Arbitrary Linear Combination Research Articles

Stability of order statistics under dependence

We precisely evaluate the upper and lower deviations of the expectation of every order statistic from an i.i.d. sample under arbitrary violations of the independence assumption, measured in scale units generated by various central absolute moments of the parent distribution of a single observation. We also determine the distributions for which the bounds are attained. The proof is based on combining the Moriguti monotone approximation of functions with the Holder inequality applied for proper integral representations of expected order statistics in the independent and dependent cases. The method allows us to derive analogous bounds for arbitrary linear combinations of order statistics.

Time-asymmetric Fokker-type action and relativistic wave equations

A relativistic two-particle system with an arbitrary linear combination of scalar and vector time-asymmetric Fokker-type interactions in a two-dimensional spacetime is considered within the framework of the front form of dynamics. It is shown that the corresponding mass-shell equation takes the form of a linear relation between the generators of the Lie algebra so(2,1). An algebraic quantization of the system is proposed and a closed form for the mass spectrum is obtained. The relativistic wave equation obtained by Barut and Rasmussen for the H atom is generalized to the case of an arbitrary linear combination of scalar and vector interactions. An extension of the results to the system in four-dimensional spacetime is suggested.

Adaptive wavelet methods for elliptic operator equations: Convergence rates

This paper is concerned with the construction and analysis of wavelet-based adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solutionuuof the equation by a linear combination ofNNwavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation touuby an arbitrary linear combination ofNNwavelets (so calledNN-term approximation), which would be obtained by keeping theNNlargest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation touuwith errorO(N−s)O(N^{-s})in the energy norm, whenever such a rate is possible byNN-term approximation. The range ofs>0s>0for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional toNN. The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization.

Quantum noise reduction and generalized two-mode squeezing in a cavity Raman laser

We study a generalized notion of two-mode squeezing for the Stokes and anti-Stokes fields in a model of a cavity Raman laser, which leads to a significant reduction in decoherence or quantum noise. The model comprises a lossless cavity with classical pump, unsaturated medium, and arbitrary homogeneous broadening and dispersion. Allowing for arbitrary linear combinations of the two modes in the definition of quadrature variables, we find that there always exists a combination of the two output modes that exhibits quadrature squeezing with noise reduction below the vacuum level. The number of noise photons for this combination mode is proportional to the square root of the number of Stokes noise photons.

Possible solution of the Coriolis attenuation problem

The most consistently useful simple model for the study of odd deformed nuclei, the particle-rotor model (strong-coupling limit of the core-particle coupling model) has nevertheless been beset by a long-standing problem: It is necessary in many cases to introduce an ad hoc parameter that reduces the size of the Coriolis interaction coupling the collective and single-particle motions. Of the numerous suggestions put forward for the origin of this supplementary interaction, none of those actually tested by calculations has been accepted as the solution of the problem. In this paper we seek a solution for the difficulty within the framework of a general formalism that starts from the spherical shell model and is capable of treating an arbitrary linear combination of multipole and pairing forces. With the restriction of the interaction to the familiar sum of a quadrupole multipole force and a monopole pairing force, we have previously studied a semimicroscopic version of the formalism whose framework is nevertheless more comprehensive than any previously applied to the problem. We obtained solutions for low-lying bands of several strongly deformed odd rare-earth nuclei and found good agreement with experiment, except for an exaggerated staggering of levels for $K=$$\frac{1}{2}$ bands, which can be understood as a manifestation of the Coriolis attenuation problem. We argue that within the formalism utilized, the only way to improve the physics is to add interactions to the model Hamiltonian. We verify that by adding a magnetic dipole interaction of essentially fixed strength, we can fit the $K=$$\frac{1}{2}$ bands without destroying the agreement with other bands. In addition we show that our solution also fits ${}^{163}$Er, a classic test case of Coriolis attenuation that we had not previously studied.

A novel construction of the response tensors for a collisionless unmagnetized plasma

The linear and nonlinear response tensors describe the response of a plasma to an electromagnetic field disturbance. A novel construction of the linear and quadratic response tensors for an unmagnetized collisionless plasma is presented here using a convenient 4-tensor notation. The construction uses the equations of charge continuity and gauge invariance, together with symmetry properties and dimensional arguments, to constrain the coefficients of an arbitrary linear combination of tensors. A new interpretation of the response tensors follows as expressions uniquely determined by fundamental physical constraints, and not simply as coefficients in a perturbation expansion of induced plasma response.

