Inconsistencies with subluminal Lorentz transformations in six-dimensional space-time

Generalized Lorentz transformations in six-dimensional space-time

We re-examined the four-dimensional spacetime formulation of invariance of electromagnetic fields between two inertial frames under Lorentz transformation, which predicts a pure electric (magnetic) field in one inertial frame is composed the Cartesian components of a pure both electric and magnetic fields in another inertial frame. This contradicts the Lorentz invariance condition which requires that the vector quantities in one inertial frame must have the same form in another inertial frame. In this work, we introduce a three-dimensional quasi-time vector to modify the classical four-dimensional spacetime (3+1) to a new six-dimensional spacetime (3+3) and derive spacetime metric equation and relativistic velocity. We use the classical vector transformation theory to derive expressions for Cartesian components of relativistic velocity and net electromagnetic force vectors. Considering two massive inertial frames form a closed system, we integrated the transformed relativistic velocity with the law of conservation of energy to prove that contrary to the common belief, the electromagnetic field that appears as a purely electric (magnetic) field in one massive inertial frame, it also appears as a pure electric (magnetic) field in another massive inertial frame under Lorentz transformation. As an application of the proposed six-dimensional spacetime theory, we prove Lorentz invariance of Maxwell’s equations with and without charge and current source. We also prove the scalar electromagnetic wave equations with and without charge and current source and the conservation laws of the continuity equations of current and densities of electromagnetic energy and linear and angular momentums between two massive inertial frames under Lorentz transformation.

So far, the direction and rate of relative motion between the Rivera and the North American plates (RIV-NAM) has been determined by the combination of two Euler poles: Rivera (RIV), with respect to Pacific (PAC), and PAC with respect to North America. Here, we estimate the relative motion of this plate pair (RIV-NAM) assuming that the horizontal projection of the direction of slip of the earthquakes occurring on the RIV-NAM boundaries reflect their relative plate motion. A catalog of earthquakes for which focal mechanisms are reported since 1976 is used in the analysis. Earthquakes were considered in the three segments of the RIV-NAM plate boundary: the subduction zone of the Rivera plate beneath the Jalisco block, the Tres Marias Escarpment and the events associated with the Tamayo Fracture Zone. The best fitting Euler pole is determined using a grid search of 64 potential poles. The slip direction predicted for each grid point is compared to the slip direction of the focal mechanisms of the earthquakes on the plate boundary. The best fitting Euler pole, determined in a root mean square sense (RMS), is located at 21.8°N, 107.6°W. A rate of rotation of 5.3°/year is estimated assuming the seismic earthquake cycle of the 1932 and 1995 great earthquakes represents a lower bound of the rate of plate motion in the subduction zone. The best fitting Euler pole shows that the subduction of the Rivera plate takes place in a direction perpendicular to the trench with a relative velocity of 4.3 cm/year, offshore Manzanillo. The rate of relative motion RIV-NAM decreases from SE to NW. North of approximately 21°N, the subduction of the Rivera plate becomes oblique to the trench and the relative velocity between the two plates decreases to an average of 1.9 cm/year. This slow rate of convergence may explain the rapid decrease of seismicity in the trench and the apparent absence of large earthquakes in this region. In the Tres Marias Escarpment, our best-fitting pole suggests that subduction stops, giving way to high-angle reverse faulting perpendicular to the Tres Marias Escarpment, in agreement with the reverse faulting earthquakes occurring here. To the north of 22.5°N, the slip predicted by the best-fitting pole suggests right-lateral faulting in a direction parallel to the Tamayo Fracture Zone, at a very low velocity (0.5–1.0 cm/year). The best fitting Euler pole determined here lies very close to the RIV-NAM plate boundary in the vicinity of the Tamayo Fracture Zone. This location of our best fitting Euler pole explains the low relative plate velocity, the relatively low level of seismic activity and the presence of a broad zone of deformation that accommodates the RIV-NAM motion.

