We prove that the average size of a mixed character sum

\begin{equation*}\sum_{1\leqslant n \leqslant x} \chi(n) e(n\theta) w(n/x)\end{equation*}

$w$

$\sqrt{x}$

$\theta$

$\chi$

$r$

$x\leqslant r$

$o(\sqrt{x})$

$\theta$

We consider induced representations $\operatorname {\mathrm {Ind}}_{\mathrm {P}(F)}^{\operatorname {\mathrm {G}}(F)} \pi $, where $\mathrm {P}$ is a maximal parabolic subgroup of a reductive group $\operatorname {\mathrm {G}}$ over a p-adic field F, and $(\pi , V)$ is a unitary supercuspidal representation of $\operatorname {\mathrm {M}}(F)$, $\operatorname {\mathrm {M}}$ being some Levi subgroup of $\mathrm {P}$. Imposing a certain ‘Heisenberg parabolic subgroup’ assumption on $\mathrm {P}$, we apply the method of Goldberg, Shahidi and Spallone to obtain an expression for a certain constant $R(\tilde {\pi })$, which captures the residue of a family $s \mapsto A(s, \pi , w_0)$ of intertwining operators associated to this situation, in terms of harmonic analysis on the twisted Levi subgroup $\tilde {\operatorname {\mathrm {M}}}(F) := \operatorname {\mathrm {M}}(F) w_0$. For $\operatorname {\mathrm {G}}$ absolutely almost simple and simply connected of type $G_2$ or $D_4$ (resp., $B_3$), and $\mathrm {P}$ satisfying the ‘Heisenberg’ condition, if the central character of $\pi $ is nontrivial (resp., trivial) on $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}(F)$, where $\operatorname {\mathrm {A}}_{\operatorname {\mathrm {M}}}$ is the connected centre of $\operatorname {\mathrm {M}}$, our formula for $R(\tilde {\pi })$ can be rewritten in terms of the Langlands parameter of $\pi $, in the spirit of a prediction of Arthur. For the same collection of $\operatorname {\mathrm {G}}$ and $\mathrm {P}$, when these central character conditions are not satisfied, Arthur’s prediction combined with our formula for $R(\tilde {\pi })$ suggests a harmonic analytic formula for a product of one or two $\gamma $-factors associated to the situation.

In this work, we study a class of elliptic problems involving nonlinear superpositions of fractional operators of the form

\begin{equation*}A_{\mu,p}u := \int_{[0,1]} (-\Delta)_{p}^{s} u \, d\mu(s),\end{equation*}

$\mu$

$[0,1]$

$p$

$p$

$p$

$p$

A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov’s result in the following form. Given $r,t\in \mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include:

• • For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline {G}$ has positive triangle density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline {G}$ has positive $K_r$-density, then $\overline {G}$ contains a biclique of size $\Omega ({n \over {\log n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2.

• • Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $\Omega ({n \over {\log n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $\Omega (n)$, where the constant $(2h-2)^{r}+1$ is optimal.

The ${n \over {\log n}}$ size of the blowups in all our results are optimal up to a constant factor.

For a complete discrete valuation field K, we show that one may always glue a separated formal algebraic space $\mathfrak {X}$ over $\mathcal {O}_K$ to a separated algebraic space U over K along an open immersion of rigid spaces $j\colon \mathfrak {X}^{\mathrm {rig}}\to U^{\mathrm {an}}$, producing a separated algebraic space X over $\mathcal {O}_K$. This process gives rise to an equivalence between such ‘gluing triples’ $(U,\mathfrak {X},j)$ and separated algebraic spaces X over $\mathcal {O}_K$, which one might interpret as a version of the Beauville–Laszlo theorem for algebraic spaces rather than coherent sheaves. Moreover, an analogous equivalence exists over any excellent base. Examples due to Matsumoto imply that the result of such a gluing might be a genuine algebraic space (not a scheme) even if U and the special fibre of $\mathfrak {X}$ are projective. The proof is a combination of the Nagata compactification theorem for algebraic spaces and of Artin’s contraction theorem. We give multiple examples and applications of this idea.

Inspired by Bhatt–Scholze [BS22], in this article, we introduce prismatic cohomology for rigid analytic spaces with l.c.i. singularities, with coefficients over Fontaine’s de Rham period ring ${\mathrm {B_{dR}^+}}$.

In this paper, we consider asymptotic behaviours of multiscale multivalued stochastic systems with small noises. First of all, for general, fully coupled systems for multivalued stochastic differential equations of slow and fast motions with small noises in the slow components, we prove an averaging principle in the strong convergence sense. Moreover, a convergence rate is given in a special case. Next, for these systems, we establish the large deviation principle by the weak convergence approach. Then, for a special case, the rate function is explicitly characterized. Finally, we explain our results with an example.

We study the sets of points where a Lévy function and a translated Lévy function share a given couple of Hölder exponents, and we investigate how their Hausdorff dimensions depend on the translation parameter.

