Legendre Transform Research Articles

Riemannian F-Manifolds, Bi-Flat F-Manifolds, and Flat Pencils of Metrics

AbstractIn this paper, we study relations between various natural structures on F-manifolds. In particular, given an arbitrary Riemannian F-manifold, we present a construction of a canonical flat F-manifold associated to it. We also describe a construction of a canonical homogeneous Riemannian F-manifold associated to an arbitrary exact homogeneous flat pencil of metrics satisfying a certain non-degeneracy assumption. In the last part of the paper, we construct Legendre transformations for Riemannian F-manifolds.

Optimal investment strategy with constant absolute risk aversion utility under an extended CEV model

In this paper, we develop an extended constant elasticity of variance (CEV) model with stochastic volatility to study an optimal investment strategy problem. This extended CEV model remedies the shortcoming of classical CEV model. In CEV model, the volatility term is an power function of stock price, which only covers firm-specific risks. Consequently, we consider the coefficient of volatility with the mean reverting process to make the volatility involve the market risks to improve the classical CEV model. For the optimal investment objective with a constant absolute risk aversion (CARA) utility function, the analytical solution under the extended CEV model cannot be obtained due to the complicated nonlinearity of the partial differential equation. In this paper we successfully employ a dual method, Legendre transformation, and an asymptotic expansion technique to approach an asymptotic solution. The numerical examples indicate the optimal strategy is an increasing function of the expectation of stock returns and correlations between the two market risks. Besides, it is a decreasing function of interest rate and risk aversion coefficient. In addition, by statistical analysis, we find that the power parameter, the expectation of stock returns and interest rate are all significant factors affecting the investment strategy.

The Hamiltonian of f(R) gravity

We derive conjugate momenta variable tensors and the Hamiltonian equation of the source-free f(R) gravity from first principles using the Legendre transformation of these conjugate momenta variable tensors, conjugate coordinates variables — fundamental metric tensor and its first ordinary partial derivatives with respect to space–time coordinates and second ordinary partial derivatives with respect to space–time coordinates — and the Lagrangian of the f(R) gravity. Interpreting the derived Hamiltonian as the energy of the f(R) gravity we have shown that it vanishes for linear Lagrangians in Ricci scalar curvature without source (e.g., Einstein–Hilbert Lagrangian without matter fields), which is the same result obtained using the stress–energy tensor equation derived from variation of the matter field Lagrangian density. The resulting Hamiltonian equation forbids any negative power law model in the dependence of the f(R) gravity on Ricci scalar curvature: f(R) = αR–r, where r and α are positive real numbers; it also forbids any polynomial equation that contains terms with negative powers of the Ricci scalar curvature including a constant term, in which cases the Hamiltonian function in the Ricci scalar and therefore the energy of the f(R) gravity would attain a negative value and would not be bounded from below. The restrictions imposed by the non-negative Hamiltonian have far-reaching consequences as a result of applying f(R) gravity to the study of black holes and the Friedmann–Lemaître–Robertson–Walker model in cosmology.

Coupling of the phase field approach to the Armstrong-Frederick model for the simulation of ductile damage under cyclic load

The present contribution proposes a thermodynamically consistent model for the simulation of the ductile damage. The model couples the phase field method of fracture to the Armstrong-Frederick plasticity model with kinematic hardening. The latter is particularly suitable for simulating the material behavior under a cyclic load. The model relies on the minimum principle of the dissipation potential. However, the application of this approach is challenging since potentials of coupled methods are defined in different spaces: The dissipation potential of the phase field model is expressed in terms of rates of internal variables, whereas the Armstrong-Frederick model proposes a formulation depending on thermodynamic forces. For this reason, a unique formulation requires the Legendre transformation of one of the potentials. The present work performs the transformation of the Armstrong-Frederick potential, such that final formulation is only expressed in the space of rates of internal variables. With the assumption for the free energy and the joint dissipation potential at hand, the derivation of evolution equations is straightforward. The application of the model is illustrated by selected numerical examples studying the material response for different load cases and sample geometries. The paper provides a comparison with the experimental results as well.

