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We show how one can construct conservation laws of Euler-Lagrange-type equations via Noether-type symmetry operators associated with what we term partial Lagrangians. This is even in the case when a system does not directly have a usual Lagrangian, e.g. scalar evolution equations. These Noether-type symmetry operators do not form a Lie algebra in general. We specify the conditions under which they do form an algebra. Furthermore, the conditions under which they are symmetries of the Euler-Lagrange-type equations are derived. Examples are given including those that admit a standard Lagrangian such as the Maxwellian tail equation, and equations that do not such as the heat and nonlinear heat equations. We also obtain new conservation laws from Noether-type symmetry operators for a class of nonlinear heat equations in more than two independent variables.

A connection between the symmetries of manifolds and differential equations is sought through the geodesic equations of maximally symmetric spaces, which have zero, constant positive or constant negative curvature. It is proved that for a space admitting so(n + 1) or so(n, 1) as the maximal isometry algebra, the symmetry of the geodesic equations of the space is given by so(n + 1) ⊕ d 2 or so(n, 1) ⊕ d 2 (where d 2 is the two-dimensional dilation algebra), while for those admitting so(n) ⊕ s R n (where ⊕ s represents semidirect sum) the algebra is sl(n + 2). A corresponding result holds on replacing so(n) by so( p, q) with p + q = n. It is conjectured that if the isometry algebra of any underlying space of non-zero curvature is h, then the Lie symmetry algebra of the geodesic equations is given by h ⊕ d 2 , provided that there is no cross-section of zero curvature at the point under consideration. If there is a flat subspace of dimension m, then the symmetry group becomes h ⊕ sl(m + 2).

SUMMARYAfter the initial seminal works of Sophus Lie on ordinary differential equations, several important results on point symmetry group analysis of ordinary differential equations have been obtained. In this review, we present the salient features of point symmetry group classification of scalar ordinary differential equations: linear nth-order, second-order equations as well as related results. The main focus here is the contributions of Peter Leach, in this area, in whose honour this paper is written on the occasion of his 65th birthday celebrations.

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The findings suggest a possible role for ultraviolet radiation and chronic immunosuppression in the induction of malignant squamous differentiation in a subset of EPCs. Further reports on this histological variant of EPC are required to determine whether a pathogenetic link does indeed exist or whether these tumours simply represent a unique variant of squamous cell carcinoma with divergent acrosyringial differentiation.

We discuss the f (R) gravity model in which the origin of dark energy is identified as a modification of gravity. The Noether symmetry with gauge term is investigated for the f (R) cosmological model. By utilization of the Noether Gauge Symmetry (NGS) approach, we obtain two exact forms f (R) for which such symmetries exist. Further it is shown that these forms of f (R) are stable.

The classical generation theorem of conservation laws from known ones for a system of differential equations which uses the action of a canonical Lie-Bäcklund generator is extended to include any Lie-Bäcklund generator. Also, it is shown that the Lie algebra of Lie-Bäcklund symmetries of a conserved vector of a system is a subalgebra of the Lie-Bäcklund symmetries of the system. Moreover, we investigate a basis of conservation laws for a system and show that a generated conservation law via the action of a symmetry operator which satisfies a commutation rule is nontrivial if the system is derivable from a variational principle. We obtain the conservation laws of a class of nonlinear diffusion-convection and wave equations in (1 + 1)-dimensions. In fact we find a basis of conservation laws for the diffusion equations in the special case when it admits proper Lie-Bäcklund symmetries. Other examples are presented to illustrate the theory.