Showing papers on "Upper and lower bounds published in 1973"
TL;DR: The number of comparisons required to select the i-th smallest of n numbers is shown to be at most a linear function of n by analysis of a new selection algorithm-PICK.
1,384 citations
IBM1
TL;DR: In this paper, it was shown that the right-hand side is a concave function of the diagonal matrix U such that the sum of the adjacency matrix of the graph plus all the elements of the sum matrix is zero.
Abstract: Let a k-partition of a graph be a division of the vertices into k disjoint subsets containing m1 ≥ m2,..., ≥mk vertices. Let Ec be the number of edges whose two vertices belong to different subsets. Let λ1 ≥ λ2, ..., ≥ λk, be the k largest eigenvalues of a matrix, which is the sum of the adjacency matrix of the graph plus any diagonal matrix U such that the suomf all the elements of the sum matrix is zero. Then Ec ≥ 1/2Σr=1k-mrλr.
A theorem is given that shows the effect of the maximum degree of any node being limited, and it is also shown that the right-hand side is a concave function of U.C omputational studies are madoef the ratio of upper bound to lower bound for the two-partition of a number of random graphs having up to 100 nodes.
693 citations
TL;DR: In this paper, the Newmark family of second-order difference approximations is compared with the original or extended Wilson and Houboult methods for the direct time integration of the spatially discretized equations of linear elastodynamics.
260 citations
TL;DR: A new lower bound on the probability of decoding error for the case of rates above capacity is presented, which forms a natural companion to Gallager's random coding bound for rates below capacity.
Abstract: A new lower bound on the probability of decoding error for the case of rates above capacity is presented. It forms a natural companion to Gallager's random coding bound for rates below capacity. The strong converse to the coding theorem follows immediately from the proposed lower bound.
241 citations
TL;DR: In this paper, an n-period single-product single-facility model with known requirements and separable piecewise concave production and storage costs is considered, and it is shown using network flow concepts that for arbitrary bounds on production and inventory in each period, there is an optimal schedule such that if, for any two periods, production does not equal zero or its upper or lower bound, then the inventory level in some intermediate period equals zero or their upper/lower bound.
Abstract: An n period single-product single-facility model with known requirements and separable piecewise concave production and storage costs is considered. It is shown using network flow concepts that for arbitrary bounds on production and inventory in each period there is an optimal schedule such that if, for any two periods, production does not equal zero or its upper or lower bound, then the inventory level in some intermediate period equals zero or its upper or lower bound. An algorithm for searching such schedules for an optimal one is given where the bounds on production are -∞, 0 or ∞. A more efficient algorithm assumes further that inventory bounds satisfy “exact requirements.”
200 citations
TL;DR: In this article, the authors consider Markovian decision processes in which the transition probabilities corresponding to alternative decisions are not known with certainty, and they consider both a game-theoretic and a Bayesian formulation.
Abstract: This paper examines Markovian decision processes in which the transition probabilities corresponding to alternative decisions are not known with certainty. The processes are assumed to be finite-state, discrete-time, and stationary. The rewards axe time discounted. Both a game-theoretic and the Bayesian formulation are considered. In the game-theoretic formulation, variants of a policy-iteration algorithm are provided for both the max-min and the max-max cases. An implicit enumeration algorithm is discussed for the Bayesian formulation where upper and lower bounds on the total expected discounted return are provided by the max-max and max-min optimal policies. Finally, the paper discusses asymptotically Bayes-optimal policies.
192 citations
TL;DR: In this paper, a variational principle is derived to obtain upper and lower bounds for the effective elastic constants of disordered materials, such as polycrystals or multiphase materials.
Abstract: Using the method of Hill a variational principle is derived to obtain upper and lower bounds for the effective elastic constants of disordered materials, such as polycrystals or multiphase materials. All bounds previously known are rederived and especially new bounds are given being closer than the ones of Hashin and Shtrikman. In detail the elastic constants of polycrystals built of cubic single crystals and of multiphase materials are considered. The analogous bounds for the dielectric constant of polycrystals are also given.
191 citations
TL;DR: The moments of the electronic density of states of a disordered system described by a tight binding model can be directly computed as a function of the overlap integrals and the local order of the system as discussed by the authors.
Abstract: The moments of the electronic density of states of a disordered system described by a tight binding model can be directly computed as a function of the overlap integrals and the local order of the system. Some of the possible techniques for estimating the density of states from its low order moments are discussed. Two methods are particularly emphasized: one using upper and lower bounds for the integrated density of states, the other using a finite continued fraction expansion of the Hilbert's transform of the density of states. The second method seems to be better due to the fact that the continued fraction coefficients rapidly converge towards their asymptotic values. In particular, their behaviour when internal singularities on a band gap appear is analysed.
