Gauss words are finite sequences of letters associated with self-intersecting closed curves in the plane. (These curves have no “triple” self-intersection). These sequences encode the order of intersections on the curves. We characterize, up to homeomorphism, all curves having a given Gauss word. We extend this characterization to the n-tuples of closed curves having a given n-tuple of words, t...

We prove that the number of near-coherence classes of non-principal ultrafilters on the natural numbers is either finite or 2c. Moreover, in the latter case the Stone-Čech compactification βω of ω contains a closed subset C consisting of 2c pairwise non-nearly-coherent ultrafilters. We obtain some additional information about such closed sets under certain assumptions involving the cardinal cha...

Genotypes of 68,230 individuals taken from 10 generations (F(4)-F(6), F(14)-F(17), F(24)-F(26)) of an experimental population of barley were determined for four esterase loci. The results show that frequencies of gametic ditypes changed significantly over generations and that striking gametic phase disequilibrium developed within a few generations for each of the six pairwise combinations of lo...

Let $kgeq 1$ be an integer, and $G=(V,E)$ be a finite and simplegraph. The closed neighborhood $N_G[e]$ of an edge $e$ in a graph$G$ is the set consisting of $e$ and all edges having a commonend-vertex with $e$. A signed Roman edge $k$-dominating function(SREkDF) on a graph $G$ is a function $f:E rightarrow{-1,1,2}$ satisfying the conditions that (i) for every edge $e$of $G$, $sum _{xin N[e]} f...

Text appearing between brackets like This ] is extracted from the rst printing; text appearing between pararentheses like f This g is to be inserted for the second printing. Chapter 1 Page 18, Replace the value of a closed-term is always a closed abstraction ] by f M] ] is a closed abstraction if it is deened g.

We give digital analogues of classical theorems of topology for continuous functions defined on spheres, for digital simple closed curves. In particular, we show the following: 1) A digital simple closed curve S of more than 4 points is not contractible, i.e., its identity map is not nullhomotopic in S ; 2) Let X and Y be digital simple closed curves, each symmetric with respect to the origin, ...

Let K be a closed convex subset of a Banach space X and let F be a nonempty closed convex subset of K. We consider complete metric spaces of self-mappings of K which fix all the points of F and are relatively nonexpansive with respect to a given convex function f on X. We prove (under certain assumptions on f) that the iterates of a generic mapping in these spaces converge strongly to a retract...

In this paper, we prove that on every Finsler n-sphere (S, F ) with reversibility λ satisfying F 2 < ( λ )g0 and l(S , F ) ≥ π(1 + 1 λ ), there always exist at least n prime closed geodesics without self-intersections, where g0 is the standard Riemannian metric on S n with constant curvature 1 and l(S, F ) is the length of a shortest geodesic loop on (S, F ). We also study the stability of thes...

Let 〈P,≤〉 be a partially ordered set. We define the poset Boolean algebra of P , and denote it by F (P ). The set of generators of F (P ) is {xp : p ∈ P}, and the set of relations is {xp ·xq = xp : p ≤ q}. We say that a Boolean algebra B is well-generated, if B has a sublattice G such that G generates B and 〈G,≤G〉 is well-founded. A wellgenerated algebra is superatomic. Theorem 1 Let 〈P,≤〉 be a...

Disjoint union is a partial binary operation returning the union of two sets if they are disjoint and undefined otherwise. A disjoint-union partial algebra of sets is a collection of sets closed under disjoint unions, whenever they are defined. We provide a recursive first-order axiomatisation of the class of partial algebras isomorphic to a disjointunion partial algebra of sets but prove that ...