We establish two complementary results about the regularity of the solution of the periodic initial value problem for the linear Benjamin–Ono equation. We first give a new simple proof of the statement that, for a dense countable set of the time variable, the solution is a finite linear combination of copies of the initial condition and of its Hilbert transform. In particular, this implies that discontinuities in the initial condition are propagated in the solution as logarithmic cusps. We then show that, if the initial condition is of bounded variation (and even if it is not continuous), for almost every time the graph of the solution in space is continuous but fractal, with upper Minkowski dimension equal to $\frac32$. In order to illustrate this striking dichotomy, in the final section, we include accurate numerical evaluations of the solution profile, as well as estimates of its box-counting dimension for two canonical choices of irrational time.

We give a notion of boundary pair $(\mathcal{B}_-,\mathcal{B}_+)$ for measured groupoids which generalizes the one introduced by Bader and Furman [BF14] for locally compact groups. In the case of a semidirect groupoid $\mathcal{G}=\Gamma \ltimes X$ obtained by a probability measure preserving action $\Gamma \curvearrowright X$ of a locally compact group, we show that a boundary pair is exactly $(B_- \times X, B_+ \times X)$, where $(B_-,B_+)$ is a boundary pair for $\Gamma$. For any measured groupoid $(\mathcal{G},\nu )$, we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to $\nu$ provide other examples of our definition. Following Bader and Furman [BF], we define algebraic representability for an ergodic groupoid $(\mathcal{G},\nu )$. In this way, given any measurable representation $\rho \,:\,\mathcal{G} \rightarrow H$ into the $\kappa$-points of an algebraic $\kappa$-group $\mathbf{H}$, we obtain $\rho$-equivariant maps $\mathcal{B}_\pm \rightarrow H/L_\pm$, where $L_\pm =\mathbf{L}_\pm (\kappa )$ for some $\kappa$-subgroups $\mathbf{L}_\pm \lt \mathbf{H}$. In the particular case when $\kappa =\mathbb{R}$ and $\rho$ is Zariski dense, we show that $L_\pm$ must be minimal parabolic subgroups.

Several classical knot invariants, such as the Alexander polynomial, the Levine-Tristram signature, and the Blanchfield pairing, admit natural extensions from knots to links, and more generally, from oriented links to so-called colored links. In this note, we explore such extensions of the Arf invariant. Inspired by the three examples stated above, we use generalized Seifert forms to construct quadratic forms and determine when the Arf invariant of such a form yields a well-defined invariant of colored links. However, apart from the known case of oriented links, these new Arf invariants turn out to be determined by the linking numbers.

In [5], a particular family of real hyperplane arrangements stemming from hyperpolygonal spaces associated with certain quiver varieties was introduced which we thus call hyperpolygonal arrangements ${\mathscr H}_n$. In this note, we study these arrangements and investigate their properties systematically. Remarkably, the arrangements ${\mathscr H}_n$ discriminate between essentially all local properties of arrangements. In addition, we show that hyperpolygonal arrangements are projectively unique and combinatorially formal.

We note that the arrangement ${\mathscr H}_5$ is the famous counterexample of Edelman and Reiner [17] of Orlik’s conjecture that the restriction of a free arrangement is again free.

We consider the two-dimensional nonlinear Schrödinger equation with point interaction and we establish a local well-posedness theory, including blow-up alternative and continuous dependence on the initial data in the energy space. We provide proof by employing Kato’s method along with Hardy inequalities with logarithmic correction. Moreover, we establish finite time blow-up for solutions with positive energy and infinite variance.

We prove a Poisson process approximation result for stabilising functionals of a determinantal point process. Our results use concrete couplings of determinantal processes with different Palm measures and exploit their association properties. Second, we focus on the Ginibre process and show in the asymptotic scenario of an increasing observation window that the process of points with a large nearest neighbour distance converges after a suitable scaling to a Poisson point process. As a corollary, we obtain the scaling of the maximum nearest neighbour distance in the Ginibre process, which turns out to be different from its analogue for independent points.

In this paper, we study the following high-order elliptic equation involving the GJMS operator:

\begin{align*}\alpha P_{\mathbb{S}^n}v_{\alpha}+2Q_{g_{\mathbb{S}^n}}=2Q_{g_{\mathbb{S}^n}}e^{nv_{\alpha}}.\end{align*}

We establish that if α > 1 and $n\geq3$ or if $\alpha\in (1-\epsilon_0, 1)$ with $n=2m\geq4$, then $v_{\alpha}\equiv0$. As an application, we present a new proof of the classical Beckner inequality.

In this article, we prove the local-in-time existence of regular solutions to dissipative Aw–Rascle system with the offset equal to gradient of some increasing and regular function of density. It is a mixed degenerate parabolic-hyperbolic hydrodynamic model, and we extend the techniques previously developed for compressible Navier–Stokes equations to show the well-posedness of the system in the $L_2-L_2$ setting. We also discuss relevant existence results for offset involving singular or non-local functions of density.

Counting independent sets in graphs and hypergraphs under a variety of restrictions is a classical question with a long history. It is the subject of the celebrated container method which found numerous spectacular applications over the years. We consider the question of how many independent sets we can have in a graph under structural restrictions. We show that any $n$-vertex graph with independence number $\alpha$ without $bK_a$ as an induced subgraph has at most $n^{O(1)} \cdot \alpha ^{O(\alpha )}$ independent sets. This substantially improves the trivial upper bound of $n^{\alpha },$ whenever $\alpha \le n^{o(1)}$ and gives a characterisation of graphs forbidding which allows for such an improvement. It is also in general tight up to a constant in the exponent since there exist triangle-free graphs with $\alpha ^{\Omega (\alpha )}$ independent sets. We also prove that if one in addition assumes the ground graph is chi-bounded one can improve the bound to $n^{O(1)} \cdot 2^{O(\alpha )}$ which is tight up to a constant factor in the exponent.

