Fermat-type equations of signature (13,13,p) via Hilbert cuspforms

In this paper we prove that for $p > 13649$ equations of the form $x^{13} + y^{13} = Cz^{p}$ have no non-trivial primitive solutions $(a,b,c)$ such that $13 \not \mid c$ for an infinite family of values for $C$ . Our method consists on relating a solution $(a,b,c)$ to the previous equation to a solution $(a,b,c_1)$ of another Diophantine equation with coefficients in $\mathbb Q (\sqrt{13})$ . Then we attach to $(a,b,c_1)$ a Frey curve $E_{(a,b)}$ defined over $\mathbb Q (\sqrt{13})$ that is not a $\mathbb Q $ -curve. We prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at $3$ . This theorem, in particular, implies modularity of $E_{(a,b)}$ . This enables us to use level lowering results and apply the modular approach via Hilbert cuspforms over $\mathbb Q (\sqrt{13})$ to prove the non-existence of $(a,b,c_1)$ and, consequently, of $(a,b,c)$ .     

A multi-parametric programming approach for constrained dynamic programming problems

In this work, we present a new algorithm for solving complex multi-stage optimization problems involving hard constraints and uncertainties, based on dynamic and multi-parametric programming techniques. Each echelon of the dynamic programming procedure, typically employed in the context of multi-stage optimization models, is interpreted as a multi-parametric optimization problem, with the present states and future decision variables being the parameters, while the present decisions the corresponding optimization variables. This reformulation significantly reduces the dimension of the original problem, essentially to a set of lower dimensional multi-parametric programs, which are sequentially solved. Furthermore, the use of sensitivity analysis circumvents non-convexities that naturally arise in constrained dynamic programming problems. The potential application of the proposed novel framework to robust constrained optimal control is highlighted.     

Parametric global optimisation for bilevel programming

We propose a global optimisation approach for the solution of various classes of bilevel programming problems (BLPP) based on recently developed parametric programming algorithms. We first describe how we can recast and solve the inner (follower’s) problem of the bilevel formulation as a multi-parametric programming problem, with parameters being the (unknown) variables of the outer (leader’s) problem. By inserting the obtained rational reaction sets in the upper level problem the overall problem is transformed into a set of independent quadratic, linear or mixed integer linear programming problems, which can be solved to global optimality. In particular, we solve bilevel quadratic and bilevel mixed integer linear problems, with or without right-hand-side uncertainty. A number of examples are presented to illustrate the steps and details of the proposed global optimisation strategy.     

Liquid crystalline glycosteroids and acyl steroid glycosides (ASG)

ABSTRACT

As part of our studies on glycolipidic liquid crystals, we have investigated some molecules comprising a steroid moiety. These systems can exhibit several types of structures depending on their polarity pattern based on the number of polar and non-polar moieties and their resulting molecular shape. Therefore, to aid describing such systems, we have proposed a specific classification based on this polarity pattern. Many compounds in this family are natural products, which possess important biological properties. Some of the compounds have bolaphilic structures, with both a steroid and a fatty alkyl chain attached to the carbohydrate moiety, such as either the β-galactoside BbGL-1 or the α-glucoside α-CAG that are found in the membranes of the pathogens Borrelia burgdorferi and Helicobacter pylori, respectively. In this account, after a brief introduction on liquid crystalline glycolipids, we focus on carbohydrate–steroid hybrids, summarising our previous work on glycosteroids prepared by the CMGL-synthon strategy, and reporting our preliminary results on the thermotropic behaviour of acyl steroid glycosides (ASG), namely cholesteryl 6-O-acyl-β-gluco- and -galacto-pyranosides.