Let M be an open Riemann surface and $n\ge 3$ be an integer. In this paper, we establish some generic properties (in Baire category sense) in the space of all conformal minimal immersions $M\to{\mathbb{R}}^n$ endowed with the compact-open topology, pointing out that a generic such immersion is chaotic in many ways. For instance, we show that a generic conformal minimal immersion $u\colon M\to {\mathbb{R}}^n$ is non-proper, almost proper, and ${\mathfrak{g}}$-complete with respect to any given Riemannian metric ${\mathfrak{g}}$ in ${\mathbb{R}}^n$. Further, its image u(M) is dense in ${\mathbb{R}}^n$ and disjoint from ${\mathbb{Q}}^3\times {\mathbb{R}}^{n-3}$, and has infinite area, infinite total curvature, and unbounded curvature on every open set in ${\mathbb{R}}^n$. In case n = 3, we also prove that a generic conformal minimal immersion $M\to {\mathbb{R}}^3$ has infinite index of stability on every open set in ${\mathbb{R}}^3$.

We derive faithful inclusions of C*-algebras from a coend-type construction in unitary tensor categories. This gives rise to different potential notions of discreteness for an inclusion in the non-irreducible case and provides a unified framework that encloses the theory of compact quantum group actions. We also provide examples coming from semi-circular systems and from factorization homology. In the irreducible case, we establish conditions under which the C*-discrete and W*-discrete conditions are equivalent.

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation

\begin{align*}\left\{\begin{array}{ll}-(a+b\int_{\Omega}|\nabla u|^2\mathrm{d}x)\Delta u+V(x)u=\mu u+\beta^*|u|^{\frac{8}{3}}u &\text{in}\ {\Omega}, \\[0.1cm]u=0&\text{on}\ {\partial\Omega}, \\[0.1cm]\int_{\Omega}|u|^2\mathrm{d}x=1, \\[0.1cm]\end{array}\right.\end{align*}

$a\geq 0$

$\Omega\subset\mathbb R^3$

$\beta^*:=\frac{b}{2}|Q|_2^{\frac{8}{3}}$

$-2\Delta u+\frac{1}{3}u-|u|^{\frac{8}{3}}u=0$

$\mathbb R^3.$

$a\searrow 0$

We compute the Fukaya category of the symplectic blowup of a compact rational symplectic manifold at a point in the following sense: suppose a collection of Lagrangian branes satisfy Abouzaid’s criterion [Abo10] for split-generation of a bulk-deformed Fukaya category of cleanly intersecting Lagrangian branes. We show (Theorem 1.1) that for a small blowup parameter, their inverse images in the blowup together with a collection of branes near the exceptional locus split-generate the Fukaya category of the blowup. This categorifies a result on quantum cohomology by Bayer [Bay04] and is an example of a more general conjectural description of the behaviour of the Fukaya category under transitions occurring in the minimal model program, namely that minimal model program transitions generate additional summands.

The Pósa–Seymour conjecture determines the minimum degree threshold for forcing the $k$th power of a Hamilton cycle in a graph. After numerous partial results, Komlós, Sárközy, and Szemerédi proved the conjecture for sufficiently large graphs. In this paper, we focus on the analogous problem for digraphs and for oriented graphs. We asymptotically determine the minimum total degree threshold for forcing the square of a Hamilton cycle in a digraph. We also give a conjecture on the corresponding threshold for $k$th powers of a Hamilton cycle more generally. For oriented graphs, we provide a minimum semi-degree condition that forces the $k$th power of a Hamilton cycle; although this minimum semi-degree condition is not tight, it does provide the correct order of magnitude of the threshold. Turán-type problems for oriented graphs are also discussed.

We develop a high-order asymptotic expansion for the mean first passage time (MFPT) of the capture of Brownian particles by a small elliptical trap in a bounded two-dimensional region. This new result describes the effect that trap orientation plays on the capture rate and extends existing results that give information only on the role of trap position on the capture rate. Our results are validated against numerical simulations that confirm the accuracy of the asymptotic approximation. In the case of the unit disk domain, we identify a bifurcation such that the high-order correction to the global MFPT (GMFPT) is minimized when the trap is orientated in the radial direction for traps centred at $0\lt r\lt r_c :=\sqrt {2-\sqrt {2}}$. When centred at position $r_c\lt r\lt 1$, the GMFPT correction is minimized by orientating the trap in the angular direction. In the scenario of a general two-dimensional geometry, we identify the orientation that minimizes the GMFPT in terms of the regular part of the Neumann Green’s function. This theory is demonstrated on several regular domains such as disks, ellipses and rectangles.

