Showing papers on "Bounded function published in 1981"
TL;DR: For systems with negligible self-gravity, the bound follows from application of the second law of thermodynamics to a gedanken experiment involving a black hole as discussed by the authors, and it is shown that black holes have the maximum entropy for given mass and size which is allowed by quantum theory and general relativity.
Abstract: We present evidence for the existence of a universal upper bound of magnitude $\frac{2\ensuremath{\pi}R}{\ensuremath{\hbar}c}$ to the entropy-to-energy ratio $\frac{S}{E}$ of an arbitrary system of effective radius $R$. For systems with negligible self-gravity, the bound follows from application of the second law of thermodynamics to a gedanken experiment involving a black hole. Direct statistical arguments are also discussed. A microcanonical approach of Gibbons illustrates for simple systems (gravitating and not) the reason behind the bound, and the connection of $R$ with the longest dimension of the system. A more general approach establishes the bound for a relativistic field system contained in a cavity of arbitrary shape, or in a closed universe. Black holes also comply with the bound; in fact they actually attain it. Thus, as long suspected, black holes have the maximum entropy for given mass and size which is allowed by quantum theory and general relativity.
1,079 citations
TL;DR: In this article, a priori bounds for positive solutions of the non-linear elliptic boundary value problem where Ω is a bounded domain in R n were derived by contradiction and used a scaling argument reminiscent to that used in the theory of Minimal Surfaces.
Abstract: We derive a priori bounds for positive solutions of the non-linear elliptic boundary value problem where Ω is a bounded domain in R n. Our proof is by contradiction and uses a scaling (“blow up”) argument reminiscent to that used in the theory of Minimal Surfaces. This procedure reduces the problem of a priori bounds to global results of Liouville type.
813 citations
TL;DR: In this article, conditions are obtained under which the stochastic equation has a strong solution in the multidimensional case where the diffusion matrix is the identity matrix and the drift vector is bounded.
Abstract: Conditions are obtained under which the stochastic equation has a strong solution. In particular, in the multidimensional case where the diffusion matrix is the identity matrix and the drift vector is bounded, these conditions are satisfied. Bibliography: 13 titles.
438 citations
TL;DR: The problem of embedding the interconnection pattern of a circuit into a two-dimensional surface of minimal area is discussed and restricted classes of graphs have to be considered in order to achieve compact embeddings.
Abstract: The problem of embedding the interconnection pattern of a circuit into a two-dimensional surface of minimal area is discussed. Since even for some natural patterns graphs containing m connections may require Ω(m2) area, in order to achieve compact embeddings restricted classes of graphs have to be considered. For example, arbitrary trees (of bounded degree) can be embedded in linear area without edges crossing over. Planar graphs can be embedded efficiently only if crossovers are allowed in the embedding.
422 citations
TL;DR: In this paper, a global inverse function theorem is established for mappings u: Ω → ℝn, Ω ⊂ n bounded and open, belonging to the Sobolev space W1.
Abstract: A global inverse function theorem is established for mappings u: Ω → ℝn, Ω ⊂ ℝn bounded and open, belonging to the Sobolev space W1.p(Ω), p > n. The theorem is applied to the pure displacement boundary value problem of nonlinear elastostatics, the conclusion being that there is no interpenetration of matter for the energy-minimizing displacement field.
329 citations
TL;DR: In this paper, the existence and multiplicity results for nonlinear elliptic equations of the type -Au = |u|''_1u + h(x) in P», u = 0 on 3s.
Abstract: This paper is concerned with existence and multiplicity results for nonlinear elliptic equations of the type -Au = |u|''_1u + h(x) in P», u = 0 on 3s. Here, s c R^ is smooth and bounded, and h e L2(Q) is given. We show that there exists pN > 1 such that for any p e (\,pN) and any h e L2(I2), the preceding equation possesses infinitely many distinct solutions. The method rests on a characterization of the existence of critical values by means of noncontractibility properties of certain level sets. A perturbation argument enables one to use the properties of some associated even functional. Several other applications of this method are also presented.
