Showing papers on "Upper and lower bounds published in 1976"
TL;DR: In this article, it was shown that phase transitions occur in (φ·φ) 3 2 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions.
Abstract: We present a new method for rigorously proving the existence of phase transitions. In particular, we prove that phase transitions occur in (φ·φ) 3 2 quantum field theories and classical, isotropic Heisenberg models in 3 or more dimensions. The central element of the proof is that for fixed ferromagnetic nearest neighbor coupling, the absolutely continuous part of the two point function ink space is bounded by 0(k−2). When applicable, our results can be fairly accurate numerically. For example, our lower bounds on the critical temperature in the three dimensional Ising (resp. classical Heisenberg) model agrees with that obtained by high temperature expansions to within 14% (resp. a factor of 9%).
484 citations
TL;DR: In this article, it was shown that one cannot artifically establish a gauge hierarchy of any desired magnitude by arbitrarily adjusting the scalar-field parameters in the Lagrangian and using the tree approximation to the potential; radiative corrections will set an upper bound on such a hierarchy.
Abstract: It is shown that one cannot artifically establish a gauge hierarchy of any desired magnitude by arbitrarily adjusting the scalar-field parameters in the Lagrangian and using the tree approximation to the potential; radiative corrections will set an upper bound on such a hierarchy. If the gauge coupling constant is approximately equal to the electromagnetic coupling constant, the upper bound on the ratio of vector-meson masses is of the order of ${\ensuremath{\alpha}}^{\ensuremath{-}\frac{1}{2}}$, independent of the sclar-field masses and their self-couplings. In particular, the usual assumption that large scalar-field mass ratios in the Lagrangian can induce large vector-meson mass ratios is false. A thus far unsuccessful search for natural gauge hierarchies is briefly discussed. It is shown that if such a hierarchy occurred, it would have an upper bound of the order of ${\ensuremath{\alpha}}^{\ensuremath{-}\frac{1}{2}}$.
341 citations
TL;DR: In this paper, a new family of upper bounds is derived using results in graph theory, and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956).
Abstract: The problem of bounding P(∪ Ai ) given P(A i) and P(A i A j) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.
316 citations
TL;DR: Conditions under which the bound on the number of iterations required for propagation algorithms when the data is represented by bit vectors and depth-first ordering of the flow graph is used are considered are considered.
Abstract: Kildall has developed data propagation algorithms for code optimization in a general lattice theoretic framework. In another direction, Hecht and Ullman gave a strong upper bound on the number of iterations required for propagation algorithms when the data is represented by bit vectors and depth-first ordering of the flow graph is used. The present paper combines the ideas of these two papers by considering conditions under which the bound of Hecht and Ullman applies to the depth-first version of Kildall's general data propagation algorithm. It is shown that the following condition is necessary and sufficient: Let ƒ and g be any two functions which could be associated with blocks of a flow graph, let x be an arbitrary lattice element, and let 0 be the lattice zero. Then (*) (∀ƒ,g,x) [ƒg(0) ≥ g(0) ∧ ƒ(x) ∧ x]. Then it is shown that several of the particular instances of the techniques Kildall found useful do not meet condition (*).
310 citations
TL;DR: An algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn2 and the average job processing time and an extremely sharp lower bound on the optimal objective value is presented.
Abstract: A branch and bound algorithm is presented for the problem of schedulingn jobs on a single machine to minimize tardiness. The algorithm uses a dual problem to obtain a good feasible solution and an extremely sharp lower bound on the optimal objective value. To derive the dual problem we regard the single machine as imposing a constraint for each time period. A dual variable is associated with each of these constraints and used to form a Lagrangian problem in which the dualized constraints appear in the objective function. A lower bound is obtained by solving the Lagrangian problem with fixed multiplier values. The major theoretical result of the paper is an algorithm which solves the Lagrangian problem in a number of steps proportional to the product ofn
2 and the average job processing time. The search for multiplier values which maximize the lower bound leads to the formulation and optimization of the dual problem. The bounds obtained are so sharp that very little enumeration or computer time is required to solve even large problems. Computational experience with 20-, 30-, and 50-job problems is presented.
