Let $r$ be any positive integer. We prove that for every sufficiently large $k$ there exists a $k$-chromatic vertex-critical graph $G$ such that $\chi (G-R)=k$ for every set $R \subseteq E(G)$ with $|R|\le r$. This partially solves a problem posed by Erdős in 1985, who asked whether the above statement holds for $k \ge 4$.

We use Stein’s method to obtain distributional approximations of subgraph counts in the uniform attachment model or random directed acyclic graph; we provide also estimates of rates of convergence. In particular, we give uni- and multi-variate Poisson approximations to the counts of cycles and normal approximations to the counts of unicyclic subgraphs; we also give a partial result for the counts of trees. We further find a class of multicyclic graphs whose subgraph counts are a.s. bounded as $n\to \infty$.

Let $\eta $ be [-11pc] [-7pc]a closed real 1-form on a closed Riemannian n-manifold $(M,g)$. Let $d_z$, $\delta _z$ and $\Delta _z$ be the induced Witten’s type perturbations of the de Rham derivative and coderivative and the Laplacian, parametrized by $z=\mu +i\nu \in \mathbb C$ ($\mu ,\nu \in \mathbb {R}$, $i=\sqrt {-1}$). Let $\zeta (s,z)$ be the zeta function of $s\in \mathbb {C}$, defined as the meromorphic extension of the function $\zeta (s,z)=\operatorname {Str}({\eta \wedge }\,\delta _z\Delta _z^{-s})$ for $\Re s\gg 0$. We prove that $\zeta (s,z)$ is smooth at $s=1$ and establish a formula for $\zeta (1,z)$ in terms of the associated heat semigroup. For a class of Morse forms, $\zeta (1,z)$ converges to some $\mathbf {z}\in \mathbb {R}$ as $\mu \to +\infty $, uniformly on $\nu $. We describe $\mathbf {z}$ in terms of the instantons of an auxiliary Smale gradient-like vector field X and the Mathai–Quillen current on $TM$ defined by g. Any real 1-cohomology class has a representative $\eta $ satisfying the hypothesis. If n is even, we can prescribe any real value for $\mathbf {z}$ by perturbing g, $\eta $ and X and achieve the same limit as $\mu \to -\infty $. This is used to define and describe certain tempered distributions induced by g and $\eta $. These distributions appear in another publication as contributions from the preserved leaves in a trace formula for simple foliated flows, giving a solution to a problem stated by C. Deninger.

We show that if one of various cycle types occurs in the permutation action of a finite group on the cosets of a given subgroup, then every almost conjugate subgroup is conjugate. As a number theoretic application, corresponding decomposition types of primes effect that a number field is determined by the Dedekind zeta function. As a geometric application, coverings of Riemannian manifolds with certain geodesic lifting behaviours must be isometric.

For given positive integers $r\ge 3$, $n$ and $e\le \binom{n}{2}$, the famous Erdős–Rademacher problem asks for the minimum number of $r$-cliques in a graph with $n$ vertices and $e$ edges. A conjecture of Lovász and Simonovits from the 1970s states that, for every $r\ge 3$, if $n$ is sufficiently large then, for every $e\le \binom{n}{2}$, at least one extremal graph can be obtained from a complete partite graph by adding a triangle-free graph into one part.

In this note, we explicitly write the minimum number of $r$-cliques predicted by the above conjecture. Also, we describe what we believe to be the set of extremal graphs for any $r\ge 4$ and all large $n$, amending the previous conjecture of Pikhurko and Razborov.

