Let H, K be a couple of vector fields class C1 in an open set U ? ?N+m, $${\cal M}$$ N-dimensional submanifold and define T}: = \left\{{z \in {\cal M}:H\left(z \right),K\left(z \right) {T_z}{\cal M}} \right\}$$ . Then the obvious property If z0 ? is interior point (relative to )of T}$$ then $$\left[{H,K} \right]\left({{z_0}} {T_{{z_0}}}{\cal admits following generalization: superdensity As coro...

Let $p(z)$ be a polynomial of degree $n$. Then the polar derivative with respect to real or complex number $\alpha$ is defined by $D_\alpha p(z)=n p(z)+(\alpha-z) p^{\prime}(z)$. Govil and Mctume [14] proved that if $n$ having all its zeros in $|z| \leq k, k \geq 1$, then for $|\alpha| 1+k+k^n$,$$\begin{aligned}\max _{|z|=1}\left|D_\alpha p(z)\right| & n\left(\frac{|\alpha|-k}{1+k^n}\right)...

We show that the series expansions of certain $q$-products have \textit{matching coefficients} with their reciprocals. Several results are associated to Ramanujan's continued fractions. For example, let $R(q)$ denote Rogers-Ramanujan fraction having well-known $q$-product repesentation $$R(q)=\dfrac{(q;q^5)_\infty(q^4;q^5)_\infty}{(q^2;q^5)_\infty(q^3;q^5)_\infty}.$$ If \begin{align*} \sum_{n=0...

The N-word is the ultimate insult that has tormented generations of African-Americans. Yet over time, N-word derivatives have become popular terms of endearment by the descendants of the very people who once had to endure the N-word. Therein lies the root of an ongoing argument in society today: Who has the ‘right’ to use the N-word and N-word derivatives?  A quota sample (N=347) of undergradua...

We bound the second eigenvalue of random d $$ -regular graphs, for a wide range degrees , using novel approach based on Fourier analysis. Let G n {G}_{n,d} be uniform graph vertices, and λ ( ) \lambda \left({G}_{n,d}\right) its largest by absolute value. For some constant c > 0 c>0 any degree with log 10 ≪ ≤ {\log}^{10}n\ll d\le cn we show that = 2 + o 1 − / \left({G}_{n,d}\right)=\left(2+o(1)\...