1,217 research outputs found

### The Natural Logarithm on Time Scales

We define an appropriate logarithm function on time scales and present its
main properties. This gives answer to a question posed by M. Bohner in [J.
Difference Equ. Appl. {\bf 11} (2005), no. 15, 1305--1306].Comment: 6 page

### Higher-Order Calculus of Variations on Time Scales

We prove a version of the Euler-Lagrange equations for certain problems of
the calculus of variations on time scales with higher-order delta derivatives.Comment: Corrected minor typo

### Complex-valued fractional derivatives on time scales

We introduce a notion of fractional (noninteger order) derivative on an
arbitrary nonempty closed subset of the real numbers (on a time scale). Main
properties of the new operator are proved and several illustrative examples
given.Comment: This is a preprint of a paper whose final and definite form will
appear in Springer Proceedings in Mathematics & Statistics, ISSN: 2194-1009.
Accepted for publication 06/Nov/201

### A General Backwards Calculus of Variations via Duality

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.Comment: Submitted to Optimization Letters 03-June-2010; revised 01-July-2010;
accepted for publication 08-July-201

### Euler-Lagrange equations for composition functionals in calculus of variations on time scales

In this paper we consider the problem of the calculus of variations for a
functional which is the composition of a certain scalar function $H$ with the
delta integral of a vector valued field $f$, i.e., of the form
$H(\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange
equations, natural boundary conditions for such problems as well as a necessary
optimality condition for isoperimetric problems, on a general time scale, are
given. A number of corollaries are obtained, and several examples illustrating
the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems
(DCDS-B); revised 10-March-2010; accepted 04-July-201

### Convolutional higher order matching pursuit

We introduce a greedy generalised convolutional algorithm to efficiently locate an unknown number of sources in a series of (possibly multidimensional) images, where each source contributes a localised and low-dimensional but otherwise variable signal to its immediate spatial neighbourhood. Our approach extends convolutional matching pursuit in two ways: first, it takes the signal generated by each source to be a variable linear combination of aligned dictionary elements; and second, it executes the pursuit in the domain of high-order multivariate cumulant statistics. The resulting algorithm adapts to varying signal and noise distributions to flexibly recover source signals in a variety of settings

### Integral Inequalities and their Applications to the Calculus of Variations on Time Scales

We discuss the use of inequalities to obtain the solution of certain
variational problems on time scales.Comment: To appear in Mathematical Inequalities & Applications
(http://mia.ele-math.com). Accepted: 14.01.201

### Transversality Conditions for Infinite Horizon Variational Problems on Time Scales

We consider problems of the calculus of variations on unbounded time scales.
We prove the validity of the Euler-Lagrange equation on time scales for
infinite horizon problems, and a new transversality condition.Comment: Submitted 6-October-2009; Accepted 19-March-2010 in revised form; for
publication in "Optimization Letters"

### R-matrix approach to integrable systems on time scales

A general unifying framework for integrable soliton-like systems on time
scales is introduced. The $R$-matrix formalism is applied to the algebra of
$\delta$-differential operators in terms of which one can construct infinite
hierarchy of commuting vector fields. The theory is illustrated by two
infinite-field integrable hierarchies on time scales which are difference
counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer
soliton systems are constructed as related finite-field restrictions.Comment: 21 page

### Necessary Optimality Conditions for Higher-Order Infinite Horizon Variational Problems on Time Scales

We obtain Euler-Lagrange and transversality optimality conditions for
higher-order infinite horizon variational problems on a time scale. The new
necessary optimality conditions improve the classical results both in the
continuous and discrete settings: our results seem new and interesting even in
the particular cases when the time scale is the set of real numbers or the set
of integers.Comment: This is a preprint of a paper whose final and definite form will
appear in Journal of Optimization Theory and Applications (JOTA). Paper
submitted 17-Nov-2011; revised 24-March-2012 and 10-April-2012; accepted for
publication 15-April-201

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