Systems with Time Delay in the Calculus of Variations: A Variational Approach

In this paper, we deal with the fixed-endpoint problem in thecalculus of variations involving a delay in the phase coordinates.Necessary conditions in the form of a maximum principle arewell known and, hence, conditions equivalent to those of Euler,Legendre, and Weierstrass. However, no results seem to existfor sufficiency, or for a corresponding Jacobi condition. Wederive necessary and sufficient conditions in terms of the firstand second variations, extending the clssical results for thedelay-free case. This is obtained directly, that is, withoutreferring to concepts such as conjugate points, fields of extremals,Riccati equations, or the Hamilton-Jacobi partial differentialequation. The first-order condtion is then characterized interms of the Euler equation, together with smoothness propertiesof solutions. Several examples illustrate the usefulness ofthe conditions obtained.     

Systems with Time Delay in the Calculus of Variations: The Method of Steps

In a recent article (see [8]), we derived necessary and sufficientconditions for minima for the fixed-endpoint problem in thecalculus of variations involving a constant delay in the phasecoordinates. These conditions are expressed, explicitly, interms of the first and second variations. The vanishing of thefirst variation is characterized in terms of an extended Euler'sequation, just as for delay-free problems, but the characterizationof the conditions on the second variation remained unsolved.In this paper we convert, through the ‘method of steps’,the delay problem into one without delay. Although this problemwill not have fixed-endpoint constraints, it allows us to introduce,in a natural way, the concept of ‘conjugate sequence’;this solves the main difficulty for delay problems, namely,to give initial conditions for existence and uniqueness of solutionsof the Hamiltonian system (which is a difference–differentialsystem with both advanced and retarded arguments). The conditionson the second variation are then characterized by an extra conditionwhich is based exclusively on a solution of a given matrix Riccatiequation.     

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

Time integration of the non-linear Galerkin method

e-mail: bosco{at}mac.cie.uva.es, frutos@mac.cie.uva.es Numerical tests are presented that, for the Kuramoto-Sivashinskyand a reaction-diffusion equation, compare the non-linear Galerkinmethod with standard pseudospectral and pure spectral Galerkindiscretizations. The results show that, for these problems,the non-linear Galerkin method is not competitive with eitherpure spectral Galerkin or pseudospectral discretizations.     

Is the weak relaxation procedure proper for noncommensurately delayed controls?

Looking for a proper relaxation procedure for optimal controlproblems with time delays in the control variables, Warga introducedin a privately circulated paper the notion of ‘weakly’relaxed controls. It has been proved that, for systems involvingtwo delays with a rational quotient (the commensurate case),one can find weakly relaxed controls that cannot be approximatedwith original controls, showing that this model may fail tobe proper. The same conclusion, for the noncommensurate case,was never established. In fact, for this case, we proved ina recent paper that (assuming the underlying control compactspace to have two points) all constant weakly relaxed controlsbelong to the (weak-star) closure of the space of ordinary controls.The purpose of this paper is to extend this result to the generalcase of time-dependent weakly relaxed controls.     

FLUSHING TIME IN THE CIENFUEGOS BAY,CUBA

We use a high‐resolution numerical model to analyze the hydrodynamical exchange processes of the Cienfuegos Bay with the Caribbean Sea. Results show that freshwater inputs and wind are the main factors controlling the water exchange in the bay. Nine simulation case studies with three wind conditions and three freshwater inputs were carried out. The main conclusion is that the Cienfuegos Bay system has a slow exchange rate, with an average flushing time of 39 days during the rainy period and 50 days during the dry one, for the most probable wind direction. Two new temporary scales for Cienfuegos Bay are discussed: the local e‐flushing time and the flushing lag. The results obtained for these new variables validate the slow exchange capacity of the bay.     

Ab initio calculation of the refractivity and hyperpolarizability second virial coefficients of neon gas

Second virial coefficients for the density dependence of a number of electric properties are calculated for neon gas. Employing an accurate CCSD(T) potential for the Ne² van der Waals dimer and interaction-induced electric dipole polarizabilities and hyperpolarizabilities obtained from CCSD response theory, we evaluated the dielectric, refractivity, Kerr and ESHG second virial coefficients using both a semiclassical and a quantum statistical approach. The results cover a wide range of temperatures and are expected to be more reliable than the available experimental and empirical data. Quantum effects are found to be important only for temperatures below 100 K. The frequency-dependence of the refractivity virial coefficient is found to be small, but not negligible. For frequencies in the visible region it accounts for a few percent of the final results. For the ESHG virial coefficient of neon, frequency dependence is found to be very important, accounting for 20–25% of the second virial coefficient at the typical frequencies employed in experiments.     

AN INDIVIDUAL‐BASED MODEL TO ESTIMATE THE DAILY ENERGETIC COST OF GREATER RHEAS AND ITS CONTRIBUTION ON POPULATION RECRUITMENT

Abstract An individual‐based model for estimating the energetic costs in Rhea americana was developed considering their sexual and seasonal differences in the behavioral activities. The model includes as variables the individual's characteristics, as well as corporal weight, the time spent on different activities, and the cost associated with each activity. We estimated the daily energetic demand of an adult rhea based on the activities individuals normally develop during postreproductive, nonreproductive, and reproductive seasons, differentiating between sexes. The time spent in each activity for one given animal was calculated from field observations of individuals and the estimations of energetic costs for each activity were obtained from specialized literature. The model built varied between sexes because males and females have different reproductive costs. Both models have the same general formulation but they differ in the cost associated with reproduction. In Greater Rheas, while males assume all of the incubation, the females only lay eggs communally in a single nest. Also the possibility that the individual reproduces or not was considered. The model does not allow to determine whether the energetic costs associated with the breeding are the reason why few individuals try to reproduce but it indicates that there is a clear difference in the daily energetic costs of individuals which reproduce and those which do not reproduce. Other activities associated with parental care posthatching, not taken into account here, would increase these differences, and would explain the low number of breeding attempts observed at wild.