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

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

Linear systems of fractional nabla difference equations

In this paper we shall consider a linear system of fractional nabla difference equations with constant coefficients. We shall construct the fundamental matrix for the homogeneous system and the causal Green’s function for the nonhomogeneous system. We employ transform methods and series methods, and we illustrate analogies with classical first-order differential or difference equations. We shall close the paper with an asymptotic result that follows from the analysis of a half-order nabla difference equation

A transform method in discrete fractional calculus

We begin with an introduction to a calculus of fractional finite differences. We extend the discrete Laplace transform to develop a discrete transform method. We define a family of finite fractional difference equations and employ the transform method to obtain solutions

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"

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

La cantidad de madera muerta y sus tasas de descomposición asociadas en reservas forestales y bosques manejados en el noroeste de Turquía

This study describes the state of coarse dead wood (CDW) in the Forest Reserve and the Managed Forest zones of northern conifer-broadleaved mixed forest. The results showed mean total CDW volumes in the ranges 30,05±11,06 m3/ha in the Forest Reserve (6,33±2,98% of the LW volume), and 9,31±2,84 m3/ha in the Managed Forest (1,96±0,84% of the LW volume). The total CDW volume was 3,22 times higher in the Forest Reserve than in the Managed Forest. The CDWlog1 and CDWsnag1 were the most abundant CDW decay classes, whilst CDWlog2 and CDWsnag2 were the lowest. Comparisons of ratios between the Managed Forest and the Forest Reserve with abundant decay classes CDWlog1 and CDWsnag1 indicated large differences. The CDWlog1 volume was 4,09 times higher, and the CDWsnag1 volume was 3,68 times greater in the Forest Reserve than in the Managed Forest. The ratio of different CWD classes in the Managed Forest to CWD classes in the Reserve Forest confirms the pattern. In both Managed and Reserve Forest zones there is balance between total CDWlogs and total CDWsnags, but the differences between total CDWlogs and total CDWsnags was not statistically significant. The total CDW volume was significantly dependent on the forest management system. The system influenced amount and diversity of CDW. In commercially managed forest the abundance and structure of CDW retained is a compromise between the needs of timber production and nature conservation.Este estudio describe el estado de la madera muerta en la zona de reserva forestal y zonas de bosques manejados de coníferas del norte de bosques mixtos de frondosas. Los resultados mostraron que la media total de los volúmenes de madera muerta es igual a 30,05 ± 11,06 m3 / ha en la Reserva Forestal (6,33 ± 2,98% del volumen de madera en pie), y 9,31 ± 2,84 m3 / ha en los bosques manejados (1,96 ± 0,84% del volumen de LW). El volumen total de madera muerta fue de 3,22 veces mayor en la Reserva Forestal de que en el bosque administrado. Las clases de decaimiento de madera muerta más abundantes eran CDWlog1 y CDWsnag1, mientras que CDWlog2 y CDWsnag2 fueron los menos abundantes. Las comparaciones de las proporciones entre el bosque manejado y la Reserva Forestal con las clases de decaimiento más abundantes (CDWlog1 y CDWsnag1) indican grandes diferencias ente las dos zonas. El volumen CDWlog1 fue 4,09 veces mayor, y el volumen CDWsnag1 fue 3,68 veces mayor en la Reserva Forestal de que en el bosque manejado. La relación de las diferentes clases de decaimiento entre los bosques manejados y la Reserva Forestal confirma el patrón. En ambos casos, bosque manejado y zonas de reserva forestal, existe un equilibrio entre CDWlogs total y CDWsnags total, pero las diferencias entre CDWlogs total y CDWsnags total no fue estadísticamente significativa. El volumen total de madera muerta depende significativamente del sistema de gestión forestal. El sistema de manejo influye sobre la cantidad y diversidad de madera muerta. En una gestión comercial de los bosques, la abundancia y estructura de madera muerta presente es un compromiso entre las necesidades de la producción de madera y la conservación de la naturaleza