266 citations
TL;DR: In this article, it was shown that the partial differential equation in Problem (I) is the classical equation of heat conduction and it is well known that under appropriate conditions on u, and 0, u(., t) + 0 as t -+ + + +a~ (see, for example, [ 91] ).
240 citations
TL;DR: In this article, the incoherent neutron scattering function for unbounded jump diffusion is calculated from random walk theory assuming a gaussian distribution of jump lengths, and the method is then applied to calculate the scattering functions for spatially bounded random jumps in one dimension.
Abstract: The incoherent neutron scattering function for unbounded jump diffusion is calculated from random walk theory assuming a gaussian distribution of jump lengths. The method is then applied to calculate the scattering function for spatially bounded random jumps in one dimension. The dependence on momentum transfer of the quasi-elastic energy broadenings predicted by this model and a previous model for bounded one-dimensional continuous diffusion are calculated and compared with the predictions of models for diffusion in unbounded media. The one-dimensional solutions can readily be generalized to three dimensions to provide a description of quasi-elastic scattering of neutrons by molecules undergoing localized random motions.
194 citations
TL;DR: In this article, it was shown that if the interval is small (approximately two standard deviations wide) then the Bayes rule against a two point prior is the unique minimax estimator under squared error loss.
Abstract: The problem of estimating a normal mean has received much attention in recent years. If one assumes, however, that the true mean lies in a bounded interval, the problem changes drastically. In this paper we show that if the interval is small (approximately two standard deviations wide) then the Bayes rule against a two point prior is the unique minimax estimator under squared error loss. For somewhat wider intervals we also derive sufficient conditions for minimaxity of the Bayes rule against a three point prior.
189 citations
TL;DR: This paper provides a recursive procedure to solve knapsack problems and differs from classical optimization algorithms of convex programming in that it determines at each iteration the optimal value of at least one variable.
Abstract: The allocation of a specific amount of a given resource among competitive alternatives can often be modelled as a knapsack problem. This model formulation is extremely efficient because it allows convex cost representation with bounded variables to be solved without great computational efforts. Practical applications of this problem abound in the fields of operations management, finance, manpower planning, marketing, etc. In particular, knapsack problems emerge in hierarchical planning systems when a first level of decisions need to be further allocated among specific activities which have been previously treated in an aggregate way. In this paper we provide a recursive procedure to solve such problems. The method differs from classical optimization algorithms of convex programming in that it determines at each iteration the optimal value of at least one variable. Applications and computational results are presented.
177 citations
TL;DR: In this article, a biorthogonality relation between the solution of the linearized equation for the stream function and the solutions of the adjoint problem is derived for flows in the half-space, y e [0, ∞).
Abstract: The expansion of an arbitrary two-dimensional solution of the linearized stream-function equation in terms of the discrete and continuum eigenfunctions of the Orr-Sommerfeld equation is discussed for flows in the half-space, y e [0, ∞). A recent result of Salwen is used to derive a biorthogonality relation between the solution of the linearized equation for the stream function and the solutions of the adjoint problem.For the case of temporal stability, the orthogonality relation obtained is equivalent to that of Schensted for bounded flows. This relationship is used to carry out the formal solution of the initial-value problem for temporal stability. It is found that the vorticity of the disturbance at t = 0 is the proper initial condition for the temporal stability problem. Finally, it is shown that the set consisting of the discrete eigen-modes and continuum eigenfunctions is complete.For the spatial stability problem, it is shown that the continuous spectrum of the Orr-Sommerfeld equation contains four branches. The biorthogonality relation is used to derive the formal solution to the boundary-value problem of spatial stability. It is shown that the boundary-value problem of spatial stability requires the stream function and its first three partial derivatives with respect to x to be specified at x = 0 for all t. To be applicable to practical problems, this solution will require modification to eliminate disturbances originating at x = ∞ and travelling upstream to x = 0.For the special case of a constant base flow, the method is used to calculate the evolution in time of a particular initial disturbance.
