Infinitesimal Transformations Research Articles

Observer-based control for nonlinear neutral type stochastic model with fractional Gaussian noise and input saturation

This paper discusses the stabilization issue for a class of nonlinear neutral-type stochastic systems subject to time-varying delays, linear fractional uncertainties, actuator faults, input saturation and fractional stochastic noise. Primarily, the derivation analysis for the stability and stabilization for the addressed systems is carried out by adopting a fractional infinitesimal operator. A fault-tolerant controller under saturation effect will be designed in order to guarantee the stabilization of the considered systems. Further, a set of stability criteria is obtained with the help of the fractional infinitesimal operator and by choosing a suitable Lyapunov-Krasovskii functional. These stability criteria are derived in the form of linear matrix inequalities that are solved by the MATLAB LMI toolbox. Finally, simulation results are provided to illustrate the effectiveness of our findings through a numerical example.

Аффинные преобразования касательного расслоения общего пространства путей

In this paper, we study infinitesimal transformations of the tangent bundle of a common path space. The general path space is a genera­liza­tion space of the affine connectivity. By affine connectivity of the com­mon path space, we construct an affine connection on the tangent bundle. For the infinitesimal transformation of the tangent bundle, a system of in­va­riance equations for the constructed affine connectivity is compiled. This system is a system of second-order differential equations with res­pect to the components of the infinitesimal transformation. The main re­sults of the article are obtained by analyzing this system taking into account the properties of homogeneous functions. It is shown that the comp­lete lift of an infinitesimal transformation of base is an infinitesimal af­fine motion of a tangent bundle if and only if the infinitesimal transfor­ma­tion of base is an affine motion in the general path space. Necessary and sufficient conditions are found that the infinitesimal transformation of a tangent bundle generated by a vertical vector field leaves the affine connec­ti­vity of the tangent bundle invariant. Conditions are given that are neces­sa­ry and sufficient so that the infinitesimal transformation of a tangent bundle with affine connectivity that preserves layers is an affine motion.

Fractional infinite time-delay evolution equations with non-instantaneous impulsive

<abstract><p>This dissertation is regarded to investigate the system of infinite time-delay and non-instantaneous impulsive to fractional evolution equations containing an infinitesimal generator operator. It turns out that its mild solution is existed and is unique. Our model is built using a fractional Caputo approach of order lies between 1 and 2. To get the mild solution, the families associated with cosine and sine which are linear strongly continuous bounded operators, are provided. It is common to use Krasnoselskii's theorem and the Banach contraction mapping principle to prove the existence and uniqueness of the mild solution. To confirm that our results are applicable, an illustrative example is introduced.</p></abstract>

Membership-Function-Dependent Design of Quantized Fuzzy Sampled-Data Controller for Semi-Markovian Jump Systems With Actuator Faults

This article concerns a fuzzy sampled-data controller for nonlinear semi-Markovian jump systems (SMJS) via the fuzzy dependent stochastic looped-Lyapunov functional (FSLF) approach. To do this, with the help of the Takagi–Sugeno (T–S) method, the stochastic jump parameters, actuator faults, and logarithmic quantizer are all considered in a unified framework. The stochastic parameters are generated by the semi-Markov process, whose transition rates depend upon the sojourn time. The nonlinear SMJS depicts real-time models subject to unpredictable structural changes, such as single-link robotic arm (S-LRA) systems. A key issue under consideration is how to design a fuzzy sampled-data controller to ensure stochastic stability for the nonlinear SMJS in the presence of actuator faults and state quantization. For this purpose, by using the weak infinitesimal operator of constructed FSLF, sufficient conditions are determined to realize the stochastic stability of the proposed nonlinear systems. Then, the stochastic stability conditions for the proposed sampled-data controller scheme can be derived under a linear matrix inequality framework. The validity of the theoretical findings is verified by Rossler’s chaotic systems. The S-LRA model is demonstrated to illustrate the applicability and efficacy of the proposed controller scheme.

