For $k \geq 2$, we determine the connected $k$-uniform hypergraphs with least distance eigenvalues in $((1-\sqrt{33})/2,0)$, the $k$-uniform hypertrees with least distance eigenvalues in $[-2k,0)$, and the $k$-uniform unicyclic hypergraphs with least distance eigenvalues in $(-k+1-\sqrt{(k-1)(k-2)},0)$, respectively, and determine the $k$-uniform hypergraphs (hypertrees, respectively) with minimum distance spread.

An injective coloring of a graph is a vertex coloring where two vertices with common neighbor receive distinct colors. The minimum integer $k$ such that $G$ has a $k$-injective coloring is called injective chromatic number of $G$ and denoted by $\chi_i(G)$. In this paper, the injective chromatic number of outerplanar graphs with maximum degree $\Delta$ and girth $g$ is studied. It is shown that every outerplanar graph $G$ has $\chi_i(G) \leq \Delta+2$, and this bound is tight. Then, it is proved that for an outerplanar graph $G$ with $\Delta = 3$, $\chi_i(G) \leq \Delta+1$ and the bound is tight for outerplanar graphs of girth $3$ and $4$. Finally, it is proved that, the injective chromatic number of $2$-connected outerplanar graphs with $\Delta = 3$, $g \geq 6$ and $\Delta \geq 4$, $g \geq 4$ is equal to $\Delta$.

In this paper, we consider the asymptotic behaviour of $\tau$-Li coefficients for the wide class of $L$-functions that contains the Selberg class, the class of all automorphic $L$-functions, the Rankin-Selberg $L$-functions, as well as products of suitable shifts of the mentioned functions. We consider both archimedean and non-archimedean contribution to the $\tau$-Li coefficients, both separately, and their joint contribution to the coefficients. We also derive the behavior of the coefficients in the case the $\tau/2$-Riemann hypothesis holds, which is the generalization of the Riemann hypothesis for the class under consideration. Finally, we conclude with some examples and numerics.

For $a,b \in \mathbb{C}$, the Lie algebra $\mathcal{W}(a,b)$ is the semidirect product of the Witt algebra and a module of the intermediate series. In this paper, all biderivations of $\mathcal{W}(a,b)$ are determined. Surprisingly, these Lie algebras have symmetric (and skew-symmetric) non-inner biderivations. As an application, commutative post-Lie algebra structures on $\mathcal{W}(a,b)$ are obtained.

We generalize, in the context of partial group action, the Kanzaki commutator theorem for Galois extensions and the structure theorem for Galois algebras given by Szeto and Xue.

We consider the maximal operators whose averages are taken over some non-smooth and non-convex hypersurfaces. For each $1 \leq i \leq d-1$, let $\phi_i \colon [-1,1] \to \mathbb{R}$ be a continuous function satisfying some derivative conditions, and let $\phi(y) = \sum_{i=1}^{d-1} \phi_i(y_i)$. We prove the $L^p$ boundedness of the maximal operators associated with the graph of $\phi$ which is a non-smooth and non-convex hypersurface in $\mathbb{R}^d$, $d \geq 3$.

In this paper, we count the number of cusps of complete Riemannian manifolds $M$ with finite volume. When $M$ is a complete smooth metric measure spaces, we show that the number of cusps in bounded by the volume $V$ of $M$ if some geometric conditions hold true. Moreover, we use the nonlinear theory of the $p$-Laplacian to give a upper bound of the number of cusps on complete Riemannian manifolds. The main ingredients in our proof are a decay estimate of volume of cusps and volume comparison theorems.

We construct a non-separable Banach space every nonzero element of which is a bounded derivative that is not Riemann integrable. This in particular improves a result presented in [3], where the corresponding space was found to be separable.

In this paper, we provide a complete function theoretic characterizations for boundedness and compactness of difference of weighted composition operators from the space of Cauchy integral transforms to logarithmic weighted-type spaces. Surprisingly, an interesting feature of these characterizations is that they are free from pseudo-hyperbolic distance between $\varphi(z)$ and $\psi(z)$, which is different from the previous characterizations of difference of weighted composition operators acting between different holomorphic function spaces.

In this paper, we prove that the escaping parameter $\mathfrak{I}$ omitted endpoints has Hausdorff measure $0$ for the gauge function $t / (\log \frac{1}{t})^s$, where $s \gt 1$. This is a counterpart result on parameter space of exponential family of Theorem 1.1 in paper [5].

We show that a solenoid is a dynamical object and we express its complexity by a number of different entropy-like quantities in Hurley's sense. Some relations between these entropy-like quantities are presented. We adopt the theory of Carathéodory dimension structures introduced axiomatically by Pesin to a case of a solenoid. Any of the above mentioned entropy-like quantities determines a particular Carathéodory structure such that its upper capacity coincides with the considered quantity. We mimic a definition of the local measure entropy, introduced by Brin and Katok for a single map, to a case of a solenoid. Lower estimations of these quantities by corresponding local measure entropies are described.

This paper is devoted to studying a semilinear parabolic equation with conical degeneration. First, we extend previous results on the vacuum isolating of solution with initial energy $J(u_0) \lt d$, where $d$ is the mountain pass level. Concretely, we obtain the explicit vacuum region, the global existence region and the blow-up region. Moreover, as far as the blow-up solution is concerned, we estimate the upper bound of the blow-up time and blow-up rate. Second, for all $p \gt 1$, we get a new sufficient condition, which demonstrates the finite time blow-up for arbitrary initial energy, and the upper bound estimate of blow-up time is obtained.

A flexible numerical method for the time-fractional telegraph equation is proposed and analyzed in this paper. The solution is discretized with a new finite difference scheme in time, and a local discontinuous Galerkin (LDG) method in space. We prove that the method is unconditionally stable and convergent with order $O(h^{k+1} + (\Delta t)^{3-\alpha})$, where $h$, $\Delta t$, $k$ are the space step size, time step size and degree of piecewise polynomial, respectively. Numerical experiments are carried out to illustrate the robustness, reliability, and accuracy of the method.

In this paper, a numerical fully discrete scheme based on the finite element approximation for the distributed order time fractional variable coefficient diffusion equations is developed and a complete error analysis is provided. The weighted and shifted Grünwald formula is applied for the time-fractional derivative and finite element approach for the spatial discretization. The unconditional stability and the global superconvergence estimate of the fully discrete scheme are proved rigorously. Extensive numerical experiments are presented to illustrate the accuracy and efficiency of the scheme, and to verify the convergence theory.