We prove that the degrees of the iterates $\deg (f^n)$ of a birational map satisfy $\liminf(\deg (f^n)) \lt + \infty$ if and only if the sequence $\deg (f^n)$ is bounded, and that the growth of $\deg (f^n)$ cannot be arbitrarily slow, unless $\deg (f^n)$ is bounded.

We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schrödinger equation\[i \partial_t u + \partial^2_x u + i {\lvert u \rvert}^{2 \sigma} \partial_x u = 0.\]The equation has a two-parameter family of solitary wave solutions of the form\[\phi_{\omega , c} (x) = \varphi_{\omega ,c} (x) \exp\left \lbrace i \frac{c}{2} x - \frac{i}{2 \sigma + 2} \int^x_{-\infty}\varphi^{2

We show Fujita’s spectrum conjecture for $\epsilon \textrm{-}\log$ canonical pairs and Fujita’s log spectrum conjecture for log canonical pairs. Then, we generalize the pseudo-effective threshold of a single divisor to multiple divisors and establish the analogous finiteness and the DCC properties.

In this paper we study the bilinear multiplier operator of the form \begin{align*}&H^t(f,g)(x)=\int_{\mathbb{R}^d}\!\int_{\mathbb{R}^d} m(t\xi,t\eta)\,\mathrm{e}^{2 \pi \mathrm{i} t|(\xi,\eta)|}\, \widehat{f}(\xi)\, \widehat{g}(\eta)\, \mathrm{e}^{2 \pi \mathrm{i}x(\xi +\eta)}\,d \xi d \eta, \\&1 \le t \le 2\end{align*}where $m$ satisfies the Marcinkiewicz–Mikhlin–Hörmander’s derivative conditions

The purpose of this article is to give a simple and explicit construction of mock modular forms whose shadows are Eisenstein series of arbitrary integral weight, level, and character. As application, we construct forms whose shadows are Hecke’s Eisenstein series of weight one associated to imaginary quadratic fields, recovering some results by Kudla, Rapoport and Yang (1999), and Schofer (2009), and

Let $M$ be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on $M$ is finite in a small interval which is optimal.

Let $X$ be a smooth projective curve of genus $g \geq 2$ over an algebraically closed field $k$ of characteristic $p \gt 0 , \mathfrak{M}^s_X (r, d)$ the moduli space of stable vector bundles of rank $r$ and degree $d$ on $X$. We study the Frobenius stratification of $\mathfrak{M}^s_X (3, d)$ in terms of Harder–Narasimhan polygons of Frobenius pull-backs of stable vector bundles and obtain the irreducibility

Let $X$ be a projective variety admitting a polarized (or more generally, int‑amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when $X$ is $\mathbb{Q}$-factorial normal, every minimal model program is equivariant relative to the monoid $\mathrm{SEnd}(X)$ of all surjective endomorphisms, up to finite index. Further, when $X$ is rationally connected and smooth

In this paper we obtain a finite set $S$ of separating invariants for the variety of Hopf algebras of a fixed dimension. In dimension $p^2$ where $p$ is a prime or when dimension is $\lt 18$, except $8, 12, 16$, these invariants determine isomorphism classes of Hopf algebras, i.e., two Hopf algebras of a given dimension are isomorphic if and only if each of the invariants in $S$ take the same values

In this short article, given a smooth diagonalizable group scheme $G$ of finite type acting on a smooth quasi-compact separated scheme $X$, we prove that (after inverting some elements of representation ring of $G$) all the information concerning the additive invariants of the quotient stack $[X/G]$ is “concentrated” in the subscheme of $G$-fixed points $X^G$. Moreover, in the particular case where

We establish $\ell^p (\mathbb{Z})$ boundedness of $r$-variational seminorm for operators of Radon type along subsets of prime numbers of the form $\lbrace p \in \mathbb{P} : \lbrace \varphi_1 (p) \rbrace \lt \psi(p) \rbrace$ As an application we obtain the corresponding pointwise ergodic theorems.