We introduce and analyze a broad class of continuous directed polymers in $\mathbb{R}^d$ driven by Gaussian environments that are white in time and spatially correlated, under Dalang's condition. Using an Itô-renormalized stochastic-heat-equation representation, we establish structural properties of the partition function, including positivity, stationarity, scaling, homogeneity, and a Chapman--Kolmogorov relation. On finite time intervals, we prove Brownian-type pathwise behavior, namely Hölder continuity and identification of the quadratic variation. We then obtain a sharp measure-theoretic dichotomy: the quenched polymer measure is singular with respect to Wiener measure if and only if $\widehat f(\mathbb{R}^d)=\infty$ (equivalently, the noise is non-trace-class), and it is equivalent otherwise. Finally, in dimension $d\ge 3$, we prove diffusive behavior at large times in the high-temperature regime. This extends the Alberts--Khanin--Quastel framework from the $1+1$ white-noise setting to higher-dimensional Gaussian environments with general spatial covariance.

For each integer $n\ge 2$, we construct infinitely many $n$-component Brunnian links of 3-balls in $S^4$. Our main tool is the third author's result on the existence of splitting spheres for the trivial two-component link of $2$-spheres in $S^{4}$; we also give a new proof of this.

In this short note, we settle a problem posed by Wickstead in ~\cite{W:24}, arising from the study of the Riesz completion of spaces of regular operators between Banach lattices.

We show that the geometric and homological finiteness properties of group pairs are invariant under a suitable notion of quasi-isometry for group pairs.

This note aims to continue our study about the applications of Poisson quasi-Nijenhuis geometry to the theory of classical completely integrable systems. More precisely, we will present new versions of the deformation and involutivity theorems, under the hypothesis that the closed 2-form triggering the deformation and the closed 3-form defining the Poisson quasi-Nijenhuis structure are factorized. These results will be supplemented by several examples of involutive Poisson quasi- Nijenhuis manifolds.

We investigate the nonlinear heat-diffusion equation \( C(u)\,\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}\!\left( K(u)\,\frac{\partial u}{\partial x} \right) \), where \( C(u) \) and \( K(u) \) are coefficients that depend on \( u \). By applying the classical Lie symmetry method, we determine the admitted Lie point symmetries and compute the corresponding infinitesimal generators according to the functional relationship between \( C(u) \) and \( K(u) \). The admitted symmetries are used to reduce the partial differential equation to ordinary differential equations and to construct invariant solutions. Particular cases of physical interest are analyzed in detail, including Storm-type materials and power-law dependence of \( C(u) \) and \( K(u) \) on \( u \). For these cases, similarity solutions are obtained.

We study a continuum model for stochastic reinforcement learning in repeated market entry games. Starting from a discrete-time microscopic learning rule, we derive a Fokker--Planck-type equation for the distribution of agents' propensities and, using a kinetic closure, obtain a nonlinear one-particle equation of a mean-field type. For the resulting Cauchy problem, we prove existence and uniqueness of solutions and analyze their long-time behavior. The PDE captures two key phenomena observed in market entry dynamics: aggregate learning (the average number of entrants approaches market capacity) and sorting (propensities concentrate near extreme behaviors). The model also yields explicit characteristic time scales, showing that aggregate learning occurs faster than sorting, in agreement with experimental and computational evidence.

In this paper, we consider the near-critical dimer model in the setup of isoradial superpositions with Temperleyan boundary conditions. We show that the centered height function converges as the mesh size tends to zero to a limiting field which agrees with the (electromagnetically tilted) sine-Gordon model, whose derivative correlations are described by Grassmann variables (or equivalently determinants involving a massive Dirac operator). This answers a longstanding question in the field. A crucial part of the work is to develop a notion of discrete massive holomorphic functions and the tools to study such functions, in particular finding an exact discrete form of the massive Cauchy--Riemann equations, which is satisfied by the inverse Kasteleyn matrix. In comparison with previous studies, a key novelty of this part of our work is that the mass is not only allowed to be non-constant but can be complex-valued.

