On the Smoothness of Value Functions

We prove that under standard Lipschitz and growth conditions, the value function of all optimal control problems for one-dimensional diffusions is twice continuously differentiable, as long as the control space is compact and the volatility is uniformly bounded below, away from zero. Under similar conditions, the value function of any optimal stopping problem is continuously differentiable. For the first problem, we provide sufficient conditions for the existence of an optimal control, which is also shown to be Markov. These conditions are based on the theory of monotone comparative statics.Super Contact; Smooth Pasting; HJB Equation; Optimal Control; Markov Control; Comparative Statics; Supermodularity; Single-Crossing; Interval Dominance Order

On the Smoothness of Value Functions

We prove that under standard Lipschitz and growth conditions, the value function of all optimal control problems for one-dimensional diffusions is twice continuously differentiable, as long as the control space is compact and the volatility is uniformly bounded below, away from zero. Under similar conditions, the value function of any optimal stopping problem is continuously differentiable. For the first problem, we provide sufficient conditions for the existence of an optimal control, which is also shown to be Markov. These conditions are based on the theory of monotone comparative statics

Pre-Inflationary Spacetime in String Cosmology

Seiberg and Witten have shown that the non-perturbative stability of string physics on conformally compactified spacetimes is related to the behaviour of the areas and volumes of certain branes as the brane is moved towards infinity. If, as is particularly natural in quantum cosmology, the spatial sections of an accelerating cosmological model are flat and compact, then the spacetime is on the brink of disaster: it turns out that the version of inflationary spacetime geometry with toral spatial sections is marginally stable in the Seiberg-Witten sense. The question is whether the system remains stable before and after Inflation, when the spacetime geometry is distorted away from the inflationary form but still has flat spatial sections. We show that it is indeed possible to avoid disaster, but that requiring stability at all times imposes non-trivial conditions on the spacetime geometry of the early Universe in string cosmology. This in turn allows us to suggest a candidate for the structure which, in the earliest Universe, forbids cosmological singularities.Comment: Final version to appear in NPB, 27 pages including 1 eps figur

Two carbon fluxes to reserve starch in potato (Solanum tuberosum L.) tuber cells are closely interconnected but differently modulated by temperature

Parenchyma cells from tubers of Solanum tuberosum L. convert several externally supplied sugars to starch but the rates vary largely. Conversion of glucose 1-phosphate to starch is exceptionally efficient. In this communication, tuber slices were incubated with either of four solutions containing equimolar [U-14C]glucose 1-phosphate, [U-14C]sucrose, [U-14C]glucose 1-phosphate plus unlabelled equimolar sucrose or [U-14C]sucrose plus unlabelled equimolar glucose 1-phosphate. 14C-incorporation into starch was monitored. In slices from freshly harvested tubers each unlabelled compound strongly enhanced 14C incorporation into starch indicating closely interacting paths of starch biosynthesis. However, enhancement disappeared when the tubers were stored. The two paths (and, consequently, the mutual enhancement effect) differ in temperature dependence. At lower temperatures, the glucose 1-phosphate-dependent path is functional, reaching maximal activity at approximately 20 °C but the flux of the sucrose-dependent route strongly increases above 20 °C. Results are confirmed by in vitro experiments using [U-14C]glucose 1-phosphate or adenosine-[U-14C]glucose and by quantitative zymograms of starch synthase or phosphorylase activity. In mutants almost completely lacking the plastidial phosphorylase isozyme(s), the glucose 1-phosphate-dependent path is largely impeded. Irrespective of the size of the granules, glucose 1-phosphate-dependent incorporation per granule surface area is essentially equal. Furthermore, within the granules no preference of distinct glucosyl acceptor sites was detectable. Thus, the path is integrated into the entire granule biosynthesis. In vitro 14C-incorporation into starch granules mediated by the recombinant plastidial phosphorylase isozyme clearly differed from the in situ results. Taken together, the data clearly demonstrate that two closely but flexibly interacting general paths of starch biosynthesis are functional in potato tuber cells

Loop Quantum Cosmology: A Status Report

The goal of this article is to provide an overview of the current state of the art in loop quantum cosmology for three sets of audiences: young researchers interested in entering this area; the quantum gravity community in general; and, cosmologists who wish to apply loop quantum cosmology to probe modifications in the standard paradigm of the early universe. An effort has been made to streamline the material so that, as described at the end of section I, each of these communities can read only the sections they are most interested in, without a loss of continuity.Comment: 138 pages, 15 figures. Invited Topical Review, To appear in Classical and Quantum Gravity. Typos corrected, clarifications and references adde

Contraints on unified models for dark matter and dark energy using H(z)

The differential age data of astrophysical objects that have evolved passivelly during the history of the universe (e.g. red galaxies) allows to test theoretical cosmological models through the predicted Hubble function expressed in terms of the redshift $z$, $H(z)$. We use the observational data for $H(z)$ to test unified scenarios for dark matter and dark energy. Specifically, we focus our analysis on the Generalized Chaplygin Gas (GCG) and the viscous fluid (VF) models. For the GCG model, it is shown that the unified scenario for dark energy and dark matter requires some priors. For the VF model we obtain estimations for the free parameters that may be compared with further analysis mainly at perturbative level.Comment: Latex file, 10 pages, 19 figures in eps format. Accepted for publication in European Journal of Physics