In this article, we study a non-uniform distribution on permutations biased by their number of records that we call record-biased permutations. We give several generative processes for record-biased permutations, explaining also how they can be used to devise efficient (linear) random samplers. For several classical permutation statistics, we obtain their expectation using the above generative processes, as well as their limit distributions in the regime that has a logarithmic number of records (as in the uniform case). Finally, increasing the bias to obtain a regime with an expected linear number of records, we establish the convergence of record-biased permutations to a deterministic permuton, which we fully characterise. This model was introduced in our earlier work [3], in the context of realistic analysis of algorithms. We conduct here a more thorough study but with a theoretical perspective.

We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci soliton.

This paper is concerned with the existence of normalized solutions for the following class of Hamiltonian elliptic systems:

\begin{align*}\left\{\begin{array}{ll}- \Delta u = \lambda u + |v|^{q-2}v \quad \text{in } \mathbb{R}^{N}, \\ - \Delta v = \lambda v + |u|^{p-2}u \quad \text{in } \mathbb{R}^{N}, \\ \displaystyle\int_{\mathbb{R}^N}(|u|^2 + |v|^2) = m,\end{array}\right.\end{align*}

$m \gt 0$

$2 \lt p,q \lt 2^{*}=2N/(N-2)$

$p,q$

$H^{1}(\mathbb{R}^N) \hookrightarrow L^{2}(\mathbb{R}^N)$

Let $r, k, n$ be integers satisfying $1\leqslant r\leqslant k\leqslant n/2$. Let ${{\mathcal{R}}}_r(n, k)$ denote the proportion of permutations $\pi \in {{\mathcal{S}}}_n$ that fix a set of size $k$ and have no cycle of length less than $r$. In this note, we determine the order of magnitude of ${{\mathcal{R}}}_r(n, k)$ uniformly for all $2\leqslant r\leqslant k\leqslant n/2$. This result generalises the corresponding estimate of Eberhard, Ford, and Green for the case $r=1$.

We prove that for any $k\geq 3$ for clause/variable ratios up to the Gibbs uniqueness threshold of the corresponding Galton-Watson tree, the number of satisfying assignments of random $k$-SAT formulas is given by the ‘replica symmetric solution’ predicted by physics methods [Monasson, Zecchina: Phys. Rev. Lett. 76 (1996)]. Furthermore, while the Gibbs uniqueness threshold is still not known precisely for any $k\geq 3$, we derive new lower bounds on this threshold that improve over prior work [Montanari and Shah: SODA (2007)]. The improvement is significant particularly for small $k$.

Confirming a conjecture of Erdős on the chromatic number of Kneser hypergraphs, Alon, Frankl and Lovász proved that in any $q$-colouring of the edges of the complete $r$-uniform hypergraph, there exists a monochromatic matching of size $\lfloor \frac {n+q-1}{r+q-1}\rfloor$. In this paper, we prove a transference version of this theorem. More precisely, for fixed $q$ and $r$, we show that with high probability, a monochromatic matching of approximately the same size exists in any $q$-colouring of a random hypergraph, already when the average degree is a sufficiently large constant. In fact, our main new result is a defect version of the Alon–Frankl–Lovász theorem for almost complete hypergraphs. From this, the transference version is obtained via a variant of the weak hypergraph regularity lemma. The proof of the defect version uses tools from extremal set theory developed in the study of the Erdős matching conjecture.

We study a size-structured tree growth model from [4–6], described by the nonlinear renewal equation $\phi(t) = \mathfrak{F} \phi_t, \ \phi_t \in L^1_\rho(\mathbb{R}_{-}),$ with reproduction, death, and growth rates $\beta$, $\mu$, and $g$. We prove that, under mild conditions on these rates, the equation generates a semiflow in $L^1_\rho(\mathbb{R}_{-})$ that is permanent and possesses a compact global attractor $\mathcal{A}$. If $\beta$ is monotone, $\mathcal{A}$ reduces to a single asymptotically stable equilibrium attracting all compact sets with positive initial data. Adapting an approach from [21], originally developed for simpler renewal equations, we investigate stability and persistence in this more complex setting via the one-dimensional recurrence $b_{n+1} = \mathfrak{F} b_n,$ thereby complementing the functional-analytic framework of [13].

