Set Of Scalar Functions Research Articles

Electromagnetic field and cylindrical compact objects in modified gravity

In this paper, we have investigated the role of different fluid parameters particularly electromagnetic field and $f(R)$ corrections on the evolution of cylindrical compact object. We have explored the modified field equations, kinematical quantities and dynamical equations. An expression for the mass function has been found in comparison with the Misner-Sharp formalism in modified gravity, after which different mass radius diagrams are drawn. The coupled dynamical transport equation have been formulated to discuss the role of thermoinertial effects on the inertial mass density of the cylindrical relativistic interior. Finally, we have presented a framework, according to which all possible solutions of the metric $f(R)$-Maxwell field equations coupled with static fluid can be written through set of scalar functions. It is found that modified gravity induced by Lagrangians $f(R)=\alpha R^2,~f(R)=\alpha R^2-\beta R$ and $f(R)=\frac{\alpha R^2-\beta R}{1+\gamma R}$ are likely to host more massive cylindrical compact objects with smaller radii as compared to GR.

Dissipative collapse of axially symmetric, general relativistic sources: A general framework and some applications

We carry out a general study on the collapse of axially (and reflection-)symmetric sources in the context of general relativity. All basic equations and concepts required to perform such a general study are deployed. These equations are written down for a general anisotropic dissipative fluid. The proposed approach allows for analytical studies as well as for numerical applications. A causal transport equation derived from the Israel-Stewart theory is applied, to discuss some thermodynamic aspects of the problem. A set of scalar functions (the structure scalars) derived from the orthogonal splitting of the Riemann tensor are calculated and their role in the dynamics of the source is clearly exhibited. The characterization of the gravitational radiation emitted by the source is discussed.

Maximum Expected Rates of Block-Fading Channels with Entropy-Constrained Channel State Feedback

We obtain the maximum average data rates achievable over block-fading channels when the receiver has perfect channel state information (CSI), and only an entropy-constrained quantized approximation of this CSI is available at the transmitter. We assume channel gains in consecutive blocks are independent and identically distributed and consider a short term power constraint. Our analysis is valid for a wide variety of channel fading statistics, including Rician and Nakagami-m fading. For this situation, the problem translates into designing an optimal entropy-constrained quantizer to convey approximated CSI to the transmitter and to define a rate-adaptation policy for the latter so as to maximize average downlink data rate. A numerical procedure is presented which yields the thresholds and reconstruction points of the optimal quantizer, together with the associated maximum average downlink rates, by finding the roots of a small set of scalar functions of two scalar arguments. Utilizing this procedure, it is found that achieving the maximum downlink average capacity C requires, in some cases, time sharing between two regimes. In addition, it is found that, for an uplink entropy constraint H <; log2 (L), a quantizer with more than L cells provides only a small capacity increase, especially at high SNRs.

Shift-invariant, DWT-based "projection" method for estimation of ultrasound pulse power spectrum.

An approach to computing estimates of the ultrasound pulse spectrum from echo-ultrasound RF sequences, measured from biological tissues, is proposed. It is computed by a "projection" algorithm based on the Discrete Wavelet Transform (DWT) using averaging over a range of linear shifts. It is shown that the robust, shift invariant estimate of the ultrasound pulse power spectrum can be obtained by the projection of RF line log spectrum on an appropriately chosen subspace of L2(R) (i.e., the space of square-integrable functions) that is spanned by a redundant collection of compactly supported, scaling functions. This redundant set is formed from the traditional (in Wavelet analysis) orthogonal set of scaling functions and also by all its linear (discrete) shifts. A proof is given that the estimate, so obtained, could be viewed as the average of the orthogonal projections of the RF line log spectrum, computed for all significant linear shifts of the RF line log spectrum in frequency domain. It implies that the estimate is shift-invariant. A computationally efficient scheme is presented for calculating the estimate. Proof is given that the averaged, shift-invariant estimate can be obtained simply by a convolution with a kernel, which can be viewed as the discretized auto-correlation function of the scaling function, appropriate to the particular subspace being considered. It implies that the computational burden is at most O(n log2 n), where n is the problem size, making the estimate quite suitable for real-time processing. Because of the property of the wavelet transform to suppress polynomials of orders lower than the number of the vanishing moments of the wavelet used, the presented approach can be considered as a local polynomial fitting. This locality plays a crucial role in the performance of the algorithm, improving the robustness of the estimation. Moreover, it is shown that the "averaging" nature of the proposed estimation allows using (relatively) poorly regular wavelets (i.e., short filters), without affecting the estimation quality. The latter is of importance whenever the number of calculations is crucial.

