Time-independent Coefficients Research Articles

Calculation of Time-Independent Maneuvering Coefficients of an Underwater Vehicle Based on Single Grid Structure

Calculation of Time-Independent Maneuvering Coefficients of an Underwater Vehicle Based on Single Grid Structure

Static and spherically symmetric wormholes in power-law [formula omitted] gravity model

By venturing into gravitational frameworks that extend beyond general relativity (GR), it becomes evident that traversable wormholes (WHs) can conceivably be constructed solely through gravitational means. In GR, the intrinsic degrees of freedom are insufficient for assembling these special kinds of objects without incorporating exotic matter forms. Nevertheless, by stretching gravity into altered frameworks, it becomes possible to satisfy the necessary stability and traversability conditions without relying on such exotic materials. The key concept lies in the view that WH solutions can appear as common features in gravity theories once extra geometrical degrees of freedom are incorporated. Interestingly, we investigate WH solutions in the context of f(R) gravity theory, using a spherically symmetric metric embracing three asymptotically flat regions. More precisely, we are focusing on f(R) gravitational theories formulated in metric formalism, presuming time-independent metric coefficients, and exploiting the well-established Morris–Thorne space–time framework. We thoroughly derived exact solutions for traversable WHs that graciously adhere to energy and pressure constraints, assuming a power-law form for f(R), denoted f(R)=αRn, where α is an arbitrary constant and n is the power-law exponent. Conclusively, our findings open up the possibility of constraining the parameters (α, n, r0) associated with WH solutions. Importantly, we found that the existence of exotic matter was necessary to satisfy the stability and traversability criteria for WHs in all the cases we analyzed.

Sharp uniqueness and stability of solution for an inverse source problem for the Schrödinger equation

This article is concerned with the uniqueness and the stability for an inverse source problem of determining a spatially varying factor f(x) of a source term given by with suitably given R(t) of the Schrödinger equation with time independent coefficients. In order to establish these results, for the Schrödinger equation, we prove (i) a logarithmic conditional stability estimate for a Cauchy problem attached with the zero Dirichlet boundary conditions on the whole boundary, (ii) the unique continuation for the Cauchy problem from an arbitrary small part of a lateral boundary. We do not assume any constraints on the geometry of the subboundary and an observation time length. The key is an integral transform with a kernel solving a null controllability problem for the one-dimensional Schrödinger equation, which transforms a solution of the Schrödinger equation to a solution of an elliptic equation.

An inverse problem for the fractional porous medium equation

We consider a time-independent variable coefficients fractional porous medium equation and formulate an associated inverse problem. We determine both the conductivity and the absorption coefficient from exterior partial measurements of the Dirichlet-to-Neumann map. Our approach relies on a time-integral transform technique as well as the unique continuation property of the fractional operator.

An inverse problem of identifying two coefficients in a time-fractional reaction diffusion system

In this paper, we aim to study an inverse problem for determining two time-independent coefficients in a fractional diffusion system from the final measurements. First, we prove the well-posedness of the state problem, and then we show some regularity results for the solution of the direct system using the Mittag-Leffler function. Then, we reformulate our inverse problem into an optimal control one. Afterwards, we establish the existence of the minimizer and prove the stability estimate for two coefficients with respect to the final data. The descent method is proposed as a numerical one based on the gradient calculus via the adjoint state and we compare it with the conjugate gradient method. Finally, we will present some numerical tests that shows the efficiency of the proposed methods.

Multistate modeling and structure selection for multitype recurrent events and terminal event data.

In cardiovascular disease studies, a large number of risk factors are measured but it often remains unknown whether all of them are relevant variables and whether the impact of these variables is changing with time or remains constant. In addition, more than one kind of cardiovascular disease events can be observed in the same patient and events of different types are possibly correlated. It is expected that different kinds of events are associated with different covariates and the forms of covariate effects also vary between event types. To tackle these problems, we proposed a multistate modeling framework for the joint analysis of multitype recurrent events and terminal event. Model structure selection is performed to identify covariates with time-varying coefficients, time-independent coefficients, and null effects. This helps in understanding the disease process as it can detect relevant covariates and identify the temporal dynamics of the covariate effects. It also provides a more parsimonious model to achieve better risk prediction. The performance of the proposed model and selection method is evaluated in numerical studies and illustrated on a real dataset from the Atherosclerosis Risk in Communitiesstudy.

Sex-specific tuning of modular muscle activation patterns for locomotion in young and older adults.

