Showing papers on "Upper and lower bounds published in 1975"
TL;DR: The minimum distance is rigorously shown to be nonzero for all transmission rates, tantamount to showing that, in the singular case of linear prediction, perfect prediction cannot be approached with bounded prediction coefficients.
Abstract: The degradation suffered when pulses satisfying the Nyguist criterion are used to transmit binary data in noise at supraconventional rates is studied. Optimum processing of the received waveforms is assumed, and attention is focused on the minimum distance between signal points as a performance criterion. An upper bound on this distance is given as a function of signaling speed. In particular, the pulse energy seems to be the minimum distance up to rates of transmission 25 percent faster than the Nyguist rate, but not beyond. Some mathematical aspects related to the above problem are also considered. In particular, the minimum distance is rigorously shown to be nonzero for all transmission rates. This is tantamount to showing that, in the singular case of linear prediction, perfect prediction cannot be approached with bounded prediction coefficients.
613 citations
TL;DR: For a matrix A which is diagonally dominant both by rows and by columns, bounds are given which can be used to give a lower bound for the smallest singular value.
361 citations
TL;DR: In this article, it was shown that Σ |e (V)|, the sum of the negative energies of a single particle in a potential V, is bounded above by (4/15 π )∫|V|5/2.
Abstract: We first prove that Σ |e (V)|, the sum of the negative energies of a single particle in a potential V, is bounded above by (4/15 π )∫|V|5/2. This in turn, implies a lower bound for the kinetic energy of N fermions of the form 3/5 (3π/4)2/3 ∫ρ 5/3, where ρ (x) is the one-particle density. From this, using the no-binding theorem of Thomas-Fermi theory, we present a short proof of the stability of matter with a reasonable constant for the bound.
352 citations
TL;DR: A new selection algorithm is presented which is shown to be very efficient on the average, both theoretically and practically.
Abstract: A new selection algorithm is presented which is shown to be very efficient on the average, both theoretically and practically. The number of comparisons used to select the ith smallest of n numbers is n + min(i,n-i) + o(n). A lower bound within 9 percent of the above formula is also derived.
319 citations
TL;DR: In this paper, it was shown that a certain commutative ring associated with a triangulation of a (d − 1)-dimensional sphere with n vertices is a Cohen-Macaulay ring.
Abstract: Let Δ be a triangulation of a (d − 1)-dimensional sphere with n vertices. The Upper Bound Conjecture states that the number of i-dimensional faces of Δ is less than or equal to a certain explicit number ci(n, d). A proof is given of a more general result. The proof uses the result, proved by G. Reisner, that a certain commutative ring associated with Δ is a Cohen-Macaulay ring.
307 citations
TL;DR: In this article, the authors studied the limiting behavior of f(n,k) and derived upper and lower bounds on these limits for all k, where n = 5 and n = 10, respectively.
Abstract: Given two random k-ary sequences of length n, what is f(n,k), the expected length of their longest common subsequence? This problem arises in the study of molecular evolution. We calculate f(n,k) for all k, where n $\leq$ 5 , and f(n,2) where n $\leq$ 10. We study the limiting behavior of $n^{-1}$f(n,k) and derive upper and lower bounds on these limits for all k. Finally we estimate by Monte-Carlo methods f(100,k), f(1000,2) and f(5000,2).
262 citations
13 Oct 1975
TL;DR: It will be shown in the sequel that the parallel arithmetic complexity of all these four problems is upper bounded by O(log2n) and the algorithms that establish this bound use a number of processors polynomial in n, disproves I. Munro's conjecture.
