Showing papers on "Upper and lower bounds published in 1969"

On Howard's upper bound for heat transport by turbulent convection

Abstract: The variational problem introduced by Howard (1963) for the derivation of an upper bound on heat transport by convection in a layer heated from below is analyzed for the case in which the equation of continuity is added as constraint for the velocity field. Howard's conjecture that the maximizing solution of the Euler equations is characterized by a single horizontal wave-number is shown to be true only for a limited range of the Rayleigh number, Ra. A new class of solutions with a multiple boundary-layer structure is derived. The upper bound for the Nusselt number, Nu, given by these solutions is Nu ≤ (Ra/1035)½ in the limit when the Rayleigh number tends to infinity. The comparison of the maximizing solution with experimental observations by Malkus (1954a) and Deardorff & Willis (1967) emphasizes the similarity pointed out by Howard.

A Fixed-point Fast Fourier Transf orrn Error Analysis

Abstract: This paper contains an analysis of the fixed-point accuracy of the sequence power of two, fast Fourier transform algorithm. This analysis leads to approximate upper and lower bounds on the root-mean-square error. Also included are the results of some accuracy experiments on a simj=O (1) ulated fixed-point machine and their comparison with the error upper n = 0, 1, * ' * , N - 1.

A general formulation of linear feedback communication systems with solutions

Abstract: The feedback coding problem for additive noise systems, in which the noise may be colored, nonstationary, and correlated between channels, is formulated in terms of arbitrary linear operations at the transmitting and receiving points. This rather general linear formulation provides a unified approach for deriving new results, as well as previous results obtained under more restrictive assumptions, in a straightforward manner. Thus the sequential form of the optimum linear feedback code with an average power constraint on the transmitter is derived for noiseless feedback but forward noise of arbitrary covariance. It is shown explicitly that noiseless feedback increases the capacity of a channel with colored noise. The noisy feedback problem is considered and upper and lower bounds on the performance presented.

An Algorithm to Determine the Reliability of a Complex System

Abstract: The method most often suggested for determining the reliability of a system is to construct a reliability network, enumerate from the network all mutually exclusive working states of the system, calculate the probability of occurrence of each working state, and sum these probabilities. For a complex system this is not a practical method for there is a very large number of working states. Esary and Proschan suggest a lower bound approximation to reliability that requires the enumeration of a much smaller set of system states. These states are called minimal cuts. An algorithm is presented to determine the set of minimal cuts and thus calculate a lower bound to system reliability. The algorithm is intended for digital-computer implementation and computational times are provided.

On Finding the Maximal Gain for Markov Decision Processes

Abstract: The method of successive approximations for solving problems on single-chain Markovian decision processes has been investigated by White and Schweitzer. This paper shows that White's scheme not only converges, but also can be modified so that monotonic upper and lower bounds on the maximal gain can be obtained.

A useful form of the Barankin lower bound and its application to PPM threshold analysis

Abstract: A form of Barankin's greatest lower bound on estimation error [7] is obtained, which is easy to compute and easy to interpret. This gives a lower bound on estimation error for non-linear modulation systems in an additive Gaussian noise back-ground. Threshold effects are included. This bound is applied to a set of pulse-position modulation waveforms designed to reduce threshold effects. It is shown that these signals do, in fact, offer reduced threshold levels (e.g., \approx 3.5 dB) with very small ( \approx \frac{1}{2} dB) degradation in large signal performance.

Intersymbol interference and probability of error in digital systems

Abstract: Intersymbol interference and additive Gaussian noise are two important sources of distortion in digital systems, and a principal goal in the analysis of such systems is the determination of the resulting probability of error Earlier related work has sought to estimate the error probability either by calculating an approximation based upon a truncated version of the random pulse train or by obtaining an upper bound which results from consideration of the worst case intersymbol interference In this paper a new upper bound is derived for the probability of error which is computationally simpler than the truncated pulse-train approximation and which never exceeds the worst case bound Moreover, the new bound is applicable in a number of cases where the worst case bound cannot be used The bound is readily evaluated and depends upon three parameters: the usual signal-to-noise ratio; the ratio of intersymbol interference power to total distortion power; and the ratio of the maximum intersymbol interference amplitude to its rms value To illustrate the utility of the bound, it is compared with the earlier methods in three cases which are representative of the most important situations occurring in practice

Decaying modes for the wave equation in the exterior of an obstacle

Abstract: In this paper we study the dependence of the set of ‘exterior’ eigenvalues {λk} of Δ on the geometry of the obstacle . In particular we show that the real eigenvalues, corresponding to purely decaying modes, depend monotonically on the obstacle , both for the Dirichlet and Neumann boundary conditions. From this we deduce, by comparison with spheres—for which the eigenvalues {λk} can be determined as roots of special functions—upper and lower bounds for the density of the real {λk}, and upper and lower bounds for λ1, the rate of decay of the fundamental real decaying mode. We also consider the wave equation with a positive potential and establish an analogous monotonicity theorem for such problems. We obtain a second proof for the above Dirichlet problem in the limit as the potential becomes infinite on . Finally we derive an integral equation for the decaying modes; this equation bears strong resemblance to one appearing in the transport theory of mono-energetic neutrons in homogeneous media, and can be used to demonstrate the existence of infinitely many modes.

