Infinitesimal Form Research Articles

Comparison and localization of invariant functions on strongly pseudoconvex domains

AbstractComparison and localization results for the Lempert function, the Carathéodory distance, and their infinitesimal forms on strongly pseudoconvex domains are obtained. Related results for visible and strongly complete domains are proved.

Symmetries and symmetry reductions of the combined KP3 and KP4 equation

To find symmetries, symmetry groups and group invariant solutions are fundamental and significant in nonlinear physics. In this paper, the finite point symmetry group of the combined KP3 and KP4 (CKP34) equation is found by means of a direct method. The related point symmetries can be obtained simply by taking the infinitesimal form of the finite point symmetry group. The point symmetries of the CKP34 equation constitute an infinite dimensional Kac-Moody–Virasoro algebra. The point symmetry invariant solutions of the CKP34 equation are obtained via the standard classical Lie point symmetry method.

Spinning particles, coadjoint orbits and Hamiltonian formalism

The extensive analysis of the dynamics of relativistic spinning particles is presented. Using the coadjoint orbits method the Hamiltonian dynamics is explicitly described. The main technical tool is the factorization of general Lorentz transformation into pure boost and rotation. The equivalent constrained dynamics on Poincare group (viewed as configuration space) is derived and complete classification of constraints is performed. It is shown that the first class constraints generate local symmetry corresponding to the stability subgroup of some point on coadjoint orbit. The Dirac brackets for second class constraints are computed. Finally, canonical quantization is performed leading to infinitesimal form of irreducible representations of Poincare group.

Anti-Microbial Resistance: A Major Global Issue

Antimicrobial Resistance (AMR) happens when infinitesimal life forms, diseases, developments, and parasites change as time goes on and now do not respond to drugs making them harder to treat and extending the risk of affliction spread, outrageous infirmity, and demise. Because of medical checks, counter-agent poisons and other antimicrobial medications become ineffective and defilements become logically problematic or hard to treat. AMR happens ordinarily after some time, regularly through innate changes. Antimicrobial safe natural elements are found in people, animals, food, plants, and the environment (in water, soil, and air). They can spread starting with one individual then onto the next or among people and animals, including from the food of animal start. The essential drivers of antimicrobial block fuse the maltreatment and maltreatment of antimicrobials; nonattendance of induction to clean water, sanitization and tidiness (WASH) for the two individuals and animals; powerless defilement and affliction expectation and control in clinical benefits workplaces and farms; defenseless permission to quality, sensible medications, vaccinations and diagnostics; nonappearance of care and data; and nonattendance of approval of sanctioning. Excessive enemy of disease use has become one of the top allies of the headway of hostile to contamination resistance. Against microbial stewardship programs appear to be useful in reducing speeds of serum poison obstacles.
 The Netherlands has a minimum speed of against contamination suggested in the Organisation for Economic co-operation and Development (OECD), at a speed of 11.4 portrayed step by step measurements Defined Daily Dose (DDD) per 1,000 people every day in 2011.
 Germany and Sweden also have lower supporting rates, with Sweden's rate had been declining beginning around 2007. Greece, France, and Belgium have high suggesting speeds of more than 28 DDD.

Fresh perspective on gauging the conformal group

We consider the construction of gauge theories of gravity that are invariant under local conformal transformations. We first clarify the geometric nature of global conformal transformations, in both their infinitesimal and finite forms, and the consequences of global conformal invariance for field theories, before reconsidering existing approaches for gauging the conformal group, namely auxiliary conformal gauge theory and biconformal gauge theory, neither of which is generally accepted as a complete solution. We then demonstrate that, provided any matter fields belong to an irreducible representation of the Lorentz group, the recently proposed extended Weyl gauge theory (eWGT) may be considered as an alternative method for gauging the conformal group, since eWGT is invariant under the full set of local conformal transformations, including inversions, as well as possessing conservation laws that provide a natural local generalisation of those satisfied by field theories with global conformal invariance, and also having an `ungauged' limit that corresponds to global conformal transformations. By contrast, although standard Weyl gauge theory also enjoys the first of these properties, it does not share the other two, and so cannot be considered a valid gauge theory of the conformal group.

A stability property in mean field type differential games

The paper is concerned with the feedback approach to the deterministic mean field type differential games. Previously, it was shown that suboptimal strategies in the mean field type differential game can be constructed based on functions of time and probability satisfying the stability condition. This property realizes the dynamic programming principle for the constant control of one player. We present the infinitesimal form of this condition involving analogs of the directional derivatives. In particular, we obtain the characterization of the value function of the deterministic mean field type differential game in the terms of directional derivatives and the set of directions feasible by virtue of the dynamics of the game.