Abstract: In this paper, an investigation of the parallel arithmetic complexity of matrix inversion, solving systems of linear equations, computing determinants and computing the characteristic polynomial of a matrix is reported. The parallel arithmetic complexity of solving equations has been an open question for several years. The gap between the complexity of the best algorithms (2n + 0(1), where n is the number of unknowns/ equations) and the only proved lower bound (2 log n (All logarithms in this paper are of base two.)) was huge. The first breakthrough came when Csanky reported that the parallel arithmetic complexity of all these four problems has the same growth rate and exhibited an algorithm that computes these problems in 2n - O(log2n) steps. It will be shown in the sequel that the parallel arithmetic complexity of all these four problems is upper bounded by O(log2n) and the algorithms that establish this bound use a number of processors polynomial in n. This disproves I. Munro's conjecture.
203 citations
TL;DR: Improved asymptotic lower bounds to the Ramsey function R(k, t), with symmetric case k = t and more general case t = t is given.
169 citations
TL;DR: By using Agmon's geodesic ideas to single out particular regions in path space, this article obtained optimal lower bounds on the leading behavior for the fall off of the ground state of multiparticle system.
Abstract: By using Agmon's geodesic ideas to single out particular regions in path space, we obtain optimal lower bounds on the leading behavior for the fall off of the ground state of multiparticle system.
143 citations
TL;DR: The most stringent bounds on P [m] and P 1 are more stringent than corresponding bounds presented in the literature, for most systems.
Abstract: Analysis of dependent probability systems of a moderate to large finite number (m) of events must often be based solely on the partial system information given by S 1, the sum of the probabilities of occurrence of the m individuaf events; and S 2, the sum of the probabilities of occurrence of each of the ( m 2) pairs of events. This information is used to develop the most stringent upper and lower bounds on the aggregated probabilities P [n], the probability of occurrence of exactly n events (n = 0, 1, …, m); and on P 1, the probability of occurrence of one or more of the m events. The most stringent bounds on P [m] and P 1 are more stringent than corresponding bounds presented in the literature, for most systems.
142 citations
TL;DR: In this paper, it was shown that for a large class of useful quantum statistical systems, the partition function is, with respect to the coupling constant, the Laplace transform of a positive measure.
Abstract: We show, by making use of the functional integral technique, that, for a large class of useful quantum statistical systems, the partition function is, with respect to the coupling constant, the Laplace transform of a positive measure. As a consequence, we derive an infinite set of monotonicly converging upper and lower bounds to it. In particular, the lowest approximation appears to be identical to the Gibbs–Bogolioubov variational bound, while the next approximations, for which we give explicit formulas for the first few ones, lead to improve the previous bound. The monotonic character of the variational successive approximations allows a new approach towards the thermodynamical limit.
01 Jan 1975
TL;DR: In this paper, a method for implicitly representing constraints of the form x≤y in linear programs when the variable y may appear in any number of such constraints is developed for problems in linear regression, plant location, and production scheduling.
Abstract: A method is developed for implicitly representing constraints of the form x≤y in linear programs when the variable y may appear in any number of such constraints. Variable x is said to have a variable upper bound (VUB). VUB constraints are common in a number of LP formulations, especially those derived from tightly formulated fixed charge integer programs. For certain of these problems the major portion of the constraints are of the VUB type. Computational experience with the method applied to problems in linear regression, plant location, and production scheduling is presented.
TL;DR: Glimm's Hamiltonian bound and Schrader's linear lower bound are obtained in theY2 Euclidean field theory from bounds of the form Z∧≦ea|∧| and (SZ)∧⩽| in the Y2 Euclidian field theory.
Abstract: We prove bounds of the formZ∧≦ea|∧| and (SZ)∧≦ea|∧| in theY2 Euclidean field theory and from this obtain Glimm's Hamiltonian bound and Schrader's linear lower bound.
TL;DR: The length l of addition chains for z is shown to be bounded from below by log 2 z + log 2 s ( z )−2.13, where s (Z) denotes the sum of the digits in the binary expansion of z and the proof will also hold for addition-subtraction chains if s (z ) is replaced by an appropriate substitute.
TL;DR: In this article, a survey of available methods for the solution of the set covering problem is presented, and the prime objective is to establish the computational efficiency and relative merits of the various algorithms that have been proposed.
