Small Extended Formulations for Cyclic Polytopes

We provide an extended formulation of size O(log n)^{\lfloor d/2 \rfloor} for the cyclic polytope with dimension d and n vertices (i,i^2,\ldots,i^d), i in [n]. First, we find an extended formulation of size log(n) for d= 2. Then, we use this as base case to construct small-rank nonnegative factorizations of the slack matrices of higher-dimensional cyclic polytopes, by iterated tensor products. Through Yannakakis's factorization theorem, these factorizations yield small-size extended formulations for cyclic polytopes of dimension d>2

Parametrization of semi-dynamical quantum reflection algebra

We construct sets of structure matrices for the semi-dynamical reflection algebra, solving the Yang-Baxter type consistency equations extended by the action of an automorphism of the auxiliary space. These solutions are parametrized by dynamical conjugation matrices, Drinfel'd twist representations and quantum non-dynamical $R$-matrices. They yield factorized forms for the monodromy matrices.Comment: LaTeX, 24 pages. Misprints corrected, comments added in Conclusion on construction of Hamiltonian

An absorbing boundary formulation for the stratified, linearized, ideal MHD equations based on an unsplit, convolutional perfectly matched layer

Perfectly matched layers are a very efficient and accurate way to absorb waves in media. We present a stable convolutional unsplit perfectly matched formulation designed for the linearized stratified Euler equations. However, the technique as applied to the Magneto-hydrodynamic (MHD) equations requires the use of a sponge, which, despite placing the perfectly matched status in question, is still highly efficient at absorbing outgoing waves. We study solutions of the equations in the backdrop of models of linearized wave propagation in the Sun. We test the numerical stability of the schemes by integrating the equations over a large number of wave periods.Comment: 8 pages, 7 figures, accepted, A &

On the Linear Extension Complexity of Regular n-gons

In this paper, we propose new lower and upper bounds on the linear extension complexity of regular $n$-gons. Our bounds are based on the equivalence between the computation of (i) an extended formulation of size $r$ of a polytope $P$, and (ii) a rank-$r$ nonnegative factorization of a slack matrix of the polytope $P$. The lower bound is based on an improved bound for the rectangle covering number (also known as the boolean rank) of the slack matrix of the $n$-gons. The upper bound is a slight improvement of the result of Fiorini, Rothvoss and Tiwary [Extended Formulations for Polygons, Discrete Comput. Geom. 48(3), pp. 658-668, 2012]. The difference with their result is twofold: (i) our proof uses a purely algebraic argument while Fiorini et al. used a geometric argument, and (ii) we improve the base case allowing us to reduce their upper bound $2 \left\lceil \log_2(n) \right\rceil$ by one when $2^{k-1} < n \leq 2^{k-1}+2^{k-2}$ for some integer $k$. We conjecture that this new upper bound is tight, which is suggested by numerical experiments for small $n$. Moreover, this improved upper bound allows us to close the gap with the best known lower bound for certain regular $n$-gons (namely, $9 \leq n \leq 13$ and $21 \leq n \leq 24$) hence allowing for the first time to determine their extension complexity.Comment: 20 pages, 3 figures. New contribution: improved lower bound for the boolean rank of the slack matrices of n-gon

On integrable boundaries in the 2 dimensional O(N)O(N) σ\sigma-models

We make an attempt to map the integrable boundary conditions for 2 dimensional non-linear O(N) $\sigma$-models. We do it at various levels: classically, by demanding the existence of infinitely many conserved local charges and also by constructing the double row transfer matrix from the Lax connection, which leads to the spectral curve formulation of the problem; at the quantum level, we describe the solutions of the boundary Yang-Baxter equation and derive the Bethe-Yang equations. We then show how to connect the thermodynamic limit of the boundary Bethe-Yang equations to the spectral curve.Comment: Dedicated to the memory of Petr Kulish, 31 pages, 1 figure, v2: conformality and integrability of the boundary conditions are distinguishe