TL;DR: In this paper, the authors generalize Arveson's extension theorem for completely positive mappings to a Hahn-Banach principle for matricial sublinear functionals with values in an injective C ∗ -algebra or an ideal in B (H ).
TL;DR: In this paper, the authors consider the more general case in which it is not assumed that the system is either stable or stably invertible, and establish local convergence for a class of adaptive control algorithms applied to general discrete, deterministic, linear, time-invariant systems.
Abstract: Recent papers have established global convergence for a class of adaptive control algorithms for discrete time linear dynamic systems. However, in most cases studied to date it has been assumed that the system is stably invertible. This assumption plays a major role in the proofs of convergence. In this paper we consider the more general case in which it is not assumed that the system is either stable or stably invertible. We establish local convergence for a class of adaptive control algorithms applied to general discrete, deterministic, linear, time-invariant systems. By convergence in this context, we mean that the system inputs and outputs remain bounded for all time and the closed loop poles are effectively assigned in the limit for a given desired trajectory.
TL;DR: Three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals, which are most suitable to linear programs in which the objectivefunction coefficients are deterministic but are likely to vary from time period to time period.
Abstract: This paper presents three algorithms for solving linear programming problems in which some or all of the objective function coefficients are specified in terms of intervals. Which algorithm is applicable depends upon a the number of interval objective function coefficients, b the number of nonzero objective function coefficients, and c whether or not the feasible region is bounded. The algorithms output all extreme points and unbounded edge directions that are “multiparametrically optimal” with respect to the ranges placed on the objective function coefficients. The algorithms are most suitable to linear programs in which the objective function coefficients are deterministic but are likely to vary from time period to time period as for example in blending problems.
28 Oct 1981
TL;DR: It is shown that the size of the ring cannot be calculated by any probabilistic algorithm in which the processes can sense termination and any algorithm may yield an incorrect value.
Abstract: Given a ring (cycle) of n processes it is required to design the processes so that they will be able to choose a leader (a uniquely designated process) by sending messages along the ring. If the processes are indistiguishable there is no deterministic algorithm, and therefore probabilistic algorithms are proposed. These algorithms need not terminate, but their expected complexity (time or number of bits of communication) is bounded by a function of n. If the processes work asynchronously then on the average O(n log2n) bits are transmitted. In the above cases the size n of the ring was assumed to be known. If n is not known it is suggested first to determine the value of n and then use the above algorithm. However, n may only be determined probabilistically and any algorithm may yield an incorrect value. In addition, it is shown that the size of the ring cannot be calculated by any probabilistic algorithm in which the processes can sense termination.
TL;DR: The complexity of decision procedures for these two problems are investigated and it is shown by reducing a bounded version of Hilbert''s Tenth problem to the finite containment problem that these two problem are extremely hard, that the complexity of each decision procedure exceeds any primitive recursive function infinitely often.
Abstract: If the reachability set of a Petri net (or, equivalently, vector addition system) is finite it can be effectively constructed. Furthermore, the finiteness is decidable. Thus, the containment and equality problem for finite reachability sets become solvable. We investigate the complexity of decision procedures for these two problems and show by reducing a bounded version of Hilbert''s Tenth problem to the finite containment problem that these two problems are extremely hard, that, in fact, the complexity of each decision procedure exceeds any primitive recursive function infinitely often. The finite containment and quality problem are thus the first uncontrived, decidable problems with provably non-primitive recursive complexity.
TL;DR: A new approach to the perfect regulation and the bounded peaking in linear multivariable control systems, based on the pole and the eigenvector assignment technique is presented, which turns out to be identical to the result obtained in the optimal regulator theory.