278 citations
TL;DR: It is shown that unless a bound on the total number of distinct symbols is assumed, every solution to the problem can consume an amount of time that is proportional to the product of the lengths of the two strings.
Abstract: The problem of finding a longest common subsequence of two strings is discussed. This problem arises in data processing applications such as comparing two files and in genetic applications such as studying molecular evolution. The difficulty of computing a longest common subsequence of two strings is examined using the decision tree model of computation, in which vertices represent “equal - unequal” comparisons. It is shown that unless a bound on the total number of distinct symbols is assumed, every solution to the problem can consume an amount of time that is proportional to the product of the lengths of the two strings. A general lower bound as a function of the ratio of alphabet size to string length is derived. The case where comparisons between symbols of the same string are forbidden is also considered and it is shown that this problem is of linear complexity for a two-symbol alphabet and quadratic for an alphabet of three or more symbols.
273 citations
TL;DR: In this article, different measures of dispersion are compared in terms of asymptotic relative efficiency, i.e., the inverse ratio of their standardized variances, and it is shown that the efficiency of a trimmed to the untrimmed standard deviation turns out not to have a positive lower bound even over the family of Tukey models.
Abstract: Measures of dispersion are defined as functionals satisfying certain equivariance and order conditions. In the main part of the paper attention is restricted to symmetric distributions. Different measures are compared in terms of asymptotic relative efficiency, i.e., the inverse ratio of their standardized variances. The efficiency of a trimmed to the untrimmed standard deviation turns out not to have a positive lower bound even over the family of Tukey models. Positive lower bounds for the efficiency (over the family of all symmetric distributions for which the measures are defined) exist if the trimmed standard deviations are replaced by pth power deviations. However, these latter measures are no longer robust, although for p <2 they are more robust than the standard deviation. The results of the paper suggest that a positive bound to the efficiency may be incompatible with robustness but that trimmed standard deviations and pth power deviations for p = 1 or 1.5 are quite satisfactory in practice.
203 citations
TL;DR: In this article, a theory for a spherical electrostatic probe in a collisionless, Maxwellian plasma containing a uniform magnetic field is presented, which yields two upper bounds and an adiabatic limit for collection of the attracted particle species (either electrons or ions).
Abstract: A theory is presented for a spherical electrostatic probe in a collisionless, Maxwellian plasma containing a uniform magnetic field. The theory yields two upper bounds and an adiabatic limit for collection of the attracted particle species (either electrons or ions). For the repelled species, it yields a lower and an upper bound. The theory is similar in concept to existing theories for cylindrical probes by Laframboise and Rubinstein. It is applicable when the ratio of probe radius to Debye length is small enough, and/or the probe potential is large enough, that no potential barriers exist near the probe. Otherwise, a theory of Sanmartin applies. The attracted‐particle current in the adiabatic limit, i.e., when mean gyroradius≪Debye length, shows negative‐resistance behavior. One of the upper bounds is based on the use of particle canonical angular momentum conservation to define allowed and forbidden regions for particle orbits, and generalizes an existing theory by Parker and Murphy to include particle thermal motion.
164 citations
TL;DR: In this paper, a lower bound for the number of nodes in a cubature formula of degree 2s-1 is derived for two-fold integrals, and a generalization to then-dimensional case is given.
Abstract: For two-fold integrals, a lower bound is derived for the number of nodes in a cubature formula of degree 2s-1. There is a formula of degree 2s-1 for which the number of nodes attains this lower bound, iff a certain condition is fullfilled. By this condition, all formulas of degree 2s-1 with that minimal number of nodes can be constructed. Examples and a generalization to then-dimensional case are given.
142 citations
TL;DR: In this paper, a lower bound on the free energy of the Ising model and the Wilson-Fisher model is given for both models, and a simple lower bound which preserves the symmetry of the Hamiltonian (the one hypercube approximation) is described.