Controlling potential difference between electrodes based on self-consistent-charge density functional tight binding.

A proper understanding and description of the electronic response of the electrode surfaces in electrochemical systems are quite important because the interactions between the electrode surface and electrolyte give rise to unique and useful interfacial properties. Atomistic modeling of the electrodes requires not only an accurate description of the electronic response under a constant-potential condition but also computational efficiency in order to deal with systems large enough to investigate the interfacial electrolyte structures. We thus develop a self-consistent-charge density functional tight binding based method to model a pair of electrodes in electrochemical cells under the constant-potential condition. The method is more efficient than the (ab initio) density functional theory calculations so that it can treat systems as large as those studied in classical atomistic simulations. It can also describe the electronic response of electrodes quantum mechanically and more accurately than the classical counterparts. The constant-potential condition is introduced through a Legendre transformation of the electronic energy with respect to the difference in the number of electrons in the two electrodes and their electrochemical potential difference, through which the Kohn-Sham equations for each electrode are variationally derived. The method is applied to platinum electrodes faced parallel to each other under an applied voltage. The electronic response to the voltage and a charged particle is compared with the result of a classical constant-potential method based on the chemical potential equalization principle.

Large deviations in continuous-time random walks.

We discuss large deviation properties of continuous-time random walks (CTRWs) and present a general expression for the large deviation rate in CTRWs in terms of the corresponding rates for the distributions of steps' lengths and waiting times. In the case of Gaussian distribution of steps' lengths the general expression reduces to a sequence of two Legendre transformations applied to the cumulant generating function of waiting times. The discussion of several examples (Bernoulli and Gaussian random walks with exponentially distributed waiting times, Gaussian random walks with one-sided Lévy and Pareto-distributed waiting times) reveals interesting general properties of such large deviations.

Open systems in classical mechanics

Generalized span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. We introduce categories LagSy and HamSy that, respectively, provide a categorical framework for the Lagrangian and Hamiltonian descriptions of open classical mechanical systems. The morphisms of LagSy and HamSy correspond to such open systems, and the composition of morphisms models the construction of systems from subsystems. The Legendre transformation gives rise to a functor from LagSy to HamSy that translates from the Lagrangian to the Hamiltonian perspective.

A new grand-canonical potential for the thermodynamic description of the reactions in solutions with constant pH

This review emphasizes the necessity to perform additional Legendre transformation of the free energy for a correct thermodynamic description of reactions in the environment with constant pH. The power of this new thermodynamic potential (Gibbs-Alberty free energy ΔGA0) is demonstrated on three examples. First example, a DFT-based study of interaction of cisplatin with glycine, demonstrates the grand canonical character of the potential, which is manifested by the fact that ΔGA0 energy is not dependent on a concrete form of molecules (protonated or deprotonated) used for energy evaluation. This independence on concrete molecular form strongly contradicts to the standard quantum–mechanical approach where only a single protonation state is used to represent each molecule. Second example, the reaction of hydroxo form of cisplatin with methionine which exhibits quite interesting pH dependency, demonstrates that it is possible to predict metal coordination sites of the amino acid based on proton (hydronium) concentration in solution: in neutral environment the metal binds to thio-group, in basic solutions to amino-group of the amino acid, and in acidic solutions to carboxyl group. This prediction is in the remarkable agreement with the pH-dependent migration of the coordination site observed experimentally in Reedijk laboratory. The final example is the combination of the approach with QM/MM MD simulations of the interaction of (hydrated) mercury cation with thymine. In this reaction, ΔGA0 changes from endergonic to exergonic when the pH passes from acidic to basic values.