Motivated by the construction of the free Banach lattice generated by a Banach space, we introduce and study several vector and Banach lattices of positively homogeneous functions defined on the dual of a Banach space E. The relations between these lattices allow us to give multiple characterizations of when the underlying Banach space E is finite-dimensional and when it is reflexive. Furthermore, we show that lattice homomorphisms between free Banach lattices are always composition operators, and study how these operators behave on the scale of lattices of positively homogeneous functions.

For $\ell \geq 3$, an $\ell$-uniform hypergraph is disperse if the number of edges induced by any set of $\ell +1$ vertices is 0, 1, $\ell$, or $\ell +1$. We show that every disperse $\ell$-uniform hypergraph on $n$ vertices contains a clique or independent set of size $n^{\Omega _{\ell }(1)}$, answering a question of the first author and Tomon. To this end, we prove several structural properties of disperse hypergraphs.

For each prime $p$, this paper constructs compact complex hyperbolic $2$-manifolds with an isometric action of $\mathbb{Z} / p \mathbb{Z}$ that is not free and has only isolated fixed points. The case $p = 2$ is special, and finding general examples for $p=2$ is related to whether or not complex hyperbolic lattices are conjugacy separable on torsion.

We show the Harris–Viehmann conjecture under some Hodge–Newton reducibility condition for a generalisation of the diamond of a non-basic Rapoport–Zink space at infinite level, which appears as a cover of the non-semi-stable locus in the Hecke stack. We show also that the cohomology of the non-semi-stable locus with coefficients coming from a cuspidal Langlands parameter vanishes. As an application, we show the Hecke eigensheaf property in Fargues’ conjecture for cuspidal Langlands parameters in the $ {\mathrm {GL}}_2$-case.

In this paper, we will investigate the following inequality

\begin{equation*}(a_{n+1}a_{n+2} - a_{n}a_{n+3})^2 -(a_{n+1}^2 -a_{n}a_{n+2})(a_{n+2}^2 - a_{n}a_{n+4}) \gt 0,\end{equation*}

which arises from the iterated Laguerre operator on functions. We will prove the sequence $\{a_n\}$ of a unified form given by Griffin, Ono, Rolen and Zagier asymptotically satisfies this inequality while the Maclaurin coefficients of the functions in Laguerre-Pólya class have not to possess this inequality. We also prove the companion version of this inequality. As a consequence, we show the Maclaurin coefficients of the Riemann Ξ-function asymptotically satisfy this property. Moreover, we make this approach effective and give the exact thresholds for the positivity of this inequalityfor the partition function, the overpartition function and the smallest part function.

Fulton’s matrix Schubert varieties are affine varieties that arise in the study of Schubert calculus in the complete flag variety. Weigandt showed that arbitrary intersections of matrix Schubert varieties, now called ASM varieties, are indexed by alternating sign matrices (ASMs), objects with a long history in enumerative combinatorics. It is very difficult to assess Cohen–Macaulayness of ASM varieties or to compute their codimension, though these properties are well understood for matrix Schubert varieties due to work of Fulton. In this paper, we study these properties of ASM varieties with a focus on the relationship between a pair of ASMs and their direct sum. We also consider ASM pattern avoidance from an algebro-geometric perspective.

Let f(z) be the normalized primitive holomorphic Hecke eigenforms of even integral weight k for the full modular group $SL(2,\mathbb{Z})$ and denote $L(s,\mathrm{sym}^{2}f)$ be the symmetric square L-function attached to f(z). Suppose that $\lambda_{\mathrm{sym}^{2}f}(n)$ be the $\mathrm{Fourier}$ coefficient of $L(s,\mathrm{sym}^{2}f)$. In this paper, we investigate the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{\mathrm{sym}^{2}f }(n) $ for $j\geqslant 3$ and obtain some new results which improve on previous error estimates. We also consider the sum $\sum\limits_{n\leqslant x}\lambda^{j}_{f }(n^{2})$ and get some similar results.

