In this paper, we give characterizations of usual wave front set and wave front set in $H^s$ in terms of wave packet transform without any restriction on basic wave packet, which give complete answers of the question raised by G. B. Folland.

Let $F$ be a non-Archimedean local field with a finite residue field. We prove that the conjecture, presented by Broussous, Sécherre, and Stevens, is verified in the essetially tame case, that is, that the Jacquet-Langlands correspondence, which was explicitly described by Bushnell and Henniart, preserves an endo-class for irreducible essentially tame representations of inner forms of $\mathrm{GL}_n(F), n \ge 1$, of parametric degree $n$. Moreover we give explicitly a parameter set for such representations of an inner form $G$ of $\mathrm{GL}_n(F)$ which contain simple characters belonging to an endo-class.

The Jorge-Meeks $n$-noid ($n\ge 2$) is a complete minimal surface of genus zero with $n$ catenoidal ends in the Euclidean 3-space $\boldsymbol{R}^3$, which has $(2\pi/n)$-rotation symmetry with respect to its axis. In this paper, we show that the corresponding maximal surface $f_n$ in Lorentz-Minkowski 3-space $\boldsymbol{R}^3_1$ has an analytic extension $\tilde f_n$ as a properly embedded zero mean curvature surface. The extension changes type into a time-like (minimal) surface.

We show that spectral Hausdorff dimensional properties of discrete Schrödinger operators with (1) Sturmian potentials of bounded density and (2) a class of sparse potentials are preserved under suitable polynomial decaying perturbations, when the spectrum of these perturbed operators have some singular continuous component.

This paper is concerned with the abstract quasilinear hyperbolic equations of Kirchhoff type with perturbation. We show the existence of the wave operators and the scattering operator for small data, and that these operators are homeomorphic with respect to a suitable metric in a neighborhood of the origin.

We show that the moduli space of semi-stable sheaves on a smooth quadric surface, having dimension $1$, multiplicity $4$, Euler characteristic $2$, and first Chern class $(2, 2)$, is the blow-up at two points of a certain hypersurface in a weighted projective space.

Let $k$ be a field of arbitrary characteristic. In 1978, Nakai proved a structure theorem for $k$-domains admitting a nontrivial locally finite iterative higher derivation when $k$ is algebraically closed. In this paper, we generalize Nakai's theorem to cover the case where $k$ is not algebraically closed. As a consequence, we obtain a cancellation theorem of the following form: Let $A$ and $A'$ be finitely generated $k$-domains with $A[x]\simeq _kA'[x]$. If $A$ and $\bar{k}\otimes _kA$ are UFDs and $\mathop{\rm trans.deg}\nolimits _kA=2$, then we have $A\simeq _kA'$. This generalizes the cancellation theorem of Crachiola.

It is proved that if two quasitoric manifolds of dimension $\le 2p^2-4$ for a prime $p$ have isomorphic cohomology rings, then they have the same $p$-local stable homotopy type.

Suppose that $f$ and $g$ are two elliptic quaternionic Möbius transformations of orders $m$ and $n$ respectively. If the hyperbolic distance $\delta(f,g)$ between ${\rm fix}(f)$ and ${\rm fix}(g)$ satisfies $$\cosh \delta(f,g) \geq\frac{\cos\frac{\pi}{m}\cos\frac{\pi}{n}+1}{\sin\frac{\pi}{m}\sin\frac{\pi}{n}},$$ then the group $\langle f , g\rangle$ is discrete non-elementary and isomorphic to the free product $\langle f \rangle* \langle g\rangle$.

The tail of the colored Jones polynomial of an alternating link is a $q$-series invariant whose first $n$ terms coincide with the first $n$ terms of the $n$-th colored Jones polynomial. Recently, it has been shown that the tail of the colored Jones polynomial of torus knots give rise to Ramanujan type identities. In this paper, we study $q$-series identities coming from the colored Jones polynomial of pretzel knots. We prove a false theta function identity that goes back to Ramanujan and we give a natural generalization of this identity using the tail of the colored Jones polynomial of Pretzel knots. Furthermore, we compute the tail for an infinite family of Pretzel knots and relate it to false theta function-type identities.

In this paper we are concerned with a homogeneous differential operator $p$ of order $m$ of which characteristic set of order $m$ is assumed to be a smooth manifold. We define the Gevrey strong hyperbolicity index as the largest number $s$ such that the Cauchy problem for $p+Q$ is well-posed in the Gevrey class of order $s$ for any differential operator $Q$ of order less than $m$. We study the case of the largest index and we discuss in which way the Gevrey strong hyperbolicity index relates with the geometry of bicharacteristics of $p$ near the characteristic manifold.