In this paper we consider the operad of holomorphic disk embeddings of the unit disk $\mathbb D \subset \mathbb C$. We introduce a suboperad $\mathbb{CE}_2^{HS}$ defined by square-integrability conditions and show that the symmetric algebra $\mathrm{Sym} A^{2}(\mathbb D)$ of the Bergman space carries a natural $\mathbb{CE}_2^{HS}$-algebra structure. Conformally flat factorization homology with coefficients in $\mathrm{Sym} A^{2}(\mathbb D)$ then yields metric-dependent invariants of two-dimensional Riemannian manifolds. Moreover, $\mathrm{Sym} A^{2}(\mathbb D)$ is identified with the ind-Hilbert space completion of the affine Heisenberg vertex operator algebra.

The coverage depth problem in DNA data storage is about computing the expected number of reads needed to recover all encoded strands. Given a generator matrix of a linear code, this quantity equals the expected number of randomly drawn columns required to obtain full rank. While MDS codes are optimal when they exist, i.e., over large fields, practical scenarios may rely on structured code families defined over small fields. In this work, we develop combinatorial tools to solve the DNA coverage depth problem for various linear codes, based on duality arguments and the notion of extended weight enumerator. Using these methods, we derive closed formulas for the simplex, Hamming, ternary Golay, extended ternary Golay, and first-order Reed-Muller codes. The centerpiece of this paper is a general expression for the coverage depth of a linear code in terms of the weight distributions of its higher-field extensions.

In this paper we combine methods from additive combinatorics and Diophantine geometry to study the generalised sum-product phenomenon in algebraic groups. As an application of this circle of ideas, we resolve a conjecture of Bremner on arithmetic progressions in coordinates of elliptic curves, along with various other generalisations studied in the literature. We also prove a uniform Bourgain--Chang-type sum-product estimate for general $1$-dimensional algebraic groups $G$ over $\mathbb{C}$. Using these ideas, we provide an alternative solution to a problem of Bays--Breuillard. Furthermore, we show an Elekes--Szabó type result in the same setting for sets with small doubling, improving upon an earlier result of Bays--Breuillard when $G$ is not $\mathbb{G}_a$. Our power saving here can be shown to be quantitatively optimal. We use a combination of deep, classical results in Diophantine geometry due to David--Philippon, Laurent and Evertse--Schmidt--Schlickewei along with the recent breakthrough work on the weak Polynomial Freiman--Ruzsa conjecture over integers due to Gowers--Green--Manners--Tao.

The spatial Muller's ratchet is a model introduced by Foutel-Rodier and Etheridge to study the impact of cooperation and competition on the fitness of an expanding asexual population. The model is an interacting particle system consisting of particles performing symmetric random walks that reproduce and die with rates that depend on the local number of particles. For each particle, we keep track of the number of deleterious mutations that it carries, and after each birth event, with some positive probability, the offspring particle can acquire an additional mutation that gives it a lower reproduction rate than its parent. We show that under an appropriate scaling, the process converges weakly to the solution of an infinite system of partial differential equations (PDEs), confirming non-rigorous computations of Foutel-Rodier and Etheridge. In the PDE limit, when the reaction term of the system of PDEs is monostable, we establish bounds on the ratio between the density of particles with a given number of mutations and the density of particles without mutations. If the reaction term satisfies a Fisher-KPP condition, we can also rigorously determine the spreading speed of the population into an empty habitat. Finally, by considering the PDE limit of a form of tracer dynamics, we answer the question of whether deleterious mutations can surf population waves in this setting.