In this paper, we study the existence of solutions to the following Hartree equation

\begin{align*}\begin{cases}-\Delta u+\lambda V(x) u+\mu u=\left(\int_{\mathbb{R}^N}\frac{|u|^p}{|x-y|^{N-\alpha}}\right)|u|^{p-2}u,\ \text{in}\ \mathbb{R}^N,\\\int_{\mathbb{R}^N}|u|^2=\omega,\end{cases}\end{align*}

Where $N\geq 3$, $\omega,\lambda \gt 0$, $p\in \left(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}\right)\setminus\left\{\frac{N+\alpha+2}{N}\right\}$ and µ will appear as a Lagrange multiplier. We assume that $0\leq V\in L^{\infty}_{loc}(\mathbb{R}^N)$ has a bottom $int V^{-1}(0)$ composed of $\ell_0$ $(\ell_{0}\geq1)$ connected components $\{\Omega_i\}_{i=1}^{\ell_0}$, where $int V^{-1}(0)$ is the interior of the zero set $V^{-1}(0)=\{x\in\mathbb{R}^N| V(x)=0\}$ of V. It is worth pointing out that the penalization technique is no longer applicable to the local sublinear case $p\in \left(\frac{N+\alpha}{N},2\right)$. Therefore, we develop a new variational method in which the two deformation flows are established that reflect the properties of the potential. Moreover, we find a critical point without introducing a penalization term and give the existence result for $p\in \left(\frac{N+\alpha}{N}, \frac{N+\alpha}{N-2}\right)\setminus\left\{\frac{N+\alpha+2}{N}\right\}$. When ω is fixed and satisfies $\omega^{\frac{-(p-1)}{-Np+N+\alpha+2}}$ sufficiently small, we construct a $\ell$-bump $(1\leq\ell\leq \ell_{0})$ positive normalization solution, which concentrates at $\ell$ prescribed components $\{\Omega_i\}^{\ell}_{i=1}$ for large λ. We also consider the asymptotic profile of the solutions as $\lambda\rightarrow\infty$ and $\omega^{\frac{-(p-1)}{-Np+N+\alpha+2}}\rightarrow 0$.

We study existence and uniqueness of spherically symmetric solutions of

\begin{equation*}S_k(D^2v)+\beta \xi\cdot\nabla v+\alpha v+\left\vert v\right\vert^{q-1}v=0\;\; \mbox{in}\;\; \mathbb{R}^n,\end{equation*}

where $\alpha,\beta$ are real parameters, $n \gt 2,\, q \gt k\geqslant 1$ and $S_k(D^2v)$ stands for the k-Hessian operator of v. Our results are based mainly on the analysis of an associated dynamical system and energy methods. We derive some properties of the solutions of the above equation for different ranges of the parameters α and β. In particular, we describe with precision its asymptotic behaviour at infinity. Further, according to the position of q with respect to the first critical exponent $\frac{(n+2)k}{n}$ and the Tso critical exponent $\frac{(n+2)k}{n-2k}$ we study the existence of three classes of solutions: crossing, slow decay or fast decay solutions. In particular, if k > 1 all the fast decay solutions have a compact support in $\mathbb{R}^n$. The results also apply to construct self-similar solutions of type I to a related nonlinear evolution equation. These are self-similar functions of the form $u(t,x)=t^{-\alpha}v(xt^{-\beta})$ with suitable α and β.

In this paper we study the class of optimal entropy-transport problems introduced by Liero, Mielke and Savaré in Inventiones Mathematicae 211 in 2018. This class of unbalanced transport metrics allows for transport between measures of different total mass, unlike classical optimal transport where both measures must have the same total mass. In particular, we develop the theory for the important subclass of semi-discrete unbalanced transport problems, where one of the measures is diffuse (absolutely continuous with respect to the Lebesgue measure) and the other is discrete (a sum of Dirac masses). We characterize the optimal solutions and show they can be written in terms of generalized Laguerre diagrams. We use this to develop an efficient method for solving the semi-discrete unbalanced transport problem numerically. As an application, we study the unbalanced quantization problem, where one looks for the best approximation of a diffuse measure by a discrete measure with respect to an unbalanced transport metric. We prove a type of crystallization result in two dimensions – optimality of a locally triangular lattice with spatially varying density – and compute the asymptotic quantization error as the number of Dirac masses tends to infinity.