Single Real Parameter Research Articles

Another convex combination of product states for the separable Werner state

In this paper, we write down the separable Werner state in a two-qubit system explicitly as a convex combination of product states, which is different from the convex combination obtained by Wootters' method. The Werner state in a two-qubit system has a single real parameter and varies from inseparable to separable according to the value of its parameter. We derive a hidden variable model that is induced by our decomposed form for the separable Werner state. From our explicit form of the convex combination of product states, we understand the following: The critical point of the parameter for separability of the Werner state comes from positivity of local density operators of the qubits.

On radial stochastic Loewner evolution in multiply connected domains

We discuss the extension of radial SLE to multiply connected planar domains. First, we extend Loewner's theory of slit mappings to multiply connected domains by establishing the radial Komatu–Loewner equation, and show that a simple curve from the boundary to the bulk is encoded by a motion on moduli space and a motion on the boundary of the domain. Then, we show that the vector-field describing the motion of the moduli is Lipschitz. We explain why this implies that “consistent,” conformally invariant random simple curves are described by multidimensional diffusions, where one component is a motion on the boundary, and the other component is a motion on moduli space. We argue what the exact form of this diffusion is (up to a single real parameter κ) in order to model boundaries of percolation clusters. Finally, we show that this moduli diffusion leads to random non-self-crossing curves satisfying the locality property if and only if κ = 6 .

ON THE OPTIMAL ESTIMATION OF ERRORS IN VARIABLES MODELS FOR ROBUST CONTROL

There exists a substantial literature dealing with the problem of errors-in-variables identification. It is known, for example, that there is an equivalence class of models that give compatible descriptions of the input-output data. In the current paper, we impose a mild restriction so as to avoid certain singular possibilities. This leads to a parameterization of the equivalence class of models via a single real parameter. We then use this result to show that there exists a model which is optimal in the sense that minimizes the maximal weighted infinity norm of the error between the chosen model and all members of the equivalence class. This model is unique and is independent of the weighting function used in the infinity norm. It is thus the natural choice to be used in applications such as robust control. The result is also compared with more conventional estimates provided by prediction error methods.

THE CONTRIBUTION OF THE VERTICAL PART OF THE REAL-TIME CONTOUR IN FINITE TEMPERATURE FIELD THEORY: FINITE TEMPERATURE FIELD THEORY

We investigate the contribution of the vertical part of the real-time contour in finite temperature field theory. For this purpose we parametrize the contour C in the complex x0-plane by a single real parameter t. We prove the existence of a nonvanishing vertical contribution.

Influence of viscoelasticity and interfacial slip on acoustic wave sensors

Acoustic wave devices with shear horizontal displacements, such as quartz crystal microbalances (QCM) and shear horizontally polarized surface acoustic wave devices, provide sensitive probes of changes at solid–solid and solid–liquid interfaces. Increasingly the surfaces of acoustic wave devices are being chemically or physically modified to alter surface adhesion or coated with one or more layers to amplify their response to any change of mass or material properties. In this work, we describe a model that provides a unified view of the modification in the shear motion in acoustic wave systems by multiple finite thickness loadings of viscoelastic fluids. This model encompasses QCM and other classes of acoustic wave devices based on a shear motion of the substrate surface and is also valid whether the coating film has a liquid or solid character. As a specific example, the transition of a coating from liquid to solid is modeled using a single relaxation time Maxwell model. The correspondence between parameters from this physical model and parameters from alternative acoustic impedance models is explicitly given. The characteristic changes in QCM frequency and attenuation as a function of thickness are illustrated for a single layer device as the coating is varied from liquid like to that of an amorphous solid. Results for a double layer structure are explicitly given and the extension of the physical model to multiple layers is described. An advantage of this physical approach to modeling the response of acoustic wave devices to multilayer films is that it provides a basis for considering how interfacial slip boundary conditions might be incorporated into the acoustic impedance used within circuit models of acoustic wave devices. Explicit results are derived for interfacial slip occurring at the substrate–first layer interface using a single real slip parameter, s, which has inverse dimensions of impedance. In terms of acoustic impedance, such interfacial slip acts as a single-loop negative feedback. It is suggested that these results can also be viewed as arising from a double-layer model with an infinitesimally thin slip layer which gives rise to a modified acoustic load of the second layer. Finally, the difficulties with defining appropriate slip boundary conditions between any two successive layers in a multilayer device are outlined from a physical point of view.

Electron scattering in a point contact of one-dimensional conductors

The scattering matrix of a point contact between one-dimensional coherent conductors is considered. It is shown that the flux conservation law, time-reversal symmetry, and an hypothesis of continuity of the wave function lead to parametrization of the scattering matrix by a single real parameter, regardless of the number of conductors connected by the contact. The condition of maximum transmission fixes this parameter and thereby uniquely defines the scattering matrix. The condition of flux conservation then reduces to the condition that the sum of the derivatives of the wave function with respect to the directions of the conductors vanish. Possible applications of the model considered to experimentally feasible arrays of one-dimensional elements are discussed.

