Showing papers on "Constant (mathematics) published in 1996"

Logarithmic sobolev inequalities for finite markov chains

Abstract: This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a self-contained development. Examples of applications include the study of a Metropolis chain for the binomial distribution, sharp results for natural chains on the box of side n in d dimensions and improved rates for exclusion processes. We also show that for most r-regular graphs the log-Sobolev constant is of smaller order than the spectral gap. The log-Sobolev constant of the asymmetric two-point space is computed exactly as well as the log-Sobolev constant of the complete graph on n points.

An analytical constant modulus algorithm

Abstract: Iterative constant modulus algorithms such as Godard (1980) and CMA have been used to blindly separate a superposition of cochannel constant modulus (CM) signals impinging on an antenna array. These algorithms have certain deficiencies in the context of convergence to local minima and the retrieval of all individual CM signals that are present in the channel. We show that the underlying constant modulus factorization problem is, in fact, a generalized eigenvalue problem, and may be solved via a simultaneous diagonalization of a set of matrices. With this new analytical approach, it is possible to detect the number of CM signals present in the channel, and to retrieve all of them exactly, rejecting other, non-CM signals. Only a modest amount of samples is required. The algorithm is robust in the presence of noise and is tested on measured data collected from an experimental set-up.

Blue screen matting

Abstract: A classical problem of imaging—the matting problem—is separation of a non-rectangular foreground image from a (usually) rectangular background image—for example, in a film frame, extraction of an actor from a background scene to allow substitution of a different background. Of the several attacks on this difficult and persistent problem, we discuss here only the special case of separating a desired foreground image from a background of a constant, or almost constant, backing color. This backing color has often been blue, so the problem, and its solution, have been called blue screen matting. However, other backing colors, such as yellow or (increasingly) green, have also been used, so we often generalize to constant color matting. The mathematics of constant color matting is presented and proven to be unsolvable as generally practiced. This, of course, flies in the face of the fact that the technique is commonly used in film and video, so we demonstrate constraints on the general problem that lead to solutions, or at least significantly prune the search space of solutions. We shall also demonstrate that an algorithmic solution is possible by allowing the foreground object to be shot against two constant backing colors—in fact, against two completely arbitrary backings so long as they differ everywhere.

Surface-Layer Fluxes, Profiles, and Turbulence Measurements over Uniform Terrain under Near-Neutral Conditions

Abstract: An atmospheric surface-layer experiment over a nearly uniform plowed field was performed to determine the constants in the flux-profile similarity formulas, particularly the von Karman constant. New instruments were constructed to minimize flow distortion effects on the turbulence measurements and to provide high-resolution gradient measurements. In addition, a hot-wire anemometer directly measured the turbulent kinetic energy dissipation rate. An average value of the von Karman constant of 0.365 ± 0.015 was obtained from 91 runs (31 h) in near-neutral stability conditions. However, four near-neutral runs when snow covered the ground gave an average value of 0.42. This result suggests that the von Karman constant depends on the roughness Reynolds number, which may resolve some of the differences in previous determinations over different surfaces. The one-dimensional Kolmogorov inertial subrange constant was found to have a value of 0.54 ± 0.03, slightly larger than previous results. The flux-prof...

Euclidean reconstruction from constant intrinsic parameters

Abstract: A new method for Euclidean reconstruction from sequences of images taken by uncalibrated cameras, with constant intrinsic parameters, is described. Our approach leads to a variant of the so called Kruppa equations. It is shown that it is possible to calculate the intrinsic parameters as well as the Euclidean reconstruction from at least three images. The novelty of our approach is that we build our calculation on a projective reconstruction obtained without the assumption on constant intrinsic parameters. This assumption simplifies the analysis, because a projective reconstruction is already obtained and we need "only" to find the correct Euclidean reconstruction among all possible projective reconstructions.

Testing for a change of the long-memory parameter

Abstract: SUMMARY Long-range dependence is often observed in long time series Correlations decay approximately like Ik I2"-2, with H E (0 5, 1), as the lag k tends to infinity The long-term features of the data are essentially characterised by the parameter H Small changes of H have strong implications for the long-term behaviour of the process In particular, rates of convergence of estimators for the mean, and for many other parameters of interest, differ for different values of H For some data sets, H appears to change with time In this paper we consider a simple test of the null hypothesis that H is constant The test is based on a functional central limit theorem for quadratic forms Critical values for the test statistic are given Simulations confirm the validity of the test A data example illustrates its practical application

Estimation of the Lipschitz constant of a function

Abstract: A number of global optimisation algorithms rely on the value of the Lipschitz constant of the objective function. In this paper we present a stochastic method for estimating the Lipschitz constant. We show that the largest slope in a fixed size sample of slopes has an approximate Reverse Weibull distribution. Such a distribution is fitted to the largest slopes and the location parameter used as an estimator of the Lipschitz constant. Numerical results are presented.