Simulating gravitational waves passing through the spacetime of a black hole

We investigate how GWs pass through the spacetime of a Schwarzschild black hole using time-domain numerical simulations. Our work is based on the perturbed $3+1$ Einstein's equations up to the linear order. We show explicitly that our perturbation equations are covariant under infinitesimal coordinate transformations. Then we solve a symmetric second-order hyperbolic wave equation with a spatially varying wave speed. As the wave speed in our wave equation vanishes at the horizon, our formalism can naturally avoid boundary conditions at the horizon. Our formalism also does not contain coordinate singularities and, therefore, does not need regularity conditions. Then, based on our code, we simulate both finite and continuous initially plane-fronted wave trains passing through the Schwarzschild black hole. We find that for the finite wave train, the wave zone of GWs is wildly twisted by the black hole. While for the continuous wave train, unlike geometric optics, GWs cannot be sheltered by the black hole. A strong beam and an interference pattern appear behind the black hole along the optical axis. Moreover, we find that the back-scattering due to the interaction between GWs and the background curvature is strongly dependent on the direction of the propagation of the trailing wavefront relative to the black hole. Finally, for a realistic input waveform generated by binary black holes, we find that the lensed waveform in the merger and ringdown phases is much longer than that of the input waveform due to the effect of back-scattering.

Locally Homogeneous Manifolds Defined by Lie Algebra of Infinitesimal Affine Transformations

This article deals with Lie algebra G of all infinitesimal affine transformations of the manifold M with an affine connection, its stationary subalgebra ℌ⊂G, the Lie group G corresponding to the algebra G, and its subgroup H⊂G corresponding to the subalgebra ℌ⊂G. We consider the center ℨ⊂G and the commutant [G,G] of algebra G. The following condition for the closedness of the subgroup H in the group G is proved. If ℌ∩ℨ+G;G=ℌ∩[G;G], then H is closed in G. To prove it, an arbitrary group G is considered as a group of transformations of the set of left cosets G/H, where H is an arbitrary subgroup that does not contain normal subgroups of the group G. Among these transformations, we consider right multiplications. The group of right multiplications coincides with the center of the group G. However, it can contain the right multiplication by element 𝒽¯, belonging to normalizator of subgroup H and not belonging to the center of a group G. In the case when G is in the Lie group, corresponding to the algebra G of all infinitesimal affine transformations of the affine space M and its subgroup H corresponding to its stationary subalgebra ℌ⊂G, we prove that such element 𝒽¯ exists if subgroup H is not closed in G. Moreover 𝒽¯ belongs to the closures H¯ of subgroup H in G and does not belong to commutant G,G of group G. It is also proved that H is closed in G if P+ℨ∩ℌ=P∩ℌ for any semisimple algebra P∈G for which P+ℜ=G.

A study of conformal almost Ricci solitons on Kenmotsu manifolds

The goal of this paper is to study conformal almost Ricci solitons within the framework of Kenmotsu manifolds. First, we demonstrate that if the potential vector field is Jacobi along the Reeb vector field, then the soliton reduces to a conformal Ricci soliton. If the manifold is [Formula: see text]-Einstein Kenmotsu manifold, we show that either the manifold is of constant scalar curvature or the potential vector field is an infinitesimal contact transformation. In addition, if we consider the soliton vector field as a contact vector field, then either the gradient of [Formula: see text] is pointwise collinear with the Reeb vector field or the manifold becomes [Formula: see text]-Einstein. Lastly, we develop an example of a conformal almost Ricci soliton on the Kenmotsu manifold.

Conformal Ricci soliton and almost conformal Ricci soliton in paracontact geometry

In this paper, we study conformal Ricci soliton and almost conformal Ricci soliton within the framework of paracontact manifolds. Here, we have shown the characteristics of the soliton vector field and the nature of the manifold if para-Sasakian metric satisfies conformal Ricci soliton. We also demonstrate the feature of the soliton vector field V and scalar curvature when the para-Sasakian manifold admitting conformal Ricci soliton and vector field is pointwise collinear with the characteristic vector field [Formula: see text]. Next, we prove that if a K-paracontact manifold confesses a gradient conformal Ricci soliton, then it is Einstein. Next, we show that a para-Sasakian metric reveals with an almost conformal Ricci soliton that is either Einstein or [Formula: see text]-Einstein metric if the soliton vector field V is an infinitesimal contact transformation. Lastly, we decorate an example of conformal Ricci soliton on para-Sasakian manifold.