Let $X, Y$ be two locally compact Hausdorff spaces and $T:C_0(X)\rightarrow C_0(Y)$ be a standard surjective ɛ-norm-additive map, i.e.

\begin{equation*}\big|\|T(f)+T(g)\|-\|f+g\|\big|\leq \varepsilon,\;{\rm for\;all}\; f, g\in C_0(X).\end{equation*}

Then there exist a homeomorphism $\varphi:Y\rightarrow X$ and a continuous function $\lambda:Y\rightarrow\lbrace\pm1\rbrace$ such that

\begin{equation*}|T(f)(y)-\lambda(y)f(\varphi(y))|\leq\frac{3}{2}\varepsilon,\;{\rm for\;all}\;y\in Y,\;f\in C_0(X).\end{equation*}

The estimate ‘$\frac{3}{2}\varepsilon$’ is optimal. And this result can be regarded as a new nonlinear extension of the Banach–Stone theorem.

This article offers an advanced and novel investigation into the intricate propagation dynamics of the Belousov–Zhabotinsky system with non-local delayed interaction, which exhibits dynamical transition structure from bistable to monostable. We first solved the enduring open problem concerning the existence, uniqueness and the speed sign of the bistable travelling waves. In the monostable case, we developed and derived new results for the minimal wave speed selection, which, as an application, further improved the existing investigations on pushed and pulled wavefronts. Our results can provide new estimate to the minimal speed as well as to the determinacy of the transition parameters. Moreover, these results can be directly applied to standard localised models and delayed reaction diffusion models by choosing appropriate kernel functions.

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $\gamma$ fraction of the vertices classified correctly, where $\gamma \in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati ((2017) Ann. Stat. 45(1) 289–315.) for non-uniform HSBMs.

Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularised adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularisation of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

In this work, we study early warning signs for stochastic partial differential equations (SPDEs), where the linearisation around a steady state is characterised by continuous spectrum. The studied warning sign takes the form of qualitative changes in the variance as a deterministic bifurcation threshold is approached via parameter variation. Specifically, we focus on the scaling law of the variance near the transition. Since we are dealing here, in contrast to previous studies, with the case of continuous spectrum and quantitative scaling laws, it is natural to start with linearisations of the drift operator that are multiplication operators defined by analytic functions. For a one-dimensional spatial domain, we obtain precise rates of divergence. In the case of the two- and three-dimensional domains, an upper bound to the rate of the early warning sign is proven. These results are cross-validated by numerical simulations. Our theory can be generically useful for several applications, where stochastic and spatial aspects are important in combination with continuous spectrum bifurcations.

We provide a general recursive method for constructing transfer systems on finite lattices. Using this, we calculate the number of homotopically distinct $N_{\infty} $ operads for dihedral groups $D_{p^n}$, $p \gt 2$ prime, and cyclic groups $C_{qp^n}$, $p \neq q$ prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful $N_\infty$ operads for these groups.

In this paper, we consider blowup of solutions to the Cauchy problem for the following biharmonic nonlinear Schrödinger equation (NLS),

\begin{equation*}\text{i } \partial_t u=\Delta^2 u-\mu \Delta u-|u|^{2 \sigma} u \quad \text{in} \,\, \mathbb{R} \times \mathbb{R}^d,\end{equation*}

where $d \geq 1$, $\mu \in \mathbb{R}$ and $0 \lt \sigma \lt \infty$ if $1 \leq d \leq 4$ and $0 \lt \sigma \lt 4/(d-4)$ if $d \geq 5$. In the mass critical and supercritical cases, we establish the existence of blowup solutions to the problem for cylindrically symmetric data. The result extends the known ones with respect to blowup of solutions to the problem for radially symmetric data.

Let Mn and Tn denote the nth Motzkin number and the nth central trinomial coefficient respectively. We prove that for any prime $p\ge 5$,

\begin{equation*}\begin{aligned}\\[-22pt]&\sum_{k=0}^{p-1}M_k^2\equiv \left(\frac{p}{3}\right)\left(2-6p\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}kM_k^2\equiv \left(\frac{p}{3}\right)\left(9p-1\right)\pmod{p^2},\\&\sum_{k=0}^{p-1}T_kM_k\equiv \frac{4}{3}\left(\frac{p}{3}\right)+\frac{p}{6}\left(1-9\left(\frac{p}{3}\right)\right)\pmod{p^2},\\[-6pt]\end{aligned}\end{equation*}

where $\left(-\right)$ is the Legendre symbol. These results confirm three supercongruences conjectured by Z.-W. Sun in 2010.

