In this work, we establish Hölder’s inequality, Minkowski’s inequality and Jensen’s inequality on time scales via the nabla integral and diamond-α dynamic integral, which is defined as a linear combination of the delta and nabla integrals. c © 2008 Published by Elsevier Ltd

We prove Euler-Lagrange and natural boundary necessary optimality conditions for problems of the calculus of variations which are given by a composition of nabla integrals on an arbitrary time scale. As an application, we get optimality conditions for the product and the quotient of nabla variational functionals.

The discrete-time, the quantum, and the continuous calculus of variations have been recently unified and extended. Two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we propose a more general approach to the calculus of variations on time scales that allows to obtain both delta and nabla r...

Abstract. The discrete, the quantum, and the continuous calculus of variations, have been recently unified and extended by using the theory of time scales. Such unification and extension is, however, not unique, and two approaches are followed in the literature: one dealing with minimization of delta integrals; the other dealing with minimization of nabla integrals. Here we review a more genera...

This paper is concerned with the efficient evaluation of higher-order derivatives of functions $f$ that are composed of matrix operations. I.e., we want to compute the $D$-th derivative tensor $\nabla^D f(X) \in \mathbb R^{N^D}$, where $f:\mathbb R^{N} \to \mathbb R$ is given as an algorithm that consists of many matrix operations. We propose a method that is a combination of two well-known tec...

Differential and integral calculus on time scales allows to develop a theory of dynamic equations in order to unify and extend the usual differential equations and difference equations. For single variable differential and integral calculus on time scales, we refer the reader to the textbooks [4, 5] and the references given therein. Multivariable calculus on time scales was developed by the aut...