Abstract: This paper is a survey of available methods for the solution of the Set Covering Problem. The prime objective has been to establish the computational efficiency and relative merits of the various algorithms that have been proposed. To this end five methods have been programmed and tested on the same set of 33 test problems some of which can be found in the literature and the rest of which were randomly generated. Some of the methods examined have—as far as the authors are aware—never been previously tested, and in some cases some surprising results have been noted. In addition, one type of tree search method has been studied in some greater detail and new techniques involving multiple dominance tests, and the calculation of a better lower bound have been suggested to limit the search. This resulting algorithm was found to be better than the original one by orders of magnitude in both computation times and numbers of tree-nodes generated, and has proved to be the most efficient of the methods tested.
TL;DR: In this paper, the classes of trajectories by which a particle can reach a given point in a given direction and obtains, for all t, the set $R(t)$ of all possible positions for P at time t were studied.
Abstract: A particle P moves in the plane with constant speed and subject to an upper bound on the curvature of its path. This paper studies the classes of trajectories by which P can reach a given point in a given direction and obtains, for all t, the set $R(t)$ of all possible positions for P at time t, thus extending the results of several recent authors.
TL;DR: A lower bound on the minimal mean-square error in estimating nonlinear Markov processes is presented, based on the Van Trees' version of the Cramer-Rao inequality.
Abstract: A lower bound on the minimal mean-square error in estimating nonlinear Markov processes is presented. The bound holds for causal and uncausal filtering. The derivation is based on the Van Trees' version of the Cramer-Rao inequality.
01 Oct 1975
TL;DR: In this article, general expressions for estimating the errors in the sum or difference far-field pattern of electrically large aperture antennas which measured by planar near-field scanning technique are derived.
Abstract: General expressions are derived for estimating the errors in the sum or difference far-field pattern of electrically large aperture antennas which measured by planar near-field scanning technique. Upper bounds are determined for the far-field errors produced by 1) the nonzero fields outside the finite scan area, 2) the inaccuracies in the positioning of the probe, 3) the distortion and non-linearities of the instrumentation which measures the amplitude and phase of the probe output, and 4) the multiple refractions. Computational errors, uncertainties in the receiving characteristics of the probe, and errors involved with measuring the input power to the test antenna are briefly discussed.
TL;DR: The rate of convergence of the nearest neighbor (NN) rule is investigated when independent identically distributed samples take values in a d -dimensional Euclidean space when the common distribution of the sample points need not be absolutely continuous.
Abstract: The rate of convergence of the nearest neighbor (NN) rule is investigated when independent identically distributed samples take values in a d -dimensional Euclidean space. The common distribution of the sample points need not be absolutely continuous. An upper bound consisting of two exponential terms is given for the probability of large deviations of error probability from the asymptotic error found by Cover and Hart. The asymptotically dominant first term of this bound is distribution-free, and its negative exponent goes to infinity approximately as fast as the square root of the number of preclassified samples. The second term depends on the underlying distributions, but its exponent is proportional to the sample size. The main term is explicitly given and depends very weakly on the dimension of the space.
01 Oct 1975
TL;DR: In this article, general expressions for estimating the errors in the sum or difference far-field pattern of electrically large aperture antennas which measured by planar near-field scanning technique are derived.
Abstract: General expressions are derived for estimating the errors in the sum or difference far-field pattern of electrically large aperture antennas which measured by planar near-field scanning technique. Upper bounds are determined for the far-field errors produced by 1) the nonzero fields outside the finite scan area, 2) the inaccuracies in the positioning of the probe, 3) the distortion and non-linearities of the instrumentation which measures the amplitude and phase of the probe output, and 4) the multiple refractions. Computational errors, uncertainties in the receiving characteristics of the probe, and errors involved with measuring the input power to the test antenna are briefly discussed.
