In this paper Lie’s formalism is applied to deduce classes of solutions of a nonlinear partial differential equation (nPDE) of second order with quadratic nonlinearity. The equation has the meaning of a field equation appearing in the formulation of kinetic models. Similarity solutions and transformations are given in a most general form derived to the first time in terms of reciprocal Jacobian...

The purpose of this note is to explain how to use the equivariant method of moving frames, [1, 10], to derive the differential invariants, invariant differential operators and invariant differential forms for surfaces in three-dimensional Euclidean space. This is, of course, a very classical problem and the results are not new; for the classical moving frame derivation, see, for instance, [2, 3...

For a vector bundle E → X with X a smooth variety over a field we prove a formula for the proper pushforward π∗ : A ∗ P(E ) → A∗X, where π : P(E ) → X denotes the associated projective bundle of lines in E . Our formula is relative in the sense that it depends on the rank of E while being independent of the dimension of X. As applications, we apply the formula to compute the pushforward to X of...

Invariant sets of dynamic systems play an important role in many situations when the dynamic behavior is constrained in some way. Knowing that a set in the state space of a system is invariant means that we have bounds on the behavior. We can verify that pre-specified bounds which originate from, for example, safety restrictions, physical constraints, or state-feedback magnitude bounds are not ...

We construct and classify the joint invariants and joint differential invariants of binary and ternary forms. In particular, we prove that the differential invariant algebra of ternary forms is generated by a single third order differential invariant. To connect our results with earlier analysis of Kogan, we develop a general method for relating differential invariants associated with different...

Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the random invariant manifold is almost surely asymptotically complete. The asymptotic dynamical behavior is thus described by a stochastic ordinary differential...

In this paper, we consider some boundary value problems (BVP) for fractional order partial differential equations ‎(FPDE)‎ with non-local boundary conditions. The solutions of these problems are presented as series solutions analytically via modified Mittag-Leffler functions. These functions have been modified by authors such that their derivatives are invariant with respect to fractional deriv...

This paper presents an invariant under scaling and linear brightness change. The invariant is based on differentials and therefore is a local feature. Rotationally invariant 2-d differential Gaussian operators up to third order are proposed for the implementation of the invariant. The performance is analyzed by simulating a camera zoom-out.