Showing papers on "Upper and lower bounds published in 1968"
TL;DR: The construction of a switching network capable of n-permutation of its input terminals to its output terminals is described and an algorithm is given for the setting of the binary cells in the network according to any specified permutation.
Abstract: In this paper the construction of a switching network capable of n!-permutation of its n input terminals to its n output terminals is described. The building blocks for this network are binary cells capable of permuting their two input terminals to their two output terminals.The number of cells used by the network is 〈n · log2n - n + 1〉 = Σnk=1 〈log2k〉. It could be argued that for such a network this number of cells is a lower bound, by noting that binary decision trees in the network can resolve individual terminal assignments only and not the partitioning of the permutation set itself which requires only 〈log2n!〉 = 〈Σnk=1 log2k〉 binary decisions.An algorithm is also given for the setting of the binary cells in the network according to any specified permutation.
488 citations
TL;DR: In this paper, it is known that there are positive constants A and B, independent of w and n, such that the modulus of continuity function w is dominated by continuity function n, and F is the family P, of polynomials of degree n-1.
Abstract: which is the precision to which F approximates K. An instance of classical interest is that in which L is C([O, 1]), K is AO-i.e. those functions bounded by one, and whose modulus of continuity is dominated by the modulus of continuity function w-and F is the family P,, of polynomials of degree n-1. For this case it is known that there are positive constants A and B, independent of w and n, such that
276 citations
TL;DR: An upper bound is derived for the probability of error of a digital communication system subject to intersymbol interference and Gaussian noise and it is shown that random binary signaling with sin x/x pulses is not so sensitive to timing error as is commonly believed, but the error probability does grow very rapidly as the system bandwidth is reduced below the Nyquist band.
Abstract: An upper bound is derived for the probability of error of a digital communication system subject to intersymbol interference and Gaussian noise. The bound is applicable to multilevel as well as binary signals and to all types of intersymbol interference. The bound agrees with the exponential portion of a normal distribution in which the larger intersymbol interference components subtract from the signal amplitude, and the smaller components add to the noise power. The results are applied to the case of random binary signaling with sin x/x pulses. It is shown that such signals are not so sensitive to timing error as is commonly believed, nor does the total signal amplitude become very large with significant probability. However, the error probability does grow very rapidly as the system bandwidth is reduced below the Nyquist band.
202 citations
TL;DR: The theory of fine-particle magnets is based on the theorem that the state of lowest free energy of a ferromagnetic particle is one of uniform magnetization for particles of less than a certain critical size.
Abstract: The theory of fine‐particle magnets is based on the theorem that the state of lowest free energy of a ferromagnetic particle is one of uniform magnetization for particles of less than a certain critical size and one of nonuniform magnetization for larger particles. The theorem is inferred from several approximate calculations and has not been proved rigorously. Rigorous statements can be made if one is content to replace equalities by inequalities and exact values of critical radii by upper and lower bounds. For a sphere of radius a with uniaxial anisotropy, it can be shown that the lowest‐free‐energy state is one of uniform magnetization if a ac1 (for low anisotropy) or ac2 (for high anisotropy), where ac0, ac1 (>ac0), and ac2(>ac0) are determined by the exchange and anisotropy constants and the spontaneous magnetization. These bounds locate the critical radius to within about 12% at low anisotropy but only to within an order of magnitude or more at high.
172 citations
170 citations
TL;DR: A quantum-mechanical form of the Cramer-Rao inequality is derived, setting a lower bound to the variance of an unbiased estimate of a parameter of a density operator.
Abstract: A quantum-mechanical form of the Cramer-Rao inequality is derived, setting a lower bound to the variance of an unbiased estimate of a parameter of a density operator. It is applied to the estimates of parameters such as amplitude, arrival time, and carrier frequency of a coherent signal as picked up by an ideal receiver in the presence of thermal noise. The estimation of parameters of a noise-like signal is also treated.
117 citations
TL;DR: In this article, a simple inequality, expressed in terms of two arbitrary distribution functions of the same normalization, is shown to be useful for computing the configurational free energy.
Abstract: A simple inequality, expressed in terms of two arbitrary distribution functions of the same normalization, is shown to be useful. By choosing various different forms for the distribution functions one can derive important results, such as the upper and lower bounds of the configurational free energy.
113 citations
TL;DR: The purpose is to give a unified development of enumerative techniques which give sharp upper bounds on Ramsey's theorem numbers and to give constructive methods for partitions to determine lower bounds on these numbers.