A Framework for Bayesian and Likelihood Approximations in Statistics

Regular Bayesian and frequentist approximations in statistics are studied within a unified framework. In particular it is shown how some higher-order likelihood-based approximations arise from their Bayesian counterparts via an unsmoothing argument. This approach serves to shed new light on these formulae and to clarify relationships between Bayesian and frequentist inferences. For example, Bayesian interpretations of higher-order approximations can give insights into the acceptability, or otherwise, of these approximations from the point of view of ‘relevance’ to the actual data observed. Furthermore, this approach is a very natural one for the study of more general ‘nonregular’ problems, models for dependent data, and approximate conditional inference. For ease of exposition the development is in terms of a single real parameter. The main analytic development proceeds in terms of signed roots of log-density ratios.

A framework for Bayesian and likelihood approximations in statistics

SUMMARY Regular Bayesian and frequentist approximations in statistics are studied within a unified framework. In particular it is shown how some higher-order likelihood-based approximations arise from their Bayesian counterparts via an unsmoothing argument. This approach serves to shed new light on these formulae and to clarify relationships between Bayesian and frequentist inferences. For example, Bayesian interpretations of higher-order approximations can give insights into the acceptability, or otherwise, of these approximations from the point of view of 'relevance' to the actual data observed. Furthermore, this approach is a very natural one for the study of more general 'nonregular' problems, models for dependent data, and approximate conditional inference. For ease of exposition the development is in terms of a single real parameter. The main analytic development proceeds in terms of signed roots of log-density ratios.

Bifurcation of quasi-periodic solutions from equilibrium points of reversible dynamical systems

It is well known that a hyperbolic equilibrium point of a finite dimensional dynamical system (the eigenvalues of the Jacobian are not on the imaginary axis) is an isolated solution in the space of bounded functions (cf. [8], [10]). Such an equilibrium point is "structurally stable", and bifurcations can occur only at those equilibrium points for which the eigenvalues have vanishing real part. Moreover, the natural question of bifurcation of oscillating solutions has been considered almost exclusively for the periodic case, very little being known about bifurcation of quasi-periodic solutions. In the linear case, quasi-periodic solutions can be obtained simply by superposition of periodic solutions with rationally independent frequencies. In the general non-linear case, however, their construction leads necessarily to the socalled problem of small divisors. In connection with the stability of an equilibrium point MOSER [17, pp. 21] and BIBIKOV [3, pp. 91] have treated analytic systems in R 2" which are either reversible or Hamiltonian. They assume that the origin is an elliptic equilibrium point (the Jacobian has only imaginary eigenvalues). Under certain conditions on the terms of order less than four of the Birkhoff normal form, they show that in every neighborhood of the origin there are quasiperiodic solutions with n independent frequencies. This set of solutions is in general not connected, but the relative measure of the points of these trajectories tends to one when the neighborhood shrinks to zero. In [1] the existence of such solutions has been extended to the more general case where eigenvalues with nonvanishing real part are allowed. In recent papers BROER [5], and BRAAKSMA and BROER [4], have constructed quasi-periodic bifurcating solutions for divergence free C l vector fields in R 3 and R 4 involving a single real parameter. We consider here reversible analytic systems in R q, which depend on a pdimensional parameter r/

Linear systems with commensurate time delays: stability and stabilization independent of delay

Notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays. Various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter. Sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays. Constructive methods for determining a stabilizing feedback are also presented. The last part of the paper deals with a standard type of observer and regulator with the requirement that the closed-loop system be stable independent of delay.

Copositive matrices and definiteness of quadratic forms subject to homogeneous linear inequality constraints

A symmetric matrix C is called copositive if the quadratic form x′Cx is nonnegative for all nonnegative values of the variables ( x 1, x 2,…, x n )= x′. A known sufficient condition for a quadratic form x′Qx to be positive unless x=0, subject to the linear inequality constraints Ax⩾0, is that there should exist a copositive matrix C such that Q− A′CA is positive definite. The main result of this paper establishes the necessity of this condition. For x'Qx to be merely nonnegative subject to Ax ⩾ 0, the situation is less straightforward. The necessity of the existence of a copositive matrix C such that Q− A′CA is positive semidefinite is proved only under various additional hypotheses regarding the size or rank of A, and counterexamples are given to show that, in general, no such matrix may exist, even when Slater's constraint qualification holds. Our approach to these existence questions also furnishes certain tests for positivity or mere nonnegativity of x′Qx subject to Ax⩾0, in which specific symmetric matrices, constructed by rational operations from A and Q and depending upon a single real parameter v, must be tested for positive definiteness or strict copositivity for large values of v. This technique is illustrated by several examples.

Propagation through a precipitation medium: Theory and measurement

The theory of propagation through a precipitation medium is based here on the differential propagation constant and two additional complex constants. The latter reduce to a single real parameter, the "canting angle," for certain simple models, for example, for a population of equioriented symmetrical scatterers. In a uniform medium the constants are closely related to the elements of an equivalent permittivity tensor. It is shown that the parameters can be determined from a sequence of depolarization measurements.

