For any positive integer n, the famous F.Smarandache function S(n) is defined as the smallest positive integer m such that n | m!. That is, S(n) = min{m : n | m!, n ∈ N}. The main purpose of this paper is using the elementary method to study the estimate problem of S (Fn), and give a sharper lower bound estimate for it, where Fn = 2 2 + 1 is called the Fermat number.

Abstract Here we provide refinements of the stability results Simons and Xin, concerning Yang–Mills fields harmonic maps respectively. The result also implies earlier Morse index estimates for both cases.

1. Integral Estimate: Ĥ(x) = − ∫ p̂(x) log p̂(x)dx 2. Re-substitution Estimate: Ĥ(x) = − 1 n ∑n i=1 log p̂(Xi) where p̂ is obtained using the samples {X1, ..., Xn}. 3. Splitting Data Estimate: Ĥ(x) = − 1 m ∑m i=1 log p̂(Xi) where p̂ is obtained using the samples {Xm+1, ..., Xn}. A cross-validation estimate can be defined similarly, e.g. the leave-one-out estimate is given as Ĥ(x) = − 1 n ∑n i=1 log p̂...

This paper investigates the relationship between inflation and growth uncertainty in Iran for the period of 1988-2008 by using quarterly data. We employ Generalized Autoregressive Conditional Heteroscedasticity in Mean (GARCH-M) model to estimate time-varying conditional residual variance of growth, as a standard measures of growth uncertainty. The empirical evidence shows that growth uncertain...

Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and M be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix M + Nn, generalizing an earlier result of Spielman and Teng for the case when x is gaussian. O...

Let x be a complex random variable with mean zero and bounded variance. Let Nn be the random matrix of size n whose entries are iid copies of x and let M be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix M +Nn, generalizing an earlier result of Spielman and Teng for the case when x is gaussian...

We show a conformal spectral estimate for the Dirac operator on a non-conformally-flat Riemannian spin manifolds of dimension n ≥ 7. The estimate is a spinorial analogue to an estimate by Aubin which solved the Yamabe problem for the above manifolds. Using Hijazi’s inequality our estimate implies Aubin’s estimate. More exactly, let M be a compact manifold of dimension n ≥ 7 equipped with a Riem...