Improved Variational Bounds on Some Bulk Properties of a Two‐Phase Random Medium

Abstract: The Hashin–Shtrikman bounds on the bulk magnetic permeability (or dielectric constant, thermal and electric conductivity, solute diffusion coefficient) for a random two‐phase material can be considerably improved by the inclusion of experimentally accessible information, such as the bulk permeability of the material at a different temperature [the main results are expressed in the inequalities (27), (45), and (51)]. These inequalities can also be written as bounds on the rate at which the bulk permeability changes with changing permeabilities of the individual phases [(29) and (30)]. In particular, application of (30) yields the upper and lower bounds (59) on the observed energy of activation for diffusion through a two‐phase medium. The inequalities (27), (45), and (51) can be expressed in a mixed form to bound, for example, the bulk permeability through use of data on the thermal conductivities of the mixed material and the individual phases. Bounds on the viscosity and elastic moduli of a two‐phase medium are also discussed briefly.

Roundoff noise in floating point fast Fourier transform computation

Abstract: A statistical model for roundoff errors is used to predict output noise-to-signal ratio when a fast Fourier transform is computed using floating point arithmetic. The result, derived for the case of white input signal, is that the ratio of mean-squared output noise to mean-squared output signal varies essentially as u = \log_{2}N where N is the number of points transformed. This predicted result is significantly lower than bounds previously derived on mean-squared output noise-to-signal ratio, which are proportional to ν2. The predictions are verified experimentally, with excellent agreement. The model applies to rounded arithmetic, and it is found experimentally that if one truncates, rather than rounds, the results of floating point additions and multiplications, the output noise increases significantly (for a given ν). Also, for truncation, a greater than linear increase with ν of the output noise-to-signal ratio is observed; the empirical results seem to be proportional to ν2, rather than to ν.

A class of upper bounds on probability of error for multihypotheses pattern recognition (Corresp.)

Abstract: A class of upper bounds on the probability of error for the general multihypotheses pattern recognition problem is obtained. In particular, an upper bound in the class is shown to be a linear functional of the pairwise Bhattacharya coefficients. Evaluation of the bounds requires knowledge of a priori probabilities and of the hypothesis-conditional probability density functions. A further bound is obtained that is independent of a priori probabilities. For the case of unknown a priori probabilities and conditional probability densities, an estimate of the latter upper bound is derived using a sequence of classified samples and Kernel functions to estimate the unknown densities.

An upper bound on moments of sequential decoding effort

Abstract: In this paper we derive an infinite tree code ensemble upper bound on the u th ( u \leq 1) moments of the computational effort connected with sequential decoding governed by the Fano \footnote[1]{algorithm}. It is shown that the u th moment of the effort per decoded branch is hounded by a constant, provided the transmission rate R_{0} satisfies inequality (2), This result, although often conjectured, has previously been shown to hold only for positive integral values of u . For a wide class of discrete memoryless channels (that includes all symmetric channels), our bounds agree qualitatively with the lower bounds of Jacobs and Berlekamp [8].

Lower Bounds for Sonic Booms in the Midfield

Abstract: This paper presents new lower bounds for sonic booms based on minimizing either the overpressure or the shock strength of the positive part of the boom wave. The results are not restricted to the limit of large distance from the aircraft. They reduce to the Jones lower bound infinitely far from the aircraft in a uniform atmosphere but they give reductions from the Jones lower bound of the order of 50% for planned supersonic transport conditions. This is because the asymptotic results of Jones are approached very slowly (as r~1/4) in a uniform atmosphere and are never reached in the real atmosphere. The direct application of these results depends upon additionally keeping the tail shock intensity less than or equal to that of the front shock.