Abstract: Approximate recursion relations which give upper and lower bounds on the free energy are described. Optimal calculations of the free energy can then be obtained by treating parameters within the renormalization equations variationally. As an example, a particularly simple lower bound approximation which preserves the symmetry of the Hamiltonian (the one-hypercube approximation) is described. The approximation is applied to both the Ising model and the Wilson-Fisher model. At the fixed point a parameter is set variationally and critical indices are calculated. For the Ising model the agreement with the exact results atd = 2 is surprisingly good, 0.1%, and is good atd=3 and evend=4. For the Wilson-Fisher model the recursion relation is reduced to a one-dimensional integral equation which can be solved numerically givingv=0.652 atd=3, or by ɛ expansion in agreement with the results of Wilson and Fisher to leading order in ɛ. The method is also used to calculate thermodynamic functions for thed = 2 Ising model; excellent agreement with the Onsager solution is found.
134 citations
TL;DR: A lower bound on the minimal mean-square error in estimating nonlinear diffusion processes is derived and the bound holds for causal and noncausal filtering.
Abstract: A lower bound on the minimal mean-square error in estimating nonlinear diffusion processes is derived. The bound holds for causal and noncausal filtering.
TL;DR: Basic limitations on the amount of protocol information that must be transmitted in a data communication network to keep track of source and receiver addresses and of the starting and stopping of messages are considered.
Abstract: We consider basic limitations on the amount of protocol information that must be transmitted in a data communication network to keep track of source and receiver addresses and of the starting and stopping of messages. Assuming Poisson message arrivals between each communicating source-receiver pair, we find a lower bound on the required protocol information per message. This lower bound is the sum of two terms, one for the message length information, which depends only on the distribution of message lengths, and the other for the message start information, which depends only on the product of the source-receiver pair arrival rate and the expected delay for transmitting the message. Two strategies are developed which, in the limit of large numbers of sources and receivers, almost meet the lower bound on protocol information.
TL;DR: It is shown theoretically that the two-pair quantization scheme has a 10-bit superiority over other above-mentioned quantization schemes in the sense of theoretically assuring that a maximum overall log spectral deviation will not be exceeded.
Abstract: The topic of quantization and bit allocation in speech processing is studied using an L 2 norm. Closed-form expressions are derived for the root mean square (rms) spectral deviation due to variations in one, two, or multiple parameters. For one-parameter variation, the reflection coefficients, log area ratios, and inverse sine coefficients are studied. It is shown that, depending upon the criterion chosen, either log area ratios or inverse sine quantization can be viewed as optimal. From a practical point of view, it is shown experimentally that very little difference exists among the various quantization methods beyond the second coefficient. Two-parameter variations are studied in terms of formant frequency and bandwidth movement and in terms of a two-pair quantization scheme. A lower bound on the number of quantization levels required to satisfy a given maximum spectral deviation is derived along with the two-pair quantization scheme which approximately satisfies the bound. It is shown theoretically that the two-pair quantization scheme has a 10-bit superiority over other above-mentioned quantization schemes in the sense of theoretically assuring that a maximum overall log spectral deviation will not be exceeded.
TL;DR: In this paper, an open-shell SCF procedure for doublet states with one vacancy in an orbital within the occupied manifold (hole states) was derived. But this procedure is not suitable for ground doublets.
Abstract: An SCF method is derived for doublet states with one vacancy in an orbital within the occupied manifold (hole states). This method gives an upper bound to an excited state energy. Hence it is a stable procedure which is bounded from below and cannot collapse to a lower energy SCF state. This new procedure is compared with several other open‐shell SCF procedures which have been advocated for the ground doublet state.
TL;DR: In this paper, a method for calculating rigorous upper and lower bounds to dynamic dipole polarizabilities for frequencies up to and beyond the first excitation threshold was described. But this method cannot be applied to the case of the two-electron atoms H(-), He, and Li(+).
Abstract: Methods are described for calculating rigorous upper and lower bounds to dynamic dipole polarizabilities for frequencies up to and beyond the first excitation threshold, even when (as usual) the field-free problem cannot be solved exactly. These methods were employed to calculate rigorous bounds to the dynamic polarizabilities of the two-electron atoms H(-), He, and Li(+), using well correlated trial wavefunctions up to 135 terms in length. The majority of previous theoretical and experimental results for these atoms can thereby be ruled out, including the highly precise Starkschall-Gordon values, which had heretofore appeared to be the most accurate available.