We derive Voronoï summation formulas for the Liouville function $\lambda(n)$, the Möbius function $\mu(n)$, and for $d^{2}(n)$, where $d(n)$ is the divisor function. The formula for $\lambda(n)$ requires explicit evaluation of certain infinite series for which the use of the Vinogradov–Korobov zero-free region of the Riemann zeta function is indispensable. Several results of independent interest are obtained as special cases of these formulas. For example, a special case of the one for $\mu(n)$ is a famous result of Ramanujan, Hardy, and Littlewood. Cohen-type and Ramanujan–Guinand-type identities are established for $\lambda(n)$ and $\sigma_a(n)\sigma_b(n)$, where $\sigma_s(n)$ is the generalized divisor function. As expected, infinite series over the non-trivial zeros of $\zeta(s)$ now form an essential part of all of these formulas. A series involving $\sigma_a(n)\sigma_b(n)$ and the product of modified Bessel functions occurring in one of our identities has appeared in a recent work of Dorigoni and Treilis in string theory. Lastly, we obtain results on oscillations of Riesz sums associated to $\lambda(n), \mu(n)$ and of the error term of Riesz sum of $d^2(n)$ under the assumption of the Riemann Hypothesis, simplicity of the zeros of $\zeta(s)$, the Linear Independence conjecture, and a weaker form of the Gonek–Hejhal conjecture.

We study a reaction–advection–diffusion model of a target–offender–guardian system designed to capture interactions between urban crime and policing. Using Crandall–Rabinowitz bifurcation theory and spectral analysis, we establish rigorous conditions for both steady-state and Hopf bifurcations. These results identify critical thresholds of policing intensity at which spatially uniform equilibria lose stability, leading either to persistent heterogeneous hotspots or oscillatory crime–policing cycles. From a criminological perspective, such thresholds represent tipping points in guardian mobility: once crossed, they can lock neighbourhoods into stable clusters of criminal activity or trigger recurrent waves of hotspot formation. Numerical simulations complement the theory, exhibiting stationary patterns, periodic oscillations and chaotic dynamics. By explicitly incorporating law enforcement as a third interacting component, our framework extends classical two-equation models. It offers new tools for analysing non-linear interactions, bifurcations and pattern formation in multi-agent social systems.

Hyperasymptotics is an analytical method that incorporates exponentially small contributions into asymptotic approximations, thereby expanding their domain of validity, improving accuracy, and providing deeper insight into the underlying singularity structures. It also allows for the computation of problem-specific invariants, such as Stokes multipliers, whose values are often assumed or remain unknown in other approaches. For differential equations, unlike standard asymptotic expansions, hyperasymptotic expansions determine solutions uniquely. In this paper, we extend the hyperasymptotic method to inverse factorial series solutions of certain higher-order linear difference equations and demonstrate that the resulting expansions also determine the solutions uniquely. We further indicate how the connection coefficients appearing in these expansions can be computed numerically using hyperasymptotic techniques. In addition, we give explicit remainder bounds for the inverse factorial series solutions. Our main tool is the Mellin–Borel transform. The expansions are expressed via universal hyperterminant functions, closely related to the hyperterminants familiar from integral and differential equation contexts. The results are illustrated by the Gauss hypergeometric function with a large third parameter and a third-order difference equation.

A trace of a sequence is generated by deleting each bit of the sequence independently with a fixed probability. The well-studied trace reconstruction problem asks how many traces are required to reconstruct an unknown binary sequence with high probability. In this paper, we study the multidimensional version of this problem for matrices and hypermatrices, where a trace is generated by deleting each row/column of the matrix or each slice of the hypermatrix independently with a constant probability. Previously, Krishnamurthy, Mazumdar, McGregor and Pal showed that $\exp (\widetilde {O}(n^{d/(d+2)}))$ traces suffice to reconstruct any unknown $n\times n$ matrix (for $d=2$) and any unknown $n^{\times d}$ hypermatrix. By developing a dimension reduction procedure and establishing a multivariate version of the Littlewood-type result that lower bounds sparse complex polynomials around $1$, we improve this upper bound by showing that $\exp (\widetilde {O}(n^{3/7}))$ traces suffice to reconstruct any unknown $n\times n$ matrix, and $\exp (\widetilde {O}(n^{3/5}))$ traces suffice to reconstruct any unknown $n^{\times d}$ hypermatrix. In contrast to the earlier bound, our new exponent is bounded away from $1$ even as $d$ becomes very large.

