Let $G$ be a finite group. The reduced power graph of $G$ is the undirected graph whose vertex set is $G$, and two distinct vertices $x$ and $y$ are adjacent if $\langle x \rangle \subset \langle y \rangle$ or $\langle y \rangle \subset \langle x \rangle$. In this paper, we give tight upper and lower bounds for the metric dimension of the reduced power graph of a finite group. As applications, we compute the metric dimension of the reduced power graph of a $\mathcal{P}$-group, a cyclic group, a dihedral group, a generalized quaternion group, and a group of odd order.

Let $M$ be a given set of positive integers. A set $S$ of nonnegative integers is said to be an $M$-set if $a,b \in S$ implies $a-b \notin M$. In an unpublished problem collection, Motzkin asked to find maximal upper asymptotic density, denoted by $\mu(M)$, of $M$-sets. The first published work on $\mu(M)$ is due to Cantor and Gordon in 1973, in which, they found the exact value of $\mu(M)$ when $|M| \leq 2$. In fact, this is the only general case, in which, we have a closed formulae for $\mu(M)$. If $|M| \geq 3$, then the exact value of $\mu(M)$ is not known for the general set $M$. In the past six decades or so, several attempts have been given to study $\mu(M)$ but $\mu(M)$ has been found exactly or estimated only in very few cases. In this paper, we study $\mu(M)$ for the families $M = \{ a,a+1,x \}$ and $M = \{ a,a+1,x,y \}$, where $y-x \leq 2$ and $y \gt x \gt a+1$. Our results in the case of $M = \{ a,a+1,x \}$ also give counterexamples to a conjecture of Carraher. Although, different counterexamples to this conjecture, were already given by Liu and Robinson in 2020. We also relate our results with the already know results for the families $M = \{ 1,2,x,x+2 \}$ and $M = \{ 2,3,x,x+2 \}$.

We consider a generalization of Piatetski–Shapiro sequences in the sense of Beatty sequences, which is of the form $(\lfloor \alpha n^c + \beta \rfloor)_{n=1}^{\infty}$ with real numbers $\alpha \geq 1$, $c \gt 1$ and $\beta$. We show there are infinitely many primes in the generalized Piatetski–Shapiro sequence with $c \in (1,14/13)$. Moreover, we prove there are infinitely many Carmichael numbers composed entirely of the primes from the generalized Piatetski–Shapiro sequences with $c \in (1,64/63)$.

Our goal is to study the syzygies of the projective embeddings defined by the ample line bundles on a fake projective plane $S$. The syzygies are studied in terms of the property $N_p$. For various kinds of ample line bundles, we give explicit lower bounds for their powers above which the property $N_p$ is satisfied.

In this paper we introduce a new numerical method for the linear complementarity problems (LCPs) arising from two-asset Black–Scholes and Heston's stochastic volatility American options pricing. Based on barycenter dual mesh, a class of finite volume method (FVM) is proposed for the spatial discretization, coupled with the backward Euler and Crank–Nicolson schemes are employed for time stepping of the partial differential equations (PDEs). Then, for the resulting time-dependent LCPs are solved by using an efficient modulus-based successive overrelaxation (MSOR) iteration method. Numerical experiments are carried out to verify the efficiency and usefulness of the proposed method.

We investigate the Cauchy problem for a $2 \times 2$-system of weakly coupled semi-linear fractional wave equations with polynomial nonlinearities posed in $\mathbb{R}^{+} \times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the fractional orders of the time derivatives, it is shown that there exists a threshold value of the dimension $N$, for which, small data-global solutions as well as finite time blowing-up solutions exist. Furthermore, we investigate the $L^{\infty}$-decay estimates of global solutions.

In this paper, the global exponential stability and periodicity are investigated for impulsive neural network models with Lipschitz continuous activation functions and piecewise alternately advanced and retarded argument of generalized argument (in short IDEPCAG). The sufficient conditions for the existence and uniqueness of periodic solutions of the model are established by applying fixed point theorem and the successive approximations method. By constructing suitable differential inequalities with piecewise alternately advanced and retarded argument, some sufficient conditions for the global exponential stability of the model are obtained. Typical numerical examples with simulations are utilized to illustrate the validity and improvement in less conservatism of the theoretical results.

We investigate the pointwise convergence of the solution to the fractional Schrödinger equation in $\mathbb{R}^2$. By establishing $H^s(\mathbb{R}^2) - L^3(\mathbb{R}^2)$ estimates for the associated maximal operator provided that $s > 1/3$, we improve the previous result obtained by Miao, Yang, and Zheng [19]. Our estimates extend the refined Strichartz estimates obtained by Du, Guth, and Li [10] to a general class of elliptic functions.

In this paper, we study the concept of approximate directional efficiency for set-valued constrained and unconstrained optimization problems. In our work, we concerned with finding conditions under which the Clarke penalization technique can be applied, and we derive some optimality conditions via variational analysis tools such as limiting normal cones and its corresponding normal coderivative.