There is increasing evidence that including sex as a biological variable is of crucial importance to promote rigorous, repeatable and reproducible science. In spite of this, the body of literature that accounts for the sex of participants in human locomotion studies is small and often produces controversial results. Here, we investigated the modular organization of muscle activation patterns for human locomotion using the concept of muscle synergies with a double purpose: i) uncover possible sex-specific characteristics of motor control and ii) assess whether these are maintained in older age. We recorded electromyographic activities from 13 ipsilateral muscles of the lower limb in young and older adults of both sexes walking (young and old) and running (young) on a treadmill. The data set obtained from the 215 participants was elaborated through non-negative matrix factorization to extract the time-independent (i.e., motor modules) and time-dependent (i.e., motor primitives) coefficients of muscle synergies. We found sparse sex-specific modulations of motor control. Motor modules showed a different contribution of hip extensors, knee extensors and foot dorsiflexors in various synergies. Motor primitives were wider (i.e., lasted longer) in males in the propulsion synergy for walking (but only in young and not in older adults) and in the weight acceptance synergy for running. Moreover, the complexity of motor primitives was similar in younger adults of both sexes, but lower in older females as compared to older males. In essence, our results revealed the existence of small but defined sex-specific differences in the way humans control locomotion and that these are not entirely maintained in older age.

Mapping method of group classification

We revisit the entire framework of group classification of differential equations. After introducing the notion of weakly similar classes of differential equations, we develop the mapping method of group classification for such classes, which generalizes all the versions of this method that have been presented in the literature. The mapping method is applied to group classification of various classes of Kolmogorov equations and of Fokker–Planck equations in the case of space dimension one. The equivalence groupoids and the equivalence groups of these classes are computed. The group classification problems for these classes with respect to the corresponding equivalence groups are reduced to finding all inequivalent solutions of heat equations with inequivalent potentials admitting Lie-symmetry extensions. This reduction allows us to exhaustively solve the group classification problems for the classes of Kolmogorov and Fokker–Planck equations with time-independent coefficients.

Light propagation in 2PN approximation in the monopole and quadrupole field of a body at rest: Initial value problem

The light trajectory in the gravitational field of one body at rest with monopole and quadrupole structure is determined in the second post-Newtonian (2PN) approximation. The terms in the geodesic equation for light rays are separated into time-independent tensorial coefficients and four kind of time-dependent scalar functions. Accordingly, the first and second integration of geodesic equation can be reduced in each case to only four kind of scalar master integrals. These integrals can be solved in closed form by recurrence relations. The 2PN terms of monopole and quadrupole contribute less than $1$ nano-arcsecond to the total light deflection. There are, however, enhanced terms in the 2PN light deflection, both in case of monopole and quadrupole. These enhanced 2PN terms are caused by the use of an impact vector which is indispensable for modeling of real astrometric measurements. In case of grazing light rays at Jupiter and Saturn, the enhanced 2PN terms, caused by the quadrupole structure of the body, amount up to 0.95 micro-arcseconds and 0.29 micro-arcseconds, respectively. Thus, the 2PN quadrupole terms are relevant for high-precision astrometry on the sub-micro-arcsecond scale of accuracy.

On the solvability of an initial-boundary value problem for a non-linear fractional diffusion equation

<abstract><p>In this paper, we consider an initial-boundary value problem for a non-linear fractional diffusion equation on a bounded domain. The fractional derivative is defined in Caputo's sense with respect to the time variable and represents the case of sub-diffusion. Also, the equation involves a second order symmetric uniformly elliptic operator with time-independent coefficients. These initial-boundary value problems arise in applied sciences such as mathematical physics, fluid mechanics, mathematical biology and engineering. By using eigenfunction expansions and Banach fixed point theorem, we establish the existence, uniqueness and regularity properties of the solution of the problem.</p></abstract>

Toward Model Reduction for Power System Transients With Physics-Informed PDE

This manuscript reports the first step towards building a robust and efficient model reduction methodology to capture transient dynamics in a transmission level electric power system. Such dynamics is normally modeled on seconds-to-tens-of-seconds time scales by the so-called swing equations, which are ordinary differential equations defined on a spatially discrete model of the power grid. Following Seymlyen (1974) and Thorpe, Seyler, and Phadke (1999), we suggest to map the swing equations onto a linear, inhomogeneous Partial Differential Equation (PDE) of parabolic type in two space and one time dimensions with time-independent coefficients and properly defined boundary conditions. We illustrate our method on the synchronous transmission grid of continental Europe. We show that, when properly coarse-grained, i.e., with the PDE coefficients and source terms extracted from a spatial convolution procedure of the respective discrete coefficients in the swing equations, the resulting PDE reproduces faithfully and efficiently the original swing dynamics. We finally discuss future extensions of this work, where the presented PDE-based modeling will initialize a physics-informed machine learning approach for real-time modeling, $n-1$ feasibility assessment and transient stability analysis of power systems.