The lie symmetry analysis of third order Korteweg-de Vries equation

This study sought to analyse the Lie symmetry of third order Korteweg-de Vries equation. Solving nonlinear partial differential equations is of great importance in the world of dynamics. Korteweg-de Vries equations are partial differential equations arising from the theory of long waves, modelling of shallow water waves, fluid mechanics, plasma fluids and many other nonlinear physical systems, and their effects are relevant in real life. In this study, Lie symmetry analysis is demonstrated in finding the symmetry solutions of the third-order KdV equation of the form. The study systematically showed the formula to find the specific solution attained by developing prolongations, infinitesimal transformations and generators, adjoint symmetries, variation symmetries, invariant transformation and integrating factors to obtain all the lie groups presented by the equation. In conclusion, infinitesimal generators, group transformations and symmetry solutions of third-order KdV equation are acquired using a method of Lie symmetry analysis. This was achieved by generating infinitesimal generators which act on the KdV equation to form infinitesimal transformations. It can be seen from the solutions of this paper that the Lie symmetry analysis method is an effective and best mathematical technique for studying linear and nonlinear PDEs and ODEs.

Dynamics of Dirac concentrations in the evolution of quantitative alleles with sexual reproduction

A proper understanding of the links between varying gene expression levels and complex trait adaptation is still lacking, despite recent advances in sequencing techniques leading to new insights on their importance in some evolutionary processes. This calls for extensions of the continuum-of-alleles framework first introduced by Kimura (1965 Proc. Natl Acad. Sci. USA 54 731–36) that bypass the classical Gaussian approximation. Here, we propose a novel mathematical framework to study the evolutionary dynamics of quantitative alleles for sexually reproducing populations under natural selection and competition through an integro-differential equation. It involves a new reproduction operator which is nonlinear and nonlocal. This reproduction operator is different from the infinitesimal operator used in other studies with sexual reproduction because of different underlying genetic structures. In an asymptotic regime where initially the population has a small phenotypic variance, we analyse the long-term dynamics of the phenotypic distributions according to the methodology of small variance (Diekmann et al 2005 Theor. Popul. Biol. 67 257–71). In particular, we prove that the reproduction operator strains the limit distribution to be a product measure. Under some assumptions on the limit equation, we show that the population remains monomorphic, that is the phenotypic distribution remains concentrated as a moving Dirac mass. Moreover, in the case of a monomorphic distribution, we derive a canonical equation describing the dynamics of the dominant alleles.

4D supersymmetric gauge theories of spacetime translations

The paper addresses the question whether in four spacetime dimensions, besides standard supergravity theories, field theories exist whose symmetries include local spacetime translations and supersymmetries generated by transformations whose commutators contain infinitesimal local spacetime translations. Several new supersymmetric free field theories are found which may provide theories of that type. It is outlined how these free field theories may be extended to globally supersymmetric theories with the desired properties. One class of such theories is constructed explicitly. These theories are similar to globally supersymmetric Yang-Mills theories in flat spacetime but have supersymmetries generated by transformations whose commutators generate infinitesimal general coordinate transformations with field dependent gauge parameters.

Preservation of adiabatic invariants and geometric numerical algorithm for disturbed nonholonomic systems

Perturbations to Mei symmetry and the numerical algorithm of disturbed nonholonomic systems are studied under total variational discretization. The discrete equations on regular lattices of nonholonomic systems in the undisturbed and the disturbed cases are presented. The determining equations of Mei symmetry are established for undisturbed and disturbed systems. The exact invariants of Noether type led by Mei symmetry for undisturbed nonholonomic systems are given under infinitesimal transformations of Lie groups. For discrete disturbed nonholonomic systems, the condition of existence of adiabatic invariants led by perturbation to Mei symmetry and their forms are presented. The numerical simulations demonstrate that the geometric numerical algorithm has a higher precision and longer time stability than the standard numerical method.

On almost ∗-Ricci soliton

Abstract. In the present paper, we prove three fundamental results concerning almost ∗-Ricci soliton in the framework of para-Sasakian manifold. The paper is organised as follows:• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V is Jacobi along Reeb vector field ξ, then g becomes a ∗-Ricci soliton.• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V as infinitesimal paracontact transformation, then V is killing and g is η-Einstein.• If a para-Sasakian metric g represents an almost ∗-Ricci soliton with potential vector field V is collinear with the Reeb vector field ξ, then λ = 0, V is strict and g is η-Einstein.

Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry

Abstract We prove that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal η \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η \eta -Ricci soliton is Einstein if its potential vector field V V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η \eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η \eta -Ricci soliton and satisfy our results. We also have studied conformal η \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.