We introduce the mean topological dimension of random bundle transformations associated with an infinite countable discrete amenable group action and show that continuous bundle random dynamical systems for amenable groups with finite fibre topological entropy have zero mean topological dimensions.

This article is concerned with the following chemotaxis-growth system

\begin{equation*}\begin{cases}u_t = \nabla\cdot \left(\nabla u - u\bf{w} \right) + \rho u - \mu u^\alpha ,&x \in \Omega ,\ t \gt 0,\\v_{1,t} = \Delta v_1 - v_1 + u,&x \in \Omega ,\ t \gt 0,\\v_{2,t} = \Delta v_2 - v_2 + v_1,&x \in \Omega ,\ t \gt 0,\\\ldots \\v_{k,t} = \Delta v_k - v_k + v_{k - 1},&x \in \Omega ,\ t \gt 0,\\{\bf{w}}_t = \Delta {\bf{w}} - {\bf{w}} + \nabla v_k,&x \in \Omega ,\ t \gt 0,\end{cases}\end{equation*}

under the homogeneous Neumann boundary condition for u, vi and the homogeneous Dirichlet boundary condition for $\bf{w}$ in a smooth bounded domain $\Omega \subset {\mathbb{R}^n}\left( {n \geqslant 1} \right),$ where ρ > 0, µ > 0, α > 1 and $i=1,\ldots,k$. We reveal that when the index α, the spatial variable n, and the number of equations k satisfy certain relationships, the global solution of the system exists and converges to the constant equilibrium state in the form of exponential convergence.

In the second part of this series of papers, we address the same evolution problem that was considered in part 1 (see [16]), namely the nonlocal Fisher-KPP equation in one spatial dimension,

\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}

$\phi *u$

$\phi (y) \equiv H\left (\frac {1}{4}-y^2\right )$

$[0,a]$

We initiate a study of large deviations for block model random graphs in the dense regime. Following [14], we establish an LDP for dense block models, viewed as random graphons. As an application of our result, we study upper tail large deviations for homomorphism densities of regular graphs. We identify the existence of a ‘symmetric’ phase, where the graph, conditioned on the rare event, looks like a block model with the same block sizes as the generating graphon. In specific examples, we also identify the existence of a ‘symmetry breaking’ regime, where the conditional structure is not a block model with compatible dimensions. This identifies a ‘reentrant phase transition’ phenomenon for this problem – analogous to one established for Erdős–Rényi random graphs [13, 14]. Finally, extending the analysis of [34], we identify the precise boundary between the symmetry and symmetry breaking regimes for homomorphism densities of regular graphs and the operator norm on Erdős–Rényi bipartite graphs.

A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimised by a random colouring in which each edge is equally likely to be red or blue. We extend this notion to an off-diagonal setting. That is, we define a pair $(H_1,H_2)$ of graphs to be $(p,1-p)$-common if a particular linear combination of the density of $H_1$ in red and $H_2$ in blue is asymptotically minimised by a random colouring in which each edge is coloured red with probability $p$ and blue with probability $1-p$. Our results include off-diagonal extensions of several standard theorems on common graphs and novel results for common pairs of graphs with no natural analogue in the classical setting.

We prove that a solution to the 3D Navier–Stokes or magneto-hydrodynamics equations does not blow up at t = T provided $\displaystyle \limsup_{q \to \infty} \int_{\mathcal{T}_q}^T \|\Delta_q(\nabla \times u)\|_\infty \, dt$ is small enough, where u is the velocity, $\Delta_q$ is the Littlewood–Paley projection and $\mathcal T_q$ is a certain sequence such that $\mathcal T_q \to T$ as $q \to \infty$. This improves many existing regularity criteria.

A random temporal graph is an Erdős-Rényi random graph $G(n,p)$, together with a random ordering of its edges. A path in the graph is called increasing if the edges on the path appear in increasing order. A set $S$ of vertices forms a temporal clique if for all $u,v \in S$, there is an increasing path from $u$ to $v$. Becker, Casteigts, Crescenzi, Kodric, Renken, Raskin and Zamaraev [(2023) Giant components in random temporal graphs. arXiv,2205.14888] proved that if $p=c\log n/n$ for $c\gt 1$, then, with high probability, there is a temporal clique of size $n-o(n)$. On the other hand, for $c\lt 1$, with high probability, the largest temporal clique is of size $o(n)$. In this note, we improve the latter bound by showing that, for $c\lt 1$, the largest temporal clique is of constant size with high probability.

We consider families of special cycles, as introduced by Kudla, on Shimura varieties attached to anisotropic quadratic spaces over totally real fields. By augmenting these cycles with Green currents, we obtain classes in the arithmetic Chow groups of the canonical models of these Shimura varieties (viewed as arithmetic varieties over their reflex fields). The main result of this paper asserts that generating series built from these cycles can be identified with the Fourier expansions of non-holomorphic Hilbert-Jacobi modular forms. This result provides evidence for an arithmetic analogue of Kudla’s conjecture relating these cycles to Siegel modular forms.