The influence of the memory effects on the Poissonian clocks with fluctuating counting rates is investigated by using the technique of characteristic functionals. A general approach for computing all cumulants of the number of counts is suggested based on an analogy with the theory of rate processes with dynamical disorder. The large time behavior of the cumulants is investigated for stationary random processes with short and long memory, respectively. For short memory in the long run all cumulants increase linearly in time and the averaged stochastic process describing the statistics of the number of counts, although generally non-Poissonian, is non-intermittent and can be used as a clock. For finite long memory described by a stationary fractal random process, even though the cumulants of the number of events increase faster than linearly in time, the fluctuations are still non-intermittent and the averaged random process is also a clock. For infinite memory, however, the fluctuations of the number of counts are intermittent and the averaged random process is not a clock any more. An alternative stochastic approach is developed based on the use of a dynamical analogue of the Porter-Thomas formula; the results are consistent with the first version of the theory. Three applications of the general theory are presented. The first application is related to the connection between the Kimura's neutral theory of molecular evolution and the Gillespie's episodic clock for the rate of amino acid substitutions through the evolutionary process. If the fluctuations of the rate of substitution have short memory or long finite memory then Gillespie's episodic clock is consistent with Kimura's theory. Only the infinite memory is not consistent with the neutral hypothesis. The second application is the study of a hopping mechanism for enhanced diffusion. A biased random walk is investigated by assuming that the distribution of the number of jumps is given by a Poissonian process with a fluctuating counting rate. If the fluctuations of the counting rate have short memory then the resulting biased diffusion is normal and obeys Einstein's linear equation for the mean square displacement of the moving particle. For long memory the mean square displacement of the moving particle increases faster than linearly in time and the diffusion is enhanced. An alternative approach for a random walk in the velocity space is developed. In this case the diffusion process is even more efficient than for a random walk in the real space. The third application is the study of Porter-Thomas relaxation for systems with dynamical disorder. It is shown that for small and moderately long times the relaxation function obeys a scaling law of the negative power law type followed by a fast decaying exponential tail which is determined by the fluctuation dynamics. This type of relaxation behavior is of interest both for nuclear and molecular physics and corresponds to a non-ideal statistical fractal.

A principle of kinematical relativity is introduced and some consequences are derived. A group of transformations having Poincare transformations as a special case is obtained for any kind of motion. A moving particle is shown to satisfy a variational «principle». Rotating frames are seen not to be «good» frames, and the clock hypothesis to be valid only for uniform rectilinear motion. Two space-time dimension transformations are discussed with further details.

Abstract. Embedding a discrete‐time autoregressive moving average (DARMA) process in a continuous‐time ARMA (CARMA) process has been discussed by many authors. These authors have considered the relationship between the autocovariance structures of continuous‐time and related discrete‐time processes. In this article, we treat the problem from a slightly different point of view. We define embedding in a more rigid way by taking account of the probability structure. We consider Gaussian processes. First we summarize the necessary and sufficient condition for a DARMA process to be able to be embedded in a CARMA process. Secondly, we show a concrete condition such that a DARMA process can be embeddable in a CARMA process. This condition is new and general. Thirdly, we show some special cases including new examples. We show how we can examine embeddability for these special cases.

A parameter estimation problem in a class of nonlinear systems is considered where the input-output relation of a nonlinear system is approximated by a polynomial model (e.g., a Volterra series). A least mean squares (LMS) type algorithm is utilized for the recursive estimation of the polynomial coefficients, and its resulting mean square error (MSE) convergence properties are investigated. Conditions for the algorithm stability (in the mean square sense) are established, steady-state MSE bounds are obtained, and the convergence rate is discussed. In addition, modeling accuracy versus steady-state performance is examined; it is found that an increase of the modeling accuracy may result in a deterioration of the asymptotic performance, that is, yielding a larger steady-state MSE. Linear system identification is studied as a special case.

Design and analysis of linear precoders under a mean square error criterion, Part I: foundations and worst case designs

Optimal linear minimum mean square error (MMSE) transceiver design techniques are proposed for Bayesian learning (BL)-based sparse parameter vector estimation in a multiple-input multiple-output (MIMO) wireless sensor network (WSN). Our proposed transceiver designs rely on majorization theory and hyperparameter estimates obtained from the BL module for minimizing the mean square error (MSE) of parameter estimation at the fusion center (FC). The linear transceiver design framework is initially proposed for the general scenario with arbitrary SNR sensor observations, followed by a special case with high-SNR sensor observations scenario. Our analysis also incorporates the channel correlation. The MMSE channel estimates are determined for the sensors (SNs), followed by a robust transceiver design procedure that is resilient to the channel state information (CSI) uncertainty arising due to the channel estimation error, an aberration that is unavoidable in practical implementations. Our simulation results demonstrate the improved performance of the proposed BL framework and optimal MMSE transceiver design in sparse parameter estimation relyingon realistic imperfect channel estimates over the benchmarks.

