A Realization of Hyperrings

The purpose of this article is to present certain results arising from a study of theory of hyperrings. By a hyperring we mean a Krasner hyperring, that is, a triple (R, +,·) is such that (R, +) is a canonical hypergroup, (R, ·) is a semigroup with a zero 0 where 0 is the scalar identity of (R, +) and · is distributive over + . In this article, we define the notions of normal, prime, maximal, and Jacobson radical of a hyperring and by considering these notions we obtain some results. We define hyperring of fractions and hyper-valuation on a hyperring. For this, as in the classical case, we need a mapping from R onto an ordered group G. Finally, we shall state and prove the Chinese Remainder Theorem for the case of hyperrings.     

Generalized Divisors and Total Reflexivity

The notion of generalized divisors on schemes is introduced by Hartshorne. It is shown that there exists a bijection between the set of all generalized divisors on a scheme X and the set of all reflexive coherent 𝒪 _X -modules which are locally free of rank one at generic points. This bijection, corresponds Cartier divisors to the set of all locally free sheaves of rank one. Our aim in this article is to study the class of generalized divisors that maps to totally reflexive coherent 𝒪 _X -modules, under this correspondence. We investigate this class of divisors, that will be called Gorenstein divisors, both over schemes and also over commutative noetherian rings. We show that this class of divisors has usual properties and fits well in the hierarchy of divisors that already exists in the literature.     

Central extensions of semidirect products and geodesic equations

We express the equation for stratified fluid and the superconductivity equation as geodesic equations on Lie group central extensions of semidirect products. We study the existence of stream functions.     

Scales for co-compact embeddings of virtually free groups

Let Γ be a group which is virtually free of rank at least 2 and let be the family of totally disconnected, locally compact groups containing Γ as a co-compact lattice. We prove that the values of the scale function with respect to groups in evaluated on the subset Γ have only finitely many prime divisors. This can be thought of as a uniform property of the family .      

Integer matrices with constraints on leading partial row and column sums

Consider the problem of finding an integer matrix that satisfies given constraints on its leading partial row and column sums. For the case in which the specified constraints are merely bounds on each such sum, an integer linear programming formulation is shown to have a totally unimodular constraint matrix. This proves the polynomial-time solvability of this case. In another version of the problem, one seeks a zero-one matrix with prescribed row and column sums, subject to certain near-equality constraints, namely, that all leading partial row (respectively, column) sums up through a given column (respectively, row) are within unity of each other. This case admits a polynomial reduction to the preceding case, and an equivalent reformulation as a maximum-flow problem. The results are developed in a context that relates these two problems to consistent matrix rounding.     

Dualities and reciprocities on graphs on surfaces

We extend the duality between acyclic orientations and totally cyclic orientations on planar graphs to dualities on graphs on orientable surfaces by introducing boundary acyclic orientations and totally bi-walkable orientations. In addition, we provide a reciprocity theorem connecting local tensions and boundary acyclic orientations. Furthermore, we define the balanced flow polynomial which is connected with tension polynomial by duality and with totally bi-walkable orientations by reciprocity.     

Holomorphic Retractions of Bounded Symmetric Domains onto Totally Geodesic Complex Submanifolds

Given a bounded symmetric domain Ω the author considers the geometry of its totally geodesic complex submanifolds S ? Ω.In terms of the Harish-Chandra realization Ω and taking S to pass through the origin 0 ∈Ω,so that S=E ∩Ωfor some complex vector subspace of Cⁿ,the author shows that the orthogonal projectionρ:Ω→ E maps Ω onto S,and deduces that S?Ω is a holomorphic isometry with respect to the Caratheodory metric.His first theorem gives a new derivation of a result of Yeung’s deduced...