Necessary condition for an Euler-Lagrange equation on time scales

We prove a necessary condition for a dynamic integrodifferential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. An example of a second order dynamic equation, which is not an Euler-Lagrange equation on an arbitrary time scale, is given. © 2014 Monika Dryl and Delfim F. M. Torres

A general delta-nabla calculus of variations on time scales with application to economics

We consider a general problem of the calculus of variations on time scales with a cost functional that is the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange delta-nabla differential equations are proved, which lead to important insights in the process of discretisation. Application of the obtained results to a firm that wants to program its production and investment policies to reach a given production rate and to maximise its future market competitiveness is discussed

Direct and Inverse Variational Problems on Time Scales: A Survey

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation (Helmholtz's problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be published in the Springer Volume 'Modeling, Dynamics, Optimization and Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted, after a revision, 19/Jan/201

Time-Fractional Optimal Control of Initial Value Problems on Time Scales

We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time derivatives and integrals are considered, on time scales, in the Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a delta-integral function of both state and control variables, with time evolving on an arbitrarily given time scale. Interpreting the Euler--Lagrange first order optimality condition with an adjoint problem, defined by means of right Riemann--Liouville fractional delta derivatives, we obtain an optimality system for the considered fractional OCP. For that, we first prove new fractional integration by parts formulas on time scales.Comment: This is a preprint of a paper accepted for publication as a book chapter with Springer International Publishing AG. Submitted 23/Jan/2019; revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial text overlap with arXiv:1508.0075

Intersyngenic variations in the esterases of axenic stocks of Paramecium aurelia

The esterase isozymes were surveyed in axenic stocks of syngens 1, 2, 4, 5, 6, and 8 of Paramecium aurelia by starch gel electrophoresis. In paramecia there appear to be four types of esterases which are clearer in axenic than in bacterized stocks. Each type differs in its substrate specificity and/or its response to the inhibitor eserine sulfate. Minor variations in type D esterases sometimes occur in different extracts of the same stock and may result from changes in the temperature of growth of the cells or growth cycle differences. Differences in the mobility of the A, B, or C (cathodal) types of esterases may occur in different syngens. They also occur for the A and B types among stocks within a syngen, but the frequency is low, except in the case of syngen 2. Since each of the types of esterases varies independently, at least four and possibly more genes appear to specify the esterases in the species complex. Some pairs of syngens vary in their electrophoretic positions for all types of esterases. Other pairs have identical zymograms. This observation suggests that some syngens may differ from each other by as many as four esterase genes, while others may not differ at all. The difference between P. aurelia and Tetrahymena pyriformis in the degree of intrasyngenic variation observed for enzymes is discussed in relation to other types of characters, the organization of the genetic material in the macronucleus, the presence of symbionts, and their breeding systems. It is suggested that enzyme variation is achieved by the action of different selective forces in these two groups of ciliated protozoa.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44170/1/10528_2004_Article_BF00485643.pd

Nucleotide sequence divergence among DNA fractions of different syngens of Tetrahymena pyriformis

The magnitude of the differences in base sequence of DNA fractions derived from different syngens of the ciliated protozoan Tetrahymena pyriformis was investigated. Each DNA was fractionated into unique and repeated sequences by hydroxylapatite chromatography, and the fractions were tested by in vitro molecular hybridization techniques. The amount of hybrid formed and the thermal stability of the hybrid molecules were examined at different incubation temperatures (50 and 65 C) for unique sequences and at 50 C for repeated sequences. The extent of the reactions involving either unique or repeated sequences was nearly complete when the two DNAs compared were derived from the same syngen. Moreover, intrasyngenic hybrids formed at 50 C (and 65 C for unique sequences) exhibit a high degree of thermal stability. In contrast, the extent of the reactions involving sequences derived from different syngens was low, as expected from the effect of mismatching on rate of reassociation, and intersyngenic hybrids formed at 50 C have low thermal stability. The reaction of unique sequences is further reduced at 65 C and the intersyngenic hybrids formed have a higher thermal stability than those formed at 50 C. The degree to which thermal stability is lowered was then used to estimate the percentage of mispaired bases. The average divergence of unique sequences between syngens is large and of the magnitude found for rodent DNAs from different genera or for Drosophila DNAs from nonsibling species. The repeated sequence fraction may contain more than one component and may be more conserved than the unique sequence fraction.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/44120/1/10528_2004_Article_BF00486091.pd

Direct and inverse variational problems on time scales: A survey

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler–Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler–Lagrange equation (Helmholtz’s problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field. © 2017, Springer International Publishing AG

The delta-nabla calculus of variations for composition functionals on time scales

Abstract We develop the calculus of variations on time scales for a functional that is the composition of a certain scalar function with the delta and nabla integrals of a vector valued field. Euler-Lagrange equations, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. Interesting corollaries and examples are presented