TL;DR: In this article, an expression for the stability factor Ns, based on the upper bound technique of limit analysis which yields a close-formed solution for sections in which the following conditions are considered: (a) log-spiral failure-plane, through and below toe; (b) non-homogeneity; (c) anisotropy; and (d) general slope.
Abstract: The upper bound technique of limit analysis has been found to be very successful in analyzing the stability of cuttings in normally consolidated clays. However, most soils in their natural states exhibit some anisotropy with respect to shear strength, and some nonhomogeneity with respect to depth. It is difficult to obtain the solution based on the classical limit equilibrium analysis with the assumed noncircular failure plane with such soil properties included. This paper establishes an expression for the stability factor Ns, based on the upper bound technique of limit analysis which yields a close-formed solution for sections in which the following conditions are considered: (a) log-spiral failure-plane, through and below toe; (b) non-homogeneity; (c) anisotropy; and (d) general slope.
TL;DR: This paper develops the most stringent upper and lower bounds on P 1, the probability of the union of the m events; and on P [m], the likelihood of the simultaneous occurrence of them events.
Abstract: For dependent probability systems of m events partially specified only by the quantities S1, the sum of the probabilities of the m individual events; S2, the sum of the probabilities of each of the (m) pairs of events and S3 the sum of the probabilities of each of the (m) combinations of three events; this paper develops the most stringent upper and lower bounds on P1, the probability of the union of the m events; and on P[m], the probability of the simultaneous occurrence of the m events.
TL;DR: The analysis includes the class of channels with spectral nulls, since band edge nulls can be associated with telephone channels that are pulsed at high rates and Forney's asymptotic formula for p is proved.
Abstract: Maximum-likelihood sequence estimation (MLSE) of data sequences is considered as an approach to increasing data rates over band-limited channels, such as exist in analog telephone facilities. Analytical prediction of the efficacy of MLSE for specific channels had been viewed as intractable until Forney gave a formula for an asymptotic upper bound on the probability of symbol error p . Forney's bound involved an infinite series, and letting the noise power No --} 0, he chose the asymptotically largest series term as an asymptotic bound on p , ignoring a very basic perplexing open question: How do we know the bounding series is not identically infinity for all N_0 \rightarrow 0? We show that under very general conditions Forney's bounding series converges for a nontrivial interval N_0 > 0 . As a corollary, we prove Forney's aforementioned asymptotic formula for p . The analysis includes the class of channels with spectral nulls, since band edge nulls can be associated with telephone channels that are pulsed at high rates.
01 Jan 1975
TL;DR: In this paper, it was shown that under very general conditions Forney's bounding series converges for a nontrivial interval No > 0 for the class of channels with spectral nulls.
Abstract: Maximum-likelihood sequence estimation (MLSE) of data sequences is considered as an approach to increasing data rates over band-limited channels, such as exist in analog telephone facilities. Analytical prediction of the efficacy of MLSE for specific channels had been viewed as intractable until Fomey gave a formula for an asymptotic upper bound on the probability of symbol error p. Fomey's bound in- volved an infinite series, and letting the noise power No + 0, he chose the asymptotically largest series term as an asymptotic bound onp, ignoring a very basic perplexing open question: How do we know the bounding series is not identically infinity for all No + O? We show that under very general conditions Forney's bounding series converges for a nontrivial interval No > 0. As a corollary, we prove Forney's afore- mentioned asymptotic formula for p. The analysis includes the class of channels with spectral nulls, since band edge nulls can be associated with telephone channels that are pulsed at high rates.
TL;DR: In this paper, it is proved that the infimum of the probability of a correct selection occurs at a point in the preference zone for which the parameters are as close together as possible Conditions are given which allow evaluation of this last infimum.
