Abstract

The purpose of this paper is to introduce a new class of admissible curves, referred to as $f$-rectifying curves, and study their geometric properties in Galilean 3-space $\mathbb{G}_3$. For some non-vanishing real-valued smooth function $f$, an $f$-rectifying curve in $\mathbb{G}_3$ is introduced as an admissible curve $\gamma$ of class at least $C^4$ such that its $f$-position vector field, given by $\gamma_f = \int f d\gamma$, lies on its rectifying planes (i.e., the planes generated by its tangent and binormal vectors). Some geometric characterizations of such curves are explored in $\mathbb{G}_3$. Moreover, they are investigated in the equiform geometry of $\mathbb{G}_3$.

Keywords: Galilean geometry; admissible curve; $f$-rectifying curve; equiform geometry.

MSC: 53A35, 53A40, 53B25, 53C40

Abstract

In the present paper, we study a Riemannian submersion $\pi$ from a Riemann soliton $(M_1,g,\xi,\lambda)$ onto a Riemannian manifold $(M_2,g^{'})$. We first calculate the sectional curvatures of any fibre of $\pi$ and the base manifold $M_2$. Using them, we give some necessary and sufficient conditions for which the Riemann soliton $(M_1,g,\xi,\lambda)$ is shrinking, steady or expanding. Also, we deal with the potential field $\xi$ of such a Riemann soliton is conformal and obtain some characterizations about the extrinsic vertical and horizontal sectional curvatures of $\pi$.

Keywords: Riemannian submersion; Riemann soliton; sectional curvature.

MSC: 53C40, 32Q15

Abstract

In this paper, firstly we study geodesic vectors for the $m$-th root homogeneous Finsler space admitting $(\alpha,\beta)$-type. Then we obtain the necessary and sufficient condition for an arbitrary non-zero vector to be a geodesic vector for the $m$-th root homogeneous Finsler metric under mild conditions. Finally, we consider a quartic homogeneous Finsler metric on a simply connected nilmanifold of dimension five equipped with an invariant Riemannian metric and an invariant vector field. We study its geodesic vectors and classify the set of all the homogeneous geodesics on $5$-dimensional nilmanifolds.

Keywords: Geodesic vectors; $m$-th root Finsler metric; quartic Finsler metric; nilpotent Lie groups; invariant metric; nilmanifolds.

MSC: 22E60, 53C30, 53C60

Abstract

The aim of this paper is to present the notions of actions and coverings of $\text{cat}^1$-groupoids and to prove the natural equivalence between their categories. Moreover, in this context, we characterize the quotient concept of $\text{cat}^1$-groupoids. Finally we extend these notions to $\text{cat}^n$-groupoids which are groupoid version of $\text{cat}^n$-groups.

Keywords: $\text{Cat}^n$-groupoid; covering; action; normality; quotient.

MSC: 20L05, 18D35, 18B40, 20M50

Abstract

In this paper, we derive some nonlinear differential equations from generating function of generalized harmonic numbers and give some identities involving generalized harmonic numbers and special numbers by using these differential equations. For example, for any positive integers $N,$ $n,$ $r,$ $\alpha $ and any integer $m\geq 2,$ \begin{align*} \dfrac{S_{1}(n+N,r+1)}{n!} &=\sum\limits_{j=0}^{n}\sum\limits_{i=0}^{n}\sum\limits_{l=0}^{i}\sum% \limits_{z=0}^{l}\sum\limits_{k=0}^{r}\left( -1\right) ^{l-z-i}\dbinom{m}{% l-z}\dbinom{i-l+m-2}{i-l}\dfrac{N^{j}\alpha ^{i}}{j!\left( n-i\right) !}\\ & \quad\times S_{1}(N,r-k+1)S_{1}\left( n-i,k\right) H(z,j-1,\alpha ) \end{align*} where $S_{1}\left( n,k\right) $ is Stirling number of the first kind.

Keywords: Generalized hyperharmonic numbers of order $r$; Daehee numbers; Stirling numbers of the first kind and the second kind; generating function.

MSC: 05A15, 05A19, 11B73

Abstract

In this paper, we initiate the study of enriched $\rho $-nonexpansive mappings in modular function spaces. First we show that in modular function spaces, every $\rho $-nonexpansive mapping is enriched $\rho $-nonexpansive mapping but not conversely and that their sets of fixed point are same. Next, we prove a $\rho $-convergence result on approximation of fixed points of enriched $\rho $-nonexpansive mappings in modular function spaces. We verify the validity of the result by an example. We construct a table to show our findings. Finally, we give one more $\rho $-convergence result under different conditions. Our results are new for $\rho $-nonexpansive mappings in modular function spaces.

Keywords: Fixed point; enriched $\rho $-nonexpansive mappings; iterative process; modular function space.

MSC: 47H09, 47H10, 54C60