CERTAIN RESULTS ON $\eta$-RICCI SOLITIONS AND ALMOST $\eta$-RICCI SOLITONS

We prove that if an $\eta$-Einstein para-Kenmotsu manifold admits a $\eta$-Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a $\eta$-Ricci soliton is Einstein if its potential vector field $V$ is infinitesimal paracontact transformation or collinear with the Reeb vector field. Further, we prove that if a para-Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits $\eta$-Ricci soliton and satisfy our results. We also have studied $\eta$-Ricci soliton in 3-dimensional normal almost paracontact metric manifolds and we show that if in a 3-dimensional normal almost paracontact metric manifold with $\alpha, \beta $ = constant, the metric is $\eta$-Ricci soliton, where potential vector field $V$ is collinear with the characteristic vector field $\xi$, then the manifold is $\eta$-Einstein manifold.

Lie Symmetry Analysis and Conservation Laws of the Axially Loaded Euler Beam

By applying the Lie symmetry method, group-invariant solutions are constructed for axially loaded Euler beams. The corresponding mathematical models of the beams are formulated. After introducing the infinitesimal transformations, the determining equations of Lie symmetry are proposed via Lie point transformations acting on the original equations. The infinitesimal generators of symmetries of the systems are presented with Maple. The corresponding vector fields are given to span the subalgebra of the systems. Conserved vectors are derived by using two methods, namely, the multipliers method and Noether’s theorem. Noether conserved quantities are obtained using the structure equation, satisfied by the gauge functions. The fluxes of the conservation laws could also be proposed with the multipliers. The relations between them are discussed. Furthermore, the original equations of the systems could be transformed into ODEs and the exact explicit solutions are provided.

Chirped self-similar pulses and envelope solutions for a nonlinear Schrödinger's in optical fibers using Lie group method

In this work, we present an application of Lie group analysis to study the generalized derivative nonlinear Schrödinger equation, which governs the evolution of a nonlinear wave and plays an important role in the propagation of short pulses in optical fiber systems. To construct Lie group reductions, we study the symmetry properties and introduce various infinitesimal operators. Further, we obtain self-similar solutions and periodic soliton solutions of the generalized derivative nonlinear Schrödinger equation. This type of solution plays a vital role in the study of the blow-up and asymptotic behavior of non-global solutions. And at the end, we present graphs for each solution by considering the physical meaning of the solutions.

Almost *-η-Ricci solitons on Kenmotsu pseudo-Riemannian manifolds

Abstract In this paper, we aim to study a special type of metric called almost * {*} -η-Ricci soliton on the special class of contact pseudo-Riemannian manifold. First, we prove that a Kenmotsu pseudo-Riemannian metric as an * {*} -η-Ricci soliton is Einstein if either it is η-Einstein or the potential vector field V is an infinitesimal contact transformation. Further, we prove that if a Kenmotsu pseudo-Riemannian manifold admits an almost * {*} -η-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present an example of * {*} -η-Ricci solitons which illustrate our results.

The theory of symmetric tensor field: From fractons to gravitons and back

We consider the theory of a symmetric tensor field in 4D, invariant under a subclass of infinitesimal diffeomorphism transformations, where the vector diff parameter is the 4-divergence of a scalar parameter. The resulting gauge symmetry characterizes the “fracton” quasiparticles and identifies a theory which depends on a dimensionless parameter, which cannot be reabsorbed by a redefinition of the tensor field, despite the fact that the theory is free of interactions. This kind of “electromagnetic gauge symmetry” is weaker that the original diffeomorphism invariance, in the sense that the most general action contains, but is not limited to, linearized gravity, and we show how it is possible to switch continuously from linearized gravity to a mixed phase where both gravitons and fractons are present, without changing the degrees of freedom of the theory. The gauge fixing procedure is particularly rich and rather peculiar, and leads to the computation of propagators which in the massive case we ask to be tachyonic-free, thus constraining the domain of the parameter of the theory. Finally, a closer contact to fractons is made by the introduction of a parameter related to the “rate of propagation”. For a particular value of this parameter the theory does not propagate at all, and we guess that, for this reason, the resulting theory should be tightly related to the fracton excitations.

Lie Symmetry Theorem for Nonshifted Birkhoffian Systems on Time Scales

The Lie theorem for Birkhoffian systems with time-scale nonshifted variational problems are studied, including free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system. First, the time-scale nonshifted generalized Pfaff-Birkhoff principle is established, and the dynamical equations for three Birkhoffian systems under nonshifted variational problems are deduced. Afterwards, in the time-scale nonshifted variational problems, by introducing the infinitesimal transformations, Lie symmetry for free Birkhoffian system, generalized Birkhoffian system and constrained Birkhoffian system are defined respectively. Then Lie symmetry theorems for three kinds of Birkhoffian systems are deduced and proved. In the end, three examples are given to explain the applications for the results.