) with primitive having sublinear growth; (c) in L1; (d) of compact support; and (e) periodic.