Probabilistic treatment of the blowing up of solutions for a nonlinear integral equation

Abstract: depends on the dimension d and power /, where G is the infinitesimal generator of a linear nonnegative contraction semigroup on the space B(Rd) of bounded measurable functions on Rd and c is a bounded nonnegative measurable function on Rd. This fact was recently proved by Fujita [2] when G is the Laplacian operator. In this paper we will give upper and lower bounds for the solution of (1.1) constructed by a probabilistic method (cf. (3.4) and (4.7)). As a corollary we shall obtain Fujita's result when G is a fractional power -(-A)", 0< a <2, of the Laplacian operator. Our method is based on probabilistic arguments relating to the branching Markov processes (cf. Ikeda-Nagasawa-Watanabe [3], Sirao [8] and Nagasawa [7]). The necessary facts of probabilistic arguments in this context will be summarized in ?2, while in ?3 and ?4 we shall give upper and lower bounds of the probabilistic solution of (1.1) and some applications.

The upper bound theorem for rigid/plastic solids generalized to include Coulomb friction

Abstract: The usual definition of a kinematically admissible velocity field is unnecessarily restrictive for the validity of the upper bound inequality of limit analysis. In consequence, it is shown that this inequality can be used to derive overestimates of quantities of interest in metal-forming processes which have not been considered hitherto. In particular, upper bounds can be obtained for the total load in the presence of Coulomb friction on the tool/workpiece interface. In illustration, the technique is applied to problems of compression and extrusion.

Upper and Lower Bounds on Transport Coefficients Arising from a Linearized Boltzmann Equation

Abstract: A systematic approach to the construction of upper and lower bounds for transport coefficients is presented for systems described by a linearized Boltzmann equation. The usefulness and efficacy of the bounds is tested by applying them to a particular Boltzmann equation whose solution is known. One easily calculable bound is found to differ from the exact result by less than 0.03%.

Resonant Scattering Theory of Association Reactions and Unimolecular Decomposition. II. Comparison of the Collision Theory and the Absolute Rate Theory

Abstract: The rate expressions derived in the preceding paper using a resonant scattering theory (RST) to describe recombination and unimolecular decomposition are compared with the absolute rate theory (ART). A one‐to‐one correspondence exists between the resonance states in RST and the activated states in ART. The “states of the activated complex” are shown to be the channel states in RST, and using the adiabatic approximation to describe the continuum states it is shown that ART does give a proper upper bound to the rate even when nonadiabatic effects are included in RST, i.e., the mean transmission coefficient κ is equal to or less than one. The collision theory gives explicit expressions for κ(kT,P) which is a function of temperature and includes the dependence on pressure. Specific expressions are given for the “tight complex,” where the “activated complex” occurs at some distorted region in configuration space, and for the “loose complex,” where the activated complex is the rotational barrier in the asymptotic channel states. Particular attention is given to the high‐pressure rate constant where the specific transmission coefficient can be simply related to the ratio of the mean widths to the mean spacings of the activated states. Criteria are given for the validity of ART, and it is shown that Light's statistical theory of reaction rates is a special microcanonical version of the ART for the “loose complex.”

Algorithm and average-value bounds for assignment problems

Abstract: A new suboptimal intermediate-speed algorithm which use n2 In n steps is developed for the assignment problem Upper and lower bounds are derived, using this algorithm and other methods, for the average values of three classes of n × n assignment problems: 1 When the elements of the matrix are random numbers uniformly distributed over the range 0 to 1, the average optimal value is smaller than 237 and larger than 1 for problems with large n Experimentally the value is about 16 2 When the elements of the matrix are random numbers such that the probability of being less than x is xk+1 (k ≠ 0), asymptotic expressions for the upper and lower bounds of the average optimal value are Cknk/(k+1) and Ck[(k+1)/k]nk/(k+1) respectively 3 When each column of the matrix is a random permutation of the integers 1 to n, asymptotic upper and lower bounds are 237n and 154n, respectively Experimentally the value is about 18n

The Determination of Optimum Gate Lengths for Time-Varying Wiener Filtering

Abstract: The response function of a time-varying filter changes with the output signal, or observation time. Most existing time-varying filter techniques involve the empirical division of a seismic trace into a number of gates (or time windows) of given length, and a time-invariant filter is determined for each such gate. Few treatments have dealt with analytical methods to establish the gate lengths according to some optimum criterion.This paper describes a technique for the determination of optimum gate lengths. It is based on the work of Berndt and Cooper, which is here applied to the calculation of time-varying Wiener filters. The Berndt and Cooper technique produces an upper bound for the mean-square error between the true and a given approximated time-varying correlation function. The minimization of this upper bound leads to a relation which enables one to establish gate lengths directly from the input trace. Thereafter, ordinary time-invariant Wiener filters can be computed for each gate. The overall filtered trace is obtained in the form of a suitably combined version of the individually filtered gates.Experimentally it is shown that, with the Berndt and Cooper technique to determine optimum gate lengths, time-varying Wiener filters can be better than a time-invariant filter.

