We show that the problem whether a given finite metric space (X, d) can be embedded into the rectilinear space R m can be formulated in terms of m-colorability of a certain hypergraph associated with (X, d). This is used to close a gap in the proof of an assertion of Bandelt and Chepoi [2] on certain critical metric spaces for this embedding problem. We also consider the question of determining...

‎It is shown that‎ ‎if $M$ is an Artinian module over a ring‎ ‎$R$‎, ‎then $M$ has Noetherian dimension $alpha $‎, ‎where $alpha $ is a countable ordinal number‎, ‎if and only if $omega ^{alpha }+2leq it{l}(M)leq omega ^{alpha‎ +1}$, ‎where $ it{l}(M)$ is‎ ‎the length of $M$‎, ‎$i.e.,$ the least ordinal number such that the interval $[0‎, ‎ it{l}(M))$ cannot be embedded in the lattice of all su...

An irreducible open 3-manifold W is R 2-irreducible if it contains no non-trivial planes, i.e. given any proper embedded plane in W some component of W ? must have closure an embedded halfspace R 2 0; 1). In this paper it is shown that if M is a connected, P 2-irreducible, open 3-manifold such that 1 (M) is nitely generated and the universal covering space f M of M is R 2-irreducible, then eith...

This paper presents the first formal measure of intelligence for agents fully embedded within their environment. Whereas previous measures such as Legg’s universal intelligence measure and Russell’s bounded optimality provide theoretical insights into agents that interact with an external world, ours describes an intelligence that is computed by, can be modified by, and is subject to the time a...

A new geometric invariant will be introduced, studied and determined on compact symmetric spaces. Introduction. We will introduce a new invariant on Riemannian manifolds, which is especially interesting on compact symmetric spaces, and we will determine the invariant for the compact symmetric spaces, thus amplifying the announcement [CN1]. A symmetric space M is defined with the point symmetry ...

let $r$ be a commutative ring with identity and $m$ be a unitary$r$-module. the primary-like spectrum $spec_l(m)$ is thecollection of all primary-like submodules $q$ such that $m/q$ is aprimeful $r$-module. here, $m$ is defined to be rsp if $rad(q)$ isa prime submodule for all $qin spec_l(m)$. this class containsthe family of multiplication modules properly. the purpose of thispaper is to intro...

In this paper we study the impact of Minkowski metric matrix on a projection in the Minkowski Space M along with their basic algebraic and geometric properties.The relation between the m-projections and the Minkowski inverse of a matrix A in the minkowski space M is derived. In the remaining portion commutativity of Minkowski inverse in Minkowski Space M is analyzed in terms of m-projections as...

We study umbilic (space-like) submanifolds of pseudo-Riemannian space forms, then define totally semi-umbilic space-like submanifold of pseudo Euclidean space and relate this notion to umbilicity. Finally we give characterization of total semi-umbilicity for space-like submanifolds contained in pseudo sphere or pseudo hyperbolic space or the light cone.A pseudo-Riemannian submanifold M in (a...

The main purpose of this paper is to introduce the compatibility of $M$-fuzzifying topologies and $M$-fuzzifying convexities, define an $M$-fuzzifying topological convex space, and give a method to generate an $M$-fuzzifying topological convex space. Some characterizations of $M$-fuzzifying topological convex spaces are presented. Finally, the notion of $M$-fuzzifying weak topologies is obtaine...