A classification of multiplication modules over multiplication rings with finitely many minimal primes is obtained. A characterization of multiplication rings with finitely many minimal primes is given via faithful, Noetherian, distributive modules. It is proven that for a multiplication ring with finitely many minimal primes every faithful, Noetherian, distributive module is a faithful multiplication module, and vice versa.

Let $M$ be an oriented smooth manifold and $\operatorname{Homeo}\!(M,\omega )$ the group of measure preserving homeomorphisms of $M$, where $\omega$ is a finite measure induced by a volume form. In this paper, we define volume and Euler classes in bounded cohomology of an infinite dimensional transformation group $\operatorname{Homeo}_0\!(M,\omega )$ and $\operatorname{Homeo}_+\!(M,\omega )$, respectively, and in several cases prove their non-triviality. More precisely, we define:

• • Volume classes in $\operatorname{H}_b^n(\operatorname{Homeo}_0\!(M,\omega ))$, where $M$ is a hyperbolic manifold of dimension $n$.

• • Euler classes in $\operatorname{H}_b^2(\operatorname{Homeo}_+(S,\omega ))$, where $S$ is an oriented closed hyperbolic surface.

We show that Euler classes have positive norms for any closed hyperbolic surface and volume classes have positive norms for all hyperbolic surfaces and certain hyperbolic $3$-manifolds; hence, they are non-trivial.

We study the vectorial length compactification of the space of conjugacy classes of maximal representations of the fundamental group $\Gamma$ of a closed hyperbolic surface $\Sigma$ in $\textrm{PSL}(2,{\mathbb{R}})^n$. We identify the boundary with the sphere ${\mathbb{P}}(({\mathcal{ML}})^n)$, where $\mathcal{ML}$ is the space of measured geodesic laminations on $\Sigma$. In the case $n=2$, we give a geometric interpretation of the boundary as the space of homothety classes of ${\mathbb{R}}^2$-mixed structures on $\Sigma$. We associate to such a structure a dual tree-graded space endowed with an ${\mathbb{R}}_+^2$-valued metric, which we show to be universal with respect to actions on products of two $\mathbb{R}$-trees with the given length spectrum.

In dimension n = 1, we obtain $L^{p_1}(\mathbb R) \times\dots\times L^{p_m}(\mathbb R)$ to $L^p(\mathbb R)$ boundedness for the multilinear spherical maximal function in the largest possible open set of indices and we provide counterexamples that indicate the optimality of our results.

We introduce and study two conditions on groups of homeomorphisms of Cantor space, namely the conditions of being vigorous and of being flawless. These concepts are dynamical in nature, and allow us to study a certain interplay between the dynamics of an action and the algebraic properties of the acting group. A group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is vigorous if for any clopen set A and proper clopen subsets B and C of A, there is $\gamma \in G$ in the pointwise stabiliser of $\mathfrak {C}\backslash A$ with $B\gamma \subseteq C$. A nontrivial group $G\leq \operatorname {Homeo}(\mathfrak {C})$ is flawless if for all k and w a nontrivial freely reduced product expression on k variables (including inverse symbols), a particular subgroup $w(G)_\circ $ of the verbal subgroup $w(G)$ is the whole group. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) flawless groups are both perfect and lawless, 3) vigorous groups are simple if and only if they are flawless, and, 4) the class of vigorous simple subgroups of $\operatorname {Homeo}(\mathfrak {C})$ is fairly broad (the class is closed under various natural constructions and contains many well known groups, such as the commutator subgroups of the Higman–Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, Röver’s group $V(\Gamma )$, and others of Nekrashevych’s ‘simple groups of dynamical origin’).

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.