Invariant Equations Research Articles

On the Approximation of Moments for Nonlinear Systems

Model reduction by moment-matching relies upon the availability of the so-called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">moment</i> . If the system is nonlinear, the computation of moments depends on an underlying specific invariance equation, which can be difficult or impossible to solve. This article presents four technical contributions related to the theory of moment matching: first, we identify a connection between moment-based theory and weighted residual methods. Second, we exploit this relation to provide an approximation technique for the computation of nonlinear moments. Third, we extend the definition of nonlinear moment to the case in which the generator is described in explicit form. Finally, we provide an approximation technique to compute the moments in this scenario. The results are illustrated by means of two examples.

A general method to construct invariant PDEs on homogeneous manifolds

Let [Formula: see text] be an [Formula: see text]-dimensional homogeneous manifold and [Formula: see text] be the manifold of [Formula: see text]-jets of hypersurfaces of [Formula: see text]. The Lie group [Formula: see text] acts naturally on each [Formula: see text]. A [Formula: see text]-invariant partial differential equation of order [Formula: see text] for hypersurfaces of [Formula: see text] (i.e., with [Formula: see text] independent variables and [Formula: see text] dependent one) is defined as a [Formula: see text]-invariant hypersurface [Formula: see text]. We describe a general method for constructing such invariant partial differential equations for [Formula: see text]. The problem reduces to the description of hypersurfaces, in a certain vector space, which are invariant with respect to the linear action of the stability subgroup [Formula: see text] of the [Formula: see text]-prolonged action of [Formula: see text]. We apply this approach to describe invariant partial differential equations for hypersurfaces in the Euclidean space [Formula: see text] and in the conformal space [Formula: see text]. Our method works under some mild assumptions on the action of [Formula: see text], namely: A1) the group [Formula: see text] must have an open orbit in [Formula: see text], and A2) the stabilizer [Formula: see text] of the fiber [Formula: see text] must factorize via the group of translations of the fiber itself.

SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS CHARACTERIZING THE DYNAMICS OF A CURRENT COLLECTION SYSTEM FOR AN ELECTRIC LOCOMOTIVE USING SHANNON WAVELET

In this paper, Shannon Wavelet Method (SWM) under certain conditions is proposed so as to numerically integrate a system of non-linear time invariant fractional differential equations and characterize the dynamics of a current collection system for an electric locomotive. Shannon wavelets operational matrices of integration and the Shannon Stretch Matrix (SSM) are utilized to tackle the fractional differential equations containing a term with a stretched argument. Numerical examples are provided to demonstrate the accuracy, efficiency and simplicity of the proposed Shannon wavelet method.

Writing motives and writing self-efficacy of Chinese students in Shanghai and Hong Kong: Measurement invariance and multigroup structural equation analyses

What motivates Chinese students to write? In the present investigation, we examined Chinese students’ writing motives and writing self-efficacy. Grades 4 and 5 Chinese students in Shanghai (n = 332) and Hong Kong (n = 305) completed a survey questionnaire assessing motives for writing using the Writing Motivation Questionnaire and writing self-efficacy based on a revised Writing Self-efficacy Scale. Using multigroup confirmatory factor analyses, measurement invariance was established by examining configural, metric, scalar and residual invariance. Latent means comparison located significant differences between the Chinese groups in relation to writing self-efficacy and writing motives derived from a focus on curiosity, competition and grades. Multigroup structural equation analyses verified a hypothesized direct-mediational model depicting the relationship between writing motives and writing self-efficacy.

A Knot Invariant Defined Based on the Skein Relation with Two Equations

Knot theory is a branch of the geometric topology, the core question of knot theory is to explore the equivalence classification of knots; In other words, for a knot, how to determine whether the knot is an unknot; giving two knots, how to determine whether the two knots are equivalent. To prove that two knots are equivalent, it is necessary to turn one knot into another through the same mark transformation, but to show that two knots are unequal, the problem is not as simple as people think. We cannot say that they are unequal because we can't see the deformation between them. For the equivalence classification problem of knots, we mainly find equivalent invariants between knots. Currently, scholars have also defined multiple knot invariants, but they also have certain limitations, and even more difficult to understand. In this paper, based on existing theoretical results, we define a knot invariant through the skein relation with two equations. To prove this knot invariant, we define a function <i>f</i>(<I>L</I>), and to prove <i>f</i>(<I>L</I>) to be a homology invariant of a non-directed link, we need to show that it remains constant under the Reideminster moves. This article first defines the <i>f_k</i>(<I>L</I>), the property of <i>f</i>(<I>L</I>) is obtained by using the properties of <i>f_k</i>(<I>L</I>). In the process of proof, the induction method has been used many times. The proof process is somewhat complicated, but it is easier to understand. And the common knot invariant is defined by one equation, which defining the knot invariant with two equations in this paper.

First Integrals of Differential Operators from SL(2,ℝ) Symmetries

The construction of first integrals for SL(2,R)-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra sl(2,R). Firstly, we provide for n=2 an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of sl(2,R), and without any kind of integration. Moreover, although there are cases when the two first integrals are functionally independent, it is proved that a second functionally independent first integral arises by a single quadrature. This result is extended for n&gt;2, provided that a solvable structure for an integrable distribution generated by the differential operator associated to the equation and one of the prolonged symmetry generators of sl(2,R) is known. Several examples illustrate the procedures.

Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities

We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe equation with isolated singularities to higher order equations. Our approach uses blow up analysis for local integral equations, and is unified for all critical elliptic equations of order smaller than the dimension. We also prove the existence of Fowler solutions to the global equations, and establish a sup ⁎ inf type Harnack inequality of Schoen for integral equations.

Local well-posedness and blow-up for a family of U(1)-invariant peakon equations

The Cauchy problem for a unified family of integrable U(1)-invariant peakon equations from the NLS hierarchy is studied. As main results, local well-posedness is proved in Besov spaces, and blow-up is established through use of an L1 conservation law.

Discrete Symmetry Transformations of Third Order Ordinary Differential Equations and Applications

Third order ordinary differential equations have already been classified by the Lie algebra they admit. Invariant equations corresponding to these Lie algebras are also available in the literature [17]. In this paper, list of all discrete symmetries corresponding to these invariant ordinary differential equations, are obtained. Some particular examples are given to show the significance of the work.

Toppling Pencils-Macroscopic Randomness from Microscopic Fluctuations.

We construct a microscopic model to study discrete randomness in bistable systems coupled to an environment comprising many degrees of freedom. A quartic double well is bilinearly coupled to a finite number N of harmonic oscillators. Solving the time-reversal invariant Hamiltonian equations of motion numerically, we show that for , the system exhibits a transition with increasing coupling strength from integrable to chaotic motion, following the Kolmogorov-Arnol’d-Moser (KAM) scenario. Raising N to values of the order of 10 and higher, the dynamics crosses over to a quasi-relaxation, approaching either one of the stable equilibria at the two minima of the potential. We corroborate the irreversibility of this relaxation on other characteristic timescales of the system by recording the time dependences of autocorrelation, partial entropy, and the frequency of jumps between the wells as functions of N and other parameters. Preparing the central system in the unstable equilibrium at the top of the barrier and the bath in a random initial state drawn from a Gaussian distribution, symmetric under spatial reflection, we demonstrate that the decision whether to relax into the left or the right well is determined reproducibly by residual asymmetries in the initial positions and momenta of the bath oscillators. This result reconciles the randomness and spontaneous symmetry breaking of the asymptotic state with the conservation of entropy under canonical transformations and the manifest symmetry of potential and initial condition of the bistable system.

On quadrilaterals in the suborbital graphs of the normalizer

n this paper‎, ‎we investigate suborbital graphs formed by $Nbig(Gamma_0(N)big)$-invariant equivalence relation induced on $hat{mathbb{Q}}$‎. ‎Conditions for being an edge are obtained as a main tool‎, ‎then necessary and sufficient conditions for the suborbital graphs to contain a circuit are investigated‎.

Perturbative-Iterative Computation of Inertial Manifolds of Systems of Delay-Differential Equations with Small Delays

Delay-differential equations belong to the class of infinite-dimensional dynamical systems. However, it is often observed that the solutions are rapidly attracted to smooth manifolds embedded in the finite-dimensional state space, called inertial manifolds. The computation of an inertial manifold yields an ordinary differential equation (ODE) model representing the long-term dynamics of the system. Note in particular that any attractors must be embedded in the inertial manifold when one exists, therefore reducing the study of these attractors to the ODE context, for which methods of analysis are well developed. This contribution presents a study of a previously developed method for constructing inertial manifolds based on an expansion of the delayed term in small powers of the delay, and subsequent solution of the invariance equation by the Fraser functional iteration method. The combined perturbative-iterative method is applied to several variations of a model for the expression of an inducible enzyme, where the delay represents the time required to transcribe messenger RNA and to translate that RNA into the protein. It is shown that inertial manifolds of different dimensions can be computed. Qualitatively correct inertial manifolds are obtained. Among other things, the dynamics confined to computed inertial manifolds display Andronov–Hopf bifurcations at similar parameter values as the original DDE model.

Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source

A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

Form-invariant solution to quantum state on the sphere

This paper investigates the quantum states that emerge from the transformation design of conformal mapping on the two-dimensional sphere. Three results are reported. First, the construction of form-invariant spherical harmonics labelled by the fractional quantum number through a scalar potential interaction is given. Second, the form-invariant equation of the charge-monopole system is studied. Rather than the half-integer classification of the monopole harmonics, the quantization of the monopole harmonics here can be any fractional number specified by the conformal index. The gauge equivalent condition of the vector potentials which result in the invariant equation shows that the monopole field and the quantization condition of the pole strength due to Dirac can be generalized to more general vector fields and values in the conformal space. Finally, we explore the quadratic conformal image of the charged particle coupling to the constant monopole field on the sphere. It is shown that the lowest order approximation of the image is the magnetic Hooke-Newton transmutation.