Application of geometrothermodynamics to the two-dimensional systems: Ideal Bose-Gas and system with strong interaction

In the framework of the method of geometrothermodynamics we studied the properties of equilibrium manifolds of the following thermodynamic systems: a two-dimensional Bose gas, a Berezinsky-Kosterlitz-Thouless system. The results are invariant under the Legendre transformations, i.e. independent of the choice of thermodynamic potential. For the systems under consideration, the corresponding metrics and scalar curvatures are calculated, and their properties are also described. Research of two-dimensional quantum thermodynamic systems is becoming more urgent. It is sufficiently to mention that such systems are related to, for example, topological insulators, graphene, systems with quantum Hall effect, etc. Two-dimensional quantum systems may have a statistical distribution different from distributions of Fermi-Dirac and Bose-Einstein. Geometric approaches in research of these thermodynamic systems certainly open the new perspective. In this paper, the thermodynamic properties of two-dimensional Bose-Gas and Berezinsky-Kosterlitz-Thouless system have been studied with the help of geometrothermodynamics. The main objective was to reproduce the Bose-Einstein condensation for the first system and find possible new phase transitions for the second. In order to study the above mentioned thermodynamic systems, we have consequently calculated the covariant metric tensors of corresponding equilibrium manifolds and their determinants, then counter-variant metric tensors, Christoffel symbols, curvature tensors and corresponding scalar curvatures. Using the thermodynamic potential, we obtained (using the MatLab system) the corresponding geometric values in a wide range of temperature and area. Explicit formulas were also obtained for each geometric quantity but due to their bulkiness we do not present them in this paper. Examples of calculated scalar curvatures for a certain range of parameters T and S are shown in the figures. The figures also show that despite the significantly different behavior of the curvatures depending on the parameters T and S, both metrics lead to the same General result regarding the location of singularities for the corresponding curvatures. Keywords: geometrothermodynamics, Legendre transformations, metric tensor, scalar curvature, two-dimensional Bose gas, Berezinsky-Kosterlitz-Thouless system

Application of geometrothermodynamics to the two-dimensional systems: Ideal Bose-Gas and system with strong interaction

In the framework of the method of geometrothermodynamics we studied the properties of equilibrium manifolds of the following thermodynamic systems: a two-dimensional Bose gas, a Berezinsky-Kosterlitz-Thouless system. The results are invariant under the Legendre transformations, i.e. independent of the choice of thermodynamic potential. For the systems under consideration, the corresponding metrics and scalar curvatures are calculated, and their properties are also described. Research of two-dimensional quantum thermodynamic systems is becoming more urgent. It is sufficiently to mention that such systems are related to, for example, topological insulators, graphene, systems with quantum Hall effect, etc. Two-dimensional quantum systems may have a statistical distribution different from distributions of Fermi-Dirac and Bose-Einstein. Geometric approaches in research of these thermodynamic systems certainly open the new perspective. In this paper, the thermodynamic properties of two-dimensional Bose-Gas and Berezinsky-Kosterlitz-Thouless system have been studied with the help of geometrothermodynamics. The main objective was to reproduce the Bose-Einstein condensation for the first system and find possible new phase transitions for the second. In order to study the above mentioned thermodynamic systems, we have consequently calculated the covariant metric tensors of corresponding equilibrium manifolds and their determinants, then counter-variant metric tensors, Christoffel symbols, curvature tensors and corresponding scalar curvatures. Using the thermodynamic potential, we obtained (using the MatLab system) the corresponding geometric values in a wide range of temperature and area. Explicit formulas were also obtained for each geometric quantity but due to their bulkiness we do not present them in this paper. Examples of calculated scalar curvatures for a certain range of parameters T and S are shown in the figures. The figures also show that despite the significantly different behavior of the curvatures depending on the parameters T and S, both metrics lead to the same General result regarding the location of singularities for the corresponding curvatures. Keywords: geometrothermodynamics, Legendre transformations, metric tensor, scalar curvature, two-dimensional Bose gas, Berezinsky-Kosterlitz-Thouless system

