Showing papers on "Upper and lower bounds published in 1970"

On Excess Over the Boundary

Abstract: A random walk, {Sn}∞n = 0 , having positive drift and starting at the origin, is stopped the first time Sn > t ≧ 0. The present paper studies the "excess," Sn - t, when the walk is stopped. The main result is an upper bound on the mean of the excess, uniform in t. Through Wald's equation, this gives an upper bound on the mean stopping time, as well as upper bounds on the average sample numbers of sequential probability ratio tests. The same elementary approach yields simple upper bounds on the moments and tail probabilities of residual and spent waiting times of renewal processes.

The lower bound conjecture for 3- and 4-manifolds

Abstract: For any closed connected d-manifold M let ](M) denote the set of vectors / (K)= (]0(K) ..... fd(K)), where K ranges over all triangulations of M and ]k(K) denotes the number of k-simplices of K. The principal results of this paper are Theorems 1 through 5 below, which, together with the Dehn-Sommerville equations reviewed in w 2, yield a characterization of ](M) for some of the simpler 3and 4-manifolds. The results for the 3and 4spheres given in Theorems 1 and 5 have immediate and obvious implications for simplicial polytopes, i.e., closed bounded convex polyhedra all of whose proper faces are simplices. In particular they provide a strong affirmative resolution in dimensions 4 and 5 of the socalled lower bound conjecture for simplicial polytopes. For a discussion of this conjecture, which in dimension 4 goes back at least to a paper by Briiclmer in 1909, and some limited results in higher dimensions the reader is referred to Section 10.3 of Griinbaum's book on polytopes [2]. Theorem 3, which is concerned with triangulations of projective 3-space, also has an immediate implication for a special subclass of the centrally symmetric simplicial polytopes. This result is stated as Theorem 6. Some special classes ~/d(n), d >~ 1, n/>0, of abstract simplicial complexes figure in the statement and proof of these theorems. For d >~ 2 each class ~,td(n) consists of certain especially simple triangulations of a class of closed d-manifolds which might be described as d-spheres with n orientable or nonorientable handles. The classes ~/d(n) may be defined inductively as follows:

Block codes for a class of constrained noiseless channels

Abstract: A class of discrete noiseless channels having upper and lower bounds on the separation between adjacent nonzero input symbols is considered. Recursion relations are derived for determining the number of input sequences which satisfy the constraints for all block lengths, and the asymptotic information rate is calculated. Applications to compaction and synchronization are discussed. An optimal algebraic block coding scheme for such channels is developed.

Adaptive delta modulation with a one-bit memory

Abstract: We propose a delta modulator which, at every sampling instant r, adapts its step-size (for a staircase approximation to the input signal) on the basis of a comparison between the two latest channel symbols, C r and C r-1 Specifically, the ratio of the modified step-size m r to the previous step size m r-1 is either +P or — Q depending on whether C r and C r-1 are equal or not (We recall that, in delta modulation, C r represents the polarity of the difference, at the sampling instant r, between the input signal X r and the latest staircase approximation to it, Y r-1 ) A simulation of the delta modulator with a band-limited speech input has revealed that PQ = 1 and P ⋍ 15 represent optimal adaptation characteristics, on the basis of signal-to-error ratios, over an important range of sampling frequencies; and that at 60 kHz, delta modulation with these adaptation parameters compares favorably with 7-bit logarithmic PCM, which reproduces speech with good telephone quality We present several graphical results from this simulation, and include an evaluation of the effect of independent channel errors on the adaptive delta modulator We proceed to suggest a heuristic theory of the delta modulator which explains the optimality of the condition PQ = 1, and develops an upper bound of 2 for the optimum value of P We conclude with a summary of results from a video simulation which revealed that aforementioned optima for P and Q apply to a video signal

Sequence-state methods for run-length-limited coding

Abstract: Methods are presented for the encoding of information into binary sequences in which the number of ZEROS occurring between each pair of successive ONES has both an upper and a lower bound. The techniques, based on the state structure of the constraints, permit the construction of short, efficient codes with favorable error-propagation-limiting properties.

