Constructive Relationships Between Algebraic Thickness and Normality

We study the relationship between two measures of Boolean functions; \emph{algebraic thickness} and \emph{normality}. For a function $f$, the algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero coefficients in the unique GF(2) polynomial representing $f$, and the normality is the largest dimension of an affine subspace on which $f$ is constant. We show that for $0 < \epsilon<2$, any function with algebraic thickness $n^{3-\epsilon}$ is constant on some affine subspace of dimension $\Omega\left(n^{\frac{\epsilon}{2}}\right)$. Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of $\Theta(\sqrt{n})$ from the best guaranteed, and when restricted to the technique used, is at most a factor of $\Theta(\sqrt{\log n})$ from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness $\Omega\left(2^{n^{1/6}}\right)$.Comment: Final version published in FCT'201

Coloniality, subalternity and narratives of resistance in an afro-ecuadorian community

Partindo-se de um campo etnográfico dedicado à realidade da comunidade afro-equatoriana “La Chiquita”, nomeadamente quanto à sua luta pelo território ancestral que é-lhe de direito face à desterritorialização promovida pelo avanço do monocultivo de palma azeiteira, propomos: (I) compreender a dimensão da «colonialidade» do Estado no interior deste conflito, incluindo-se o ethos disciplinar e institucional que interpreta a negritude; e (II) identificar e analisar as diversas formas de resistência desta comunidade em resposta a tal estrutura sociopolítica vertical, trazendo à tona as narrativas do seu contexto desde uma crítica centrada na epistemologia e na sociologia jurídica.Departing from an ethnographic fieldwork dedicated to the reality of the afro-ecuadorian community of “La Chiquita”, in particular on its struggle concerning their ancestral territorial integrity in response of a growing dispossession process orchestrated by oil palm monocultures in Ecuador, we propose: (I) comprehend the “coloniality” of the State within this conflict, including the disciplinary and institutional ethos in interpreting blackness; and (II) identify and analyze the various forms of resistance of this community in response to such vertical sociopolitical structure, bringing out their narratives in context from a criticism focused on epistemology and the sociology of law

Integrable discrete systems on R and related dispersionless systems

The general framework for integrable discrete systems on R in particular containing lattice soliton systems and their q-deformed analogues is presented. The concept of regular grain structures on R, generated by discrete one-parameter groups of diffeomorphisms, through which one can define algebras of shift operators is introduced. Two integrable hierarchies of discrete chains together with bi-Hamiltonian structures are constructed. Their continuous limit and the inverse problem based on the deformation quantization scheme are considered.Comment: 19 page

On the Complexity of Computing Two Nonlinearity Measures

We study the computational complexity of two Boolean nonlinearity measures: the nonlinearity and the multiplicative complexity. We show that if one-way functions exist, no algorithm can compute the multiplicative complexity in time $2^{O(n)}$ given the truth table of length $2^n$, in fact under the same assumption it is impossible to approximate the multiplicative complexity within a factor of $(2-\epsilon)^{n/2}$. When given a circuit, the problem of determining the multiplicative complexity is in the second level of the polynomial hierarchy. For nonlinearity, we show that it is #P hard to compute given a function represented by a circuit

The extended Toda hierarchy

Using construction of logarithm of a difference operator, we present the Lax pair formalism for certain extension of the continuous version of the classical Toda lattice hierarchy, provide a well defined notion of tau function for its solutions, and give an explicit formulation of the relationship between the CP1 topological sigma model and the extended Toda hierarchy. We also establish an equivalence of the latter with certain extension of the nonlinear Schrodinger hierarchy

Enumeration of hypermaps and Hirota equations for extended rationally constrained KP

We consider the Hurwitz Dubrovin--Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also known as the total descendant potential) associated with this Dubrovin--Frobenius manifold is a tau function of a rational reduction of the Kadomtsev--Petviashvili hierarchy. This statement was conjectured by Liu, Zhang, and Zhou. We also provide a partial enumerative meaning for this partition function associating one particular set of times with enumeration of rooted hypermaps.Comment: 39 page

Para una Socio-antropología Jurídica poscolonial: del "ethos" occidental a las narrativas de resistencia afroecuatorianas (Toward a Post-colonial Socio-anthropology of law: from the western "ethos" to the Afro-Ecuadorian narratives of resistance)

Partiendo de un trabajo de campo etnogr&aacute;fico dedicado a la realidad de la comunidad afro-ecuatoriana La Chiquita, especialmente respecto a su lucha por el territorio ancestral a que tienen derecho, frente a la desterritorializaci&oacute;n promovida por el avance del monocultivo de palma aceitera, nos proponemos: I) comprender la dimensi&oacute;n de la colonialidad del Estado al interior de este conflicto, incluyendo el ethos disciplinar e Institucional que interpreta la negritud; y II) identificar y analizar las diversas formas de resistencia de esta comunidad en respuesta a esa estructura sociopol&iacute;tica vertical, incorporando al an&aacute;lisis las narrativas de su contexto desde una cr&iacute;tica centrada en la epistemolog&iacute;a y en la socio-antropolog&iacute;a jur&iacute;dica. Departing from an ethnographic fieldwork dedicated to the reality of the Afro-Ecuadorian community of La Chiquita, in particular on its struggle concerning the integrity of the ancestral territory in response of a growing dispossession process orchestrated by oil palm monocultures in Ecuador, we propose: (I) an overview to comprehend the coloniality of the State within this conflict, including the disciplinary and Institutional ethos in interpreting blackness; and (II) identify and analyze the various forms of resistance of this community in response to such vertical sociopolitical structure, exposing their narratives in such a context from a criticism focused on epistemology and the sociology of law. DOWNLOAD THIS PAPER FROM SSRN: http://ssrn.com/abstract=3133055</a

The multicomponent 2D Toda hierarchy: Discrete flows and string equations

The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov--Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov--Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations.Comment: 27 pages. In the revised paper we improved the presentatio