Uncertainties in field-line tracing in the magnetosphere. Part I: the axisymmetric part of the internal geomagnetic field

Abstract. The technique of tracing along magnetic field lines is widely used in magnetospheric physics to provide a "magnetic frame of reference'' that facilitates both the planning of experiments and the interpretation of observations. The precision of any such magnetic frame of reference depends critically on the accurate representation of the various sources of magnetic field in the magnetosphere. In order to consider this important problem systematically, a study is initiated to estimate first the uncertainties in magnetic-field-line tracing in the magnetosphere that arise solely from the published (standard) errors in the specification of the geomagnetic field of internal origin. Because of the complexity in computing these uncertainties for the complete geomagnetic field of internal origin, attention is focused in this preliminary paper on the uncertainties in magnetic-field-line tracing that result from the standard errors in just the axisymmetric part of the internal geomagnetic field. An exact analytic equation exists for the magnetic field lines of an arbitrary linear combination of axisymmetric multipoles. This equation is used to derive numerical estimates of the uncertainties in magnetic-field-line tracing that are due to the published standard errors in the axisymmetric spherical harmonic coefficients (i.e. gn0 ± δgn0). Numerical results determined from the analytic equation are compared with computational results based on stepwise numerical integration along magnetic field lines. Excellent agreement is obtained between the analytical and computational methods in the axisymmetric case, which provides great confidence in the accuracy of the computer program used for stepwise numerical integration along magnetic field lines. This computer program is then used in the following paper to estimate the uncertainties in magnetic-field-line tracing in the magnetosphere that arise from the published standard errors in the full set of spherical harmonic coefficients, which define the complete (non-axisymmetric) geomagnetic field of internal origin. Numerical estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, calculated here for the axisymmetric part of the internal geomagnetic field, should be regarded as "first approximations'' in the sense that such estimates are only as accurate as the published standard errors in the set of axisymmetric spherical harmonic coefficients. However, all procedures developed in this preliminary paper can be applied to the derivation of more realistic estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, following further progress in the determination of more accurate standard errors in the spherical harmonic coefficients.

Computation of infinite integrals involving Bessel functions of arbitrary order by the [formula omitted

The D- transformation due to the author is an effective extrapolation method for computing infinite oscillatory integrals of various kinds. In this work two new variants of this transformation are designed for computing integrals of the form ∫ a ∞ g(t) C v(t) dt , where g( x) is a nonoscillatory function and C v(x) may be an arbitrary linear combination of the Bessel functions of the first and second kinds J v ( x) and Y v ( x), of arbitrary real order v. When applied to such integrals, the D- transformation and its new variants are observed to produce very accurate results. It is also seen that their performance is very similar to that of the modified W-transformation due to the author, as extended in a recent work by Lucas and Stone with C v(x) = J v(x) . The present paper is concluded by stating the relevant convergence and stability results and by appending a numerical example.

Uncertainties in field-line tracing in the magnetosphere. Part I: the axisymmetric part of the internal geomagnetic field

The technique of tracing along magnetic field lines is widely used in magnetospheric physics to provide a "magnetic frame of reference'' that facilitates both the planning of experiments and the interpretation of observations. The precision of any such magnetic frame of reference depends critically on the accurate representation of the various sources of magnetic field in the magnetosphere. In order to consider this important problem systematically, a study is initiated to estimate first the uncertainties in magnetic-field-line tracing in the magnetosphere that arise solely from the published (standard) errors in the specification of the geomagnetic field of internal origin. Because of the complexity in computing these uncertainties for the complete geomagnetic field of internal origin, attention is focused in this preliminary paper on the uncertainties in magnetic-field-line tracing that result from the standard errors in just the axisymmetric part of the internal geomagnetic field. An exact analytic equation exists for the magnetic field lines of an arbitrary linear combination of axisymmetric multipoles. This equation is used to derive numerical estimates of the uncertainties in magnetic-field-line tracing that are due to the published standard errors in the axisymmetric spherical harmonic coefficients (i.e. gn0 ± δgn0). Numerical results determined from the analytic equation are compared with computational results based on stepwise numerical integration along magnetic field lines. Excellent agreement is obtained between the analytical and computational methods in the axisymmetric case, which provides great confidence in the accuracy of the computer program used for stepwise numerical integration along magnetic field lines. This computer program is then used in the following paper to estimate the uncertainties in magnetic-field-line tracing in the magnetosphere that arise from the published standard errors in the full set of spherical harmonic coefficients, which define the complete (non-axisymmetric) geomagnetic field of internal origin. Numerical estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, calculated here for the axisymmetric part of the internal geomagnetic field, should be regarded as "first approximations'' in the sense that such estimates are only as accurate as the published standard errors in the set of axisymmetric spherical harmonic coefficients. However, all procedures developed in this preliminary paper can be applied to the derivation of more realistic estimates of the uncertainties in magnetic-field-line tracing in the magnetosphere, following further progress in the determination of more accurate standard errors in the spherical harmonic coefficients.