01 Nov 1976
TL;DR: In this article, the authors derived a broad range of theoretical results concerning the performance and limitations of a class of analog adaptive filters and proved the exponential convergence to zero of the norm ||r(t)|| with weak nondegeneracy requirements on x(t).
Abstract: We derive a broad range of theoretical results concerning the performance and limitations of a class of analog adaptive filters. Applications of these filters have been proposed in many different engineering contexts which have in common the following idealized identification problem: A system has a vector input x(t) and a scalar output z(t)=h'x(t), where h is an unknown time-invariant coefficient vector. From a knowledge of x(t) and z(t) it is required to estimate h. The filter considered here adjusts an estimate vector h^(t) in a control loop, thus d/dt h^= KF[z(t)-z^(t)]x(t) where z^(t) = h^'x(t), F is a suitable, in general nonlinear, function, and K is the loop gain. The effectiveness of the filter is determined by the convergence properties of the misalignment vector, r = h - h^. With weak nondegeneracy requirements on x(t) we prove the exponential convergence to zero of the norm ||r(t)||. For all values of K, we give upper and lower bounds on the convergence rate which are tight in that both bounds have similar qualitative dependence on K. The dependence of these bounds on K is unexpected and important since it reveals basic limitations of the filters which are not predicted by the conventional approximate method of analysis, the "method of averaging." By analyzing the effects of an added forcing term u(t) in the control equation we obtain upper bounds to the effects on the convergence process of various important departures from the idealized model as when noise is present as an additional component of z(t), the coefficient vector h is time-varying, and the integrators in a hardware implementation have finite memory.
TL;DR: A consistent estimator is discussed which is computationally more efficient than estimators based on Parzen's estimation and its relation between the distance of a sample from the decision boundary and its contribution to the error is derived.
Abstract: The L^{ \alpha} -distance between posterior density functions (PDF's) is proposed as a separability measure to replace the probability of error as a criterion for feature extraction in pattern recognition. Upper and lower bounds on Bayes error are derived for \alpha > 0 . If \alpha = 1 , the lower and upper bounds coincide; an increase (or decrease) in \alpha loosens these bounds. For \alpha = 2 , the upper bound equals the best commonly used bound and is equal to the asymptotic probability of error of the first nearest neighbor classifier. The case when \alpha = 1 is used for estimation of the probability of error in different problem situations, and a comparison is made with other methods. It is shown how unclassified samples may also be used to improve the variance of the estimated error. For the family of exponential probability density functions (pdf's), the relation between the distance of a sample from the decision boundary and its contribution to the error is derived. In the nonparametric case, a consistent estimator is discussed which is computationally more efficient than estimators based on Parzen's estimation. A set of computer simulation experiments are reported to demonstrate the statistical advantages of the separability measure with \alpha = 1 when used in an error estimation scheme.
TL;DR: A lower bound on finding the mode of a multiset as a function of the actual multiplicity is given, and it is demonstrated that the bound can be achieved to within a multiplicative constant.
Abstract: In this paper the problem of sorting multisets is considered. An information theoretic lower bound on the number of three branch comparisons is obtained, and it is shown that this bound is asymptotically attainable. It is shown that the multiplicities of a set can only be obtained by comparisons if the total order is discovered in the process. A lower bound on finding the mode of a multiset as a function of the actual multiplicity is given, and it is demonstrated that the bound can be achieved to within a multiplicative constant. The determination of the intersection of two multisets is also discussed, and partial results, including a generalization of Reingold’s result for determining whether or not two sets have a nonempty intersection, are obtained.
IBM1
TL;DR: In this paper, it was shown that there is a positive lower bound,c, to the uniform error in any scheme designed to recover all functions of a certain smoothness from their values at a fixed finite set of points.
Abstract: It is shown that there is a positive lower bound,c, to the uniform error in any scheme designed to recover all functions of a certain smoothness from their values at a fixed finite set of points. This lower bound is essentially attained by interpolation at the points by splines with canonical knots. Estimates ofc are also given.
TL;DR: In this paper, an improved lower bound for the smallest singular value of a matrix was obtained for a wider class of matrices, under the assumption that A is strictly diagonally dominant.
