Let $G$ be a connected graph and $D^{L}(G) = \operatorname{Tr}(G) - D(G)$ be the distance Laplacian matrix of $G$, where $\operatorname{Tr}(G)$ and $D(G)$ are diagonal matrix with vertex transmissions of $G$ and distance matrix of $G$, respectively. The $D^{L}$-spectral radius of $G$ is defined as the largest absolute value of the distance Laplacian eigenvalues of $G$. In this paper, we characterize the unique extremal graphs which maximize the $D^{L}$-spectral radius among the complements of trees and unicyclic graphs, respectively.

Let $k \geq 1$ be an integer, and let $(U_{n})$ be the Lucas sequence of the first kind defined by \[ U_{0} = 0, \quad U_{1} = 1 \quad \textrm{and} \quad U_{n} = kU_{n-1} + U_{n-2} \quad \textrm{for $n \geq 2$}. \] It is well known that $(U_{n})$ is periodic modulo any integer $m \geq 2$, and we let $\pi(m)$ denote the length of this period. A prime $p$ is called a $k$-Wall–Sun–Sun prime if $\pi(p^{2}) = \pi(p)$.

Let $f(x) \in \mathbb{Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over $\mathbb{Q}$. We say $f(x)$ is monogenic if $\Theta = \{ 1, \theta, \theta^{2}, \ldots, \theta^{N-1} \}$ is a basis for the ring of integers $\mathbb{Z}_{K}$ of $K = \mathbb{Q}(\theta)$, where $f(\theta) = 0$. If $\Theta$ is not a basis for $\mathbb{Z}_{K}$, we say that $f(x)$ is non-monogenic.

Suppose that $k \not\equiv 0 \pmod{4}$ and that $\mathcal{D} := (k^{2}+4)/\gcd(2,k)^{2}$ is squarefree. We prove that $p$ is a $k$-Wall–Sun–Sun prime if and only if $\mathcal{F}_{p}(x) = x^{2p}-kx^{p}-1$ is non-monogenic. Furthermore, if $p$ is a prime divisor of $k^{2}+4$, then $\mathcal{F}_{p}(x)$ is monogenic.

We construct new symplectic $4$-manifolds with non-negative signatures and with the smallest Euler characteristics, using fake projective planes, Cartwright–Steger surfaces and their normal covers and product symplectic $4$-manifolds $\Sigma_{g} \times \Sigma_{h}$, where $g \geq 1$ and $h \geq 0$, along with exotic symplectic $4$-manifolds constructed in [7, 12]. In particular, our constructions yield to (1) infinitely many irreducible symplectic and infinitely many non-symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1) \mathbb{CP}^{2} \# (2n-1) \overline{\mathbb{CP}}^{2}$ for each integer $n \geq 9$, (2) infinite families of simply connected irreducible nonspin symplectic and such infinite families of non-symplectic $4$-manifolds that have the smallest Euler characteristics among the all known simply connected $4$-manifolds with positive signatures and with more than one smooth structure. We also construct a complex surface with positive signature from the Hirzebruch's line-arrangement surfaces, which is a ball quotient.

This paper presents a Roe-type numerical scheme for the model of fluid flows in a nozzle with variable cross-section. The proposed scheme is built using the Roe method combined with stationary contact jumps at interfaces. The scheme is proven to capture smooth stationary waves precisely and preserve the positivity of the fluid's density. The numerical tests show that this approach can give considered accuracy to the exact solutions, except where the exact solution crosses a sonic surface.

We estimate the effects of discretization and dilation/translation errors on a familiar class of Bessel families $\{\psi_{(Q)}\}_{Q \in \mathcal{D}_{d}}$ indexed over $\mathcal{D}_{d}$, the dyadic cubes in $\mathbf{R}^{d}$. We show that, if the $\psi_{(Q)}$s satisfy an appropriately scaled uniform Hölder-$\alpha$ condition for $0 \lt \alpha \leq 1$, replacing each $\psi_{(Q)}$ by its averages over arbitrary (but sufficiently small) dyadic subcubes of $Q$ yields a family $\{\widetilde{\psi}_{(Q)}\}_{Q \in \mathcal{D}_{d}}$ arbitrarily close to $\{\psi_{(Q)}\}_{Q \in \mathcal{D}_{d}}$ in the sense of Bessel bounds. For $\alpha = 1$ we get the same result by replacing each $\{\psi_{(Q)}\}_{Q \in \mathcal{D}_{d}}$ with averages over arbitrary identical subcubes of $Q$. We show that, when $\alpha = 1$, the signal-processing properties of $\{\widetilde{\psi}_{(Q)}\}_{Q \in \mathcal{D}_{d}}$ are not strongly affected when subjected to small errors in dilation and translation.

We introduce notions of continuous orbit equivalence and strong (respective, weak) continuous orbit equivalence for automorphism systems of étale equivalence relations, and characterize them in terms of the semi-direct product groupoids, as well as their reduced groupoid $C^{*}$-algebras and the associated $C^{*}$-automorphism systems of group actions or coactions on them. In particular, we study topological rigidity of expansive automorphism actions on compact (connected) metrizable groups.

We give Schwarz lemma at the boundary for holomorphic mappings between $p$-unit ball $B_{p}^{n} \subset \mathbb{C}^{n}$ and $B_{p}^{N} \subset \mathbb{C}^{N}$, where $p \geq 2$. When $p = 2$, this result reduces to that of Liu, Chen and Pan [21] between the Euclidean unit balls, and our method is new. By generalizing pluriharmonic Schwarz lemma of Chen and Gauthier [5] from $p = 2$ to $p \geq 2$, we obtain the boundary Schwarz lemma for pluriharmonic mappings between $p$-unit balls.

In this paper, we first introduce the pseudo-entropy of free semigroup actions and show that it is equal to the topological entropy of free semigroup actions defined by Bufetov [9]. Second, for free semigroup actions, the concepts of chain recurrence and chain recurrence time, chain mixing and chain mixing time are introduced, and upper bounds for these recurrence times are calculated. Furthermore, the lower box dimension and the chain mixing time provide a lower bound on topological entropy of free semigroup actions. Third, the structure of chain transitive systems of free semigroup actions is discussed. Our analysis generalizes the results obtained by Misiurewicz [21], Richeson and Wiseman [23], and Bufetov [9] etc.

In this paper, utilizing Bregman projections which are different from the sunny generalized nonexpansive retractions and generalized metric projection in Banach spaces, we introduce some new parallel extragradient methods for finding the solution of the multiple-sets split equilibrium problem and the solution of the multiple-sets split variational inequality problem in $p$-uniformly convex and uniformly smooth Banach spaces. Moreover, we introduce a $\Delta$-Lipschitz-type condition on the equilibrium bifunctions to prove strongly convergent of the generated iterates in parallel extragradient methods. To illustrate the usability of our results and also to show the efficiency of the proposed methods, we present some comparative examples with several existing schemes in the literature in finite and infinite dimensional spaces.

We present Tseng's forward-backward-forward method with extrapolation from the past for pseudo-monotone variational inequalities in Hilbert spaces. In addition, we propose a variable stepsize scheme of the extrapolated Tseng's algorithm governed by the operator which is pseudo-monotone, Lipschitz continuous and sequentially weak-to-weak continuous. We also investigate the algorithm's adaptive stepsize scenario, which arises when it is impossible to calculate the Lipschitz constant of a pseudo-monotone operator correctly. Finally, we prove a weak convergence theorem and conduct a numerical experiment to support it.