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...

The family of rings the form&#x0D; \frac{\mathbb{Z}_{4}\left \langle x,y \right \rangle}{\left x^2-a,y^2-b,yx-xy-2(c+dx+ey+fxy) \rangle}&#x0D; is investigated which contains generalized Hamilton quaternions over $\Z_4$. These are local order 256. This has 256 contained in 88 distinct isomorphism classes. Of non-isomorphic rings, 10 minimal reversible nonsymmetric and 21 abelian reflexive nonsem...

We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians curves over an algebraically closed non-Archimedean field. Let J be Jacobian curve X, and let $$W_{d} \subset J$$ locus effective divisor classes degree d. show that pair $$(J^{an},W_d^{an})$$ is d-connected, thus in particular inclusion analytification theta $$\Theta ^{an}$$ into $$J^{an}$$ satisfies $$...

Graph homomorphisms from the $${\mathbb {Z}}^d$$ lattice to {Z}}$$ are functions on whose gradients equal one in absolute value. These height corresponding proper 3-colorings of and, two dimensions, 6-vertex model (square ice). We consider uniform model, obtained by sampling uniformly such a graph homomorphism subject boundary conditions. Our main result is that delocalizes having no translatio...

In [FGRS1,FGRS2] the relationship between universal and elementary theory of a group ring $R[G]$ corresponding associated $G$ $R$ was examined. Here we assume that is commutative with identity $1 \ne 0$. Of course, these are relative to an appropriate logical language $L_0,L_1,L_2$ for groups, rings respectively. Axiom systems were provided in [FGRS1]. [FGRS1] it proved if elementarily equivale...

The paper discusses the asymptotic depth of a reversible circuit consisting of NOT, CNOT and 2-CNOT gates. Reversible circuit depth function $D(n, q)$ for a circuit implementing a transformation $f\colon \mathbb Z_2^n \to \mathbb Z_2^n$ is introduced as a function of $n$ and the number of additional inputs $q$. It is proved that for the case of implementing a permutation from $A(\mathbb Z_2^n)$...