Peter Leach

The Riccati and Ermakov-Pinney hierarchies

Abstract Rota-Baxter operators or relations were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. In this paper, we commence to study the Rota-Baxter operators of weight zero on pre-Lie algebras. Such operators satisfy (the operator form of) the classical Yang-Baxter equation on the sub-adjacent Lie algebras of the pre-Lie algebras. We not only study the invertible Rota-Baxter operators on pre-Lie algebras, but also give some interesting construction of Rota-Baxter operators. Furthermore, we give all Rota-Baxter operators on 2-dimensional complex pre-Lie algebras and some examples in higher dimensions.

ASPECTS OF PROPER DIFFERENTIAL SEQUENCES OF ORDINARY DIFFERENTIAL EQUATIONS

We define a proper differential sequence of ordinary differential equations and introduce a method for deriving an alternate sequence of integrals for such a sequence. We describe some general properties, illustrated by several examples.

Nonlinear ordinary differential equations : a discussion on symmetries and singularities

Two essential methods, the symmetry analysis and the singularity analysis, for the study of the integrability of nonlinear ordinary differential equations is the purpose of this work. The main similarities and the differences of these two different methods are discussed.

On the Hojman conservation quantities in Cosmology

Abstract We discuss the application of the Hojmans Symmetry Approach for the determination of conservation laws in Cosmology, which has been recently applied by various authors in different cosmological models. We show that Hojmans method for regular Hamiltonian systems, where the Hamiltonian function is one of the involved equations of the system, is equivalent to the application of Noethers Theorem for generalized transformations. That means that for minimally-coupled scalar field cosmology or other modified theories which are conformally related with scalar-field cosmology, like f ( R ) gravity, the application of Hojmans method provide us with the same results with that of Noethers Theorem. Moreover we study the special Ansatz. ϕ ( t ) = ϕ ( a ( t ) ) , which has been introduced for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz, the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of Noetherian conservation laws for the free particle. On the other hand, for f ( T ) teleparallel gravity, it is not the existence of Hojmans conservation laws which provide us with the special function form of f ( T ) functions, but the requirement that the reduced second-order differential equation admits a Jacobi Last multiplier, while the new conservation law is nothing else that the Hamiltonian function of the reduced equation.

A NOVEL RICCATI SEQUENCE

Hierarchies of evolution partial differential equations have become well-established in the literature over the last thirty years. More recently sequences of ordinary differential equations have been introduced. Of these perhaps the most notable is the Riccati Sequence which has beautiful singularity, symmetry and integrability properties. We examine a variation of this sequence and find that there are some remarkable changes in properties consequential upon this variation.

Properties of the Calogero-Degasperis-Ibragimov-Shabat differential sequence

We present a differential sequence based upon the Calogero-Degasperis-Ibragimov-Shabat Equation and determine first integrals and the general solution. Under suitable transformations eachmember of the differential sequence can be recast as a product of two factors and we report some of the properties of the factored form.

The algebraic properties of the space- and time-dependent one-factor model of commodities

Abstract We consider the one-factor model of commodities for which the parameters of the model depend upon the stock price or upon the time. For that model we study the existence of group-invariant transformations. When the parameters are constant, the one-factor model is maximally symmetric. That also holds for the time-dependent problem. However, in the case for which the parameters depend upon the stock price (space) the one-factor model looses the group invariants. For specific functional forms of the parameters the model admits other possible Lie algebras. In each case we determine the conditions which the parameters should satisfy in order for the equation to admit Lie point symmetries. Some applications are given and we show what should be the precise relation amongst the parameters of the model in order for the equation to be maximally symmetric. Finally we discuss some modifications of the initial conditions in the case of the space-dependent model. We do that by using geometric techniques.

Solution of the Master Equation for Quantum Brownian Motion Given by the Schrödinger Equation

We consider the master equation of quantum Brownian motion, and with the application of the group invariant transformation, we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schrodinger Equation.

Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility

We perform a classification of the Lie point symmetries for the Black--Scholes--Merton Model for European options with stochastic volatility,

The Globalisation of Applied Mathematics

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