Orbifold-induced μ term and electroweak symmetry breaking

It is known that a Higgs μ term can be naturally generated through the Kähler potential in orbifold string models in which one of the three compactified complex planes has order two. In this class of models explicit expressions for both the μ parameter and the soft SUSY-breaking parameters can be obtained under the assumption that the goldstino is an arbitrary linear combination of the fermionic partners of the dilaton S and all the moduli T i , U i . We apply this picture to the MSSM and explore the consistency of the obtained boundary conditions with radiative gauge symmetry breaking. We find that consistency with the measured value of the top-quark mass can only be achieved if the goldstino has a negligible dilatino component and relevant components along the T 3, U 3 moduli associated to the order-two complex plane.

Algebraic and spectral properties of some quasiorthogonal polynomials encountered in quantum radiation

The nodal structure of the wavefunctions of a large class of quantum-mechanical potentials is often governed by the distribution of zeros of real quasiorthogonal polynomials. It is known that these polynomials (i) may be described by an arbitrary linear combination of two orthogonal polynomials {Pn(x)} and (ii) have real and simple zeros. Here, the three term recurrence relation, the second order differential equation and the distribution of zeros of quasiorthogonal polynomials of the classical class (i.e., when Pn(x) is a Jacobi, Laguerre or Hermite polynomial) are derived and analyzed. Specifically, the exact values of the Newton sum rules and the WKB density of zeros of these polynomials are found.

A Generalization of Dirichlet′s Integral

Using the Weyl integral transform representation of fractional calculus, we generalize Dirichlet′s iterated integral identity to include functions whose dummy argument is an arbitrary positive linear combination.

Multilayer neural networks for reduced-rank approximation

This paper is developed in two parts. First, the authors formulate the solution to the general reduced-rank linear approximation problem relaxing the invertibility assumption of the input autocorrelation matrix used by previous authors. The authors' treatment unifies linear regression, Wiener filtering, full rank approximation, auto-association networks, SVD and principal component analysis (PCA) as special cases. The authors' analysis also shows that two-layer linear neural networks with reduced number of hidden units, trained with the least-squares error criterion, produce weights that correspond to the generalized singular value decomposition of the input-teacher cross-correlation matrix and the input data matrix. As a corollary the linear two-layer backpropagation model with reduced hidden layer extracts an arbitrary linear combination of the generalized singular vector components. Second, the authors investigate artificial neural network models for the solution of the related generalized eigenvalue problem. By introducing and utilizing the extended concept of deflation (originally proposed for the standard eigenvalue problem) the authors are able to find that a sequential version of linear BP can extract the exact generalized eigenvector components. The advantage of this approach is that it's easier to update the model structure by adding one more unit or pruning one or more units when the application requires it. An alternative approach for extracting the exact components is to use a set of lateral connections among the hidden units trained in such a way as to enforce orthogonality among the upper- and lower-layer weights. The authors call this the lateral orthogonalization network (LON) and show via theoretical analysis-and verify via simulation-that the network extracts the desired components. The advantage of the LON-based model is that it can be applied in a parallel fashion so that the components are extracted concurrently. Finally, the authors show the application of their results to the solution of the identification problem of systems whose excitation has a non-invertible autocorrelation matrix. Previous identification methods usually rely on the invertibility assumption of the input autocorrelation, therefore they can not be applied to this case.

Linear and Finite-Amplitude Localized Baroclinic Instability

Abstract The linear and finite-amplitude dissipative dynamics of unstable, zonally localized baroclinic disturbances is investigated in cases where the supercriticality varies in the zonal direction. The zonal confinement occurs due to O(1) variations of the frictional influence on the current's instability. A two-layer f-plane model is used. No meridional shear is present in the basic shear flow. When the basic current is equal and opposite in the two layers, two zonally localized modes with the same growth rate and opposite symmetries exist for all unstable parameter values. Thus, an infinite family of unstable modes formed from an arbitrary linear combination of these two modes exists. This degeneracy persists in finite amplitude. Hence, the phase of individual crests in the disturbance is a function of initial conditions even for dissipative localized instabilities. The presence of a mean barotropic flow reduces the growth rates of the localized disturbances and expunges the symmetry properties of the...

