Liouville closed $H$-fields are ordered differential fields where the ordering and derivation interact in a natural way and every linear differential equation of order $1$ has a nontrivial solution. (The introduction gives a precise definition.) For a Liouville closed $H$-field $K$ with small derivation we show that $K$ has the Intermediate Value Property for differential polynomials if and only if $K$ is elementarily equivalent to the ordered differential field of transseries. We also indicate how this applies to Hardy fields.

For every $\mathcal{U} \subset \mathrm{Diff}^\infty_{\mathrm{vol}}(\mathbb{T}^2)$ there is a measure of finite support contained in $\mathcal{U}$ which is uniformly expanding.

The Ramsey theory of the space of equivalence relations with infinite quotients defined on the set $\mathbb{N}$ of natural numbers is an interesting field of research. We view this space as a topological Ramsey space $(\mathcal{E}_\infty,\leq, r)$ and present a game theoretic characterization of the Ramsey property of subsets of $\mathcal{E}_{\infty}$. We define a notion of coideal and consider the Ramsey property of subsets of $\mathcal{E}_\infty$ localized on a coideal $\mathcal{H}\subseteq \mathcal{E}_{\infty}$. Conditions a coideal $\mathcal{H}$ should satisfy to make the structure $(\mathcal{E}_{\infty},\mathcal{H},\leq, r)$ a Ramsey space are presented. Forcing notions related to a coideal $\mathcal{H}$ and their main properties are analyzed.

The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.

We prove $L^p$ bounds for the maximal operators associated to an Ahlfors-regular variant of fractal percolation. Our bounds improve upon those obtained by I. Łaba and M. Pramanik and in some cases are sharp up to the endpoint. A consequence of our main result is that there exist Ahlfors-regular Salem Cantor sets of any dimension $> 1/2$ such that the associated maximal operator is bounded on $L^2(\mathbb{R})$. We follow the overall scheme of Łaba–Pramanik for the analytic part of the argument, while the probabilistic part is instead inspired by our earlier work on intersection properties of random measures.

We study tilings of polygons $R$ with arbitrary convex polygonal tiles. Such tilings come in continuous families obtained by moving tile edges parallel to themselves (keeping edge directions fixed). We study how the tile shapes and areas change in these families. In particular we show that if $R$ is convex, the tile shapes can be arbitrarily prescribed (up to homothety). We also show that the tile areas and tile “orientations” determine the tiling. We associate to a tiling an underlying bipartite planar graph $\mathcal{G}$ and its corresponding Kasteleyn matrix $K$. If $\mathcal{G}$ has quadrilateral faces, we show that $K$ is the differential of the map from edge intercepts to tile areas, and extract some geometric and probabilistic consequences.

We address in this paper Fano manifolds with positive higher Chern characters, which are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. For instance, they should be covered by higher dimensional rational varieties, and families of higher Fano manifolds over higher dimensional bases should admit meromorphic sections (modulo the Brauer obstruction). Aiming at finding new examples of higher Fano manifolds, we investigate positivity of higher Chern characters of rational homogeneous spaces. We determine which rational homogeneous spaces of Picard rank 1 have positive second Chern character, and show that the only rational homogeneous spaces of Picard rank 1 having positive second and third Chern characters are projective spaces and quadric hypersurfaces. We also classify Fano manifolds of large index having positive second and third Chern characters. We conclude by discussing conjectural characterizations of projective spaces and complete intersections in terms of these higher Fano conditions.

In this note we discuss some geometric analogs of the classical harmonic functions on $\mathbb{R}^n$ and their associated evolutions.

In this paper we characterize viscosity solutions to nonlinear parabolic equations (including parabolic Monge–Ampère equations) by asymptotic mean value formulas. Our asymptotic mean value formulas can be interpreted from a probabilistic point of view in terms of dynamic programming principles for certain two-player, zero-sum games.

We prove some results on the fibers and images of rational maps from a hyper-Kähler manifold. We study in particular the minimal genus of fibers of a fibration into curves. The last section of this paper is devoted to the study of the rational map defined by a linear system on a hyper-Kähler fourfold satisfying numerical conditions similar to those considered by O'Grady in his study of fourfolds numerically equivalent to $\mathrm{K}3^{[2]}$. We extend his results to this more general context.

We survey some of the recent developments in the study of the compactness and uniqueness problems for some classes of conformally compact Einstein manifolds.

