Related papers: Realisation and Splitting for Poincar\'e Duality P…
We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincar\'e…
We propose matching pairs of half-BPS boundary conditions related by IR dualities of 3d $\mathcal{N}=2$ gauge theories. From these matching pairs we construct duality interfaces. We test our proposals by anomaly matching and the computation…
Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in…
The canonical formalism of three dimensional gravity coupled with the Dirac field is considered. We introduce complex variables to simplify the Dirac brackets of canonical variables and examine the canonical structure of the theory. We…
We give necessary and sufficient conditions for differentiating under the integral sign an integral that depends on a parameter. The conditions require the equality of two iterated integrals and depend on being able to integrate every…
Parafermions of order two and three are shown to be the fundamental tool to construct superspaces related to cubic and quartic extensions of the Poincar\'e algebra. The corresponding superfields are constructed, and some of their main…
In this paper, we study simplicial hyperplane arrangements in real projective $3$-space. We give a necessary condition for the characteristic polynomial to have only real roots, valid also for non-simplicial arrangements. As application, we…
The first nonzero eigenvalue of the Neumann Laplacian is shown to be minimal for the degenerate acute isosceles triangle, among all triangles of given diameter. Hence an optimal Poincar\'{e} inequality for triangles is derived. The proof…
We present mechanisms for generating conical singularities both in three and four-dimensions in the systems with copies of scalar or chiral multiplets coupled to $N=2$ or $N=1$ supergravity. Our mechanisms are useful for supersymmetry…
We briefly recall a fundamental exterior differential system introduced by the author and then apply it to the case of three dimensions. Here we find new global tensors and intrinsic invariants of oriented Riemaniann 3-manifolds. The system…
The present work establishes necessary and sufficient conditions for a nonlinear system with two inputs to be described by a specific triangular form. Except for some regularity conditions, such triangular form is flat. This may lead to the…
We give sufficient conditions for three- or four-dimensional truncated Poincare-Dulac normal forms of resonance degree two to be meromorphically nonintegrable when the Jacobian matrices have a zero and pair of purely imaginary eigenvalues…
We extend the classification of mixed states of quantum systems composed of arbitrary number of subsystems of arbitrary dimensions. This extended classification is complete in the sense of partial separability and gives 1+18+1 partial…
We classify pro-$p$ Poincar\'e duality pairs in dimension two. We then use this classification to build a pro-$p$ analogue of the curve complex and establish its basic properties. We conclude with some statements concerning separability…
We prove that subsets of ${\Bbb R}^d$, $d \ge 4$ of large enough Hausdorff dimensions contain vertices of an equilateral triangle. It is known that additional hypotheses are needed to assure the existence of equilateral triangles in two…
The interior structure of arbitrary sets of quaternion units is analyzed using general methods of the theory of matrices. It is shown that the units are composed of quadratic combinations of fundamental objects having a dual mathematical…
The conditions for superintegrable systems in two-dimensional Euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third-order integral are studied. It is shown that only…
In this Letter we find the new criteria of separability of multipartite qubit density matrixes. Especially, we discuss in detail the criteria of separability for tripartite qubit density matrixes. We find the sufficient and necessary…
We investigate some connections between two different ways of defining Poincar\'e Duality, and relate them geometrically to the level curve mapping.
We develop a general theory for the existence of extremal K\"ahler metrics of Poincar\'e type in the sense of Auvray, defined on the complement of a toric divisor of a polarized toric variety. In the case when the divisor is smooth, we…