Turbulent Flamelet Propagation

Abstract A formalism for a flamelet's evolution of its spatial distribution is derived from a field equation which is slightly more general than Williams' field equation. Unlike Williams' field equation, the field equation used here, though non-linear, has the property that an arbitrary linear combination of interface solutions (Heavyside type of functions) is also a solution. We therefore can describe the location of the flamelet with two interfaces rather than one, both moving relative to the flow in the same direction. The volume between these two interfaces is on average conserved; this makes it possible to define a probability density for the spatial distribution of the flamelet, and thereby derive equations describing the evolution of the spatial distribution of folds and wrinkles of the flame front. Three main conclusions are reached in this paper using this formalism, through the exact analytical study of a flamelet in an arbitrary l-d velocity field, and through the numerical study of a flamelet ...

Realisation of Arbitrary Linear Combinations of Voltages with a Single Operational Amplifier

A method for obtaining any arbitrary linear combination of voltages using a single operational amplifier and resistances is described. The usual textbook circuit for this purpose has limitations on the weights and one normally believes that some classes of linear combinations require more than one operational amplifier for realisation. This is shown to be untrue. An example of such a case is presented.

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

AbstractThe eigenvalue problem for arbitrary linear combinations kα + μα† of a boson annihilation operator α and a boson creation operator α† is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ  μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ⩾ 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α† to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α† are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ  μ/k as a parameter.

Anisotropic, time-dependent solutions in maximally Gauss-Bonnet extended gravity

In an arbitrary number of dimensions, we find the full exact anisotropic, time-dependent, diagonal-metric solutions to maximally Gauss-Bonnet extended gravity theory. This class of theories, for which the lagrangian is an arbitrary linear combination of dimensionally extended Euler forms, is the most general gravitational theory in which the field equations contain no more than second derivatives of the metric. We show that the space-time exponentially approaches an asymptotic state of constant, anisotropic curvature and prove three theorems concerning two generic types of singularities. The first theorem gives conditions for the existence of Kasner-like curvature singularities. For these the metric diverges as t Pi where ∑ p i = 2 k max- 1 and k max is the highest power of the curvature in the lagrangian. Other critical point singularities can arise from the polynomial nature of the theory. The remaining theorems demonstrate that the generic solution is extendible at all of these other critical points and that the generic critical points occur at moments of external volume density of space-time. We give an explicit coordinate transformation which produces a smooth extension through the critical point. The space-time may therefore alternately expand and contract for many cycles before expanding forever or contracting to a singularity. Many particular cases are treated in detail including several power series solutions, the generalized Kasner solution to general relativity with or without cosmological constant, the perturbative solution for quadratic string gravity, and five-dimensional extended gravity.

An Inverse Convolution Method for Regular Parabolic Equations

In this paper, the problem of determining an unknown boundary control of a parabolic distributed parameter system, evolving over finite or infinite time, from incomplete, approximate interior temperature measurements is investigated. First, an exact solution is obtained for square integrable boundary controls associated with a general class of parabolic equations on a finite spatial interval under the assumption that the interior temperature is known for all times. Then, it is shown that this exact solution can be used to develop a stable algorithm for the numerical solution of this problem, without the introduction of standard regularization techniques. This algorithm assumes only knowledge of a finite, discrete set of approximate temperature readings. One advantage of this inversion process is the availability of a priori error bounds based on the measurement errors and frequency of sampling that are obtained in this paper. Another useful feature is that it encompasses boundary controls which are arbitrary linear combinations of surface temperature and flux. Numerical results are presented.

BAMBI — A transient 2D-MESFET model with general boundary conditions including Schottky and current controlled contacts

Boundary conditions using a current-dependent carrier recombination velocity distribution are developed for modelling Schottky contacts by computer-aided physical simulation. In addition, a boundary condition in the form of an arbitrary linear combination of voltage and current at the contact is presented. Thus MESFET devices with simple circuits connected to device terminals can be simulated by solving additional equations. As an example the switching behaviour of a MESFET with a drain resistor is investigated.