On communication of analog data from a bounded source space

Abstract: We consider the problem of the transmission of discrete-time analog data with a variety of fidelity criteria. The outputs of the analog source are assumed to belong to a bounded set. Bounds on the minimum achievable average distortion for memoryless sources are derived both for the case where the coding delay is infinite (an extension of the Shannon Theory) and also for some cases where the coding delay is finite. Several examples are given, for which the upper and lower bounds coincide. Further, we discuss the case where the assumption of the existence of a probabilistic model for the source is dropped. We adopt as our fidelity criterion the supremum over all possible source-output n-sequences x, of the conditional expectation of the distortion given x (“guaranteed distortion”). The Shannon Theory is not directly applicable in determining the minimum guaranteed distortion. We do obtain results for two important cases. Some generalizations and applications are also discussed.

Optimum Passive Bearing Estimation

Abstract: This paper studies the minimum bearing error attainable with a linear passive array. Signal and noise are stationary Gaussian processes with arbitrary power spectra, and the noise is assumed to be statistically independent from hydrophone to hydrophone. The Cramer‐Rao technique is used to set a lower bound on the rms hearing error and the results are compared with the bearing error of a slightly modified split beam tracker. The latter reaches the lower bound for a two‐element array and comes very close to reaching it for a linear array with an arbitrary number of equally spaced hydrophones. Thus, the modified split beam tracker is very nearly optimal. The dependence of the rms error on signal‐to‐noise ratio is the same for the split beam tracker and for the Cramer‐Rao lower bound. Parts of the results are extended to arrays operating in a noise field containing a directional component. [Work supported by Office of Naval Research through prime contract with General Dynamics/Electric Boat under the SUBIC program.]

Improved Equation for Lower Bounds to Eigenvalues; Bounds for the Second‐Order Perturbation Energy

Abstract: Using the bounding theorems given by Eqs. (1) and (2), an improved equation for lower bounds to eigenvalues is obtained [Eq. (8)]. This improved result is a generalization of an earlier lower‐bound equation given by Gay and another given by Miller. A numerical application to the ground state of the two‐electron atom is presented. Upper and lower bounds for the second‐order perturbation energy are also presented, and a general upper bound for inverse operators is obtained.

An application of rate-distortion theory to a converse to the coding theorem

Abstract: A lower bound to the information rate R(D) for a discrete memoryless source with a fidelity criterion is presented for the case in which the distortion matrix contains the same set of entries, perhaps permuted, in each column. A necessary and sufficient condition for R(D) to equal this bound is given. In particular, if the smallest column element is zero and occurs once in each row, then there is a range of D, 0 \leq D \leq D_{1} , in which equality holds. These results are then applied to the special case of d_{ij}= 1 - \delta_{ij} , for which the average distortion is just the probability of incorrectly reproducing the source output. We show how to construct R(D) for this case, from which one can solve for the minimum achievable probability of error when transmitting over a channel of known capacity.

Some Theorems on Sorting

Abstract: It is a “well-known fact” that a lower bound for the average number of comparisons required to sort a table of N items is $\log _2 N!$, where the average is taken over all possible permutations of the table. In this paper a somewhat better lower bound is obtained, which in a way provides considerable insight into the theoretical limitations on methods of sorting by comparison.

Optimization Procedure for the Analysis of Coherent Structures

Abstract: The determination of the reliability level at which to manufacture the components of a coherent structure so that the system reliability h(p) is at a certain level and the overall system cost is minimized is considered. The cost of utilizing component ci at reliability level pi, Ci(pi), is assumed to be a convex increasing function of pi with a continuous first derivative and Ci'(qi)>0 where qi is the lower bound on the reliability level for component ci. Since for most coherent structures the constraint set defines a nonconvex set, any mathematical programming procedure blindly applied to the program converges to a local optimum rather than a global optimum. However, in certain cases, the global optimum can be found for the series and parallel (SP) type of systems. The key to the solution is to optimize each module separately and then to substitute a component for each module where the cost function for the component is the value of the objective function for the module. As long as the cost function for each module maintains the convexity property with In R or In(1 - R) as the argument (R being the reliability of the module), the optimization procedure can continue and a global optimum found.

A class of upper-bounds on probability of error for multi-hypotheses pattern recognition

Abstract: A class of upper bounds on the probability of error for the general multihypotheses pattern recognition problem is obtained. In particular, an upper bound in the class is shown to be a linear functional of the pairwise Bhattacharya coefficients. Evaluation of the bounds requires knowledge of a-priori probabilities and of the hypothesis-conditional probability density functions. A further bound is obtained that is independent of apriori probabilities. For the case of unknown apriori probabilities and conditional probability densities, an estimate of the latter upper bound is derived using a sequence of classified samples and Kernel functions to estimate the unknown densities.