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...

Let $\mathbb{F}_p$ be a finite field and $u$ be an indeterminate. This article studies $(1-2u^k)$-constacyclic codes over the ring $\mathcal{R}=\mathbb{F}_p+u\mathbb{F}_p+u^2\mathbb{F}_p+u^{3}\mathbb{F}_{p}+\cdots+u^{k}\mathbb{F}_{p}$ where $u^{k+1}=u$. We illustrate the generator polynomials and investigate the structural properties of these codes via decomposition theorem.

a {em roman dominating function} on a graph $g = (v ,e)$ is a function $f : vlongrightarrow {0, 1, 2}$ satisfying the condition that every vertex $v$ for which $f (v) = 0$ is adjacent to at least one vertex $u$ for which $f (u) = 2$. the {em weight} of a roman dominating function is the value $w(f)=sum_{vin v}f(v)$. the roman domination number of a graph $g$, denoted by $gamma_r(g)$, equals the...

let $mathcal {a}$ be an abelian category with enough projective objects and $mathcal {x}$ be a full subcategory of $mathcal {a}$. we define gorenstein projective objects with respect to $mathcal {x}$ and $mathcal{y}_{mathcal{x}}$, respectively, where $mathcal{y}_{mathcal{x}}$=${ yin ch(mathcal {a})| y$ is acyclic and $z_{n}yinmathcal{x}}$. we point out that under certain hypotheses, these two g...

We introduce a family of mathematical objects called $\mathcal{P}$-schemes, where $\mathcal{P}$ is a poset of subgroups of a finite group $G$. A $\mathcal{P}$-scheme is a collection of partitions of the right coset spaces $H\backslash G$, indexed by $H\in\mathcal{P}$, that satisfies a list of axioms. These objects generalize the classical notion of association schemes as well as the notion of $...

Phaseless reconstruction from space-time samples is a nonlinear problem of recovering a function $x$ in a Hilbert space $\mathcal{H}$ from the modulus of linear measurements $\{\lvert \langle x, \phi_i\rangle \rvert$, $ \ldots$, $\lvert \langle A^{L_i}x, \phi_i \rangle \rvert : i \in\mathscr I\}$, where $\{\phi_i; i \in\mathscr I\}\subset \mathcal{H}$ is a set of functionals on $\mathcal{H}$, a...

 The category of the title is called $mathcal{W}$. This has all free objects $F(I)$ ($I$ a set). For an object class $mathcal{A}$, $Hmathcal{A}$ consists of all homomorphic images of $mathcal{A}$-objects. This note continues the study of the $H$-closed monoreflections $(mathcal{R}, r)$ (meaning $Hmathcal{R} = mathcal{R}$), about which we show ({em inter alia}): $A in mathcal{A}$ if and  only if...

A unital $C^*$ -- algebra $mathcal A,$ endowed withthe Lie product $[x,y]=xy- yx$ on $mathcal A,$ is called a Lie$C^*$ -- algebra. Let $mathcal A$ be a Lie $C^*$ -- algebra and$g,h:mathcal A to mathcal A$ be $Bbb C$ -- linear mappings. A$Bbb C$ -- linear mapping $f:mathcal A to mathcal A$ is calleda Lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

Let $\Gamma$ be a $d$-bounded distance-regular graph with $d\geq 3$. Suppose that $P(x)$ is a set of all strongly closed subgraphs containing $x$ and that $P(x,i)$ is a subset of $P(x)$ consisting of the elements of $P(x)$ with diameter $i$. Let ${\mathcal{L}}'(x,i)$ be the set generated by the join of the elements in $P(x,i)$. By ordering ${\mathcal{L}}'(x,i)$ by inclusion or reverse inclusion...

The problem of minimax detection of Gaussian random signal vector in White Gaussian additive noise is considered. It is supposed that an unknown vector $\boldsymbol{\sigma}$ of the signal vector intensities belong to the given set ${\mathcal E}$. It is investigated when it is possible to replace the set ${\mathcal E}$ by a smaller set ${\mathcal E}_{0}$ without loss of quality (and, in particul...