We study the representation of non-weakly compact operators between $AL$-spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between $C(L)$-spaces where $L$ is extremally disconnected. We also characterize the weak essential norm for operators between $AL$-spaces in terms of factorizations of the identity on $\ell_1$. As a consequence, we deduce that the weak Calkin algebra $\mathscr{B}(E)/\mathscr{W}(E)$ admits a unique algebra norm for every $AL$-space $E$. By duality, similar results are obtained for $C(K)$-spaces. In particular, we prove that for operators $T: L_{\infty}[0,1] \to L_{\infty}[0,1]$ the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.

We classify all Polish semigroup topologies on the symmetric inverse monoid $I_{\mathbb N}$ on the natural numbers $\mathbb N$. This result answers a question of Elliott et al. There are countably infinitely many such topologies. Under containment, these Polish semigroup topologies form a join-semilattice with infinite descending chains, no infinite ascending chains, and arbitrarily large finite anti-chains. Also, we show that the monoid $I_{\mathbb N}$ endowed with any second countable $T_1$ semigroup topology is homeomorphic to the Baire space $\mathbb N^{\mathbb N}$.

We provide four equivalent combinatorial conditions for a simple assembly graph (rigid vertex graph where all vertices are of degree 1 or 4) to have the largest number of Hamiltonian sets of polygonal paths relative to its size. These conditions serve to prove the conjecture that such a maximum, which is equal to $F_{2n+1}-1$, where $F_k$ denotes the $k$th Fibonacci number, is achieved only for special assembly graphs, called tangled cords.

The doubly degenerate nutrient taxis system(0.1)

\begin{equation} \left \{ \begin{aligned} &u_{t}=\nabla \cdot (uv\nabla u)-\chi \nabla \cdot (u^{\alpha }v\nabla v)+\ell uv,&x\in \Omega ,\, t\gt 0,\\[5pt] & v_{t}=\Delta v-uv,&x\in \Omega ,\, t\gt 0,\\ \end{aligned} \right . \end{equation}

$\Omega \subset \mathbb{R}^3$

$\alpha \gt 0,\chi \gt 0$

$\ell \gt 0$

$\alpha \in (\frac {3}{2},\frac {19}{12})$

$(u_\infty , 0)$

$t\rightarrow \infty$

$u_{\infty }$

$v_0$

$u_0$

We identify the size of the largest connected component in a subcritical inhomogeneous random graph with a kernel of preferential attachment type. The component is polynomial in the graph size with an explicitly given exponent, which is strictly larger than the exponent for the largest degree in the graph. This is in stark contrast to the behaviour of inhomogeneous random graphs with a kernel of rank one. Our proof uses local approximation by branching random walks going well beyond the weak local limit and novel results on subcritical killed branching random walks.

In this paper, we consider the time-dependent Born–Oppenheimer approximation (BOA) of a classical quantum molecule involving a possibly large number of nuclei and electrons, described by a Schrödinger equation. In the spirit of Born and Oppenheimer’s original idea, we study quantitatively the approximation of the molecular evolution. We obtain an iterable approximation of the molecular evolution to arbitrary order, and we derive an effective equation for the reduced dynamics involving the nuclei equivalent to the original Schrödinger equation and containing no electron variables. We estimate the coefficients of the new equation and find tractable approximations for the molecular dynamics going beyond the one corresponding to the original Born and Oppenheimer approximation.

Recent research has demonstrated the importance of spatial diffusion and environmental heterogeneity in influencing the transmission dynamics of infectious diseases. At the same time, human mobility patterns have been shown to exhibit scale-free, nonlocal dynamics characterized by an anomalous Lévy process diffusion, which is mathematically represented by nonlocal equations involving fractional Laplacian operators. To investigate the effects of environmental heterogeneity and long-range geographical disease transmission, we propose a time-periodic susceptible-infectious-susceptible (SIS) epidemic model that incorporates anomalous diffusion and spatial heterogeneity. The key issues of this paper include the existence and stability of both disease-free and endemic periodic equilibria, as well as the impact of diffusion rates and fractional powers on the spatial distribution of these periodic states. Our analytical findings indicate that spatio-temporal heterogeneity promotes disease persistence and that the fractional power can modulate the transmission threshold.