Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$, respectively. For any real $\alpha \in [0,1]$, Nikiforov defined the matrix $A_{\alpha}(G)$ as \[ A_{\alpha}(G) = \alpha D(G) + (1-\alpha) A(G). \] In this paper, we generalize some previous results about the $A_{1/2}$-spectral radius of bicyclic graphs with a given degree sequence. Furthermore, we characterize all extremal bicyclic graphs which have the largest $A_{\alpha}$-spectral radius in the set of all bicyclic graphs with prescribed degree sequences.

We prove a new case of mixed discrete joint universality theorem on approximation of certain target couple of analytic functions by the shifts of a pair consisting of the function $\varphi(s)$ belonging to wide class of Matsumoto zeta-functions and the periodic Hurwitz zeta-function $\zeta(s,\alpha;\mathfrak{B})$. We work under the condition that the common difference of arithmetical progression $h > 0$ is such that $\exp \{2\pi/h\}$ is a rational number and the parameter $\alpha$ is a transcendental number. The essential difference from the result in our previous article [10] is that here we do not study the class of partial zeta-functions $\varphi_{h}(s)$, but work with the class of the original functions $\varphi(s)$.

The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.

In this paper, we investigate the least squares solution with the minimal norm to the system (1.1) over reduced biquaternion via complex representation of reduced biquaternion tensors and the Moore–Penrose inverse of tensors. Besides, we establish some necessary and sufficient conditions for the solvability to the above system and give an expression of the general solution to the system when the solvability conditions are met. Moreover, the algorithm and numerical example are presented to verify the main results of this paper.

Motivated by a theorem of Gordon and Hedenmalm in 1999, the study of composition operators acting on various scales of function spaces of Dirchlet series has arisen intensive interest. In this paper, we characterize the boundedness of composition operators induced by specific Dirichlet series symbols from Bergman space to Hardy space of Dirichlet series.

For the two-dimensional nonlinear system \[ u' = a(t) |v|^{1/\alpha} \operatorname{sgn}v, \quad v' = - b(t) |u|^{\alpha} \operatorname{sgn}u \] with $\alpha \gt 0$, $a,b \in C[t_{0},\infty)$, $a(t) \geq 0$ ($t \geq t_{0}$), new oscillation criteria and nonoscillation criteria are given in both cases $\int_{t_{0}}^{\infty} a(s) \, ds = \infty$ and $\int_{t_{0}}^{\infty} a(s) \, ds \lt \infty$. One of the main results is an analogue of the Hartman–Wintner oscillation theorem. Our results generalize Li and Yeh's results for second order half-linear scalar equations.

We prove the existence of singular limit solutions for a nonlinear elliptic Emden–Fowler system with Navier boundary conditions by using the nonlinear domain decomposition method and the Pohozaev identity.

In this paper, we study an optimal control problem in which their cost function is interval-valued. Also, a stochastic differential equation governs their state space. Moreover, we introduce a generalized version of Bellman's optimality principle for the stochastic system with an interval-valued cost function. Also, we obtain the Hamilton–Jacobi–Bellman equations and their control decisions. Two numerical examples happen in finance in which their cost function are interval-valued functions, illustrating the efficiency of the discussed results. The obtained results provide significantly reliable decisions compared to the case where the conventional cost function is applied.