110 citations
92 citations
TL;DR: In this paper, the maximum value unity for the parameter jl is associated with the uniform bar and the lower bound on permissible values of jL depends on the ratio r of the end mass to the total mass of the bar.
Abstract: The maximum value unity for the parameter jl is associated with the uniform bar. For this case f = 0, as may be observed from Eq. (15), and the frequency is determined from Eq. (14). The lower bound on permissible values of jEt depends on the ratio r of the end mass to the total mass of the bar. This bound may be obtained from Eqs. (14) and (15) by letting f = 1. For the simpler case /c0 = 0, its value, say ju = #*, is found to be p* = /3r tanh/3 (16) As # -*• #*, the solution approaches the result obtained in the prior unconstrained problem. Numerical results are developed for the bar with a specific uniformly distributed mass KQ = 0.82, and zero end mass (r = 0). For this case, the range of jE is 0 ^ JE ^ 1. A set of typical designs is shown in Fig. 1. The curve of Fig. 2 summarizes the effect of variation in ft on the response frequency /3 for the set of designs within a specified total mass for the bar. The same results might be plotted in a form to indicate the mass required for a bar that vibrates at specified lowest mode frequency, as a function of the design constraint jE. For the present example, the savings in mass of material associated with optimum design ranges between 0 and 32% of the equal response-frequency uniform bar. Roughly twothirds of this saving is achieved in the optimum design associated with the constraint value ft = 0.5. Summary (F For practical reasons, structures are seldom made in the form of continuously varying shapes, such as those which
64 citations
TL;DR: In this article, a series solution to the first-passage problem in random vibration is derived which is valid for any type of response process and for both single and double-sided barriers.
TL;DR: In this article, a variational method is used to give rigorous upper bounds on the bound state energies, and a rigorous lower bound on the ground state energy is also obtained, and Faddeev's equations are solved keeping only the S-wave in two-body partial wave expansion.
TL;DR: In this paper, simple consistent estimates for the parameters of a multivariate exponential distribution are derived for the mean squared error of the estimates, and a lower bound for their efficiency is derived.
Abstract: Simple consistent estimates are derived for the parameters of a multivariate exponential distribution. The mean squared error of the estimates is computed, and a lower bound for their efficiency is derived. A precise lower bound for the efficiency is computed in the bivariate case. It does not seem feasible to compute the efficiency precisely in the multivariate case. Some attention is given to the behavior of the estimates when observations are rounded prior to analysis.
TL;DR: This paper is concerned with the transmission of a discrete, independent letter information source over a discrete channel and a distortion function is defined between source output letters and decoder output letters.
Abstract: This paper is concerned with the transmission of a discrete, independent letter information source over a discrete channel A distortion function is defined between source output letters and decoder output letters and is used to measure the performance of the system for each transmission The coding block length is introduced as a variable and its influence upon the minimum attainable transmission distortion is investigated The lower bound to transmission distortion is found to converge to the distortion level d c (C is the channel capacity) algebraically as a/n The nonnegative coefficient a is a function of both the source and channel statistics, which are interrelated in such a way as to suggest the utility of this coefficient as a measure of “mismatch” between source and channel, the larger the mismatch the slower the approach of the lower bound to the asymptote d c For noiseless channels a = ∞ and for this case the lower bound is shown to converge to d c as a 1 (ln n)/n For noisy channels the upper bound to transmission distortion is found to converge to the asymptote d c algebraically as b[(ln n)/n]1/2 For noiseless channels, the upper bound converges to d c as a 1 (ln n)/n
TL;DR: In this article, the problem of the effective conductivity of a composite material whose local conductivity is a function of position is treated, using the analogy between this problem and the diffusion of ions in a periodic potential.
Abstract: The problem of the effective conductivity of a composite material whose local conductivity is a function of position is treated. Using the analogy between this problem and the diffusion of ions in a periodic potential, previously obtained variational results are used to obtain upper and lower bounds for the effective conductivity. These bounds are shown to be the conductivities obtained in certain commonly used equivalent circuit approximations. Although the discussion in the paper is in terms of the electrical conductivity, the theory is equally applicable to other transport coefficients, such as the heat conductivity and the diffusion constant, as well as to a class of response functions such as the magnetic permeability and dielectric constant.
01 Apr 1968
TL;DR: In this paper, the authors formulated the problem of finding the number of ways of dissecting an n-brick into 2-dimers in the 3-dimensional case, and showed that this problem increases exponentially with n.