Information rates of autoregressive processes

Abstract: The rate distortion function R(D) is calculated for two time-discrete autoregressive sources--the time-discrete Gaussian autoregressive source with a mean-square-error fidelity criterion and the binary-symmetric first-order Markov source with an average probability-of-error per bit fidelity criterion. In both cases it is shown that R(D) is bounded below by the rate distortion function of the independent-letter identically distributed sequence that generates the autoregressive source. This lower bound is shown to hold with equality for a nonzero region of small average distortion. The positive coding theorem is proved for the possibly nonstationary Gaussian autoregressive source with a constraint on the parameters. Finally, it is shown that the rate distortion function of any time-discrete autoregressive source with a difference distortion measure can be bounded below by the rate distortion function of the independent-letter identically distributed generating sequence with the same distortion measure.

A new approach for evaluating the error probability in the presence of intersymbol interference and additive Gaussian noise

Abstract: The determination of the error probability of a data transmission system in the presence of intersymbol interference and additive gaussian noise is a major goal in the analysis of such systems. The exhaustive method for finding the error probability calculates all the possible states of the received signal using an N-sample approximation of the true channel impulse response. This method is too time-consuming because the computation involved grows exponentially with N. The worst-case sequence bound avoids the lengthy computation problem but is generally too loose. In this paper, we have developed a new method∗ which yields the error probability in terms of the first 2k moments of the intersymbol interference. A recurrence relation for the moments is derived. Therefore, a good approximation to the error probability of the true channel can be obtained by choosing N large enough, and the amount of computation involved increases only linearly with N. The series expansion is shown to be absolutely convergent, and an upper bound on the series truncation error is given. In order to show the improvement provided in this new method, it is compared with the Chernoff bound technique in three representative cases. An order of magnitude improvement in accuracy is obtained.

Slowly varying discrete system xi+1=Aixi

Abstract: If all eigenvalues of all matrices Ai (i=0, 1, 2, ...) lie in a disc of radius <1 and if the matrices Ai, vary sufficiently slowly, then the system xi+1=Aixi is exponentially stable. An explicit upper bound on the variation rate is given.

A Reliability Bound for Systems of Maintained, Interdependent Components

Abstract: In this article the minimal cut lower bound on the reliability of a coherent system, derived in Esary-Proschan [6] for the case of independent components not subject to maintenance, is shown to hold under a variety of component maintenance policies and in several typical cases of component dependence. As an example, the lower bound is obtained for the reliability of a “two out of three” system in which each component has an exponential life length and an exponential repair time. The lower bound is compared numerically with the exact system reliability; for realistic combinations of failure rate, repair rate, and mission time, the discrepancy is quite small.

Performance limitations and error calculations for parameter estimation

Abstract: Error calculations cannot be carried out precisely when parameters are estimated which affect the observation nonlinearly. This paper summarizes the available approaches to studying performance and compares the resulting answers for a specific case. It is shown that the familiar Cramer-Rao lower bound on rms error yields an accurate answer only for large signal-to-noise ratios (SNR). For low SNR, lower bounds on rms error obtained by Ziv and Zakai give easily calculated and fairly tight answers. Rate distortion theory gives a lower bound on the error achievable with any system. The Barankin lower bound does not appear to give useful information as a computational tool. A technique for approximating the error can be used effectively for a large class of systems. With numerical integration, an upper bound obtained by Seidman gives a fairly tight answer. Recent work by Ziv gives bounds on the bias of estimators but, in general, these appear to be rather weak. Tighter results are obtained for maximum-likelihood estimators with certain symmetry conditions. Applying these techniques makes it possible to locate the threshold level to within a few decibels of channel signal-to-noise ratio. Further, these calculations can be easily carried out for any system.

Upper bounds on free energies in terms of hard-sphere results

Abstract: The Gibbs-Bogoliubov inequality is used to develop a first-order perturbation theory that provides an upper bound on the free energy. Charged systems as well as a system of Lennard-Jones particles are discussed, and detailed numerical estimates of the bounds are presented.

Plane Stress Limit Analysis by Finite Elements

Abstract: A numerical method is presented for the determination of lower bounds on the yield-point load of plane stress problems. In this method, a finite element technique is used to construct a parametric family of piecewise quadratic, equilibrium stress fields. The best lower bound is then found by maximizing the load, subject to the yield constraints, by means of the sequential unconstrained minimization technique. Because all conditions of the Lower Bound theorem are met exactly, the resulting solutions are true bounds. Results are given for square slabs with various cutouts and compared to upper bounds and complete elastic plastic finite element solutions.