Terminal value problem for nonlinear parabolic equation with Gaussian white noise

<abstract><p>In this paper, We are interested in studying the backward in time problem for nonlinear parabolic equation with time and space independent coefficients. The main purpose of this paper is to study the problem of determining the initial condition of nonlinear parabolic equations from noisy observations of the final condition. The final data are noisy by the process involving Gaussian white noise. We introduce a regularized method to establish an approximate solution. We establish an upper bound on the rate of convergence of the mean integrated squared error. This article is inspired by the article by Tuan and Nane <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>.</p></abstract>

On strong solutions of Itô’s equations with σ∈Wd1 and b∈Ld

We consider Itô uniformly nondegenerate equations with time independent coefficients, the diffusion coefficient in Wd,loc1 and the drift in Ld. We prove the unique strong solvability for any starting point and prove that, as a function of the starting point, the solutions are Hölder continuous with any exponent <1. We also prove that if we are given a sequence of coefficients converging in an appropriate sense to the original ones, then the solutions of approximating equations converge to the solution of the original one.

Solution of the System of Two Coupled First-Order ODEs with Second-Degree Polynomial Right-Hand Sides

The explicit solution x n t , $x_{n}\left (t\right ) ,$ n = 1,2, of the initial-values problem is exhibited of a subclass of the autonomous system of 2 coupled first-order ODEs with second-degree polynomial right-hand sides, hence featuring 12 a priori arbitrary (time-independent) coefficients: x ̇ n = c n 1 x 1 2 + c n 2 x 1 x 2 + c n 3 x 2 2 + c n 4 x 1 + c n 5 x 2 + c n 6 , n = 1 , 2 . $$ \dot{x}_{n}=c_{n1}\left( x_{1}\right)^{2}+c_{n2}x_{1}x_{2}+c_{n3}\left( x_{2}\right)^{2}+c_{n4}x_{1}+c_{n5}x_{2}+c_{n6}~,~~~n=1,2~. $$ The solution is explicitly provided if the 12 coefficients cnj (n = 1,2; j = 1,2,3,4,5,6) are expressed by explicitly provided formulas in terms of 10 a priori arbitrary parameters; the inverse problem to express these 10 parameters in terms of the 12 coefficients cnj is also explicitly solved, but it is found to imply—as it were, a posteriori—that the 12 coefficients cnj must then satisfy 4 algebraic constraints, which are explicitly exhibited. Special subcases are also identified the general solutions of which are completely periodic with a period independent of the initial data (“isochrony”), or are characterized by additional restrictions on the coefficients cnj which identify particularly interesting models.

Solutions to linear dissipative quantum systems

We use the characteristic function of the Wigner function (its double Fourier transform) to give solution to any generic open quantum linear systems (systems whose Hamiltonian is at most quadratic). The solution is carried out in terms of the application of the transition matrix of the dynamical evolution in the Fourier space. We address two cases: the time-independent coefficients for which we give the solutions for several dissipative models of the quantum harmonic oscillator and the one-dimensional free particle. In the latter, we also derive a heuristic model for a pure damped motion with suppression of diffusion. For the time-dependent coefficient problem, we give some particular cases that are integrable and derive a second order approximation to the generic case in which all parameters are time-dependent. We additionally explore the solutions of the system when the diffusion processes of the dissipative model lie beyond the weak coupling limit.

Modeling the resistance mechanism of passive knee joint flexion and extension for use in rehabilitation equipment.