Abstract: The dimer problem, which in the three-dimensional case is one of the classical unsolved problems of solid-state chemistry, can be formulated mathematically as follows. We define a brick to be a d-dimensional (d ≥ 2) rectangular parallelopiped with sides whose lengths are integers. An n-brick is a brick whose volume is n; and a dimer is a 2-brick. The problem is to determine the number of ways of dissecting an n-brick into dimers; and since this is only possible when n is even we confine attention hereafter to n-bricks with n even. Consider an n-brick with sides of length a1, a2, …, ad, where n = a1a2 … ad, and write a = (a1, a2, …, ad). Let fa denote the number of ways of dissecting this brick into ½n dimers. On the basis of physical and heuristic arguments chemists have known for many years that fa increases more or less exponentially with n; and recently a rigorous proof (1) of this fact has been given in the following form: if ai → ∞ for all i = 1, 2, …, d, then n−1 logfa tends to a finite limit, which we denote by λd. The principal outstanding problem for chemists is to determine the numerical value of λ3, or failing an exact determination to estimate λ3 or to find upper and lower bounds for it.
TL;DR: The problem of optimizing the expurgated upper bound to the probability of error associated with transmission over discrete memoryless channels is investigated and a general sufficient condition is found under which, for a given value of the parameter, the channel input distribution that leads to the optimal exponent corresponds to a constant memoryless source.
Abstract: In this paper, we investigate the problem of optimizing the expurgated upper bound to the probability of error associated with transmission over discrete memoryless channels. We find a general sufficient condition under which, for a given value of the parameter \varrho \varepsilon [1, \infty) , the channel input distribution that leads to the optimal exponent corresponds to a constant memoryless source. We then derive a necessary and sufficient condition that the above property holds for all 1 \leq \varrho (even then, different values of o would, in general, induce different optimal input distributions). Finally, we define a class of equidistant channels that includes all binary input channels, and show that for this class and all \varrho \varepsilon [1, \infty) the optimal expurgated exponent is attained by the uniform distribution over the inputs.
01 Nov 1968
TL;DR: In this article, a formulation of the interrelationship among mutual coupling element efficiency, active impedance, and element radiation patterns for infinite linear (uniformly spaced) arrays is presented.
Abstract: A formulation is presented of the interrelationship among mutual coupling element efficiency, active impedance, and element radiation patterns for infinite linear (uniformly spaced) arrays. Numerical results are obtained for element efficiency and mutual coupling when the array elements are elementary dipoles. A new lower upper bonnd is obtained on element efficiency. This upper bound is expressed directly in terms of the element patterns in the open-circuit array environment.
TL;DR: In this paper, upper and lower bounds for the response of a system initially in equilibrium to a weak damped-harmonic perturbation are derived for the case where the only information required in the construction of these error bounds are equilibrium properties of the unperturbed system, in particular, the equilibrium moments of the spectral density corresponding to the perturbations.
Abstract: Upper and lower bounds are derived for the response of a system initially in equilibrium to a weak damped‐harmonic perturbation. The only information required in the construction of these error bounds are equilibrium properties of the unperturbed system, in particular, the equilibrium moments of the spectral density corresponding to the perturbation are used. The error bounds are shown to be the most precise possible, given only this equilibrium information. As an example, error bounds are obtained for the shape of a nuclear‐resonance spectrum due to dipolar broadening in solids, and good agreement is obtained with nuclear‐magnetic resonance experiments on CaF2. The error bounds are applicable to many other areas of spectroscopy and nonequilibrium statistical mechanics.
TL;DR: Variational upper and lower bounds for the dynamic polarizability at imaginary frequencies were derived in this article, and some theoretical applications were given for the theoretical application of dynamic polarization at imaginary frequency.
Abstract: Variational upper and lower bounds are derived for the dynamic polarizability at imaginary frequencies, and some theoretical applications are given.
TL;DR: In this paper, a rigorous upper bound for the heat capacity at constant volume was derived for the Ising model and an analogous upper bound was obtained for the adiabatic compressibility.
Abstract: A rigorous upper bound for the heat capacity at constant volume ${C}_{V}$ is used to provide an alternative derivation of a result due to Rice: A locus of points of infinite ${C}_{V}$ is in general incompatible with thermodynamic stability. An analogous upper bound is obtained for the adiabatic compressibility. The bounds are extended to multicomponent stystems, where they suggest that ${C}_{V}$ should not diverge along a continuous line of critical points or plait points. They make plausible the absence of an infinite heat capacity in a certain class of "decorated" Ising models (including Syozi's model for a dilute ferromagnet) and in the spherical model. Possible implications for fluids or ferromagnets containing impurities are briefly discussed.
