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

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

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

Beyond a Call to Action

This paper is intended as a critique and development of morality in literature, seeking to prove that literature can have deep effects on a reader\u27s moral character. The stakes for such research are rather high: especially during the pandemic, our culture is heavily informed by social and mass media, and it is hard to imagine a good future for the world if these mediums cannot shake the status quo. Though this paper takes a narrower scope of investigation than moral progress itself, the reader should keep in mind that all of our practices of communication can and should be informed by literary tradition, among other practices. For art is our name for the most refined and deliberate artifacts of human expression, capable of great scale, subtlety, and mass dissemination. Morality in communication is explored here through literature, but I hope the reader will make an attempt to apply any knowledge gleaned to as diverse a range of their practices as is possible. The basic issue of my research is that conventional calls to action, for example Sarah McLachlan\u27s famous SPCA commercials, are not terribly effective. This is obvious insofar as there is a saturation of these calls to action, and a shocking lack of action or concern from many, but I will also seek to justify this in the theories of Immanuel Levinas, then develop a solution from a more philosophical framing of the problem, with Louis Althusser\u27s writings on ideology, as well as case studies from literature. Eventually, this all leads to the question: how do we address ideology in art? Which will hopefully be somewhat answered by the development of the question, and further addressed by the case studies comprising the latter half of this writing

Score-matching estimators for continuous-time point-process regression models

We introduce a new class of efficient estimators based on score matching for probabilistic point process models. Unlike discretised likelihood-based estimators, score matching estimators operate on continuous-time data, with computational demands that grow with the number of events rather than with total observation time. Furthermore, estimators for many common regression models can be obtained in closed form, rather than by iteration. This new approach to estimation may thus expand the range of tractable models available for event-based data

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

Avoidance Control on Time Scales

We consider dynamic systems on time scales under the control of two agents. One of the agents desires to keep the state of the system out of a given set regardless of the other agent's actions. Leitmann's avoidance conditions are proved to be valid for dynamic systems evolving on an arbitrary time scale.Comment: Revised edition in JOTA format. To appear in J. Optim. Theory Appl. 145 (2010), no. 3. In Pres

Necessary optimality conditions for infinite horizon variational problems on time scales

We prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems of the calculus of variations on time scales. Here the Lagrangian depends on the independent variable, an unknown function and its nabla derivative, as well as a nabla indefinite integral that depends on the unknown function