Ruelle gave an explicit second-order expansion at $c=0$ of the Hausdorff dimension of the Julia set of the quadratic family $f_c(z)=z^2+c$. McMullen later extended this result to polynomial perturbations of $z^d$ for arbitrary degree $d\geq 2$. In this paper we study an analogue of this problem for skew products in $\mathbb C^2$. Since holomorphic dynamical systems in higher dimensions are non-conformal, we replace the Hausdorff dimension by the \emph{volume dimension}, a dynamically defined notion we introduced in our earlier work and characterized as the zero of a natural pressure function. We consider families of holomorphic skew products of the form \[ f_t(z,w)=(z^d, w^d+t(c_1 (z) w^{d-1} +c_2(z)w^{d-2} + \cdots+c_d(z))). \] Our main result gives an explicit second-order expansion of the volume dimension of the Julia set $J(f_t)$ as $t\to0$ in terms of the coefficients $c_k(z)$.

In this article, we consider a generalisation of the spatial Muller's ratchet introduced by Foutel-Rodier and Etheridge. This particle system is a spatial model of an asexual population, with birth and death rates that depend on the local population density. Particles live in discrete demes and migrate to neighbouring demes. Each particle carries some number of mutations (its `type'), and additional mutations can occur during birth events. Mutations are assumed to be deleterious, i.e.~carrying a higher number of mutations results in a lower birth rate. Our main result shows that this interacting particle system can be constructed even when the total initial number of particles is infinite. We also prove moment bounds on the local density of particles; these bounds are a crucial ingredient of the proof of a law of large numbers result for the particle system in the companion article. The construction of the particle system uses a sequence of approximating processes. Proving weak convergence of this sequence of processes is non-trivial because the particle system is non-monotone and interactions are non-local in type space. The uniqueness of the limit relies on a delicate coupling argument.

We investigate the consequences of timelike sectional curvature bounds in Lorentzian length spaces for the existence and structure of the space of directions at a point. It is established that, under upper timelike sectional curvature bounds, the space of directions exists and is itself a metric space with curvature bounded above by $-1$. Furthermore, the metric cone over the space of directions, which canonically models the tangent space at a given point, is shown to constitute a Lorentzian length space with timelike sectional curvature bounded above by $0$. To do this, we introduce the notion of $ε$-$μ$ timelike sectional curvature bounds, which are compatible with pre-existing synthetic curvature conditions. These results extend the comparison-geometric framework to the Lorentzian setting, providing a synthetic characterization of geodesics, tangent cones, and curvature under causal constraints.

We give examples of K3 surfaces over $\mathbb{Q}$ of degree $10$ whose geometric Picard group has rank~$1$. These K3 surfaces are intersections in $\mathbb{P}^9$ of three hyperplanes, one quadric and the image of the Plücker embedding of the Grasmannian $\mathrm{Gr}(2,5)$. We also give an example of a K3 surface of degree $6$ over~$\mathbb{Q}$ whose Picard rank is $1$.

We provide simple ``left-inverse characterizations'' of the recently introduced singularity invariants $c(Z)$, $w(Z)$, and ${\rm HRH}(Z)$ of an equidimensional variety $Z$. Combining this with a trace morphism, we establish descent results of these invariants for finite surjective morphisms.

A random variable $ξ$ has a {\it light-tailed} distribution (for short: is light-tailed) if it possesses a finite exponential moment, $\E \exp (λξ) <\infty$ for some $λ>0$, and has a {\it heavy-tailed} distribution (is heavy-tailed) if $\E \exp (λξ) = \infty$, for all $λ>0$. In \cite{LSK1}, the authors presented a particular example of a light-tailed random variable that is the minimum of two independent heavy-tailed random variables. In \cite{FKT}, it was shown that any light-tailed random variable with right-unbounded support may be represented as the minimum of two independent heavy-tailed random variables, with further generalisations of the result in a number of directions. We analyse an ``inverse'' question. Namely, we obtain necessary and sufficient conditions on the distribution of a heavy-tailed random variable, say $ξ_1$, that allow to find another independent heavy-tailed random variable, say $ξ_2$, such that their minimum $\min (ξ_1,ξ_2)$ is light-tailed. We also provide a number of extensions of this result

