One-loop effective potential in M4 x T2 with and without 't Hooft flux

We review the basic notions of compactification in the presence of a background flux. In extra-dimentional models with more than five dimensions, Scherk and Schwarz boundary conditions have to satisfy 't Hooft consistency conditions. Different vacuum configurations can be obtained, depending whether trivial or non-trivial 't Hooft flux is considered. The presence of the "magnetic" background flux provide, in addition, a mechanism for producing four-dimensional chiral fermions. Particularizing to the six-dimensional case, we calculate the one-loop effective potential for a U(N) gauge theory on M4 x T2. We firstly review the well known results of the trivial 't Hooft flux case, where one-loop contributions produce the usual Hosotani dynamical symmetry breaking. Finally we applied our result for describing, for the first time, the one-loop contributions in the non-trivial 't Hooft flux case

Nonlinear energy-maximising optimal control of wave energy systems: A moment-based approach

Linear dynamics are virtually always assumed when designing optimal controllers for wave energy converters (WECs), motivated by both their simplicity and computational convenience. Nevertheless, unlike traditional tracking control applications, the assumptions under which the linearization of WEC models is performed are challenged by the energy-maximizing controller itself, which intrinsically enhances device motion to maximize power extraction from incoming ocean waves. \GSIn this article, we present a moment-based energy-maximizing control strategy for WECs subject to nonlinear dynamics. We develop a framework under which the objective function (and system variables) can be mapped to a finite-dimensional tractable nonlinear program, which can be efficiently solved using state-of-the-art nonlinear programming solvers. Moreover, we show that the objective function belongs to a class of generalized convex functions when mapped to the moment domain, guaranteeing the existence of a global energy-maximizing solution and giving explicit conditions for when a local solution is, effectively, a global maximizer. The performance of the strategy is demonstrated through a case study, where we consider (state and input-constrained) energy maximization for a state-of-the-art CorPower-like WEC, subject to different hydrodynamic nonlinearities

On the approximation of moments for nonlinear systems

Model reduction by moment-matching relies upon the availability of the so-called moment. If the system is nonlinear, the computation of moments depends on an underlying specific invariance equation, which can be difficult or impossible to solve. This note presents four technical contributions related to the theory of moment matching: first, we identify a connection between moment-based theory and weighted residual methods. Second, we exploit this relation to provide an approximation technique for the computation of nonlinear moments. Third, we extend the definition of nonlinear moment to the case in which the generator is described in explicit form. Finally, we provide an approximation technique to compute the moments in this scenario. The results are illustrated by means of two examples

Energy-maximising moment-based constrained optimal control of ocean wave energy farms

Successful commercialisation of wave energy technology inherently incorporates the concept of an array of wave energy converters (WECs). These devices, which constantly interact via hydrodynamic effects, require optimised control that can guarantee maximum energy extraction from incoming ocean waves while ensuring, at the same time, that any physical limitations associated with device and actuator systems are being consistently respected. This paper presents a moment-based energy-maximising optimal control framework for WECs arrays subject to state and input constraints. The authors develop a framework under which the objective function (and system variables) can be mapped to a finite-dimensional tractable quadratic program (QP), which can be efficiently solved using state-of-the-art solvers. Moreover, the authors show that this QP is always concave, i.e. existence and uniqueness of a globally optimal solution is guaranteed under this moment-based framework. The performance of the proposed strategy is demonstrated through a case study, where (state and input constrained) energy-maximisation for a WEC farm composed of CorPower-like WEC devices is considered

Model reduction by moment matching: beyond linearity a review of the last 10 years

We present a review of some recent contributions to the theory and application of nonlinear model order reduction by moment matching. The tutorial paper is organized in four parts: 1) Moments of Nonlinear Systems; 2) Playing with Moments: Time-Delay, Hybrid, Stochastic, Data-Driven and Beyond; 3) The Loewner Framework; 4) Applications to Optimal Control and Wave Energy Conversion

Symmetry breaking from Scherk-Schwarz compactification

We analyze the classical stable configurations of an extra-dimensional gauge theory, in which the extra dimensions are compactified on a torus. Depending on the particular choice of gauge group and the number of extra dimensions, the classical vacua compatible with four-dimensional Poincar\'e invariance and zero instanton number may have zero energy. For SU(N) on a two-dimensional torus, we find and catalogue all possible degenerate zero-energy stable configurations in terms of continuous or discrete parameters, for the case of trivial or non-trivial 't Hooft non-abelian flux, respectively. We then describe the residual symmetries of each vacua.Comment: 24 pages, 1 figure, Section 4 modifie

Constructing Lifshitz solutions from AdS

Under general assumptions, we show that a gravitational theory in d+1 dimensions admitting an AdS solution can be reduced to a d-dimensional theory containing a Lifshitz solution with dynamical exponent z=2. Working in a d=4, N=2 supergravity setup, we prove that if the AdS background is N=2 supersymmetric, then the Lifshitz geometry preserves 1/4 of the supercharges, and we construct the corresponding Killing spinors. We illustrate these results in examples from supersymmetric consistent truncations of type IIB supergravity, enhancing the class of known 4-dimensional Lifshitz solutions of string theory. As a byproduct, we find a new AdS4 x S1 x T(1,1) solution of type IIB.Comment: 29 pages, no figures; v2 minor corrections, a reference adde