In this paper, we will introduce a new geometric constant $L_{\mathrm{YJ}}(\lambda ,\mu ,X)$ based on an equivalent characterization of inner product space, which was proposed by Moslehian and Rassias. We first discuss some equivalent forms of the proposed constant. Next, a characterization of uniformly non-square is given. Moreover, some sufficient conditions which imply weak normal structure are presented. Finally, we obtain some relationship between the other well-known geometric constants and $L_{\mathrm{YJ}}(\lambda ,\mu ,X)$. Also, this new coefficient is computed for X being concrete space.

In this research paper, we deal with the problem of determining the function $\chi :G\to \mathbb{R}$, which is the solution to the maximum functional equation (MFE) $max\{\chi \left(xy\right),\chi \left(x{y}^{-1}\right)\}=\chi \left(x\right)\chi \left(y\right),$ when the domain is a discretely normed abelian group or any arbitrary group G. We also analyse the stability of the maximum functional equation $max\{\chi \left(xy\right),\chi \left(x{y}^{-1}\right)\}=\chi \left(x\right)+\chi \left(y\right)$ and its solutions for the function $\chi :G\to \mathbb{R}$, where G be any group and also investigate the connection of the stability with commutators and free abelian group K that can be embedded into a group G.

We introduce a new geometric constant $C^{p}_{NJ}(\zeta,\eta,X)$ in Banach spaces, which is called the skew generalized von Neumann-Jordan constant. First, the upper and lower bounds of the new constant are given for any Banach space. Then we calculate the constant values for some particular spaces. On this basis, we discuss the relation between the constant $C^{p}_{NJ}(\zeta,\eta,X)$ and the convexity modules $\delta_X(\varepsilon)$, the James constant $J(X)$. Finally, some sufficient conditions for the uniform normal structure associated with the constant $C^{p}_{NJ}(\zeta,\eta,X)$ are established.

In this paper, we propose an innovative approach to fractional-order dynamics by introducing a 10-dimensional (10D) chaotic system that leverages the intrinsic memory characteristic of the Grünwald–Letnikov (G-L) derivative. We utilize Lyapunov exponents as a quantitative measure to characterize hyperchaotic behavior, and classify the nature of the suggested 10D fractional-order system (FOS). While several methods exist for calculating Lyapunov exponents (LEs) through the utilization of integer-order systems, these approaches are not applicable for FOS due to its non-local nature. Initially, the system dynamics are thoroughly examined through Lyapunov exponents and bifurcation analysis, considering the influence of both state variables and fractional orders. To assess the hyperchaotic behavior of the proposed model, sensitivity analyses are conducted by exploring changes in state variables under two distinct initial conditions, along with time history simulations for various parameter settings. Furthermore, we examine the impact of different fractional-order sets on the system’s dynamics. A comprehensive performance comparison is conducted between the proposed 10-dimensional fractional-order hyperchaotic system and several existing hyperchaotic systems. This comparison utilizes advanced metrics, including the Kolmogorov–Sinai (KS) entropy, Kaplan–Yorke dimension, the Perron effect analysis, and the 0-1 test for chaos. Simulation outcomes reveal that the proposed system surpasses existing algorithms, delivering improved precision and accuracy.