Dynamic scaling in vacancy-mediated disordering

We consider the disordering dynamics of an interacting binary alloy with a small admixture of vacancies which mediate atom-atom exchanges. Starting from a perfectly phase-segregated state, the system is rapidly heated to a temperature in the disordered phase. A suitable disorder parameter, namely, the number of broken bonds, is monitored as a function of time. Using Monte Carlo simulations and a coarse-grained field theory, we show that the late stages of this process exhibit dynamic scaling, characterized by a set of scaling functions and exponents. We discuss the universality of these exponents and comment on some subtleties in the early stages of the disordering process.

The theory of primary measurements after Poincaré-Ehrenfest for the nonstationary Schrödinger equation

We define a primary measurement as an association by which a solution of the Schrodinger equation is linked to a set of scalar functions depending only on time and subject to a linear autonomous differential equation. Necessary and sufficient conditions are given for the existence of primary measurements. In fact, we study the Poincare integral invariants for the nonstationary Shorodinger equation. The principal object of the theory is an inversion of the classical Ehrenfest theorem. In this way, we obtain a parametrization of primary measurements and investigate the structure of the space of parameters by the methods of bifurcation theory (bifurcation diagrams are supplied when the number of parameters is ≤3). Bibliography: 43 titles.

A forecasting method for time series with fractal geometry and its application

This paper deals with the prediction of time series that have fractal geometry. We also describe the prediction error, the estimation of fractal dimension of the times series, and several applications. First, we assume that the time series is represented by a convolution of the input signal and the impulse response function, expanded by using a set of scaling functions. Then, a prediction method is derived by using the fact that the impulse response retains self-similarity even if the time scale is expanded, due to the fractal geometry of the time series. We demonstrate the ability of the prediction method for various fractal dimensions by showing that the one-step-ahead prediction error is very small, and also that the n-step-ahead prediction error can be made small by using adaptive correction of the data in which the prediction is successively used as an observation. We also describe the estimation of the fractal dimension by using the covariance matrix and the shape of the spectrum of the time series. As an application, the option price of the stock index estimated by the method is compared to the price obtained by conventional estimation, and our strategy is shown to give a greater profit. © 1998 Scripta Technica, Electron Comm Jpn Pt 3, 82(3): 31–39, 1999

Adaptive multiwavelet initialization

The multiwavelet concept uses a set of scaling functions and a set of wavelets to generate an orthogonal multichannel multiresolution pyramid decomposition for finite energy signals. The multiple scaling functions extract the lowpass information of the input data set, and the multiwavelets extract the bandpass information. When the lowpass filters associated with the scaling functions are cascaded, a multiresolution pyramid decomposition/reconstruction system is formed with each pyramid convolution operator having several inputs and several outputs. However, there is only one data set available to initialize this process. This paper addresses the question of how to modify the data set using prefilters so that its decomposition contains the most useful information for the chosen application. The best prefilters are determined by the minimization of the energy of preselected decomposition components. The resulting decomposition is, therefore, signal adaptive, and under appropriate conditions, perfect reconstruction of the input data set can be achieved with a proper postfiltering. The detailed discussions in the paper are focussed on the two-wavelet and two-scaling function case; the general multiwavelet case is addressed at the end of the paper. A compression example is provided to demonstrate the performance of the optimally initialized multiwavelet method and to compare it with a single wavelet method and another multiwavelet initialization method proposed by other authors.

Forecasting of time series with fractal geometry by using scale transformations and parameter estimations obtained by the wavelet transform

This paper deals with the prediction of time series derived from the combination of two basic methods for the fractal time series. The first basic method is the estimation of parameters, which utilizes the fact that by applying the Wavelet transform to the fractal times series, the coefficients are represented by the variance and fractal dimension. The second method is the prediction of time series based upon a procedure in which the impulse response retains self-similarity even if the time scale is expanded due to the fractal geometry of the time series, if the time series is modeled by the convolution of the input signal and impulse response function expanded by using a set of scaling functions. We introduce a model to generate time series having long-term and short-term correlation by using a system in which the fractal time series is passed through an auto-regressive moving average ARMA filter. Then, a method is presented to estimate simultaneously the coefficients of the ARMA filter and the variance and dimension of the fractal time series. As an application, the stock price is predicted, showing that the error of one-step-ahead prediction is very small and that the error of n-step-ahead prediction can be made small by using adaptive correction of the data. As a result, our strategy is proven to be effective. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(8): 20–30, 1997

Differential invariants of a Euclidean algebra

Functional bases of second-order differential invariants of a Euclidean algebra and a conformal algebra are found for a set of scalar functions depending on n variables.