Although Flory's approximation describes very well the radius r of a polymer, it disagrees sharply with an exact e expansion (derived from an analogy to n = 0 spins). Here the polymer is described explicitly qs a critical cluster of n = 0 spins. While ordinary clusters resemble trees bifurcating into b branches, a cluster of n = 0 spins constitutes a single line, weighted by b phantom bifurcati...

In this paper, using the best proximity theorems for an extensionof Brosowski's theorem. We obtain other results on farthest points. Finally, wedene the concept of e- farthest points. We shall prove interesting relationshipbetween the -best approximation and the e-farthest points in normed linearspaces (X; ||.||). If z in W is a e-farthest point from an x in X, then z is also a-best approximati...

i n s u p p o r t i n g e n v i r o n m e n t a l s u s t a i n a b i l i t y, m a n a g i n g p r o d u c t r e t u r n s h a s b e c o m e a v e r y i m p o r t a n t a n d c h a l l e n g i n g i s s u e. r e s p o n d i n g t o t h i s t r e n d, r e s e a r c h e r s i n m a n y p a r t s o f t h e w o r l d h a v e c o n d u c t e d n u m e r o u s s t u d i e s i n r e v e r s e l o g i ...

One can prove the following four propositions: (1) For all points p, q of E2 T and for every subset P of E2 T such that P is an arc from p to q holds P is compact. (2) For every real number r holds 0 ¬ r and r ¬ 1 iff r ∈ the carrier of I. (3) For all points p1, p2 of E 2 T and for all real numbers r1, r2 such that (1 − r1) · p1 + r1 · p2 = (1 − r2) · p1 + r2 · p2 holds r1 = r2 or p1 = p2. (4) ...

Given a prime p, an elliptic curve E/Fp over the finite field Fp of p elements and a binary linear recurrence sequence (u(n))∞n=1 of order r, we study the distribution of the sequence of points r−1 ∑ j=0 u(n+ j)Pj , n = 1, . . . , N, on average over all possible choices of Fp-rational points P1, . . . , Pr on E . For a sufficiently large N we improve and generalise a previous result in this dir...

the genus eremostachys (lamiaceae), consists of about 60 species which 15 species of eremostachys grow wildly in the north-west of iran. the rhizomes and roots of e. laciniata have traditionally been used as a local analgesic and for the treatment of rheumatic diseases and pain in the feet. this experiment was conducted in the north-west of iran in 2012-2013, to evaluate morphological diversity...

As in the previous chapter, let us denote points in R by p = (x, y, z) and points in R by u = (u, v). An edge is then defined as the convex hull of (or, equivalently, the line segment between) two distinct points and a triangle as the convex hull of three non-collinear points. We will denote edges and triangles in R with capital letters and those in R with small letters, for example, e = [u1,u2...

a r t i c l e i n f o a b s t r a c t The class of swapped or OTIS (optical transpose interconnect system) networks, built of n copies of an n-node cluster by connecting node i in cluster j to node j in cluster i for i = j, has been studied extensively. One problem with such networks is that node i of cluster i has no intercluster link. This slight asymmetry complicates a number of algorithms a...

We consider the following interpolation problem. Suppose one is given a finite set E ⊂ R, a function f : E → R, and possibly the gradients of f at the points of E. We want to interpolate the given information with a function F ∈ C(R) with the minimum possible value of Lip(∇F ). We present practical, efficient algorithms for constructing an F such that Lip(∇F ) is minimal, or for less computatio...

We consider the following interpolation problem. Suppose one is given a finite set E ⊂ R, a function f : E → R, and possibly the gradients of f at the points of E as well. We want to interpolate the given information with a function F ∈ C(R) with the minimum possible value of Lip(∇F ). We present practical, efficient algorithms for constructing an F such that Lip(∇F ) is minimal, or for less co...