The purpose of this study was to model the resistance mechanism of Passive Knee Joint Flexion and Extension to create a similar torque mechanism in rehabilitation equipment. In order to better model the behavior of passive knee tissues, it is necessary to exactly calculate the two coefficients of elasticity of time-independent and time-dependent parts. Ten healthy male volunteers (mean height 176.4+/-4.59 cm) participated in this study. Passive knee joint flexion and extension occurred at velocities of 15, 45, and 120 (degree/s), and in five consecutive cycles and within the range of 0 to 100° of knee movement on the sagittal plane on Cybex isokinetic dynamometer. To ensure that the muscles were relaxed, the electrical activity of knee muscles was recorded. The elastic coefficient, (KS) increased with elevating the passive velocity in flexion and extension. The elastic coefficient, (KP) was observed to grow with the passive velocity increase. While, the viscous coefficient (C) diminished with passive velocity rise in extension and flexion. The heightened passive velocity of the motion resulted in increased hysteresis (at a rate of 42%). The desired of passive velocity is lower so that there is less energy lost and the viscoelastic resistance of the tissue in the movement decreases. The Coefficient of Determination, R2 between the model-responses and experimental curves in the extension was 0.96 < R2 < 0.99 and in flexion was 0.95 < R2 < 0.99. This modeling is capable of predicting the true performance of the components of passive knee movement and we can create a resistance mechanism in the rehabilitation equipment to perform knee joint movement. Quantitative measurements of two elastic coefficients of Time-independent and Time-dependent parts passive knee joint coefficients should be used for better accurate simulation the behavior of passive tissues in the knee which is not seen in other studies.

Numerical solutions of the wave equation using time-independent shearlet coefficients

In this paper, we present a new function (a shearlet) and calculate shearlet coefficients which yield a solution to the wave equation. In our method, the effect of the parameter time which makes the calculations complicated, is reduced.

Lower complexity of motor primitives ensures robust control of high-speed human locomotion

Walking and running are mechanically and energetically different locomotion modes. For selecting one or another, speed is a parameter of paramount importance. Yet, both are likely controlled by similar low-dimensional neuronal networks that reflect in patterned muscle activations called muscle synergies. Here, we challenged human locomotion by having our participants walk and run at a very broad spectrum of submaximal and maximal speeds. The synergistic activations of lower limb locomotor muscles were obtained through decomposition of electromyographic data via non-negative matrix factorization. We analyzed the duration and complexity (via fractal analysis) over time of motor primitives, the temporal components of muscle synergies. We found that the motor control of high-speed locomotion was so challenging that the neuromotor system was forced to produce wider and less complex muscle activation patterns. The motor modules, or time-independent coefficients, were redistributed as locomotion speed changed. These outcomes show that humans cope with the challenges of high-speed locomotion by adapting the neuromotor dynamics through a set of strategies that allow for efficient creation and control of locomotion.

Strong unique continuation for second-order hyperbolic equations with time-independent coefficients

In this paper we prove that if $u$ is a solution to second order hyperbolic equation $\partial^2_tu+a(x)\partial_tu-(div_x\left(A(x)\nabla_x u\right)+b(x)\cdot\nabla_x u+c(x)u)=0$ and $u$ is flat on a segment $\{x_0\}\times (-T,T)$ then $u$ vanishes in a neighborhood of $\{x_0\}\times (-T,T)$.

Semi-Closed Form Prices of Barrier Options in the Time-Dependent CEV and CIR Models

The authors continue a series of articles where prices of the barrier options written on the underlying, whose dynamics follow a one-factor stochastic model with time-dependent coefficients and the barrier, are obtained in semi-closed form; see Carr and Itkin (2020) and Itkin and Muravey (2020). This article extends this methodology to the Cox–Ingersoll–Ross model for zero-coupon bonds and to the constant elasticity of variance model for stocks, which are used as the corresponding underlying for the barrier options. The authors describe two approaches. One is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so it is named “the method of Bessel potentials.” The authors also propose a general scheme for constructing the potential method for any linear differential operator with time-independent coefficients. The second approach is the method of generalized integral transform, which is also extended to the Bessel process. In all cases, a semi-closed solution means that first, we need to numerically solve a linear Volterra equation of the second kind, and then the option price is represented as a one-dimensional integral. The authors demonstrate that their method is computationally more efficient than both the backward and forward finite difference methods, while providing better accuracy and stability. Also, it is shown that these methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities. TOPICS:Options, statistical methods Key Findings • This article extends this methodology to the CIR model for zero-coupon bonds and to the CEV model for stocks, which are used as the corresponding underlying for the barrier options. • One approach is a generalization of the method of heat potentials (for the heat equation) to the Bessel process, so we call it “the method of Bessel potentials.” The second approach is the method of generalized integral transform, which is also extended to the Bessel process. • These methods do not duplicate, but instead they complement each other, as one provides very accurate results at shorter maturities and the other provides such results at longer maturities.