The Hamilton–Jacobi characteristic equations for topological invariants: Pontryagin and Euler classes

By using the Hamilton–Jacobi [HJ] framework the topological theories associated with Euler and Pontryagin classes are analyzed. We report the construction of a fundamental HJ differential where the characteristic equations and the symmetries of the theory are identified. Moreover, we work in both theories with the same phase space variables and we show that in spite of Pontryagin and Euler classes share the same equations of motion their symmetries are different. In addition, we report all HJ Hamiltonians and we compare our results with other formulations reported in the literature.

Caregiver abuse screen for older adults: Urdu translation, validation, factorial equivalence, and measurement invariance

Treating the elder people in an inhumane way is termed as elder abuse. Caregiver Abuse Screen (CASE) is a measure of elder abuse that needed to be translated in the Urdu language for its validation for the Pakistani population owing to its brevity and sound psychometric properties. For Urdu translation, the standard back-translation procedure was adopted. In a purposive dyadic sample of primary caregivers (N = 500; men = 128 and women = 372) of older adults (men = 263 and women = 237) from Lahore and Sargodha districts, the CFA of Urdu Caregiver Abuse Screen (UCASE) revealed a single factor solution with 8 indicators, which demonstrated configural, metric, and scalar invariance as well as comparable latent mean scores across both genders. The caregivers’ UCASE score had a significant positive relationship with depression and a non-significant relationship with the accommodation scores of the older adults, which provided evidence for the convergent and discriminant validities, respectively. Evidence of the concurrent validity was established as the caregivers who acknowledged to be abusive/neglectful scored significantly higher on UCASE as compared to the caregivers who reported not to be abusive/neglectful. Furthermore, the caregivers of the ill older adults scored significantly higher on UCASE as compared to the caregivers of healthy older adults. These pieces of evidence suggest that the UCASE is a reliable and valid screening tool for the caregiver abuse for both men and women.

De Sitter field equations from quadratic curvature gravity: A group theoretical approach

In this paper, the linearized field equations related to the quadratic curvature gravity theory have been obtained in the four-dimensional de Sitter (dS) space–time. The massless spin-2 field equations have been written in terms of the Casimir operators of dS group making use of the ambient space notations. By imposing some simple constraints, arisen from group theoretical interpretation of the field equations, a new four-dimensional Gauss–Bonnet (GB-)like action has been introduced with the related field equations transforming according to the unitary irreducible representations (UIRs) of dS group. Since, the field equations transform according to the UIRs of dS group, the GB-like action, we just obtained, is expected to be a successful model of modified gravity. For more clarity, the gauge invariant field equations have been solved in terms of a gauge-fixing parameter [Formula: see text]. It has been shown that the solution can be written as the multiplication of a symmetric rank-2 polarization tensor and a massless minimally coupled scalar field on dS space. The Krein–Gupta–Bleuler quantization method has been utilized and the covariant two-point function has been calculated in terms of the massless minimally coupled scalar two-point function, using the ambient space notations. It has been written in terms of dS intrinsic coordinates from the ambient space counterpart. The two-point functions are dS invariant and free of any theoretical problems. It means that the proposed model is a successful model of modified gravity and it can produce significant results in the contexts of classical theory of gravity and quantum gravity toy models.

Psychometric Properties of the Copenhagen Burnout Inventory in an Academic Healthcare Institution Sample in the U.S.

The Copenhagen Burnout Inventory (CBI) has demonstrated good psychometric properties among respondents in many different countries, but minimal research exists using the CBI in a U.S.-based sample. The current study represents a secondary analysis of existing CBI data from 1,679 academic health center employees at one mid-size teaching hospital in the southeastern region of the U.S. Analyses assessed CBI scale reliability, confirmatory factorial validity, discriminant validity against a measure of meaningful work, and test invariance for professional role sub-groups (physicians, nurses/physician assistants, and other hospital staff), gender groups, and different age groups. Results provided evidence for good reliability and discriminant validity as well as construct validity supporting the CBI proposed three-factor structure. Configural and metric variance equivalence were demonstrated across the range of employee types, and across age and gender groups. Scalar invariance equivalence was not established, suggesting further research may be needed to support group mean comparisons using the CBI.

Impulse Delay in the Cardiac Conduction System

The physiology or otherwise of blood circulation is predicated on the electrical conduction of the heart. As a rule electrical impulse suffusing the cardiac cells, just like all time-dependent phenomena, transmits with a modicum of time delay. Such delay may be physiological (benign) or pathological; the later is seen as a cardiac liability. This paper treated impulse conduction delay in the cardiac system. A set of matrices resulting from the graph theoretic description of the conduction system was generated and fitted into a continuous time invariant state-space delay equation, and a state-transition matrix solution was sought. An input control-based minimization scheme by which ensuing deleteriousness of pathological delay could be assuaged was proposed.

Massive scalar field on (A)dS space from a massless conformal field in R6

We show how the equations for the scalar field (including the massive, massless, minimally, and conformally coupled cases) on de Sitter and anti-de Sitter [(A)dS] spaces can be obtained from both the SO(2, 4)-invariant equation □ϕ = 0 in R6 and two geometrical constraints defining the (A)dS space. Thus, apart from the equation in R6, the results only follow from the geometry.