has at least n+ 1 distinct prime divisors. (b) When a = 3, determine all the positive integers n for which the assertion in (a) is false. 2. ABCD is a square; points U and V are situated on the respective sides BC and CD. Prove that the perimeter of triangle CUV is equal to twice the sidelength of the square if and only if ∠UAV = 45◦. 3. Let f(x) be a convex increasing realvalued function defin...

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

We say a pair of integers (a, b) is findable if the following true. For any δ > 0 there exists p0 such that for prime p ≥ and red-blue colouring ℤ/pℤ in which each colour has density at least δ, we can find an arithmetic progression length + b inside whose first elements are red last blue. Szemeredi’s Theorem on progressions implies (0, k) (1, find-able k. prove (2, also However, same not true ...

Let G be a (p, q) graph. Let f : V (G) → {1, 2, . . . , k} be a map. For each edge uv, assign the label gcd (f(u), f(v)). f is called k-prime cordial labeling of G if |vf (i) − vf (j)| ≤ 1, i, j ∈ {1, 2, . . . , k} and |ef (0) − ef (1)| ≤ 1 where vf (x) denotes the number of vertices labeled with x, ef (1) and ef (0) respectively denote the number of edges labeled with 1 and not labeled with 1....

assume that $a$‎, ‎$b$ are banach algebras and that $m:atimes brightarrow b$‎, ‎$m^prime:atimes arightarrow b$ are bounded bilinear mappings‎. ‎we study the relationships between arens regularity of $m$‎, ‎$m^prime$ and the banach algebras $a$‎, ‎$b$‎. ‎for a banach $a‎$‎-bimodule $b$‎, ‎we show that $b$ factors with respect to $a$ if and only if $b^{**}$ is unital as an $a^{**}‎$‎-module‎. ‎le...

When we speak about a function field of one variable over a field K, we mean a finitely generated regular extension F of K of transcendence degree 1. We briefly recall the definitions of the main objects attached to F/K and their properties. See the books [Che51] or [Sti93] for details. A more comprehensive survey can be found [FrJ08, Sections 3.1-3.2]. A K-place of F is a place φ: F → K̃ ∪ {∞} ...

A generalized mover-stayer model is described for conditionally Markov processes under panel observation. Marginally the model represents a mixture of nested continuous-time Markov processes in which sub-models are defined by constraining some transition intensities to zero between two or more states of a full model. A Fisher scoring algorithm is described which facilitates maximum likelihood e...

We consider a boundary-value problem for the nonlinear integrodifferential equation $$ {u}^{\prime \prime }-m\left(\underset{0}{\overset{l}{\int }}{u}^{\prime 2} dx\right){u}^{{\prime\prime} }=f\left(x,u,{u}^{\prime}\right),\kern1em m(z)\ge \upalpha >0,\kern1em 0\le z<\infty, simulating static state of Kirchhoff beam. The is reduced to integral equation, which solved by using Picard iterative m...