The aim of this note is to investigate f -vectors of simplicial balls, especially the relations between interior and boundary faces. For a simplicial ball B we denote by fi(B) the number of i-dimensional faces. The boundary ∂B of B is a simplicial sphere with face numbers fi(∂B). We also define fi(int B) := fi(B) − fi(∂B) although the interior int B of B is not a polyhedral complex. For simplic...

‎Let Γa be a graph whose each vertex is colored either white or black‎. ‎If u is a black vertex of Γ such that exactly one neighbor‎ ‎v of u is white‎, ‎then u changes the color of v to black‎. ‎A zero forcing set for a Γ graph is a subset of vertices Zsubseteq V(Γ) such that‎ if initially the vertices in Z are colored black and the remaining vertices are colored white‎, ‎then Z changes the col...

‎Let $mathcal A$ and $mathcal B$ be unital rings‎, ‎and $mathcal M$‎ ‎be an $(mathcal A‎, ‎mathcal B)$-bimodule‎, ‎which is faithful as a‎ ‎left $mathcal A$-module and also as a right $mathcal B$-module‎. ‎Let ${mathcal U}=mbox{rm Tri}(mathcal A‎, ‎mathcal M‎, ‎mathcal‎ ‎B)$ be the triangular ring and ${mathcal Z}({mathcal U})$ its‎ ‎center‎. ‎Assume that $f:{mathcal U}rightarrow{mathcal U}$ is...

800x600 to analyse the effect of congenital unilateral ptosis on the ocular higher order aberrations (hoa) and to compare these eyes with normal fellow eyes this study has been performed. in this observational comparative case series, 16 eyes of 16 patients less than 15 years old with congenital unilateral upper eyelid ptosis were included. corrected distance visual acuity (cdva), corneal topog...

Assume that $mathbb{D}$ is the open unit disk. Applying Ozaki's conditions, we consider two classes of locally univalent, which denote by $mathcal{G}(alpha)$ and $mathcal{F}(mu)$ as follows begin{equation*}  mathcal{G}(alpha):=left{fin mathcal{A}:mathfrak{Re}left( 1+frac{zf^{prime prime }(z)}{f^{prime }(z)}right) <1+frac{alpha }{2},quad 0<alphaleq1right}, end{equation*} and begin{equation*}  ma...

high yield photolytic reduction of a series of thiaphosphoranes z p sr to give z p h with uv-light-induced tri-n-butyltin hydride through phosphoranyl radical z p is described. this reaction is inhibited by a-methylstyrene and initiated by a radical initiator such as azobisisobutyronitrile

High yield photolytic reduction of a series of thiaphosphoranes Z p SR to give Z p H with UV-light-induced tri-n-butyltin hydride through phosphoranyl radical Z p is described. This reaction is inhibited by a-methylstyrene and initiated by a radical initiator such as azobisisobutyronitrile

‎we determine the‎ ‎form of polynomially bounded solutions to the loewner differential ‎equation that is satisfied by univalent subordination chains of the‎ ‎form $f(z,t)=e^{int_0^t a(tau){rm d}tau}z+cdots$‎, ‎where‎ ‎$a:[0,infty]rightarrow l(mathbb{c}^n,mathbb{c}^n)$ is a locally‎ ‎lebesgue integrable mapping and satisfying the condition‎ ‎$$sup_{sgeq0}int_0^inftyleft|expleft{int_s^t‎ ‎[a(tau)...