Abstract: Let $\pi_1,\cdots, \pi_k$ be $k$ populations with $\pi_i$ characterized by a scalar $\lambda_i \in \Lambda$, a specified interval on the real line The object of the problem is to make some inference about $\pi_{(k)}$, the population with largest $\lambda_i$ The present work studies rules which select a random number of populations between one and $m$ where the upper bound, $m$, is imposed by inherent setup restrictions of the subset selection and indifference zone approaches A selection procedure is defined in terms of a set of consistent sequences of estimators for the $\lambda_i$'s It is proved the infimum of the probability of a correct selection occurs at a point in the preference zone for which the parameters are as close together as possible Conditions are given which allow evaluation of this last infimum The number of non-best populations selected, the total number of populations selected, and their expectations are studied both asymptotically and for fixed $n$ Other desirable properties of the rule are also studied
TL;DR: The lower bound for discrimination information in terms of variation, derived recently by Kullback [7] for the distribution-free case, is sharpened and under a restriction, a lower bound is derived that is sharper than all other existing bounds.
Abstract: The lower bound for discrimination information in terms of variation, derived recently by Kullback [7] for the distribution-free case, is sharpened. Furthermore, under a restriction, a lower bound is derived that is sharper than all other existing bounds.
TL;DR: The problem of meeting stream dissolved oxygen standards while optimizing some objective is treated and constraint elimination techniques are developed that can further reduce the number of constraints necessary.
Abstract: The problem of meeting stream dissolved oxygen standards while optimizing some objective is treated. New properties of the oxygen sag equation allow the constraint set of such mathematical programs to be described to a high degree of accuracy by linear inequalities; except for upper and lower bounds on pollutant discharges, three linear constraints at most are required per reach. Constraint elimination techniques are developed that can further reduce the number of constraints necessary. As a means of highlighting the potential power of these techniques to large-scale models they are applied to two well-known examples from the literature.
TL;DR: In this paper, a simple method is described which can be used to generate complemen-tary bivariational principles yielding upper and lower bounds to the quantity Q ═ ∫ s p (s ) n 0 ( s ) d s, where p(s ) is the (vector) solution of a linear integral equation of Fredholm type p 0 (s) = p( s ).
Abstract: A simple method is described which can be used to generate complementary bivariational principles yielding upper and lower bounds to the quantity Q ═ ∫ s p ( s ) n 0 ( s ) d s , where p ( s ) is the (vector) solution of a linear integral equation of Fredholm type p 0 ( s ) = p ( s ) — λ ∫ s K ( s, s9 ) p ( s9 ) d s9 and n 0 ( s ) and p 0 ( s ) are given functions. The method involves a generalization, requiring two approximating functions, of results obtained from a study of the particular case n 0 ( s ) = p 0 ( s ), a classical variational problem occurring in transport theory and other fields of applied mathematics. The bounds are compared with those of other authors and some further generalizations are indicated.
TL;DR: In this paper, detailed finite element calculations and experimental work on plate structures are reported, together with the concept of local kinematic determinacy, lead to a correction factor on the upper bound life prediction.
TL;DR: In this article, a lower bound of O(p \cdot (\log p)^{d - 1} ), where p is the array size and d the dimensionality, is derived for this measure; and an extendible allocation scheme which achieves this lower bound is exhibited.
Abstract: Schemes which allocate storage for extendible arrays cannot utilize storage as efficiently as can their nonextendible counterparts. Relative to formal notions of array scheme and (extendible) array realization, a formal way of gauging efficiency of storage utilization by extendible array realizations is proposed; a lower bound of $O(p \cdot (\log p)^{d - 1} )$, where p is the array size and d the dimensionality, is derived for this measure; and an extendible allocation scheme which achieves this lower bound is exhibited. Certain seminorms on Euclidean spaces can be used to construct extendible array realizations. It is shown that for realizations so constructed, the lower bound on storage utilization efficiency is $O(p^d )$. In the opposite direction, certain restrictions on the patterns of expansions of arrays can be used to circumvent the lower bound: When arrays are constrained to expand according to some fixed finite set of patterns, then one can devise extendible realizations which (a) utilize storag...