In this manuscript, we establish local Schauder estimates for flat viscosity solutions, that is, solutions with sufficiently small norms, to a class of fully nonlinear elliptic partial differential equations of the form \[ F(D^{2} u, x) + \langle \mathfrak{B}(x), D u \rangle = f(x) \quad \text{in} \quad \mathrm{B}_1 \subset \mathbb{R}^{n}, \] where the operator \(F\) is differentiable, though not necessarily convex or concave. In addition, we impose suitable Dini-type continuity assumptions on the data. Our methodology is based on geometric tangential techniques, combined with compactness and perturbative arguments. This approach is strongly motivated by recent advances in the theory of nonlinear elliptic equations and free boundary problems. As a byproduct of our analysis, we also obtain an Evans-Krylov type estimate. Our results can be viewed as an extension of the work by dos Prazeres and Teixeira (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 15 (2016), 485-500), now within the framework of linear drift terms and Dini continuity assumptions. Finally, we apply our results to characterize the nodal sets of flat viscosity solutions of non-convex, fully nonlinear, uniformly elliptic PDEs.

It is known that a point in three-dimensional Euclidean space whose coordinates are equal to the cosines of the angles $\angle BDC, \angle ADC, \angle ADB$, where the point $D$ lies in the plane of a given triangle $ABC$, lies on the surface $\mathbb{BP}\subset [-1,1]^3$, given by the equation $1+2x_1x_2x_3-x_1^2-x_2^2-x_3^2 = 0$. It should be emphasized that the set of corresponding points essentially depends on the shape of triangle $ABC$. In this paper, we solve the following problem: For a fixed triangle $ABC$, for each point $U \in \mathbb{BP}$, determine the number of points $D$ from the plane of the triangle with the condition $U=(\cos \angle BDC, \cos \angle ADC, \cos \angle ADB)$. The problem of determining such points $D$ is known as the Snellius-Pothenot problem.

Given an o-minimal structure, we show that every definable (in this structure) mapping that is Lipschitz with respect to the inner metric can be approximated by $\mathscr{C}^1$ mappings that are Lipschitz with respect to the inner metric with arbitrarily close bounds for the derivative. When the o-minimal structure admits $\mathscr{C}^\infty$ cell decomposition, we show that the approximation can be required to be $\mathscr{C}^\infty$ and we extend this result to outer Lipschitz mappings. The proof involves the construction of partitions of unity with sharp bounds for the derivative, which can be useful for other approximation problems.

For solving constrained (pseudo)-monotone variational inequality, we prove that the upper bound of stepsize $\frac{1}{2L}$ established for the Popov's algorithm and the forward-reflected-backward algorithm is tight. For unconstrained case, we can enlarge the upper bound to $\frac{1}{\sqrt{3}L}$ and show that this upper bound is also tight. The convergence analysis is carried out by using a new Lyapunov-type function.

Sylvester showed that the partition of an integer into a set of positive integers can be represented as a sum of the polynomial term and quasiperiodic components called the Sylvester waves. The wave itself is a weighted sum of the polynomial terms multiplied by the periodic functions. The integer weights are found to be a sum of partitions into a smaller set of integers implying the recursive structure of integer partitions.

Let $f(x)=x^{2p}+ax^p+b^p$, where $p$ is a prime and $a,b\in {\mathbb Z}$ with $ab\ne 0$. If $f(x)$ is irreducible over ${\mathbb Q}$, we say that $f(x)$ is monogenic if $\{1,θ,θ^2,\ldots ,θ^{2p-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. In this article, we give a characterization of the monogenic trinomials $f(x)$ according to their Galois groups. These results extend prior investigations of the authors.

We confirm Martin's conjecture for a broad subclass of weakly quasi-o-minimal theories.

