Let {Zk}k ≥ 1 denote a sequence of independent Bernoulli random variables defined by P(Zk = 1) 1/k 1− 0) (k and put Tn ≔ ∑1 ≤ k n kZk. It is known that Tn/n convergesweakly to real variable D with density proportional the Dickman function, delay-differential equation uϱ ′ (u) + ϱ(u − 0 (u > initial condition ϱ(u) 1(0 u 1). Improving on earlier work, we propose asymptotic formulae remainders for...

We introduce the class of $(M, k)$-quasi-$*$-paranormal operators on a Hilbert space $H$. This extends classes $*$-paranormal and $k$-quasi-$*$-paranormal operators. An operator $T$ complex is called if there exists $M>0$ such that \begin{equation*} \sqrt{M}\left\Vert T^{k+2}x\right\Vert \left\Vert T^{k}x\right\Vert \geq T^{\ast }T^{k}x\right\Vert ^{2} \end{equation*} for all $x\in H.$ In prese...

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 H,K be subgroups of G. We investigate the intersection properties of left and right cosets of these subgroups. If H and K are subgroups of G, then G can be partitioned as the disjoint union of all left cosets of H, as well as the disjoint union of all right cosets of K. But how do these two partitions of G intersect each other? Definition 1. Let G be a group, and H a subgroup of G. A left t...

Let 2[n] and ( $$\matrix{{\left[ n \right]} \cr i } $$ ) be the power set collection of all i-subsets {1, 2, …, n}, respectively. We call t (t ≥ 2) families $${{\cal A}_1},{{\cal A}_2}, \ldots ,{{\cal A}_t} \subseteq {2^{\left[ \right]}}$$ cross-intersecting if Ai ∩ Aj ≠ ∅ for any $${A_i} \in {{\cal A}_i}$$ $${A_j} A}_j}$$ with j. show that, k +l, l r 1, c > 0 $${\cal A} \left( {\matrix{{\left[...

In this short article, we observe that the path of particle of mass $m$ moving along $mathbf{r}= mathbf{r}(t)$ under pseudo-force $mathbf{A}(t)$, $t$ denotes the time, is given by $mathbf{r}_d= int(frac{dmathbf{r}}{dt} mathbf{A}(t)) dt +mathbf{c}$. We also observe that the effective force $mathbf{F}_e$ on that particle due to pseudo-force $mathbf{A}(t)$, is given by $ mathbf{F}_e= mathbf{F} mat...

The aim of this paper is to investigate harmonic Stieltjes constants occurring in the Laurent expansions function \[ \zeta_{H}\left( s,a\right) =\sum_{n=0}^{\infty}\frac{1}{\left( n+a\right) ^{s}}\sum_{k=0}^{n}\frac{1}{k+a},\text{ }\operatorname{Re}\left( s\right) >1, \] which we call Hurwitz zeta function. In particular evaluation formulas for $\gamma_{H}\left( m,1/2\right) $ and m,1\right) ar...

We show that the diameter of connected $k$-colorable graphs with minimum degree $\geq \delta$ and order $n$ is at most $\left(3-\frac{1}{k-1}\right)\frac{n}{\delta}-1$, while for $k=3$, it $\frac{57n}{23\delta}+O\left(1\right)$.