Contact polarizations and associated metrics in geometric thermodynamics

In this work we show that a Legendre transformation is nothing but a mere change of contact polarization from the point of view of contact geometry. Then, we construct a set of Riemannian and pseudo-Riemannian metrics on a contact manifold by introducing almost contact and para-contact structures and we analyze their isometries. We show that it is not possible to find a class of metric tensors which fulfills two properties: on the one hand, to be polarization independent i.e. the Legendre transformations are the corresponding isometries and, on the other, that it induces a Hessian metric into the corresponding Legendre submanifolds. This second property is motivated by the well known Riemannian structures of the geometric description of thermodynamics which are based on Hessian metrics on the space of equilibrium states and whose properties are related to the fluctuations of the system. We find that to define a Riemannian structure with such properties it is necessary to abandon the idea of an associated metric to an almost contact or para-contact structure. We find that even extending the contact metric structure of the thermodynamic phase space the thermodynamic desiderata cannot be fulfilled.

Extended ⋁-systems and trigonometric solutions to the WDVV equations

Rational solutions of the Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations of associativity are given in terms of configurations of vectors, which satisfy certain algebraic conditions known as ⋁-conditions [A. P. Veselov, Phys. Lett. A 261, 297–302 (1999)]. The simplest examples of such configurations are the root systems of finite Coxeter groups. In this paper, conditions are derived that ensure that an extended configuration—a configuration in a space one-dimension higher—satisfies these ⋁-conditions. Such a construction utilizes the notion of a small orbit, as defined in Serganova [Commun. Algebra, 24, 4281–4299 (1996)]. Symmetries of such resulting solutions to the WDVV equations are studied, in particular, Legendre transformations. It is shown that these Legendre transformations map extended-rational solutions to trigonometric solutions, and for certain values of the free data, one obtains a transformation from extended ⋁-systems to the trigonometric almost-dual solutions corresponding to the classical extended affine Weyl groups.

Boundary conditions in topological AdS4/CFT3

We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional mathcal{N} = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.

Alignment of time series with use of the generalized Legendre's transformation and methods of idempotenty mathematics

Alignment of time series [time-series smoothing] identification of the main tendency of development (временнго a trend) by "cleaning" of a time series of the accidental deviations distorting this tendency. At a research of time series of economy (bioinformation science) apply for detection of patterns [1-3]. 
 In this work it is offered to use for this purpose Legendre's transformation well-known in physics and mathematics. Its direct application to poorly regular objects is difficult therefore in work its idempotent analog is defined previously and on its basis the concept of the TRACK for a time series is defined. 
 In recent years within the international center "Cuofus Li" the new field of mathe-matics idempotent or "tropical" mathematics gained intensive development that is reflected in works of the academician V.P. Maslov and his pupils: G.L. Litvinov, A.N. Sobolevsky, etc. 
 The purpose of this work to be beyond duality of the theory of linear vector spaces, using similar concepts of duality of conformally flat Riemannian geometry and of idempotent algebra.
 By analogy with the polar transformation of a conformally flat Riemannian metrics entered in E.D. Rodionov and V.V. Slavsky's works the abstract idempotent analog of transformation of Legendre is under construction. In the MATLAB system the program complex for calculation the TRACK of a time series is created. It is in-vestigated its opportunities for digital processing of time series.

Explicit form of yield conditions dual to a class of dissipation potentials dependent on three invariants

In the paper, a pragmatic approach to finding the dual formulation for isotropic perfectly plastic materials given a dissipation potential dependent on three cylindrical invariants and involving the Ottosen shape function is proposed and illustrated by examples. The main goal is to provide instructions on how to perform the Legendre transformation used when passing from a dissipation potential to its conjugate yield condition and offer some suggestions regarding calibration for particular potentials dependent on the trace of the strain rate tensor and the product of the norm of its deviator and the Ottosen shape function, which covers a wide class of engineering materials. The classic framework for constitutive modelling of thermodynamically consistent materials within the small deformation theory is used. First, general formulae connecting a dissipation potential dependent on three invariants of the strain rate tensor to the coupled yield condition are derived. Then, they are narrowed down for the aforementioned case of dissipation functions dependent on the Lode angle in a way proposed by Ottosen. Finally, three examples are given involving classical potentials: Beltrami’s, Drucker–Prager’s and Mises–Schleicher’s generalised potential using the shape function. Detailed calculations exposing the introduced technique are performed. Also, a method of the calibration of such potentials leading to explicit mathematical formulae is demonstrated, based on the typical tests located on the tension and compression meridians.

