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Menger's basis property is a generalization of $\sigma$-compactness and admits an elegant combinatorial interpretation. We introduce a general combinatorial method to construct non $\sigma$-compact sets of reals with Menger's property.…
A real symmetric n times n matrix is called copositive if the corresponding quadratic form is non-negative on the closed first orthant. If the matrix fails to be copositive there exists some non-negative certificate for which the quadratic…
We compactify M(atrix) theory on Riemann surfaces Sigma with genus g>1. Following [1], we construct a projective unitary representation of pi_1(Sigma) realized on L^2(H), with H the upper half-plane. As a first step we introduce a suitably…
For positive integers $n,d$, let $\lambda(n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex. We show that…
In this paper, for any compact Lie group $G$, we show that the space of $G$-invariant Riemannian metrics with positive scalar curvature (PSC) on any closed three-manifold is either empty or contractible. In particular, we prove the…
We define a 3-algebra with structure constants being symmetric in the first two indices. We also introduce an invariant anti-symmetric tensor into this 3-algebra and call it a symplectic 3-algebra. The general N=5 superconformal…
Given a Riemannian compact manifold (M,g) of dimension n>4, we have proven in [1] under some conditions that the equation : Pg(u) = Bu +Au2+Cu (1) where Pg is the GJMS-operator, n = dim(M) > 2k, A, B and C are smooth positive functions on…
Recently Chen and Movassagh proposed the quantum Merkle tree, which is a quantum analogue of the well-known classical Merkle tree. It gives a succinct verification protocol for quantum state commitment. Although they only proved security…
An unorthodox unified theory is developed to describe external and internal attributes and symmetries of fundamental fermions, quarks and leptons. Basic ingredients of the theory are an algebra which consists of all the…
In this paper, we present a computational approach to certify almost sure reachability for discrete-time polynomial stochastic systems by turning drift--variant criteria into sum-of-squares (SOS) programs solved with standard semidefinite…
Two theorems involving curl eigenfields on the 3--sphere are obtained using angular momentum theory. Spinor hyperspherical harmonics are shown to form an explicit, convenient basis. In particular, a spin--one vector calculus is reviewed. An…
We employ the methods of discrete (Lorentzian) Regge calculus for analysing Lorentzian quantum cosmology models with a special focus on discrete analogues of the no-boundary proposal for the early universe. We use a simple 4-polytope, a…
We construct positive solutions of the semilinear elliptic problem $\Delta u+ \lambda u + u^p = 0$ with Dirichet boundary conditions, in a bounded smooth domain $\Omega \subset \R^N$ $(N\geq 4)$, when the exponent $p$ is supercritical and…
We prove that the basis of cluster monomials of a skew-symmetric cluster algebra A of finite type is the atomic basis of A. This means that an element of A is positive if and only if it has a non-negative expansion in the basis of cluster…
An effective approach for solving the three-dimensional Dirac equation for spherically symmetric local interactions, which we have introduced recently, is reviewed and consolidated. The merit of the approach is in producing Schrodinger-like…
Let Omega be a quasisimple classical group in its natural representation over a finite vector space V, and let Delta be its normaliser in the general linear group. We construct the projection from Delta to Delta/Omega and provide fast,…
Refining a basic result of Alexander, we show that two flag simplicial complexes are piecewise linearly homeomorphic if and only if they can be connected by a sequence of flag complexes, each obtained from the previous one by either an edge…
A novel, nonperturbative, way to generate chiral symmetry breaking within the linear sigma model for 3 flavours is discussed. After spontaneous chiral symmetry breaking in the vacuum at the tree level the scalar nonet obtains mass, while…
In 1995, Jockusch constructed an infinite family of centrally symmetric $3$-dimensional simplicial spheres that are cs-$2$-neighborly. Here we generalize his construction and show that for all $d\geq 3$ and $n\geq d+1$, there exists a…
The classical limit of polymer quantum theories yields a one parameter family of `effective' theories labeled by \lambda. Here we consider such families for constrained theories and pose the problem of taking the `continuum limit', \lambda…