In optimal control, extending the class of admissible controls is a common strategy to guarantee the existence of optimal solutions. However, such extensions may introduce a gap between the infimum of the original problem and the minimum of the extended one, especially in the presence of endpoint constraints. Since Warga's seminal work, normality of first-order necessary conditions for extended minimizers has been recognized as a sufficient condition to avoid this phenomenon, though it is far from being necessary. In this paper, we consider impulsive extensions of control-affine systems with unbounded controls. We establish that a notion of \textit{higher-order normality}, based on iterated Lie brackets of the systems vector fields, suffices to prevent an infimum gap. The key novelty of this manuscript consists in showing that this holds under a local topology defined by the $L^1$-distance between controls, rather than the more common $L^\infty$-distance between trajectories. Among the reasons that motivate the interest in this issue, let us mention that a counterexample by R. B. Vinter shows that for a different extension -- based on convexification of the velocity set -- a local extended minimizer that is normal with respect to the $L^1$-norm of the controls may still exhibit a gap. Our method relies on set-separation techniques. Such an approach makes it possible to derive higher-order conditions and to exploit the corresponding notion of higher-order normality.

In this paper, we investigate rigid-ruling folding motions of crease-rule patterns, that is, conjugacy-preserving isometries of developable semi-discrete conjugate nets. We derive two conditions for the rigid-ruling foldability of pairs of curves and consider two applications. First, we introduce computations that enable the sequential construction of rigid-ruling foldable crease-rule patterns. Second, we examine combinations of planar and constant fold-angle creases. In particular, we show that constant fold-angle creases are only compatible with other constant fold-angle creases, and we provide a characterization of rigid-ruling foldable combinations of planar and constant fold-angle creases.

In a previous paper we began our analysis on the role of non self-adjoint Hamiltonians in connection with the Heisenberg dynamics for quantum systems. Here, motivated by the growing interest on this topic and on some recent results on dynamical systems, we continue this analysis focusing on what we believe is an unexplored (or, at least, not so explored! aspect of Heisenberg dynamics, related to the need for using vectors which are {\em brute-force normalized}. Our main interest is on conserved quantities, and on conditions which guarantee that some observables of the system, or their mean values, do not evolve in time.

We establish the existence of nontrivial nonnegative weak solutions to the following equation \begin{equation*} -Δ_γu + V(z)u = Q(z)f(u), \quad z\in \mathbb{R}^N, \end{equation*} where $Δ_γ$ denotes the so-called Grushin-type operator in $\mathbb{R}^N$. The potentials $V$ and $Q$ are assumed to be controlled below and above, respectively, by functions of type $(1+|z|)^a$, $a\in\mathbb{R}$. The main result is the embedded of the space $E_V^γ$ into the weighted Lebesgue space $L_Q^p(\mathbb{R}^N)$, under suitable conditions. Finally, we derive regularity results for the obtained weak solutions.

We introduce a stochastic nonlocal reaction--diffusion model arising in tumour dynamics. Spatial dispersal is described by the fractional Laplacian, accounting for anomalous diffusion and long--range relocation events. The system is perturbed by multiplicative fractional Brownian motion (fBm) with Hurst parameter $H>1/2$, which we interpret as temporally correlated fluctuations in the tumour microenvironment and host response. We first establish well--posedness and identify parameter regimes leading to global--in--time solutions or finite--time blow--up under general multiplicative fractional noise. We then focus on linear multiplicative noise and, via a Doss--Sussmann transformation, derive sharper results: explicit lower and upper bounds for the blow--up time together with quantitative estimates of the blow--up probability, clarifying how noise intensity can accelerate progression or, on favourable paths, enhance suppression consistent with extinction (loss of viability). Finally, one--dimensional simulations illustrate the interplay between anomalous diffusion, fractional noise, and the nonlocal reaction mechanism in shaping the long--time dynamics.