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We introduce novel polyhedral approximation hierarchies for the cone of nonnegative forms on the unit sphere in Rn and for its (dual) cone of moments. We prove computable quantitative bounds on the speed of convergence of such hierarchies. We also introduce a novel optimization-free algorithm for building converging sequences of lower bounds for polynomial minimization problems on spheres. Finally, some computational results are discussed, showcasing our implementation of these hierarchies in the programming language Julia. [ABSTRACT FROM AUTHOR]

We prove that the determinantal complexity of a hypersurface of degree [Formula omitted] is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the [Formula omitted] permanent is 7. We also prove that for [Formula omitted], there is no nonsingular hypersurface in [Formula omitted] of degree d that has an expression as a determinant of a [Formula omitted] matrix of linear forms, while on the other hand for [Formula omitted], a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5., Author(s): Jarod Alper [sup.1] , Tristram Bogart [sup.2] , Mauricio Velasco [sup.2] Author Affiliations: (1) 0000 0001 2180 7477grid.1001.0Mathematical Sciences Institute, Australian National University, 0200, Canberra, ACT, Australia (2) 0000000419370714grid.7247.6Departamento [...]

We prove new mixing rate estimates for the random walks on homogeneous spaces determined by a probability distribution on a finite group $G$. We introduce the switched random walk determined by a finite set of probability distributions on $G$ , prove that its long-term behaviour is determined by the Fourier joint spectral radius of the distributions, and give Hermitian sum-of-squares algorithms for the effective estimation of this quantity. [ABSTRACT FROM AUTHOR]

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We bound the Pythagoras number of a real projective subvariety: the smallest positive integer r such that every sum of squares of linear forms in its homogeneous coordinate ring is a sum of at most r squares. Enhancing existing methods, we exhibit three distinct upper bounds involving known invariants. In contrast, our lower bound depends on a new invariant of a projective subvariety called the quadratic persistence. Defined by projecting away from points, this numerical invariant is closely related to the linear syzygies of the variety. In addition, we classify the projective subvarieties of maximal and almost-maximal quadratic persistence, and determine their Pythagoras numbers. [ABSTRACT FROM AUTHOR]

We propose reducible algebraic curves as a mechanism to construct partial maximum distance separable codes geometrically. We obtain new general existence results, new explicit constructions, and improved estimates on the smallest field sizes over which such codes can exist. Our results are obtained by combining ideas from projective algebraic geometry, combinatorics, and probability theory. [ABSTRACT FROM AUTHOR]

We define a statistical measure of the typical size of words of low weight in a linear code over a finite field. We prove that the dual toric codes coming from polytopes of degree one are characterized, among all dual toric codes, by being extremal with respect to this measure. We also give a geometric interpretation of the minimum distance of dual toric codes and characterize its extremal values. Finally, we obtain exact formulas for the parameters of both primal and dual toric codes associated to polytopes of degree one. [ABSTRACT FROM AUTHOR]

We prove that linearizing multilinear optimization problems leads to new functorial operations on real convex sets. These operations are convex analogues of hom functors, tensor products, symmetric powers, exterior powers, and general Schur functors on vector spaces and lead to novel constructions even for polyhedra. We discuss their general theory and introduce mechanisms to compute them or approximate them in ways amenable to efficient computation. [ABSTRACT FROM AUTHOR]

Si reconocemos que las transformaciones políticas conducen a las transformaciones en los espacios mediáticos y tecnológicos -y no al contrario, en una deificación tecnológica- las redes sociales serían una manifestación del surgimiento de los nuevos actores del siglo XXI, que tienen una forma diferente de organización -y expresión- política. Son actores que, además, responden a la irrupción de lo cultural en lo político o que plantean la 'lucha' cultural como una dimensión política. Lo cultural incluye aquello que se expresa fuertemente en las redes sociales: género, derechos, ambiente e incluso arte, pero también una postura ya no solo nacional, sino global y mundial con respecto a lo político. [ABSTRACT FROM AUTHOR]