In this paper,the notion of Essentially weighted Rationalized Toeplitz Hankel operator is introduced and some properties of the set of all such kind of operators are discussed. Precisely if W-ERTHO($L^2\left(\beta \right)$) is the set of all Essentially weighted Rationalized Toeplitz Hankel operators, we throw some light on this set and attempt to investigate the properties of W-ERTHO($L^2\left(\beta \right)$).

At this work, our main objective is to present the idea of hemi-slant Riemannian submersions from almost contact metric manifolds as a natural generalization of anti-invariant Riemannian submersions, semi-invariant Riemannian submersions and slant Riemannian submersions. We mostly examined on hemi-slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. During this way, we tend to study and investigate integrability conditions, the geometry of leaves of distributions which are emerged from the definition of the submersion. Besides, we tend to get new conditions for these submersions to be totally geodesic. Finally, we construct some quality examples of such submersion.

We examine the generators of linear groups associated with the roots of Fibonacci polynomials, matrix representations of the generators, some elements, and group structures. From the viewpoint of the roots of Fibonacci polynomials, we obtain the features of various classes of linear groups under certain circumstances. Furthermore, we analyze the relationships between the roots of Fibonacci polynomials and the new generating matrices. We get some amazing relationships between the roots of Fibonacci polynomials and the extended modular group, the extended Hecke groups, and the extended generalized Hecke groups with geometric viewpoints.

We complete the construction of the biassociahedra $KK$, construct the free matrad ${\mathcal{H}}_{\infty }$, realize ${\mathcal{H}}_{\infty }$ as the cellular chains of $KK$, and define an $A_{\infty }$-bialgebra as an algebra over ${\mathcal{H}}_{\infty }$. We construct the bimultiplihedra $JJ$, construct the relative free matrad $r{\mathcal{H}}_{\infty }$ as a ${\mathcal{H}}_{\infty }$-bimodule, realize $r{\mathcal{H}}_{\infty }$ as the cellular chains of $JJ$, and define a morphism of $A_{\infty }$-bialgebras as a bimodule over ${\mathcal{H}}_{\infty }$. We prove that the homology of every $A_{\infty }$-bialgebra over a commutative ring with unity admits an induced $A_{\infty }$-bialgebra structure. We extend the Bott-Samelson isomorphism to an isomorphism of $A_{\infty }$-bialgebras and determine the $A_{\infty }$-bialgebra structure of $H_{*}\left(\mathrm{\Omega }\mathrm{\Sigma }X;\mathrm{\mathbb{Q}}\right)$. For each $n\ge 2$, we construct a space $X_n$ and identify an induced nontrivial $A_{\infty }$-bialgebra operation ${\omega }_2^n:H^{*}{\left(\mathrm{\Omega }{X}_n;{\mathrm{\mathbb{Z}}}_2\right)}^{\otimes 2}\to H^{*}{\left(\mathrm{\Omega }{X}_n;{\mathrm{\mathbb{Z}}}_2\right)}^{\otimes n}$.

The aim of this paper is to give a new characterization of discrete valuation rings, namely, it is shown that if $(D,\mathfrak{m}\mathfrak{)}$ is a local (Noetherian) ring such that depth and ${\mathfrak{m}}^k$ is principal ideal for some integer $k\ge 1$, then $D$ is a discrete valuation ring.

In the present paper, we consider a subclass of close-to-star functions associated to exponential function, in the open unit disc $E=\{z\in \mathrm{ℂ}:|z|<1\}$. Here the main purpose is to establish the upper bound for the third Hankel determinant for this class. Also, we investigate the same bounds for two-fold and three-fold symmetric functions.

The dimensions of zero and first regular cohomologies of a biparabolic subalgebra $B$ of some simple Lie algebra are calculated. Namely, it is proved that if $S$ and $T$ are subsets of simple roots such as $B=H\oplus L^{R_{+}^S}\oplus L^{R_{-}^T}$, where $H$ is a splitting Cartan subalgebra and $R_{+}^S$ and $R_{-}^T$ are the positive (negative) roots generated by $S$ (by $T$ respectively) then the dimension $d_0$ of the center of $B$ is equal to the number of simple roots which is not contained in $S\cup T$. If $n=a_0+a_1+\dots +a_r=b_0+b_1+\dots +b_s$ where $a_i,b_i\in N$ are ordered partititions of $n$ and $B$ is the corresponding biparabolic subalgebra of $sl\left(n\right)$, then the dimension of outer derivations of $B$ is equal to $(r+s-d_0)d_0$.