Sufficient Conditions for the Admissibility Under Squared Errors Loss of Formal Bayes Estimators

This paper is concerned with the problem of finding reasonably explicit sufficient conditions for the almost admissibility of formal Bayes estimators (for definitions, see Section 2), where the underlying distribution is assumed known up to a single real parameter, and a real function of this parameter is being estimated with squared error as loss. The parameter space is assumed to be a possibly unbounded interval. These conditions are derived in Section 3. They are similar, in appearance, to results obtained by Karlin [3] when the underlying distribution is a member of the family of one parameter exponential distributions and the mean of this distribution is being estimated. The results of Section 3 should be viewed as a refinement of a heuristic argument given by Stein ([6] and [7] pages 233-240). In Section 2, some preliminary results and definitions are given. The results of Section 3 are applied in Section 4 to problems involving either the one dimensional exponential family or the estimation of a function of a single location parameter. Some of the results are known, in at least a similar form, while others are new. A counterexample based on a one dimensional location parameter problem is given in Section 5. It suggests that conditions of the type obtained here may even be necessary.

Fiducial consistency and group structure

SUMMARY A consistency criterion is proposed for fiducial probability distributions: if a fiducial density function is used to modulate a prior density function, the resulting frequency function should be that obtained by a Bayesian analysis. For a real variable and a real parameter, the fulfilment of this criterion is equivalent to a condition examined by Lindley (1958) and it requires the parameter to be essentially a location parameter. For several variables and the same number of parameters, the criterion if applied to fiducial probability derived from a pivotal quantity produces a simple differential equation condition on the pivotal quantity. If one-dimensional fiducial distributions can be generated in an arbitrary direction from any point in the parameter space, then the fulfilment of the criterion requires the model to be of transformation-parameter form; for such models a variety of properties concerned with consistency and interpretation are available (Fraser, 1961). The fiducial method of inference was introduced by Fisher (1930) and has been discussed and developed in many of his subsequent papers. The development is primarily in terms of examples, with but little attention to precise governing principles. In this paper the method will be interpreted in terms of the pivotal quantities that are a common feature in the examples. A pivotal quantity is a function of a sufficient statistic and a parameter that has a fixed frequency distribution. For a single real variable and a single real parameter the distribution function is a pivotal quantity and it has the uniform distribution on (0, 1) regardless of the parameter value. In some statistical problems there may be a natural choice for the pivotal quantity based, perhaps on the physical origin of the problem. The quantity may then express a position, a probability or error position for an outcome relative to the parameter value. In other problems there may be a natural choice based on mathematical properties of the specification. In this paper a pivotal quantity will be assumed given as part of the statistical problem: the distribution of the pivotal variable will be interpreted as describing the error distribution or error pattern, and the pivotal equation as describing the way that the probability mass of the error pattern is applied to the sample space. The fiducial method as presented by Fisher involves a preliminary reduction to an exhaustive statistic. If the exhaustive statistic is of the same dimension as the parameter it is used as the basic variable in the pivotal quantity. If the exhaustive statistic is of higher dimension, then an ancillary statistic is needed such that conditionally the exhaustive statistic has the same dimension as the parameter and can be used as basic variable in the pivotal quantity.

Minimum Principle for Multi-Channel Scattering

The usual variational principles of scattering theory are simply stationary principles. In a recent series of papers, all restricted to single- channel scattering, conditions have been established under which variational principles can be found which are much more powerful in that the functional that represents the variational estimate is not simply stationary in the neighborhood of the cxact scattering solution but is rather an extremum. A lower bound has been obtained on the single real parameter, tann/sub c/, which charactertzes the scattering in the (uncoupled) channel c. The conditions previously established allowed for composite bound states, for the Pauli principle, for arbitrary angular momenta, and for long-range and in particular Coulomb potentials; at nonzero energies, the various potentials had to be truncated. The present paper deals with the extension to multi-channel scattering in which each of the open channels contains only two systems. The bounds are now on linear combinations of the elements of the reactance matrix or of the derivative matrix. The potentials must be truncated in a fashion very similar to that used in WignerEisenbud theory. (auth)

Bhattacharyya Bounds without Regularity Assumptions

In [1] a method for removing the regularity conditions from the Cramer-Rao Inequality was given and applied to the estimation of a single real parameter. It was noted there that the method would extend to problems more general than estimating a single real parameter. However, the method extends also for the estimation of a single real parameter and produces analogues of the Bhattacharyya bounds with and without nuisance parameters.

Minimum Variance Estimation Without Regularity Assumptions

Following the essential steps of the proof of the Cramer-Rao inequality [1, 2] but avoiding the need to transform coordinates or to differentiate under integral signs, a lower bound for the variance of estimators is obtained which is (a) free from regularity assumptions and (b) at least equal to and in some cases greater than that given by the Cramer-Rao inequality. The inequality of this paper might also be obtained from Barankin's general result [3]. Only the simplest case--that of unbiased estimation of a single real parameter--is considered here but the same idea can be applied to more general problems of estimation.