184 citations
TL;DR: In this article, a lower bound of n(lg2 n?2) multiplications and divisions are shown to be necessary to compute the set of elementary symmetric functions inn indeterminates.
Abstract: n(lg2 n?2) multiplications and divisions are necessary to compute the set of elementary symmetric functions inn indeterminates. This lower bound and similar ones for the computational complexity of various evaluation and interpolation problems are obtained by introducing ideas and results from algebraic geometry.
176 citations
TL;DR: In this article, the authors investigated the problem of designing a compensating control for a linear multivariable system so that the impulse response matrix of the resulting closed-loop system coincides with a prespecified linear model; this is the model following problem; necessary and sufficient conditions for a solution to exist are given.
Abstract: This paper investigates the problem of designing a compensating control for a linear multivariable system so that the impulse response matrix of the resulting closed-loop system coincides with the impulse response matrix of a prespecified linear model; this is the model following problem. A new formulation of the problem is developed, and necessary and sufficient conditions for a solution to exist are given. An upper bound is determined for the number of integrators needed to construct the compensating control and, if the open-loop plant in question possesses a left-invertible transfer matrix, this bound is shown to be as small as possible. The relationship between the internal structure of a model following system and the model being followed is explained, and a description is given of the possible distributions of system eigenvalues which can be achieved while maintaining a model following configuration. This leads to a statement of necessary and sufficient conditions for the existence of a solution to the problem which results in a stable compensated system.
168 citations
TL;DR: A unified formulation for lower bounds on the minimum number of processors and on time is presented and these lower bounds are sharper than previously known values and provide a general framework that gives insight for deriving simplified expressions.
Abstract: Two problems of importance for the scheduling of multiprocessing systems composed of identical units are discussed in this paper. 1) Given a partially ordered set of computations represented by the vertices of an acyclic directed graph with their associated execution times, find the minimum number of processors in order to execute them in a time not exceeding the length of the critical path of this graph. 2) Determine the minimum time to process this set of computations when a fixed number of processors is available. A unified formulation for lower bounds on the minimum number of processors and on time is presented. These lower bounds are sharper than previously known values and provide a general framework that gives insight for deriving simplified expressions. A new upper bound on the minimum number of processors is presented, which is sharper than the known bounds. The computational aspects of these bounds are also discussed.
TL;DR: In this paper, a drift kinetic equation is derived which contains higher order effects and the upper bound appropriate to the drift ordering is only imposed so that the result is quite general and can be reduced to previous drift equations.
Abstract: A drift kinetic equation is derived which contains higher order effects. The upper bound appropriate to the drift ordering is only imposed so that the result is quite general and can be reduced to previous drift equations. The differential geometry of the magnetic field enters only trivially and a single recursion suffices to obtain the desired result.
TL;DR: A formulation of the traveling salesman problem with more than one salesman with computational advantages over other formulations is offered and certain subtours may satisfy the constraints, thus reducing the search.
Abstract: A formulation of the traveling salesman problem with more than one salesman is offered. The particular formulation has computational advantages over other formulations. Experience is obtained with an exact branch and bound algorithm employing both upper and lower bounds mean run time for 55 city problems is one minute. Due to the special formulation, certain subtours may satisfy the constraints, thus reducing the search. A very good initial tour and upper bound are employed. The determination of these as well as the pathology of the formulation and the algorithm are discussed. No increase in computation time over the one-salesman case is experienced.
Journal Article•
TL;DR: The random coding bound of information theory provides a well-known upper bound to the probability of decoding error for the best code of a given rate and block length, which is constructed by upper bounding the average error probability over an ensemble of codes as mentioned in this paper.
Abstract: The random coding bound of information theory provides a well-known upper bound to the probability of decoding error for the best code of a given rate and block length. The bound is constructed by upperbounding the average error probability over an ensemble of codes. The bound is known to give the correct exponential dependence of error probability on block length for transmission rates above the critical rate, but it gives an incorrect exponential dependence at rates below a second lower critical rate. Here we derive an asymptotic expression for the average error probability over the ensemble of codes used in the random coding bound. The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.
TL;DR: A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound c can be assigned to the modulus of the complex numbers involved in the computation.
Abstract: A lower bound for the number of additions necessary to compute a family of linear functions by a linear algorithm is given when an upper bound c can be assigned to the modulus of the complex numbers involved in the computation. In the case of the fast Fourier transform, the lower bound is (n/2) log2n when c = 1.
TL;DR: A matrix formulation of the reliability analysis and reliability-based design of structures is developed and can be used by designers to choose among alternative designs that satisfy all existing code regulations, and to design systems in the absence of formal code regulations.