Criteria of Accuracy of Approximate Wavefunctions

Abstract: The accuracy with which a trial function φ approximates the true wavefunction ψ is quantitatively assessed by the overlap integral S = 〈φ | ψ〉. Upper and lower bounds to S therefore furnish direct criteria of accuracy of the approximation φ and also of the associated physical properties. The available literature on overlap estimates is assembled and critically discussed from a unified point of view, based upon a method of determinantal inequalities. In particular, the relationships among the various approaches are pointed up, several results are extended or generalized, and some new results are obtained. Finally, the various formulas are illustrated by numerical applications to some simple soluble problems.

On the Addition of Binary Numbers

Abstract: An upper bound is derived for the time required to add numbers modulo 2n, using circuit elements with a limited fan-in and unit delay, and assuming that all numbers have the usual binary encoding. The upper bound is within a factor (1 + e) of Winograd's lower bound (which holds for all encodings), where e→0 as n→∞, and only O(n log n) circuit elements are required.

Bounds for some equilibrium properties of an electron gas.

Abstract: Upper and lower bounds are derived for the average potential energy and Helmholtz free energy of an electron gas with uniform positive background. In the ground-state limit, upper and lower bounds are given for the average kinetic energy, average potential energy, and total ground-state energy. Inequalities are derived for the static form factor $S(k)$ and wave-number-dependent dielectric function $\ensuremath{\epsilon}(k, 0)$, making use of exact sum rules for the Fourier-transformed density-density commutator and of the assumption that $S(k)\ensuremath{\le}1$. Comparison is made with the exact behavior of these quantities for small $k$. The sum rules are used to construct an approximate nonlinear integral equation for the ground-state static form factor of the electron gas.

Upper bound on the efficiency of dc-constrained codes

Abstract: We derive the limiting efficiencies of dc-constrained codes. Given bounds on the running digital sum (RDS), the best possible coding efficiency η, for a K-ary transmission alphabet, is η = log 2 λ max /log 2 K, where λ max is the largest eigenvalue of a matrix which represents the transitions of the allowable states of RDS. Numerical results are presented for the three special cases of binary, ternary and quaternary alphabets.

Inequalities for van der Waals Force Constants and Quantum Mechanical Sums

Abstract: Quantum mechanical oscillator strength sums S(k) are used to find rigorous upper and lower bounds to van der Waals coefficients for two‐ and three‐body interactions and to the sums themselves. It is shown that any two sums of any multipole order give a bound to any other sum and either a bound or an estimate for the like force constant for that multipole. An average energy function is defined in terms of the sums; a particular limiting case of this function gives an excellent approximation for force constants (rms deviation for 19 cases: 0.25%) and depends solely on the value and slope of S(k) at k = − 2. The results are tested on data for 18 dipole cases: H, He, Ne, Ar, Kr, Xe, Li, Na, K, Rb, Cs, Hg, H2, N2, CH4, He(21S), He(23S), He+; and 1 quadrupole case: H.

Error Performance of Multiphase DPSK with Noise and Interference

Abstract: This paper is a sequel to a previous paper which considered the error performance of binary DPSK transmission for the case of noise and general cochannel interferences. The present work extends that analysis to include multiphase signaling. Whereas the binary analysis produced exact expressions for the probability of error, the extension to M -ary DPSK considered here leads to upper and lower bounds to the symbol error probability. The bounds have a ratio of 2. However, by a heuristic argument it is possible to estimate the proximity of the error probability to one of the bounds, and so the uncertainty in the approximation is small enough to make the results quite usable for systems design. Computations are displayed for quaternary ( M = 4 ) single difference (conventional) DPSK.

Error bounds for polynomial spline interpolation

Abstract: New upper and lower bounds for the L2 and L- norms of derivatives of the error in polynomial spline interpolation are derived. These results improve corresponding results of Ahlberg, Nilson, and Walsh, cf. (1), and Schultz and Varga, cf. (5). 1. Introduction. In this paper, we derive new bounds for the L2 and LX norms of derivatives of the error in polynomial spline interpolation. These bounds improve and generalize the known error bounds, cf. (1) and (5), in the following important ways: (1) these bounds can be explicitly calculated and are not merely asymptotic error bounds such as those given in (1) and (5); (2) explicit lower bounds are given for the error for a class of functions; (3) the degree of regularity required of the func- tion, f, being interpolated is extended, i.e., in L1) and (5) we demand that the mth or 2mth derivative of f be in L2, if we are interpolating by splines of degree 2m - 1, while here we demand only that some pth derivative of f, where m ? p ? 2m, be in L2; and (4) bounds are given for high-order derivatives of the interpolation errors. 2. Notations. Let - o < a < b < o and for each positive integer, m, let Km(a, b) denote the collection of all real-valued functions u(x) defined on (a, b) such that u E Cm'-(a, b) and such that Dm-lu is absolutely continuous, with Dmu E L2 (a, b), where Du _ du/dx denotes the derivative of u. For each nonnegative integer, M,