Forward and Inverse Dynamics of a Six-Axis Accelerometer Based on a Parallel Mechanism.

The solution of the dynamic equations of the six-axis accelerometer is a prerequisite for sensor calibration, structural optimization, and practical application. However, the forward dynamic equations (FDEs) and inverse dynamic equations (IDEs) of this type of system have not been completely solved due to the strongly nonlinear coupling relationship between the inputs and outputs. This article presents a comprehensive study of the FDEs and IDEs of the six-axis accelerometer based on a parallel mechanism. Firstly, two sets of dynamic equations of the sensor are constructed based on the Newton–Euler method in the configuration space. Secondly, based on the analytical solution of the sensor branch chain length, the coordination equation between the output signals of the branch chain is constructed. The FDEs of the sensor are established by combining the coordination equations and two sets of dynamic equations. Furthermore, by introducing generalized momentum and Hamiltonian function and using Legendre transformation, the vibration differential equations (VDEs) of the sensor are derived. The VDEs and Newton–Euler equations constitute the IDEs of the system. Finally, the explicit recursive algorithm for solving the quaternion in the equation is given in the phase space. Then the IDEs are solved by substituting the quaternion into the dynamic equations in the configuration space. The predicted numerical results of the established FDEs and IDEs are verified by comparing with virtual and actual experimental data. The actual experiment shows that the relative errors of the FDEs and the IDEs constructed in this article are 2.21% and 7.65%, respectively. This research provides a new strategy for further improving the practicability of the six-axis accelerometer.

Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates

Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebański’s first and second heavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4 + 4-dimensional integrable generalisation of the general heavenly equation are found. These are obtained by means of (partial) Legendre transformations. As a particular application, we prove that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors. This generalises the known link between a particular class of self-dual Einstein spaces and the dispersionless Hirota equation encoding three-dimensional Einstein–Weyl geometries.

The Lagrangian and Hamiltonian for RLC Circuit: Simple Case

The Lagrangian and Hamiltonian for series RLC circuit has been formulated. We use the analogical concept of classical mechanics with electrical quantity. The analogy is as follow mass, position, spring constant, velocity, and damping constant corresponding with inductance, charge, the reciprocal of capacitance, electric current, and resistance respectively. We find the Lagrangian for the LC, RL, RC, and RLC circuit by using the analogy and find the kinetic and potential energy. First, we formulate the Lagrangian of the system. Second, we construct the Hamiltonian of the system by using the Legendre transformation of the Lagrangian. The results indicate that the Hamiltonian is the total energy of the system which means the equation of constraints is time independent. In addition, the Hamiltonian of overdamping and critical damping oscillation is distinguished by a certain factor.

Analytical mechanics allows novel vistas on mathematical epidemic dynamics modeling

This contribution aims to shed light on mathematical epidemic dynamics modelling from the viewpoint of analytical mechanics. To set the stage, it recasts the basic SIR model of mathematical epidemic dynamics in an analytical mechanics setting. Thereby, it considers two possible re-parameterizations of the basic SIR model. On the one hand, it is proposed to re-scale time, while on the other hand, to transform the coordinates, i.e.\ the independent variables. In both cases, Hamilton's equations in terms of a suited Hamiltonian as well as Hamilton's principle in terms of a suited Lagrangian are considered in minimal and extended phase and state space coordinates, respectively. The corresponding Legendre transformations relating the various options for the Hamiltonians and Lagrangians are detailed. Ultimately, this contribution expands on a multitude of novel vistas on mathematical epidemic dynamics modelling that emerge from the analytical mechanics viewpoint. As result, it is believed that interesting and relevant new research avenues open up when exploiting in depth the analogies between analytical mechanics and mathematical epidemic dynamics modelling.

Markov chains under nonlinear expectation

AbstractIn this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.