A continuous open finite-to-one surjection f preserves Cantor rank. If the domain of f is a Hausdorff compact space, Cantor degree variations are bounded by the maximal size of the finite fibres. If the fibres are infinite, we show inequalities involving the maximal and minimal rank of the fibres. A closed preorder on a denumerable Hausdorff compact space is the intersection of clopen preoders....

A ring is called a Gelfand ring (pm ring ) if each prime ideal is contained in a unique maximal ideal. For a Gelfand ring R with Jacobson radical zero, we show that the following are equivalent: (1) R is Artinian; (2) R is Noetherian; (3) R has a finite Goldie dimension; (4) Every maximal ideal is generated by an idempotent; (5) Max (R) is finite. We also give the following resu1ts:an ideal...

Let C be a general curve of genus g, embedded in Pr via a general linear series of degree d. In this paper, we prove the Maximal Rank Conjecture, which determines the Hilbert function of C ⊂ Pr.

We consider the problem of matching applicants to posts where applicants have preferences over posts. Thus the input to our problem is a bipartite graph G = (A ∪ P , E), where A denotes a set of applicants, P is a set of posts, and there are ranks on edges which denote the preferences of applicants over posts. A matching M in G is called rank-maximal if it matches the maximum number of applican...

We derive the formulas of the maximal andminimal ranks of four real matrices $X_{1},X_{2},X_{3}$ and $X_{4}$in common solution $X=X_{1}+X_{2}i+X_{3}j+X_{4}k$ to quaternionmatrix equations $A_{1}X=C_{1},XB_{2}=C_{2},A_{3}XB_{3}=C_{3}$. Asapplications, we establish necessary and sufficient conditions forthe existence of the common real and complex solutions to the matrixequations. We give the exp...

Let $ G = \prod_{i 1}^{\mathsf r} G_i be a product of simple real algebraic groups rank one and \Gamma an Anosov subgroup with respect to minimal parabolic subgroup. For each \mathsf v in the interior positive Weyl chamber, let \mathscr R_ v\subset \Gamma\backslash denote Borel subset all points recurrent \exp (\mathbb R_+ v) $-orbits. maximal horospherical N $, we show that $-action on {\maths...

conclusions large-scale studies on the healthy black african population could assess the relationship between diastolic dyssynchrony and changes in myocardial performance related to racial differences. results the frequency of some dyssynchrony markers was similar to that of other studies. the difference between maximal time-to-peak systolic velocity and minimal time-to-peak systolic velocity w...

The rank of a graph is defined to be the its adjacency matrix. A called reduced if it has no isolated vertices and two with same set neighbors. G said maximal any containing as proper induced subgraph higher rank. main intent this paper present some results on graphs. First, we introduce characterization trees (a tree not subtree rank). Next, give near-complete ‘generalized friendship graphs.’ ...