For an integer n ≥ 2 we denote by P (n) the largest prime factor of n. We obtain several upper bounds on the number of solutions of congruences of the form n! + 2 − 1 ≡ 0 (mod q) and use these bounds to show that lim sup n→∞ P (n! + 2 − 1)/n ≥ (2π + 3)/18.

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....

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...

The N 4 0 0 is an endogenous event-related brain potential (ERP) that is sensitive to semantic processes during language comprehension. The general question we address in this paper is which aspects of the comprehension process are manifest in the N 4 0 0 . The focus is on the sensitivity of the N 4 0 0 to the automatic process of lexical access, or to the controlled process of lexical integrat...

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...

Fermat’s Last Theorem (FLT) is the assertion that for n ≥ 3, the equation X + y = Z has no solutions in integers X,Y, Z with XY Z 6= 0. It was proven by Fermat for n = 4 and by Euler for n = 3, cf. [Weil, page 104]. To prove it in general, then, it suffices to prove it when n is any prime p ≥ 5. For fixed p, the ”first case” of FLT is the assertion that there are no integer solutions with XY Z ...

Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces is bounded by an explicitly given polynomial in $n$. The result also holds all spanning Euler characteristic. Previously known bounds were exponential.

The purpose of this paper is to obtain the structure of certain near-rings satisfying the following conditions: (i) d(I) ⊆ Z(N), (ii) d(−I) ⊆ Z(N), (iii) d([x, y]) = 0, (iv) d([x, y]) = [x, y], (v) d(x ◦ y) = 0, (vi) d(x ◦ y) = x ◦ y for all x, y ∈ I , with I is a semigroup ideal and d is a semiderivation associated with an automorphism. Furthermore; an example is given to illustrate that the 3...