We study the relationship between the enumerative geometry of rational curves in local geometries and various versions of maximal contact logarithmic curve counts. Our approach is via quasimap theory, and we show versions of the [vGGR19] local/logarithmic correspondence for quasimaps, and in particular for normal crossings settings, where the Gromov-Witten theoretic formulation of the correspondence fails. The results suggest a link between different formulations of relative Gromov-Witten theory for simple normal crossings divisors via the mirror map. The main results follow from a rank reduction strategy, together with a new degeneration formula for quasimaps.

We consider steady-state diffusion in a bounded planar domain with multiple small targets on a smooth boundary. Using the method of matched asymptotic expansions, we investigate the competition of these targets for a diffusing particle and the crucial role of surface reactions on the targets. We start from the classical problem of splitting probabilities for perfectly reactive targets with Dirichlet boundary conditions and improve some earlier results. We discuss how this approach can be generalised to partially reactive targets characterised by a Robin boundary condition. In particular, we show how partial reactivity reduces the effective size of the target. In addition, we consider more intricate surface reactions modelled by mixed Steklov-Neumann or Steklov-Neumann-Dirichlet problems. We provide the first derivation of the asymptotic behaviour of the eigenvalues and eigenfunctions for these spectral problems in the small-target limit. Finally, we show how our asymptotic approach can be extended to interior targets in the bulk and to exterior problems where diffusion occurs in an unbounded planar domain outside a compact set. Direct applications of these results to diffusion-controlled reactions are discussed.

We establish sufficient conditions for the existence of ground states of the following normalized nonlinear Schrödinger–Newton system with a point interaction:

\begin{equation*}\begin{cases}- \Delta_\alpha u + \omega u=w u+\beta u |u|^{p - 2}&\text{on} ~ \mathbb{R}^2;\\- \Delta w=2 \pi |u|^2&\text{on} ~ \mathbb{R}^2;\\\|u\|_{L^2}^2 = c,\end{cases}\end{equation*}

where $p \gt 2$; $\alpha, \beta \in \mathbb{R}$; $c \gt 0$ and $- \Delta_\alpha$ denotes the Laplacian of point interaction with s-wave scattering length $(- 2 \pi \alpha)^{- 1}$, the unknowns being $u \colon \mathbb{R}^2 \to \mathbb{C}$, $w \colon \mathbb{R}^2 \to \lbrack0, \infty\lbrack$ and the Lagrange multiplier $\omega \in \mathbb{R}$. Additionally, we show that critical points of the corresponding constrained energy functional are naturally associated with standing waves of the evolution problem

\begin{equation*}\mathrm{i} \psi' (t)=- \Delta_\alpha \psi (t)-(\log |\cdot| \ast |\psi (t)|^2)\psi (t)-\beta\psi (t)|\psi (t)|^{p - 2}.\end{equation*}

We investigate uniqueness of solution to the heat equation with a density $\rho$ on complete, non-compact weighted Riemannian manifolds of infinite volume. Our main goal is to identify sufficient conditions under which the solution $u$ vanishes identically, assuming that $u$ belongs to a certain weighted Lebesgue space with exponential or polynomial weight, $L^p_{\phi}$. We distinguish between the cases $p \gt 1$ and $p = 1$ which required stronger assumptions on the manifold and the density function $\rho$. We develop a unified method based on a conformal transformation of the metric, which allows us to reduce the problem to a standard heat equation on a suitably weighted manifold. In addition, we construct explicit counterexamples on model manifolds which demonstrate optimality of our assumptions on the density $\rho$.

This note establishes sharp time-asymptotic algebraic rate bounds for the classical evolution problem of Fujita, but with sublinear rather than superlinear exponent. A transitional stability exponent is identified, which has a simple reciprocity relation with the classical Fujita critical blow-up exponent.

This paper is concerned with a duality between $r$-regular permutations and $r$-cycle permutations, and a monotone property due to Bóna-McLennan-White on the probability $p_r(n)$ for a random permutation of $\{1,2,\ldots, n\}$ to have an $r$th root, where $r$ is a prime. For $r=2$, the duality relates permutations with odd cycles to permutations with even cycles. For the general case where $r\geq 2$, we define an $r$-enriched permutation as a permutation with $r$-singular cycles coloured by one of the colours $1, 2, \ldots, r-1 $. In this setup, we discover a bijection between $r$-regular permutations and enriched $r$-cycle permutations, which in turn yields a stronger version of an inequality of Bóna-McLennan-White. This leads to a fully combinatorial understanding of the monotone property, thereby answering their question. When $r$ is a prime power $q^l$, we further show that $p_r(n)$ is monotone. In the case that $n+1 \not\equiv 0 \pmod q$, the equality $p_r(n)=p_r(n+1)$ has been established by Chernoff.

We investigate axisymmetric surfaces in Euclidean space that are stationary for the energy $E_\alpha=\int_\Sigma |p|^\alpha\, d\Sigma$. Using a phase plane analysis, we classify these surfaces under the assumption that they intersect the rotation axis orthogonally. We also provide applications of the maximum principle, characterizing closed stationary surfaces and compact stationary surfaces with circular boundary in the case $\alpha=-2$. Finally, we prove that helicoidal stationary surfaces must in fact be rotational surfaces.