In this paper, we define a totally umbilic hypersurface of first order and show that a totally umbilic hypersurface of first order in an Einstein manifold has a parallel second fundamental form. Furthermore we prove that a complete, simply connected and totally umbilic hypersurface of first order in a space of constant curvature is a Riemannian product of Einstein manifolds. Finally we show a proper example which is a totally umbilic hypersurface of first order but not a totally umbilic hypersurface.

In this paper we consider the tight closure of an ideal relative to a module whose its zero submodule has a primary decomposition.

We introduce and investigate envelope of submodules. We show that the proper envelopes of certain submodules is a union of prime submodules.

In this paper, we define and enumerate two tree-like hypergraph structures which we call them (d, k)-trees and d-trees, where $d{\geq}2$ and k > 0 are integers. These new definitions generalize traditional and HP-hypertrees.

For real parameters ${\alpha}$ and ${\beta}$ such that ${\alpha}$ < 1 < ${\beta}$, we denote by $\mathcal{P}({\alpha},{\beta})$ the class of analytic functions p, which satisfy p(0) = 1 and ${\alpha}$ < ${\Re}\{p(z)\}$ < ${\beta}$ in ${\mathbb{D}}$, where ${\mathbb{D}}$ denotes the open unit disk. Let ${\mathcal{A}}$ be the class of analytic functions in ${\mathbb{D}}$ such that f(0) = 0 = f'(0) - 1. For $f{\in}{\mathcal{A}}$, ${\mu}{\in}{\mathbb{C}}{\backslash}\{0\}$ and ${\nu}{\in}{\mathbb{C}}$, let $I_{{\mu},{\nu}:{\mathcal{A}}{\rightarrow}{\mathcal{A}}$ be an integral operator defined by $$I_{{\mu},{\nu}[f](z)}=\({\frac{{\mu}+{\nu}}{z^{\nu}}}{\int}^z_0f^{\mu}(t)t^{{\nu}-1}dt\)^{1/{\mu}}$$. In this paper, we find some sufficient conditions on functions to be in the class $\mathcal{P}({\alpha},{\beta})$. One of these results is applied to the integral operator $I_{{\mu},{\nu}}$ of two classes of starlike functions which are related to the class $\mathcal{P}({\alpha},{\beta})$.

In this paper, we study planar normal section curves. We have interpreted curvatures of normal section curves. On the other hand we have investigated sufficient and necessary conditions for a normal section curve to be biharmonic.

The main motivation of this article is to generalized the concept of fuzzy ideals, (${\alpha},{\beta}$)-fuzzy ideals, (${\in},{\in}{\vee}q_k$)-fuzzy ideals of semigroups. By using the concept of $q^{\delta}_K$-quasi-coincident of a fuzzy point with a fuzzy set, we introduce the notions of (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy left ideal, (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy right ideal of a semigroup. Special sets, so called $Q^{\delta}_k$-set and $[{\lambda}^{\delta}_k]_t$-set, condition for the $Q^{\delta}_k$-set and $[{\lambda}^{\delta}_k]_t$-set-set to be left (resp. right) ideals are considered. We finally characterize different classes of semigroups (regular, left weakly regular, right weakly regular) in term of (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy left ideal, (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy right ideal and (${\in},{\in}{\vee}q^{\delta}_k$)-fuzzy ideal of semigroup S.

Hurwitz zeta function occurs in various parts of mathematics. In particular, it plays an important role in some area of number theory. In this paper, using a certain transformation formula, we find some identities of relations between ${\zeta}(2n+1)$ and ${\zeta}(2n+1,{\alpha})$ for special values of ${\alpha}$.

The notion of $I_{BE}$-energetic subsets in BE-algebras is introduced, and several properties are investigated. Characterizations of $I_{BE}$-energetic subsets are discussed, and conditions for a subset to be an $I_{BE}$-energetic subset are provided.

A new class of functions called $R_{\theta}$-supercontinuous functions is introduced. Their basic properties are studied and their place in the hierarchy of strong variants of continuity, which already exist in the literature, is elaborated. The class of $R_{\theta}$-supercontinuous functions properly contains the class of $R_z$-supercontinuous functions [39] which in turn properly contains the class of $R_{cl}$-supercontinuous functions [43] and so includes all cl-supercontinuous (clopen continuous) functions ([38], [34]) and is properly contained in the class of $R_{\delta}$-supercontinuous functions [24].

In this paper, by means of the $S{\breve{a}}l{\breve{a}}gean$ operator, we introduce a new subclass $\mathcal{B}^{m,n}_{\Sigma}({\gamma};{\varphi})$ of analytic and bi-univalent functions in the open unit disk $\mathbb{U}$. For functions belonging to this class, we consider Fekete-$Szeg{\ddot{o}}$ inequalities.

In this paper, we give some relations related with a spacelike principal-direction curve and a spacelike binormal-direction curve of a timelike Frenet curve. The Darboux-direction curve and the Darboux-rectifying curve of a timelike Frenet curve in Minkowski 3-space $E^3_1$ are introduced and some characterizations related with these associated curves are given. Later we define the spacelike V-direction curve which is associated with a timelike curve lying on a timelike oriented surface in $E^3_1$ and present some results together with the relationships between the curvatures of this associated curve.

By applying a classical approach for scalar rational interpolation problem, certain type of minimal interpolation problem for rational matrix functions is solved. It is shown that an appropriately defined matrix plays a key role in solving the minimal problem.

Elastica is known as classical curve that is a solution of variational problem, which minimize a thin inextensible wire's bending energy. Studies on elastica has been conducted in Euclidean space firstly, then it has been extended to Riemannian manifold by giving different characterizations. In this paper, we focus on energy of the elastic curve in a Lie group. We attepmt to compute its energy by using geometric description of the curvature and the torsion of the trajectory of the elastic curve of the trajectory of the moving particle in the Lie group. Finally, we also investigate the relation between energy of the elastic curve and energy of the same curve in Frenet vector fields in the Lie group.

In this note, we seek traveling wave solutions of a shallow water model in a one dimensional space by a simple but rigorous calculation. From the profile equation of traveling wave solutions, we need to investigate the phase portrait of a one dimensional ordinary differential equation $\tilde{u}^{\prime}=F(\tilde{u})$ connecting two end states of the traveling wave solution.

In this paper we define $AP_c$ and $AP_{cc}$ spaces which are stronger than the property of approximation by points(AP). We investigate operations on their subspaces and study function theorems on $AP_c$ and $AP_{cc}$ spaces. Using those results, we prove that every continuous image of a countably compact Hausdorff space with AP is AP. Finally, we prove a theorem that every compact ${\kappa}$-WAP space is ${\kappa}$-pseudoradial, and prove a theorem that the product of a compact ${\kappa}$-radial space and a compact ${\kappa}$-WAP space is a ${\kappa}$-WAP space.

Several properties of the sequence generated from a kind of Danzer matrices were examined and proved using already known facts about the Chebyshev polynomials. Asymptotic behavior of our interest sequence also discussed.

Let P be a finitely generated projective module over a commutative ring R with identity. If P has finite rank, then it will be shown that the map ${\varphi}:Aut_R(P){\rightarrow}U(R)$ defined by ${\varphi}({\alpha})={\det}({\alpha})$ is locally surjective and $Ker({\varphi})=SL_R(P)$.