Asymptotic regulation of minimum phase nonlinear systems using output feedback

Abstract: We consider a single-input/single-output (SISO) nonlinear system which has a well-defined normal form with asymptotically stable zero dynamics. We allow the system's equation to depend on constant uncertain parameters and disturbance inputs which do not change the relative degree. Our goal is to design an output feedback controller which regulates the output to a constant reference. The integral of the regulation error is augmented to the system equation, and a robust output feedback controller is designed to bring the state of the closed-loop system to a positively invariant set. Once inside this set, the trajectories approach a unique equilibrium point at which the regulation error is zero. We give regional as well as semiglobal results.

Application of the Nosé−Hoover Chain Algorithm to the Study of Protein Dynamics

Abstract: We have implemented the Nose−Hoover chain (NHC) method combined with an explicit reversible integrator into the MD module of AMBER (i.e., SANDER) to study the dynamics and the structure of biopolymers. We have implemented both constant temperature (NVT) and constant temperature and pressure (NTP) methods. We have studied the structure and dynamics of the antifreeze protein (AFP) in the gas phase and when solvated by water. Single and multiple chains of thermostats attached to the solute, solvent, and simulation box (in the case of constant-pressure simulations) were examined. The simulation results from constant energy, Berendsen constant temperature and pressure and Nose−Hoover chain NVT and NTP methods indicate all these methods can evolve the system to equilibrium at a comparable rate. The NHC method controls temperature better over the other methods. In particular, separate thermostat chains can eliminate the cold solute−hot solvent problem. For the constant temperature and pressure simulations, the N...

Tracer Dispersion in a Self-Organized Critical System.

Abstract: We have studied experimentally transport properties in a slowly driven granular system which recently was shown to display self-organized criticality [Frette et al., Nature (London) 379, 49 (1996)]. Tracer particles were added to a pile and their transit times measured. The distribution of transit times is a constant with a crossover to a decaying power law. The average transport velocity decreases with system size. This is due to an increase in the active zone depth with system size. The relaxation processes generate coherently moving regions of grains mixed with convection. This picture is supported by considering transport in a 1D cellular automaton modeling the experiment.

On Unapproximable Versions of NP -Complete Problems

Abstract: We prove that all of Karp's 21 original $NP$-complete problems have a version that is hard to approximate. These versions are obtained from the original problems by adding essentially the same simple constraint. We further show that these problems are absurdly hard to approximate. In fact, no polynomial-time algorithm can even approximate $\log^{(k)}$ of the magnitude of these problems to within any constant factor, where $\logk$ denotes the logarithm iterated $k$ times, unless $NP$ is recognized by slightly superpolynomial randomized machines. We use the same technique to improve the constant $\epsilon$ such that MAX CLIQUE is hard to approximate to within a factor of $n^\epsilon$. Finally, we show that it is even harder to approximate two counting problems: counting the number of satisfying assignments to a monotone 2SAT formula and computing the permanent of $-1$, $0$, $1$ matrices.

Constant mean curvature surfaces via an integrable dynamical system

Abstract: It is shown that the equation which describes constant mean curvature surfaces via the generalized Weierstrass - Enneper induction has Hamiltonian form. Its simplest finite-dimensional reduction is the integrable Hamiltonian system with two degrees of freedom. This finite-dimensional system admits -action and classes of -equivalence of its trajectories are in one-to-one correspondence with different helicoidal constant mean curvature surfaces. Thus the interpretation of well known Delaunay and do Carmo - Dajczer surfaces via an integrable finite-dimensional Hamiltonian system is established.

Time dependent deformation during indentation testing

Abstract: Constant loading rate/load indentation tests (1/P dP/dt) and constant rate of loading followed by constant load (CRL/Hold) indentation creep tests have been conducted on high purity electropolished indium. It is shown that for a material with a constant hardness as a function of depth, a constant (1/P dP/dt) load-time history results in a constant indentation strain rate (1/h dh/dt). The results of the two types of tests are discussed and compared to data in the literature for constant stress tensile tests. The results from the constant (1/P dP/dt) experiments appear to give the best correlation to steady-state uniaxial data.

Analytical Solutions for Two-Dimensional Transport Equation with Time-Dependent Dispersion Coefficients

Abstract: Analytical solutions to advection-dispersion equations are of continuous interest because they present benchmark solutions to problems in hydrogeology, chemical engineering, and fluid mechanics. In this paper, we examine solutions to two-dimensional advection-dispersion equation with time-dependent dispersion coefficients. The time- and space-dependent nature of the dispersion coefficient in subsurface contaminant transport problems has been demonstrated in the literature in both field and laboratory scale studies. Analytical solutions given in this paper could be used to mdoel the transport of solute in hydrogeologic systems characterized by dispersion coefficients that may vary as a function of travel time from the input source. In particular, in this paper we develop instantaneous and continuous point-source solutions for constant, linear, asymptotic, and exponentially varying dispersion coefficients. The relationship between the proposed general solution and the particular solutions given in the relevant literature are discussed. Examples are included to demonstrate the effect of time-dependent dispersion coefficients on solute transport.

