In this paper, we introduce a new class $\mathcal{R}^{\alpha}_{m}(h)$ of functions $F=f\ast\psi$, defined in the open unit disc $E$ with $F(0)=F^{\prime}(0)-1=0$ and satisfying condition \begin{equation*} F^{\prime}(z)+\alpha zF^{\prime\prime}(z)=\left(\frac{m}{4}+\frac{1}{2}\right)p_{_{1}}(z)-\left(\frac{m}{4}-\frac{1}{2}\right)p_{_{2}}(z), \end{equation*} for $\alpha\geq0,\,m\geq2$ $p_{_{i}}\...

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....

This paper establishes the first four moment expansions to the order o(a^−1) of S_{t_{a}}^{prime }/sqrt{t_{a}}, where S_{n}^{prime }=sum_{i=1}^{n}Y_{i} is a simple random walk with E(Yi) = 0, and ta is a stopping time given by t_{a}=inf left{ ngeq 1:n+S_{n}+zeta _{n}>aright}‎ where S_{n}=sum_{i=1}^{n}X_{i} is another simple random walk with E(Xi) = 0, and {zeta _{n},ngeq 1} is a sequence of ran...

We examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b′ = 0′). This extends Csima’s result that every CAD theor...

Let H H be a connected Hopf algebra acting on an A"> A encoding="application/x-tex">A </inline-formul...

In this paper, we define a subclass of analytic functions by denote $T_{\beta}H\left( z,C_{n}^{\left( \lambda \right) }\left( t\right) $ satisfying the following subordinate condition \begin{equation*} \left( 1-\beta \frac{zf^{^{\prime }}\left( z\right) }{f\left( }\right) +\beta 1+\frac{zf^{^{\prime \prime }{f^{^{\prime \prec \frac{1}{\left( 1-2tz+z^{2}\right) ^{\lambda }}, \end{equation*} wher...

Abstract We investigate the equation of form $$(a(t)u^{\prime }(t))^{\prime }+Aa(t)u^{\prime }(t)+f(t,u(t))=0$$ ( a t ) u ′ + A f ,<...

Yes, but such a formula is complicated. For example, is there a polynomial f ∈ Z[x] for which f(n) = pn? f(x) = anx n + · · ·+ a1x+ a0 f(a0) = ana0 n + · · ·+ a1a0 + a0 so a0 | f . Suppose q is prime and f(n) = q. Then q | f(n + kq) for each k ∈ Z. So, in particular, we see that if f(m) is prime for each positive integer m, then f is a constant. In particular, f(x) = q for some prime q. The pol...