TL;DR: In this paper, a lower bound for the asymptotic power of Savage's test for scale change of any positive random variable possessing a finite second moment was shown. But this lower bound was only applicable to positive random variables with increasing failure rate averages (IFRA).
Abstract: The rapid development of non-parametric rank tests was generated, in part, by the result of Hodges and Lehmann [6] which stated that the asymptotic relative efficiency (ARE) of the Wilcoxon test to the classical two-sample $t$-test was always $\geqq.86$. They also conjectured that the ARE of the normal scores test to the $t$-test was always greater than or equal to one. In 1958, Chernoff and Savage [3] proved the validity of this conjecture using variational methods. In this paper we give a simple proof of their result. Recently Doksum [4] has shown that the Savage test [11] maximizes the minimum asymptotic power for testing for scale change over the family of distributions with increasing failure rate averages (IFRA) [2]. The technique of our proof enables us to obtain a lower bound for the asymptotic power of Savage's test for scale change of any positive random variable possessing a finite second moment. When the positive random variables are restricted to be IFRA, Doksum's [4] results follow.
TL;DR: It is shown that the non-linearities of the lower-bound expressions can be exploited by choosing `symmetric sum' operators to optimize the lower bound.
Abstract: A formula is presented giving a rigorous lower bound to the true quantum-mechanical expectation value of a positive operator F >or= 0 in terms of some approximate wave function . This bound is compared with previous results of Bazley and Fox and of Jennings and Wilson as well as with a much more general lower-bound expression which is often applicable even when matrix elements of F2 do not exist. It is shown that the non-linearities of the lower-bound expressions can be exploited by choosing `symmetric sum' operators to optimize the lower bound. As an illustrative application lower bounds are calculated for various powers of the nuclear-electronic and interelectronic distances r1, r12 in the normal helium atom using a simple screened hydrogenic approximation.
TL;DR: If in addition one knows an accurate energy for the ground state, then it is shown that one can construct a sequence of upper and lower bounds to the overlap between the approximate function and the true (but unknown) ground‐state wavefunction.
Abstract: A sequence of variational principles for upper and lower bounds to the energy of the ground state of a quantum‐mechanical system is constructed. The first approximations in the sequence are simply the Ritz upper bound and the Temple lower bound. Higher approximations give improved upper and lower bounds, at the cost of computing additional integrals of the form 〈φ | Hm | φ〉 over the approximate wavefunction. As in the Temple formula, the lower bounds to the ground‐state energy also depend on a knowledge of the energy of the first excited state. If in addition one knows an accurate energy for the ground state (say from experiment), then it is shown that one can construct a sequence of upper and lower bounds to the overlap between the approximate function and the true (but unknown) ground‐state wavefunction. The first approximation in this sequence is the Eckart lower bound to this overlap, and the higher approximations provide improved upper and lower bounds to the overlap.
TL;DR: In this paper, a method of determining functions of the sample data which can be used to provide lower tolerance bounds of given confidence or a service life with specified assurance of no failures among a given number of items to be produced is presented.
Abstract: If the distribution of life length is a member of a specified subset of distributions which have increasing failure rates, we find a method of determining functions of the sample data which can be used to provide lower tolerance bounds of given confidence or a service life with specified assurance of no failures among a given number of items to be produced. The method of finding such functions of the observations is shown to be optimal in a sense and the calculation of a lower bound for the probability of no failure in the future production is carried out when such functions are used. The confidence in the tolerance bound and the assurance of no failure in a production lot of specified size are compared when using bounds obtained from these functions.
TL;DR: An upper bound on the maximum attainable minimum distance for a k -ary code for a certain class of distance functions is obtained, which includes the α th power of the Lee distance.
Abstract: We obtain an upper bound on the maximum attainable minimum distance for a k -ary code for a certain class of distance functions. This class includes the α th power of the Lee distance (0 α ≤ 1).
TL;DR: In this paper, upper and lower bounds for the effective conductivity of composite materials are derived for several simple geometrical arrangements and for certain limiting cases, for two-phase composite materials.
Abstract: Previously derived upper and lower bound expressions for the effective conductivity of composite materials are applied to two‐phase materials. Results are presented for several simple geometrical arrangements and for certain limiting cases.