Abstract: This paper presents a new approach to the perfect regulation (p.r.) and the bounded peaking (b.p.) in linear multivariable control systems, based on the pole and the eigenvector assignment technique. Roughly speaking the p.r. represents an ideal control action which reduces the settling time to 0, while the b.p. simply means the bounded overshoot in this ideal situation. Simple frequency domain characterizations of the p.r. and the b.p. are derived. They reveal some invariance properties of the p.r. and the b.p. that provide a powerful tool for achieving the p.r. and the b.p. The existence condition for the p.r. is derived, which turns out to be identical to the result obtained in the optimal regulator theory. This result is extended to a more general control objective of attaining the p.r. for one output while keeping the b.p. for another output. A condition on which the p.r. is realized by an output feedback is also derived. A simple design algorithm is proposed for achieving the p.r. which is essentially the eigenvector assignment procedure. Finally, an application to a real system is discussed.
TL;DR: In this article, the asymptotic normality of these statistics is established under certain regularity conditions, and the statistics are used to construct consistent estimators of various conditional quantities.
Abstract: Let (Xi , Yi )(i = 1, 2, …, n) be independent identically distributed as (X, Y). Then the rth ordered X variate is denoted by Xr:n and the associated Y variate, the concomitant of the rth order statistic, by Y [r:n]. This paper considers statistics of the form and more generally of the form , where J is a bounded smooth function and may depend on n. Under certain regularity conditions, the asymptotic normality of these statistics is established. These statistics are used to construct consistent estimators of various conditional quantities, for example E(Y | X = x), P(Y ∈ A | X = x) and var(Y | X = x).
IBM1
TL;DR: This paper examines a number of programming primitives in query languages for relational databases and shows that equality cannot be simulated using all the other primitives, generic variables can be simulated with only ranked variables, and that with bounded loops one can determine the isomorphism class of the database when generic variables are allowed, but not otherwise.
Abstract: This paper examines a number of programming primitives in query languages for relational databases The basic framework is a language based on relational algebra, whose variables take relations as values The primitives considered are (i) looping, (ii) counters, (iii) generic (or unranked) variables, (iv) equality, (v) bounded looping (which corresponds to counting the number of tuples in a relation), and (vi) isomorphism class (which corresponds to knowing the isomorphism class of the data base) A comparison diagram is given relating all combinations of these six primitives, and several of the resulting classes of queries are characterized by their complexity or algebraic properties It is shown, for example, that equality cannot be simulated using all the other primitives, that generic variables (with loops) cannot be simulated with only ranked variables and all the other primitives, and that with bounded loops one can determine the isomorphism class of the database when generic variables are allowed, but not otherwise
TL;DR: An algorithm for a simpler problem, namely, the partitioning of Tinto the maximum number of connected components whose weight is bounded below, is presented and combined with the technique of binary search yields an alternative algorithm for the max-rain k-partition problem with complexity dependent on the range of the given weights.
Abstract: The max-rain k-partition algorithm may be formulated as follows: Given a tree T with n edges and a nonnegative weight associated with each vertex, assign a cut to each of k distinct edges of T so as to maximize the weight of the lightest resulting connected subtree. An algorithm for this problem is presented which initially assigns all k cuts to one edge incident with a terminal vertex of T; thereafter the cuts are shifted from edge to adjacent edge on the basis of local information. An efficient implementation with complexity O(k 2. rd(T) + kn), where rd(T) is the number of edges in the radius of T, is described. An algorithm for a simpler problem, namely, the partitioning of Tinto the maximum number of connected components whose weight is bounded below, is then described. Combined with the technique of binary search, it yields an alternative algorithm for the max-rain k-partition problem with complexity dependent on the range of the given weights.
01 Jan 1981
TL;DR: In this article, the authors characterize combinatorial optimization problems which can be solved approximately by polynomially bounded algorithms and prove that there is no fast approximation scheme unless their algorithmic ideas apply.
