Korean Journal of Mathematics

$\ast$-Ricci-Yamabe Soliton and Contact Geometry

2025-08-19T15:59:25+09:00

It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the $\ast$-Ricci soliton, the notion of $\ast$-Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A $(2n + 1)$-dimensional non-Sasakian $N(k)$-contact metric manifold admitting $\ast$-Ricci-Yamabe soliton is completely classified. Further, it is proved that if a Sasakian 3-manifold $M$ admits $\ast$-Ricci-Yamabe soliton $(g,V,\lambda,\alpha,\beta)$ under certain conditions on the soliton vector field $V$, then $M$ is $\ast$-Ricci flat, positive Sasakian and the transverse geometry of $M$ is Fano. In addition, the Sasakian 3-metric $g$ is homothetic to a Berger sphere and the soliton is steady. Also, the potential vector field $V$ is an infinitesimal automorphism of the contact metric structure.

Solution and stability of a functional equation deriving from additive, quadratic and quartic in quasi-Banach spaces

2025-06-19T21:35:29+09:00

In this paper, we investigate the general solution of the following functional equation
$$f(x+3y)+f(x-3y)=9(f(x+y)+f(x-y))+12f(y)-12f(2y)+4f(3y)-16f(x)$$
and discuss its Hyers-Ulam stability in quasi-Banach spaces.

Existence and Uniqueness of Positive Solutions for a Class of Fractional Differential Equations with a Parameter

2025-08-26T23:54:28+09:00

In this paper, we use the features of generalized concave operators to verify the uniqueness of positive solutions and establish the existence of positive solutions for a certain class of fractional differential equations.

On zero-dimensional spaces of closed subsets

2025-08-25T19:19:34+09:00

In this paper, we study some basic properties related to separation axioms of the space of closed subsets of a zero-dimensional topological space.
Thus we characterize the hyperspace of a zero-dimensional topological space via the notions of normality and partition. Then we establish five equivalent conditions characterizing when the hyperspace of a compact Hausdorff space is zero-dimensional. Furthermore, we give some examples related to our results.

Inducing the sum of the Fibonacci sequences from the Moment of Inertia of an Object

2025-08-25T19:27:15+09:00

By employing the Parallel Axis Theorem for a thin rod, we derive a refined identity that is applicable to an arbitrary sequence. Through the substitution of diverse general sequences within this identity, we establish novel sums of Fibonacci sequences.

A Reexamination of Semigroup Homomorphisms Arising from the Product Structure in Pure Quartic Number Fields

2025-09-22T19:23:27+09:00

Given a real parameter $\alpha$ and a semigroup $S$, we consider a functional equation arising from the multiplicative structure of the quartic number field $\mathbb{Q}(\sqrt[4]{\alpha})$. A recent study by Mouzoun and Zeglami [Bol. Soc. Mat. Mex. 28:73 (2022)] investigated solutions to this equation, yet specific results were found to be incorrect. In this work, we provide a rigorous reexamination of the equation, identify the inaccuracies, and introduce refined conditions that ensure its correct formulation. Our results offer a precise characterization of semigroup homomorphisms associated with the product structure in pure quartic number fields, thereby contributing to the broader study of functional equations in algebraic systems.

Affine relations in the card game SET and related games

2025-09-12T16:47:00+09:00

The game of SET is a popular card game that involves finding particular visual patterns. In this paper, we introduce a new game rule using a SET card deck, and show that the game is equivalent to finding an affine relation in the affine space $AG(4,3)$. Furthermore, we observe that similar approaches can be applied to other SET-like card games related to finite affine spaces over other finite fields.

Functional inequalities and pairs of hom-derivations and homomorphisms

2025-08-26T23:47:50+09:00

In this paper, we introduce and solve the following additive-additive $(s,t)$-functional inequality
\begin{eqnarray}\label{0.1} && \left\|2 g\left(\frac{x+y}{2}\right)-g(x)-g(y) \right\| +\left\| 2h\left(\frac{x+y}{2}\right)+ 2h \left(\frac{x-y}{2}\right)- 2h (x) \right\| \\ && \le \left\|s\left( g\left(x+y\right) -g(x) -g(y) \right)\right\|+ \left\|t \left( h(x+y) + h(x-y) -2 h(x) \right) \right\| , \nonumber
\end{eqnarray}
where $s$ and $t$ are fixed nonzero complex numbers with $|s|+|t| <1$.
We define a pair of hom-derivation and homomorphism in complex Banach algebras, and using the direct method and the fixed point method, we prove the Hyers-Ulam stability of pairs of hom-derivations and homomorphisms in complex Banach algebras, associated to the additive-additive $(s,t)$-functional inequality (\ref{0.1}) and the following functional inequality
\begin{eqnarray}\label{0.2} \| g(xy)-g(x) h(y) - h(x) g(y) \| +\| h(xy) - h(x) h(y) \| \le \varphi(x,y).
\end{eqnarray}