The Value Function as a Decision Support Tool in Unmanned Vehicle Operations

General problems of optimal trajectory generation and of optimal space-time rendezvous for autonomous underwater vehicles affected by time-varying fluid flows are formulated and solved in the framework of dynamic programming. The optimal solutions include optimal trajectories, as well as departure times and positions.The approach consists in using the principle of optimality (PO) to embed, for example, an optimal time to reach a target problem from some fixed position and time into a more general problem of finding the optimal time to reach a target from any point and time. The solution of this general problem is given by the value function, the solution of a Hamilton-Jacobi-Bellman equation (HJBE) which expresses the PO in an infinitesimal form. The HJBE is solved using an efficient parallel numerical solver.The problems of interest are solved either by minimizing the value function over one or more variables (e.g., time) or by using level sets of the value function to coordinate departure times for multiple vehicles to rendezvous at a given target.The paper presents a description and an illustration of the approach and briefly discusses how value-function-based calculations provide a very effective way to solve complex motion planning and coordination problems. The discussion is aided by examples modeling real operational scenarios using current velocity forecasts from a state-of-the-art model of the Sado river estuary in Portugal.

АСИМПТОТИКА ФУНКЦИИ ЦЕНЫ В МОДЕЛЯХ ЭКОНОМИЧЕСКОГО РОСТА

Asymptotic behavior of the value function is studied in an infinite horizon optimal control problem with an unlimited integrand index discounted in the objective functional. Optimal control problems of such type are related to analysis of trends of trajectories in models of economic growth. Stability properties of the value function are expressed in the infinitesimal form. Such representation implies that the value function coincides with the generalized minimax solution of the Hamilton–Jacobi equation. It is shown that that the boundary condition for the value function is substituted by the property of the sublinear asymptotic behavior. An example is given to illustrate construction of the value function as the generalized minimax solution in economic growth models.

A binary infinitesimal form of the Teichmüller metric

Let S be a hyperbolic Riemann surface of finite analytic type. We give a binary infinitesimal form of the Teichmullermetric on the Teichmuller space T(S) of S by use of the fundamental Reich-Strebel inequalities. As an application, we reduce the Teichmuller disk problem to the infinitesimal setting. Moreover, we define the “angle” between geodesic rays in T(S) and give an explicit formula.

On the kernel of the regulator map

By using the infinitesimal methods due to Bloch, Green, and Griffiths in [1,4], we construct an infinitesimal form of the regulator map and verify that its kernel is ΩC/Q1, which suggests that Question 1.1 seems reasonable at the infinitesimal level.

Thermodynamic framework for the ground state of a simple quantum system.

The ground state of a two-level system (associated with probabilities p and 1-p, respectively) defined by a general Hamiltonian H[over ̂]=H[over ̂]_{0}+λV[over ̂] is studied. The simple case characterized by λ=0, whose Hamiltonian H[over ̂]_{0} is represented by a diagonal matrix, is well established and solvable within Boltzmann-Gibbs statistical mechanics; in particular, it follows the third law of thermodynamics, presenting zero entropy (S_{BG}=0) at zero temperature (T=0). Herein it is shown that the introduction of a perturbation λV[over ̂] (λ>0) in the Hamiltonian may lead to a nontrivial ground state, characterized by an entropy S[p] (with S[p]≠S_{BG}[p]), if the Hermitian operator V[over ̂] is represented by a 2×2 matrix, defined by nonzero off-diagonal elements V_{12}=V_{21}=-z, where z is a real positive number. Hence, this new term in the Hamiltonian, presenting V_{12}≠0, may produce physically significant changes in the ground state, and especially, it allows for the introduction of an effective temperature θ (θ∝λz), which is shown to be a parameter conjugated to the entropy S. Based on this, one introduces an infinitesimal heatlike quantity, δQ=θdS, leading to a consistent thermodynamic framework, and by proposing an infinitesimal form for the first law, a Carnot cycle and thermodynamic potentials are obtained. All results found are very similar to those of usual thermodynamics, through the identification T↔θ, and particularly the form for the efficiency of the proposed Carnot Cycle. Moreover, S also follows a behavior typical of a third law, i.e., S→0, when θ→0.