TL;DR: The table is obtained by combining the best of the existing bounds on d_{max} (n,k) with the minimum distances of known codes and a variety of code-construction techniques.
Abstract: This paper presents a table of upper and lower bounds on d_{max} (n,k) , the maximum minimum distance over all binary, linear (n,k) error-correcting codes. The table is obtained by combining the best of the existing bounds on d_{max} (n,k) with the minimum distances of known codes and a variety of code-construction techniques.
TL;DR: The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.
Abstract: The random coding bound of information theory provides a well-known upper bound to the probability of decoding error for the best code of a given rate and block length. The bound is constructed by upper-bounding the average error probability over an ensemble of codes. The bound is known to give the correct exponential dependence of error probability on block length for transmission rates above the critical rate, but it gives an incorrect exponential dependence at rates below a second lower critical rate. Here we derive an asymptotic expression for the average error probability over the ensemble of codes used in the random coding bound. The result shows that the weakness of the random coding bound at rates below the second critical rate is due not to upperbounding the ensemble average, but rather to the fact that the best codes are much better than the average at low rates.
TL;DR: A new nonparametric method of estimating the Bayes risk using an unclassified test sample set as well as a classified design sample set is introduced, which provides unbiased estimates of the k -NN classification error, thus providing an upper bound on the Baye error.
Abstract: A new nonparametric method of estimating the Bayes risk using an unclassified test sample set as well as a classified design sample set is introduced. The classified design set is used to obtain nonparametric estimates of the conditional Bayes risk of classification at each point of the unclassified test set. The average of these risk estimates is the error estimate. For large numbers of design samples the new error estimate has less variance than does an error-count estimate for classified test samples using the optimum Bayes classifier. The first application of the nonparametric method uses k -nearest neighbor ( k -NN) estimates of the posterior probabilities to form the risk estimate. A large-sample analysis is made of this estimate. The expected value of this estimate is shown to be a lower bound on the Bayes error. A simple modification provides unbiased estimates of the k -NN classification error, thus providing an upper bound on the Bayes error. The second application of the method uses Parzen approximation of the density functions to obtain estimates of the risk and subsequently the Bayes error. Results of experiments on simulated data illustrate the small-sample behavior.
TL;DR: In this article, an experimental study is made of nonlinear interactions in a laminar free shear layer, where two disturbances (f1 and f2) excited by sound, amplify and grow independently for small amplitudes.
Abstract: An experimental study is made of nonlinear interactions in a laminar free shear layer. Two disturbances (f1 and f2), excited by sound, amplify and grow independently for small amplitudes. At larger amplitudes the disturbances interact to generate fluctuations of sum and difference frequencies (f2 ± f1). Harmonics and subharmonics of f1 and f2 are also generated and all fluctuations interact to generate additional fluctuations of the form (nf2/m) ± (pf1/q); n, p = 1,2,3,…, m, q = 1,2. Nonlinear mode competition suppresses the growth of f1 or f2, depending on their relative amplitudes, and contributes to finite amplitude equilibration. An upper bound on the modal integral of total u′r.m.s.2 fluctuation energy is found. Fluctuation energy tends to be distributed among all possible frequency components, and its upper bound does not increase as the number of components increases.
TL;DR: In this paper, the performance analysis of the Viterbi algorithm maximum likelihood detector (MLD) and the decision-feedback equalizer (DFE) was studied on the √f channel representative of coaxial cables and some wire pairs, and it was shown that even the MLD experiences a substantial penalty in S/N ratio relative to the isolated pulse bound on this channel.
Abstract: In a companion paper,1 a geometric approach to the study of intersymbol interference was introduced. In the present paper this approach is applied to the performance analysis of the Viterbi algorithm maximum likelihood detector (MLD) of Forney.2–4 It is shown that a canonical relationship exists between the minimum distance, which Forney has shown determines the performance of the MLD, and the performance and tap-gains of the decision-feedback equalizer (DFE). Upper and lower bounds on the minimum distance are derived, as is an iterative technique for computing it exactly. The performances of the MLD, DFE, and zero-forcing equalizer (ZFE) are compared on the √f channel representative of coaxial cables and some wire pairs. One important conclusion is that, previous statements notwithstanding,2.4 even the MLD experiences a substantial penalty in S/N ratio relative to the isolated pulse bound on this channel of practical interest.
TL;DR: In this paper, a variational analysis of the ground state energy of 4He trimers with pair potential coupling constant is presented, and a lower bound of −0.05 to − 0.2 K is given for the ground states of spin 1/2 trimers.