Containment control of leader-following multi-agent systems with Markovian switching network topologies and measurement noises

The Proportionate Normalized Least Mean Squares (PNLMS) algorithm has been quite successful in combining higher convergence rates with low to moderate complexity that at the same time avoids numerical difficulties in fixed-point implementations. While the algorithm is stable in the mean square and l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -sense for time-invariant matrices, the treatment of time-variant matrices requires additional approximations. These approximations are discarded in this paper which allows us to analyse the robustness in terms of l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -stability for actually time-variant matrix step-sizes. This provides important results, as the algorithm in its variants also occurs in other fields of adaptive filtering such as cascaded filter structures. By simulations as well as by theoretical analysis, we demonstrate that in general, even small variations of the matrix step-size are sufficient for the algorithm to loose its robustness. Only in special cases, where specific constraints are imposed additionally, robustness can be guaranteed.

Optimal receiver diversity combining employing linear channel estimation is examined. Based on the statistical properties of least-squares (LS) and minimum mean square error (MMSE) channel estimation, an optimal diversity receiver for wireless systems employing practical linear channel estimation on Rician fading channels is proposed. The new receiver structure includes the conventional maximal ratio combining receiver as a special case. Exact analytical expressions for the symbol error rates (SERs) of LS and MMSE channel estimation aided optimal diversity combining are derived. It is shown that, if an optimal detector is used, an MPSK wireless system with MMSE channel estimation has the same SER when the MMSE channel estimation is replaced by LS estimation. This is an interesting counterexample to the common perception that channel estimation with smaller mean square error leads to smaller SER. Extensive simulation results validate the theoretical results.

A description is given of a general closure principle involving the minimization of the mean square error. The procedure based upon this principle can be applied to the truncation of the BBGKY hierarchy at various stages and to the approximation of unwanted terms arising in the equation of motion method by linear combinations of the observables to be retained. On a general level the significance of the closure principle is described in terms of the geometry of function space, and several useful general properties of the principle are derived. A discussion is devoted to the relation between the closure error (i.e., the least mean square error) and the error in the end result (e.g., the free energy, the radial distribution function, etc.); however, the results, while providing some insight, are not sufficiently refined to provide upper bounds to errors in all problems of statistical mechanics where the method is applicable. On the level of specific application it is shown that the principle yields results identical to the random phase approximation and to the linearized version of the Kirkwood superposition approximation in two special cases. Later sections of the paper describe in greater than usual generality, the formalism connecting thermodynamic properties and other equilibrium properties with the microscopic equations of motion in which closure approximations have been introduced. Two illustrative examples of the application of the over-all method were made to the case of a classical system of electrons in a uniform background of compensating charge, one leading to the well-known results of Debye and the other to a more accurate and elaborate theory developed in quantitative detail elsewhere.

We consider the polynomial regression model. In this model a hierarchical structure, or natural ordering, in the parameter space can be assumed. Maximum likelihood estimators may be found for the parameters of each order model in the hierarchy. We introduce the class of estimators given by weighted combinations of these maximum likelihood estimators under certain restrictions. This class is obtained by considering a Bayes estimator class and contains the subset regression estimator as a special case. The optimal weights which minimize the predictive mean squared error are obtained exactly, using an alternative method to that of Kanda (1985). The estimated weights which minimize the estimated predictive mean square error in a similar way to the Mallow’s Cp-statistic are also exactly presented and some numerical examples are shown.

This paper is concerned with the mean square exponential stability and stabilization problems of Markovian jump systems with time delay. New Lyapunov functionals are proposed by choosing distinct Lyapunov matrices for different system modes and introducing a triple-integral term. Some delay-dependent conditions, including some existing results as their special cases, are derived under which the resulting closed-loop system is mean square exponentially stable with a decay rate. The design of the feedback gain matrices is accomplished by solving linear matrix inequalities. Finally, the effectiveness and performance of the obtained results are demonstrated by numerical examples.