Complex dynamics with three oligopolists

Abstract: The adjustment process by three Cournot oligopolists is studied. An iso-elastic demand function and constant marginal costs are assumed. The system can easily result in chaotic behaviour, and a muc ...

Reference Model for Smooth Growth of Software Systems

Abstract: This note presents a model of smooth software system evolution. The model assumes constant effort per release and takes into account the growth of system complexity.

Multiple positive solutions of nonhomogeneous semilinear elliptic equations in ℝN

Abstract: We consider the following problemwhere for all ≦f(x,u)≦c1up-1 + c2u for all x ∈ℝN,u≧0 with c1>0,c2∈(0, 1), 2

Proof of lyapunov exponent pairing for systems at constant kinetic energy

Abstract: We present a proof that a system consisting of any finite number of particles that move under the action of a scalar potential at constant kinetic energy exhibits conjugate pairing of Lyapunov exponents; that is, the Lyapunov exponents come in pairs, which sum to the same constant. This result generalizes previous results, because it is independent of the size of the system. @S1063-651X~96!51206-7#

Ericksen's bar revisited : Energy wiggles

Abstract: This paper addresses the non-uniqueness pointed out by Ericksen in his classical analysis of the equilibrium of a one-dimensional elastic bar with non-convex energy. According to Ericksen, for the bar in a hard device, the piecewise constant functions delivering the global minimum of the energy can have an arbitrary numberN of discontinuities in strain (phase-boundaries). Following some previous work in this area, we regularize the problem in order to resolve this degeneracy. We add two non-local terms to the energy density: one depends on the high (second) derivatives of the displacement, the other contains low (zero) derivatives. The low-derivative term (scaled with a constant β) introduces a strong non-locality, and simulates a three-dimensional interaction with the loading device, forcing the formation of layered microstructures in the process of energy minimization. The high-derivative (strain-gradient) term (scaled with a different constant α), represents a surface energy contribution which penalizes the formation of phase interfaces and prevents the infinite refinement of microstructures. In our description we consider the positions of interfaces as variables. This singles out in a natural way an infinite number of finite-dimensional subspaces, where all the essential nonlinearity is concentrated. In this way we can calculate explicitly the local minimizers (metastable states) and their energy, which turns out to be a multi-valued function of the interface positions and the imposed overall straind. Our approach thus gives an explicit framework for the study of the rich variety of finite-scale equilibrium microstructures for the bar and their stability properties. This allows for the study of a number of properties of phase transitions in solids; in particular their hysteretic behavior. Among our goals is the investigation of the phase diagram of the system, described by the functionN(d, α, β) giving the number of phase-boundaries in the absolute minimizer. We observe the somewhat counterintuitive effect that the energy at the global minimum, as a function of the overall strain, generically develops non-smooth oscillations (wiggles).

Stability criteria for homogeneously stressed materials and the calculation of elastic constants.

Abstract: Second-order elastic constants cannot be applied directly to the study of the mechanical stability of a stressed material. We derive general expressions for stability criteria by constructing appropriate thermodynamic potentials. For a system under isotropic initial stress, elastic stiffness coefficients which govern stress-strain relations can be used as stability criteria. However, for a system under anisotropic initial stress, stability criteria are different from either elastic constants or elastic stiffness coefficients. We show that stability conditions in the constant pressure ensemble are stronger than in the constant volume ensemble, i.e., a state can be stable in the constant volume ensemble but unstable in the constant pressure ensemble. From the general formalism simple fluctuation formulas for the stability criteria are deduced for systems in the canonical ensemble, for arbitrary stress. \textcopyright{} 1996 The American Physical Society.

Near fields of the constant current thin circular loop antenna of arbitrary radius

Abstract: Assuming a known (constant) current distribution on the thin circular loop antenna of arbitrary radius in free space, an exact integration of the vector potential is performed without recourse to approximations. The only restrictions on the solution variables are that the observation point distance must be greater than the loop radius and that the polar angle must run between 0 and /spl pi/. The resulting vector potential infinite series solution possesses a real part composed of linear combinations of complete elliptic integrals of the first and second kind and an imaginary part composed of elementary functions. Thus, it is possible to obtain an exact solution which is valid everywhere that r>a and 0/spl les//spl theta//spl les//spl pi/. The electromagnetic field components of the constant current circular loop antenna are then determined by direct series differentiation. These solutions are valid in the near and induction fields, converging rapidly there, and are also valid in the far field, although many terms of the series are needed for convergence.