Abstract: Variational calculations of the ground state energy of 4He trimers are reported and are compared with the Hall‐Post‐Stenschke lower bound on the ground state energy. The variational result for the ground state energy for a range of model pair potentials is in the range −0.05 to −0.2 K; the effect of the triplet nonadditive term is estimated to be less than 1% of the contribution of the pair potentials. A study of the variation of the ground state energy of 3 bosons with pair potential coupling constant is also reported for a Lennard‐Jones and a Morse potential model. There is no bound spin 3/2 3He trimer; a lower bound is given for the ground state energy of the spin 1/2 trimer and analogy with the boson results is used to argue that the spin 1/2 trimer is probably not bound. This paper contains computational applications of formal results derived elsewhere.
30 Apr 1973
TL;DR: A general model for studying bilinear multiplication is proposed and defended in order to provide a common framework for discussing a wide class of problems and derive some general results which unify and extend numerous known results.
Abstract: Although general theories are beginning to emerge in the area of automata based complexity theory, there are very few general methods or even general problem formulations in the area of arithmetic complexity. In this paper we propose and defend a general model for studying bilinear multiplication in order to provide a common framework for discussing a wide class of problems. At the heart of a number of problems in minimizing the number of multiplications required to perform a calculation is a problem in matrix algebra relating to the expansion of a given set of matrices as linear combinations of rank one matrices. In this paper we make a systematic attack on this problem and derive some general results which unify and extend numerous known results. Among the new results given here to illustrate the strength of this approach is a new lower bound on the number of multiplications required for n by n matrix multiplication of 3n2-3n+1 which is independent of the subset of the reals with respect to which multiplication is regarded as free. An even sharper bound is obtained if this set is restricted to the integers.
TL;DR: In this article, upper and lower bounds on the spectral and maximum norms of stiffness, flexibility and mass matrices generated from regular and irregular meshes of finite elements were established for second and fourth order problems in one, two and three dimensions discretized with linear, triangular and tetrahedronal elements.
TL;DR: Sharp upper and lower bounds of the Chebyshev type are established for the probability of error due to intersymbol interference and additive Gaussian noise in a digital communication system.
Abstract: Sharp upper and lower bounds of the Chebyshev type are established for the probability of error due to intersymbol interference and additive Gaussian noise in a digital communication system. The results are in relatively closed form, and the only statistical knowledge assumed about the interference is the peak eye opening and the variance. The bounds apply to correlated and uncorrelated signals and for any signal to noise ratio.
TL;DR: In this article, upper and lower bounds on the adiabatic bulk modulus are calculated using energy principles for two cases: high-frequency and low-frequency case, deformation on the scale of a single grain, and temperature may vary from grain to grain.
Abstract: Upper and lower bounds on the adiabatic bulk modulus are calculated using energy principles for two cases. In the high-frequency case, deformation is adiabatic on the scale of a single grain, and temperature may vary from grain to grain. Bounds are found to be given by the Voigt and Reuss averages of the components of the adiabatic stiffness and compliance tensors. At lower frequencies, deformation is adiabatic overall, but temperature of adjacent grains has sufficient time to reach an equilibrium value. Bounds at high frequencies for olivine at room temperature are found to be nearly equal numerically to the low-frequency bounds.
TL;DR: In this article, a generalization of Bargmann's inequality for the Z-th partial wave of the Schrodinger equation with a spherically symmetric potential function in the n-dimensional space is presented.
Abstract: An inequality is derived which gives an upper bound of the number of bound states in the Z-th partial wave (Z=0, !,-••) of the two-body Schrodinger equation with a spherically symmetric potential function in the n -dimensional space (re —1,2 •••). This is a generalization of Bargmann's inequality for the case n = 3. The generalization is straightforwa rd for the case Z^l with n= 2 and Zj^O with ra^3. After a mathematically rigorous justification of his heuristic argument, Schwinger's method in his simple proof of Bargmann's inequality is employed here. Newton's result for the case / = —___, n— 3, A which is equivalent to the case /=0, n=29 is reobtained. in 1952 the so-called "Bargmann's inequality" which gives an upper bound for the number of bound states produced, in the two-body system, by a spherically symmetric potential. His method of proof is to count the number of zeros of the zero-energy solution, with a smoothness condition at the origin, of the partial-wave Schrodinger equation and to make use of the well-known correspondence between the number of zeros and that of bound states. Avoiding this highly complicated method, Schwinger [~2~] has elegantly rederiverl, in a heuristic manner, Bargmann's inequality and its versions, by transforming the zero-energy "hound state11 problem to an integral equation and applying Mercer's theorem to the positive integral operator thus obtained. Soon after, an inequality for the case I = — —— LJ