Poisson Yang-Baxter maps with binomial Lax matrices

A construction of multidimensional parametric Yang-Baxter maps is presented. The corresponding Lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the Sklyanin bracket. These maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings. An interesting family of quadrirational symplectic YB maps on $\mathbb{C}^4 \times \mathbb{C}^4$ with $3\times 3$ Lax matrices is also presented.Comment: 22 pages, 3 figure

Robin problems with indefinite linear part and competition phenomena

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite potential. The reaction term involves competing nonlinearities. More precisely, it is the sum of a parametric sublinear (concave) term and a superlinear (convex) term. The superlinearity is not expressed via the Ambrosetti-Rabinowitz condition. Instead, a more general hypothesis is used. We prove a bifurcation-type theorem describing the set of positive solutions as the parameter $\lambda > 0$ varies. We also show the existence of a minimal positive solution $\tilde{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \tilde{u}_\lambda$

Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential

We study perturbations of the eigenvalue problem for the negative Laplacian plus an indefinite and unbounded potential and Robin boundary condition. First we consider the case of a sublinear perturbation and then of a superlinear perturbation. For the first case we show that for $\lambda<\widehat{\lambda}_{1}$ ($\widehat{\lambda}_{1}$ being the principal eigenvalue) there is one positive solution which is unique under additional conditions on the perturbation term. For $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. In the superlinear case, for $\lambda<\widehat{\lambda}_{1}$ we have at least two positive solutions and for $\lambda\geq\widehat{\lambda}_{1}$ there are no positive solutions. For both cases we establish the existence of a minimal positive solution $\bar{u}_{\lambda}$ and we investigate the properties of the map $\lambda\mapsto\bar{u}_{\lambda}$

A wideband CPW ring power combiner with low insertion loss and high port isolation

In this paper we present a coplanar waveguide (CPW)-based ring power combiner that exhibits less than 0.8 dB insertion loss, better than 15 dB port match and higher than 22 dB isolation loss over the frequency range from 50 GHz to 100 GHz. Compared with the conventional 2-way Wilkinson combiner, the proposed ring power combiner replaces the resistor between the two input ports with two quasi quarter-wave CPWs, a 180º CPW phase inverter, and two resistors that lead to frequency-insensitive port isolation and wideband port match. The power combiner is realized using an electron beam-based GaAs MMIC process along with simple electron beam airbridge technology. These results agree well with 3D full-wave simulations

On Quadrirational Yang-Baxter Maps

We use the classification of the quadrirational maps given by Adler, Bobenko and Suris to describe when such maps satisfy the Yang-Baxter relation. We show that the corresponding maps can be characterized by certain singularity invariance condition. This leads to some new families of Yang-Baxter maps corresponding to the geometric symmetries of pencils of quadrics.Comment: Proceedings of the workshop "Geometric Aspects of Discrete and Ultra-Discrete Integrable Systems" (Glasgow, March-April 2009

Yang-Baxter maps and multi-field integrable lattice equations

A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice equations introduced by Adler and Yamilov and which are related to the nonlinear superposition formulae for the B\"acklund transformations of the nonlinear Schr\"odinger system and specific ferromagnetic models.Comment: 16 pages, 4 figures, corrected versio

Multi-parametric linear programming under global uncertainty

Multi-parametric programming has proven to be an invaluable tool for optimisation under uncertainty. Despite the theoretical developments in this area, the ability to handle uncertain parameters on the left-hand side remains limited and as a result, hybrid, or approximate solution strategies have been proposed in the literature. In this work, a new algorithm is introduced for the exact solution of multi-parametric linear programming problems with simultaneous variations in the objective function's coefficients, the right-hand side and the left-hand side of the constraints. The proposed methodology is based on the analytical solution of the system of equations derived from the first order Karush–Kuhn–Tucker conditions for general linear programming problems using symbolic manipulation. Emphasis is given on the ability of the proposed methodology to handle efficiently the LHS uncertainty by computing exactly the corresponding nonconvex critical regions while numerical studies underline further the advantages of the proposed methodology, when compared to existing algorithms

A new integrable system related to the Toda lattice

A new integrable lattice system is introduced, and its integrable discretizations are obtained. A B\"acklund transformation between this new system and the Toda lattice, as well as between their discretizations, is established.Comment: LaTeX, 14 p

Multi-parametric mixed integer linear programming under global uncertainty

Major application areas of the process systems engineering, such as hybrid control, scheduling and synthesis can be formulated as mixed integer linear programming (MILP) problems and are naturally susceptible to uncertainty. Multi-parametric programming theory forms an active field of research and has proven to provide invaluable tools for decision making under uncertainty. While uncertainty in the right-hand side (RHS) and in the objective function's coefficients (OFC) have been thoroughly studied in the literature, the case of left-hand side (LHS) uncertainty has attracted significantly less attention mainly because of the computational implications that arise in such a problem. In the present work, we propose a novel algorithm for the analytical solution of multi-parametric MILP (mp-MILP) problems under global uncertainty, i.e. RHS, OFC and LHS. The exact explicit solutions and the corresponding regions of the parametric space are computed while a number of case studies illustrates the merits of the proposed algorithm