Abstract: We characterize those combinatorial optimization problems which can be solved approximately by polynomially bounded algorithms. Using slight modifications of the Sahni and Ibarra and Kim algorithms for the knapsack problem we prove that there is no fast approximation scheme unless their algorithmic ideas apply. Hence we show that these algorithms are not only the origin but also prototypes for all polynomial or fully polynomial approximation schemes.
TL;DR: In this paper, it was shown that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to: euXx + b(x)ux = f(x), for O O, b and f smooth, e in (0, 1], and u(O) and u (I) given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by Ch 2 with the constant C independent of h and e).
Abstract: It will be proven that an exponential tridiagonal difference scheme, when applied with a uniform mesh of size h to: euXx + b(x)ux = f(x) for O O, b and f smooth, e in (0, 1], and u(O) and u(I) given, is uniformly second-order accurate (i.e., the maximum of the errors at the grid points is bounded by Ch2 with the constant C independent of h and e). This scheme was derived by El-Mistikawy and Werle by a C' patching of a pair of piecewise constant coefficient approximate differential equations across a common grid point. The behavior of the approximate solution in between the grid points will be analyzed, and some numerical results will also be given.
TL;DR: In this paper, Davie and Kaijser generalized Littlewood and Hardy's results to multilinear bilinear forms on lp spaces and proved that these results can be generalized to multilevel forms.
TL;DR: In this paper, the Holder continuity of bounded weak solutions of quasilinear parabolic systems with main part in diagonal form is proved via a parabolic hole-filling technique.
Abstract: The Holder continuity of bounded weak solutions of quasilinear parabolic systems with main part in diagonal form is proved via a “parabolic hole-filling technique”.
TL;DR: In this paper, it was shown that all solutions of (1) are attracted towards the set {(u, v): u* + v* < 1} for any constant a where u is the solution of a single differential equation.
TL;DR: In this paper, the large order behavior of φ44 euclidean perturbative quantum field theory was bound rigorously, and it was shown that the Schwinger functions at ordern are bounded by KNN!, which implies a finite radius of convergence for the Borel transform of the perturbation series.
Abstract: We bound rigorously the large order behaviour of φ44 euclidean perturbative quantum field theory, as the simplest example of renormalizable, but non-super-renormalizable theory. The needed methods are developed to take into account the structure of renormalization, which plays a crucial role in the estimates. As a main thorem, it is shown that the Schwinger functions at ordern are bounded byKnn!, which implies a finite radius of convergence for the Borel transform of the perturbation series.
TL;DR: The results are shown to extend Huber's solution for the \epsilon -contaminated classes, which is contained as a special case, and a minimum-distance property of the least favorable density pair is made explicit.
Abstract: Least favorable pairs of probability density functions and robust tests are obtained for classes of density functions specified as bands with upper and lower bounds. The results are shown to extend Huber's solution for the \epsilon -contaminated classes, which is contained as a special case. A minimum-distance property of the least favorable density pair is also made explicit.
TL;DR: In this article, the problem of studying the boundedness properties of complete minimal surfaces in 1R was approached via the consideration of a certain gradient flow, and it was soon realized that the basic technique (Lemma 3 below) could be successfully employed to study boundedness of arbitrary complete submanifolds.
Abstract: This paper is a natural outgrowth of [9], where the problem of studying the boundedness properties of complete minimal surfaces in 1R" was approached via the consideration of a certain gradient flow. After the completion of [9] it was soon realized that the basic technique (Lemma 3 below) could be successfully employed to study boundedness of arbitrary complete submanifolds. In this regard, the method can be used to prove the following three theorems.
TL;DR: In this paper, a weak canonical form for vector spaces of m x n matrices all of rank at most r is derived, and it is shown that m and n are bounded by functions of r and these bounds are tight.
Abstract: A weak canonical form is derived for vector spaces of m x n matrices all of rank at most r. This shows that the structure of such spaces is controlled by the structure of an associated 'primitive' space. In the case of primitive spaces it is shown that m and n are bounded by functions of r and that these bounds are tight.