TL;DR: A general lower bound on the minimal number of additions in monotone, rational computations is proved, which implies that anymonotone rational computation of the n th degree convolution at least requires n 2 − 2 n + 1 additions.
[...]
TL;DR: It is shown that the minimum number of comparisons necessary for the computation of the kth element of a totally ordered set of size n, V_k (n), is bounded below by $n - k + (k - 1)\lceil \log _2 (n/( k - 1)) \rceil $.
Abstract: In this paper we show that the minimum number of comparisons necessary for the computation of the kth element of a totally ordered set of size n, $V_k (n)$, is bounded below by $n - k + (k - 1)\lceil \log _2 (n/(k - 1)) \rceil $. For $3 < k < n/4$, this bound is an improvement on the best lower bound presently known. A new algorithm which yields an upper bound that is better than the currently known bound for a large range of values of n will also be presented.
TL;DR: In this article, the authors consider two variants of the Cramer-Rao inequality for estimating the parameters of canonical states, in particular, the canonical parameters of a Lie group, and show that these bounds are globally attainable only for canonical states for which there exist e¢ cient measurements or quasimeasurements.
Abstract: We consider two variants of a quantum-statistical generalization of the Cramer-Rao inequality that establishes an invariant lower bound on the mean square error of a generalized quantum measurement. The proposed complex variant of this inequality leads to a precise formulation of a generalized uncertainty principle for arbitrary states, in contrast to Helstroms variant [1] in which these relations are obtained only for pure states. A notion of canonical states is introduced and the lower mean square error bound is found for estimating of the parameters of canonical states, in particular, the canonical parameters of a Lie group. It is shown that these bounds are globally attainable only for canonical states for which there exist e¢ cient measurements or quasimeasurements.
TL;DR: In this article, a lower bound on the dependence of two discrete random variables, the common information, with important operational significance, was established for all joint distributions of two n-ary variables, and the bound holds with equality for an n-parameter family of joint distributions.
Abstract: Recently, Wyner has defined a new measure of dependence of two discrete random variables, the common information, with important operational significance. The principal result of this paper is a lower bound on this quantity, for all joint distributions of two n-ary variables. For each n, the bound holds with equality for an n-parameter family of joint distributions. The bound depends on the maximum over all row permutations of the trace of the joint distribution matrix. A bound for joint distributions of zero trace is also obtained and the cases of equality are characterized. The common information of L-shaped distributions is determined.
TL;DR: In this paper, upper and lower bounds that depend on the absolute moment of the intersymbol interference random variable, the second moment, the fourth moment, and an "exponential moment" are found by analytical, graphical, or iterative approaches.
Abstract: A method for the evaluation of upper and lower bounds to the error probability of a linear pulse-amplitude modulation (PAM) system with bounded intersymbol interference and additive Gaussian noise is obtained via an isomorphism theorem from the theory of moment spaces. These upper and lower bounds are seen to be equivalent to upper and lower envelopes of some compact convex body generated from a set of kernel functions. Depending on the selection of these kernels and their corresponding moments, different classes of bounds are obtained. In this paper, upper and lower bounds that depend on the absolute moment of the intersymbol interference random variable, the second moment, the fourth moment, and an "exponential moment" are found by analytical, graphical, or iterative approaches. We study in detail the exponential moment case and obtain a family of new upper and a family of new lower bounds. Within each family, expressions for these bounds are given explicitly as a function of an arbitrary real-valued parameter. For two channels of interest, upper and lower bounds are evaluated and compared. Results indicate these bounds to be tight and useful.
TL;DR: It is believed that this example is a “canonical” example for interval orders, so that the computation of its dimension should have significant impact on the problem of computing the dimension of interval orders in general.
TL;DR: In this article, the maximum reduction of area obtainable, when directly drawing section from round bar, is calculated from a lower bound analysis; this reduction is less than that occurring when drawing round rod from round bars.
TL;DR: In this paper, a first-order definability of the Ramified Analytical Hierarchical Hierarchy is given, and a degree theoretic definition of the ramified analytical hierarchy is obtained.