A binary infinitesimal form of teichmüller metric and angles in an asymptotic teichmüller space

The geometry of Teichmüller metric in an asymptotic Teichmüller space is studied in this article. First, a binary infinitesimal form of Teichmüller metric on AT(X) is proved. Then, the notion of angles between two geodesic curves in the asymptotic Teichmüller space AT(X) is introduced. The existence of such angles is proved and the explicit formula is obtained. As an application, a sufficient condition for non-uniqueness geodesics in AT(X) is obtained.

Laplace type invariants for variable coefficient mKdV equations

We consider a class of variable-coefficient mKdV equations. We derive the equivalence transformations in the infinitesimal form and we employ them to construct differential invariants of the respective equivalence algebra. Operators of invariant differentiation are also constructed. Applications, similar to Laplace invariants, are presented.

Foundations of higher-order variational theory on Grassmann fibrations

A setting for higher-order global variational analysis on Grassmann fibrations is presented. The integral variational principles for one-dimensional immersed submanifolds are introduced by means of differential 1-forms with specific properties, similar to the Lepage forms from the variational calculus on fibred manifolds. Prolongations of immersions and vector fields to the Grassmann fibrations are defined as a geometric tool for the variations of immersions, and the first variation formula in the infinitesimal form is derived. Its consequences, the Euler–Lagrange equations for submanifolds and the Noether theorem on invariant variational functionals are proved. Examples clarifying the meaning of the Noether theorem in the context of variational principles for submanifolds are given.

Derivation of quantum Chernoff metric with perturbation expansion method

We investigate a measure of distinguishability defined by the quantum Chernoff bound, which naturally induces the quantum Chernoff metric over a manifold of quantum states. Based on a quantum statistical model, we alternatively derive this metric by means of perturbation expansion. Moreover, we show that the quantum Chernoff metric coincides with the infinitesimal form of the quantum Hellinger distance, and reduces to the variant version of the quantum Fisher information for the single-parameter case. We also give the exact form of the quantum Chernoff metric for a qubit system containing a single parameter.

Generalisations of the strain-energy function of linear elasticity to model biological soft tissue

Strain measures consistent with the linear, infinitesimal form of the strain-energy function are obtained within the context of isotropic, homogeneous, compressible, and non-linear elasticity. It will be shown that there are two distinct families of such measures. One family has already been much studied in the literature, the most important member being the strains whose principal values are a function only of the corresponding principal stretches. The second family of strains appears new. The motivation for studying such strains is the intuitive expectation that, for at least moderate deformations, a good fit with experimental data from material characterisation tests will be obtained with the corresponding strain-energy functions. In particular, there is the expectation that such models could prove useful for the modelling of biological soft tissue, whose physiological response is characterised by moderate strains. It will be shown that this is indeed the case for simple tension tests on porcine brain tissue.

$${\mathcal{W}}$$ -Symmetry of the Adèlic Grassmannian

We give a geometric construction of the W_{1+infty} vertex algebra as the infinitesimal form of a factorization structure on an adelic Grassmannian. This gives a concise interpretation of the higher symmetries and Backlund-Darboux transformations for the KP hierarchy and its multicomponent extensions in terms of a version of "W_{1+infty}-geometry": the geometry of D-bundles on smooth curves, or equivalently torsion-free sheaves on cuspidal curves.

Finite Symmetry Transformation Groups and Some Exact Solutions to (2+1)-Dimensional Cubic Nonlinear Schrödinger Equantion

Making use of the direct method proposed by Lou et al. and symbolic computation, finite symmetry transformation groups for a (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation and its corresponding cylindrical NLS equations are presented. Nine related linear independent infinitesimal generators can be obtained from the finite symmetry transformation groups by restricting the arbitrary constants in infinitesimal forms. Some exact solutions are derived from a simple travelling wave solution.

Bounded Motions of the Dynamical Systems Described by Differential Inclusions

The boundedness of the motions of the dynamical system described by a differential inclusion with control vector is studied. It is assumed that the right-hand side of the differential inclusion is upper semicontinuous. Using positionally weakly invariant sets, sufficient conditions for boundedness of the motions of a dynamical system are given. These conditions have infinitesimal form and are expressed by the Hamiltonian of the dynamical system.

The bilinear negative Kadomtsev–Petviashvili system and its Kac–Moody–Virasoro symmetry group

The symmetry transformation group of the bilinear negative Kadomtsev–Petviashvili system is studied by means of a direct method. The Kac–Moody–Virasoro-type Lie point symmetry algebra is found to be a special infinitesimal form of the symmetry group.