Let  be a commutative ring and  be a unitary  module. We define a semi prime sub module of a module and consider various properties of it. Also we define semi-radical of a sub module of a module and give a number of its properties. We define modules which satisfy the semi-radical formula and present the existence of such a module.

Assuming Martin’s Axiom, we provide an example of two Fréchet-Urysohn 〈α4〉-spaces, whose product is a non-Fréchet-Urysohn 〈α4〉-space. This gives a consistent negative answer to a question raised by T. Nogura.

Quantifying the risk carried by an aggregate position Sd := ∑d i=1Xi comprising many risk factors Xi is fundamental to both insurance and financial risk management. Fréchet inequalities quantify the worst-case risk carried by the aggregate position given distributional information concerning its composing factors but without assuming independence. This marginal factor modeling of the aggregate ...

let $r$ be a commutative ring with identity and $m$ be a finitely generated unital $r$-module. in this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. moreover, we investigate some conditions which imply that the module $m$ is the direct sum of some cyclic modules and free modules. then some properties of fitting ideals of...

A Fréchet space, X , with a sequence of generating seminorms {‖·‖k} ∞ k=1 is called tame in case there exists an increasing function σ : N → N, such that for every continuous linear operator T from X into itself, there exists an N0 and C > 0 such that ‖T (x)‖n ≤ C‖x‖σ(n), ∀x ∈ X and n ≥ N0. This property does not depend upon the choice of fundamental system of semi-norms for X and is a property...

We describe two (1 + ε)-approximation algorithms for computing the Fréchet distance between two homeomorphic piecewise linear surfaces R and S of genus zero and total complexity n, with Fréchet distance δ. 1. A 2 (( n+ Area(R)+Area(S) (εδ)2 )2) time algorithm if R and S are composed of fat triangles (triangles with angles larger than a constant). 2. An O(D/(εδ)2) · n + 2O(D/(εδ)) time algorithm...

In this paper we consider Bernstein’s Lethargy Theorem (BLT) in the context of Fréchet spaces. Let X be an infinite-dimensional Fréchet space and let V = {Vn} be a nested sequence of subspaces of X such that Vn ⊆ Vn+1 for any n ∈ N and X = ⋃∞ n=1 Vn. Let en be a decreasing sequence of positive numbers tending to 0. Under an additional natural condition on sup{dist(x, Vn)}, we prove that there e...

let r be a ring, m a right r-module and (s,≤) a strictly ordered monoid. in this paper we will show that if (s,≤) is a strictly ordered monoid satisfying the condition that 0 ≤ s for all s ∈ s, then the module [[ms,≤]] of generalized power series is a uniserial right [[rs,≤]] ]]-module if and only if m is a simple right r-module and s is a chain monoid.

We study the decision tree complexity of the discrete Fréchet distance (decision version) under the L1 and L∞ metrics over R. While algorithms for the Euclidean (L2) discrete Fréchet distance were studied extensively, the problem in other metrics such as L1 and L∞ seems to be much less investigated. For the L1 discrete Fréchet distance in R we present a 2d-linear decision tree with depth O(n lo...

Suppose a two-member team is required to navigate through dangerous territory (a plane with polygonal obstacles) along two preselected polygonal paths. The team could consist of police officers, soldiers, or even robots on the surface of Mars. Team members are allowed to coordinate their movements, and it is imperative to keep the team close together during the mission so that if an emergency o...