Locating New Facilities with Respect to Existing Facilities

Abstract: An efficient algorithm is presented for optimally locating new facilities with respect to existing facilities when movements between facilities are rectilinear. A cost-minimization objective is assumed. The algorithm has been programmed for a digital computer and large problems (500 existing and 100 new facilities) have been solved that illustrate the efficiency of the algorithm. An example problem is presented. Upper and lower bounds on the optimal value for direct (straight-line) movements are derived. Experience has shown that these bounds are relatively close.

A Branch-and-Bound Algorithm for Flow Shop Scheduling Problems

Abstract: This article is concerned with the solution of the flow shop scheduling problem in which all jobs have the same machine ordering. A branch-and-bound algorithm is developed for finding the sequence of J jobs to be processed on M machines which minimizes the schedule time. Thib algorithm consists of branching and bounding processes, but without the backtracking process which guarantees optimality. The procedure employed is that in constructing a subset of feasible sequences, a node representing a partial sequence is branched. Selection of the node depends on the lower-bound concept as a decision rule. This lower bound is based on resolving the conflict of jobs on the last machine. By using this algorithm, the number of explored nodes is considerably reduced and, hence, the computational effort involved in obtaining an optimal or near-optimal solution is decreased. High quality of solutions is obtained. Computationally, this algorithm extends the size of problems that can reasonably be solved.

Jacobi’s bound for the order of systems of first order differential equations

Abstract: Let Au , An be a system of differential polynomials in the differential indeterminates ya\ , yw, and let ^ be an irreducible component of the differential variety -ef(Ai,, A") If dim ~# = 0, there arises the question of securing an upper bound for the order of ~# in terms of the orders rii of the polynomials Ai in yw It has been conjectured that the Jacobi number

A Generalized Machine-Scheduling Algorithm

Abstract: It is proposed to treat a general machine-job-scheduling problem using a branch-and-bound method. Here the general problem is that in which the routing of any job through the machines is specified in advance but is independent of the routing of any other job. In addition there is no requirement for the job to visit all machines. Since any two operations to be performed on the same machine cannot be performed simultaneously, the set of all schedules can be divided into two subsets-one in which the pair of operations is performed in one order and the second in which the order is reversed. This division corresponds to branching in the method. For each of the new subsets formed by branching a lower bound on the duration of all schedules in the subset is calculated and the schedule with minimum duration is found by successive subdivision. An illustrative example is solved and the method is compared to other published general methods.

Approximations to System Reliability Using a Modular Decomposition

Abstract: Since the computation of system reliability is often a difficult task, approximation procedures are needed. Esary and Proschan have developed a procedure for finding a lower bound estimate of the system reliability of a coherent structure. In this paper, a new lower bound procedure based on the modular decomposition of a coherent structure is proposed. It is shown that this procedure provides a sharper lower bound estimate of the system reliability of a coherent structure than the Esary Proschan procedure and is computationally more efficient.

Effective Permittivity of a Polycrystalline Dielectric

Abstract: We use statistical variational principles to determine upper and lower bounds for the effective permittivity of a polycrystalline dielectric. We indicate how to derive bounds containing permittivity correlation functions of arbitrary order, and we obtain explicit expressions for bounds depending on one‐ and two‐point correlation functions and for bounds containing one‐, two‐, and three‐point correlation functions. We prove that for two classes of polycrystal, the effective permittivity may be exactly determined, and we use these exact expressions to show that we have obtained the best possible upper and lower bounds.

Note---The Distribution Problem with Upper and Lower Bounds on the Node Requirements

Abstract: This paper considers a distribution model with upper and lower bounds on the number of units shipped from an origin or to a destination. Our problem differs from the classical distribution model in which the node shipping amounts are, by contrast, specified exactly. This generalization of the classical distribution problem not only makes the model more versatile from a theoretical standpoint but also makes the model more usable from an applications viewpoint.