Given rational univariate polynomials $f$ and $g$ such that $\gcd (f,g)$ and $f/\gcd(f,g)$ are relatively prime, we show that $g$ is non-negative at all the real roots of $f$ if and only if $g$ is a sum of squares of rational polynomials modulo $f$. We complete our study by exhibiting an algorithm that produces a certificate that a polynomial $g$ is non-negative at the real roots of a non-zero polynomial $f$ when the above assumption is satisfied.

We define the Bernstein–Sato ideal associated to a tuple of ideals and we relate it to the jumping points of the corresponding mixed multiplier ideals.

Considering singular Sturm–Liouville differential expressions of the type \[ \tau_{\alpha} = -(d/dx)x^{\alpha}(d/dx) + q(x), \quad x \in (0,b), \, \alpha \in \mathbb{R}, \] we employ some Sturm comparison-type results in the spirit of Kurss to derive criteria for $\tau_{\alpha}$ to be in the limit-point and limit-circle case at $x=0$. More precisely, if $\alpha \in \mathbb{R}$ and, for $0 < x$ sufficiently small, \[ q(x) \geq [(3/4)-(\alpha/2)]x^{\alpha-2}, \] or, if $\alpha\in (-\infty,2)$ and there exist $N\in\mathbb{N}$ and $\varepsilon>0$ such that, for $0 < x$ sufficiently small, \begin{align*} & q(x)\geq[(3/4)-(\alpha/2)]x^{\alpha-2} - (1/2) (2 - \alpha) x^{\alpha-2} \sum_{j=1}^{N}\prod_{\ell=1}^{j}[\ln_{\ell}(x)]^{-1} \\ & \quad\quad\quad +[(3/4)+\varepsilon] x^{\alpha-2}[\ln_{1}(x)]^{-2}, \end{align*} then $\tau_{\alpha}$ is nonoscillatory and in the limit-point case at $x=0$. Here iterated logarithms for $0 < x$ sufficiently small are of the form \[ \ln_1(x) = |\ln(x)| = \ln(1/x), \qquad \ln_{j+1}(x) = \ln(\ln_j(x)), \quad j \in \mathbb{N}. \] Analogous results are derived for $\tau_{\alpha}$ to be in the limit-circle case at $x=0$. We also discuss a multi-dimensional application to partial differential expressions of the type \[ - \operatorname{div} |x|^{\alpha} \nabla + q(|x|), \quad \alpha \in \mathbb{R}, \, x \in B_n(0;R) \backslash \{0\}, \] with $B_n(0;R)$ the open ball in $\mathbb{R}^n$, $n\in \mathbb{N}$, $n \geq 2$, centered at $x=0$ of radius $R \in (0, \infty)$.

We show that there exist inequivalent representations of the dual space of $\mathbb{C}[0,1]$ and of $L_p[\mathbb{R}^n]$ for $p \in [1,\infty)$. We also show how these inequivalent representations reveal new and important results for both the operator and the geometric structure of these spaces. For example, if $\mathcal{A}$ is a proper closed subspace of $\mathbb{C}[0,1]$, there always exists a closed subspace $\mathcal{A}^\bot$ (with the same definition as for $L_2[0,1]$) such that $\mathcal{A} \cap \mathcal{A}^\bot = \{0\}$ and $\mathcal{A} \oplus \mathcal{A}^\bot = \mathbb{C}[0,1]$. Thus, the geometry of $\mathbb{C}[0,1]$ can be viewed from a completely new perspective. At the operator level, we prove that every bounded linear operator $A$ on $\mathbb{C}[0,1]$ has a uniquely defined adjoint $A^*$ defined on $\mathbb{C}[0,1]$, and both can be extended to bounded linear operators on $L_2[0,1]$. This leads to a polar decomposition and a spectral theorem for operators on the space. The same results also apply to $L_p[\mathbb{R}^n]$. Another unexpected result is a proof of the Baire one approximation property (every closed densely defined linear operator on $\mathbb{C}[0,1]$ is the limit of a sequence of bounded linear operators). A fundamental implication of this paper is that the use of inequivalent representations of the dual space is a powerful new tool for functional analysis.

Modulation theory is a well-known tool to describe the long-time evolution and stability of small-amplitude, oscillating solutions to dispersive nonlinear partial differential equations. There have been a number of approaches to deriving envelope equations for weakly nonlinear waves. Here we review a systematic method based on Hamiltonian transformation theory and averaging Hamiltonians. In the context of the modulation of two- or three-dimensional deep-water surface waves, this approach leads to a Dysthe equation that preserves the Hamiltonian character of the water wave problem. An explicit calculation of the third-order Birkhoff normal form that eliminates all non-resonant cubic terms yields a non-perturbative procedure for the reconstruction of the free surface. We also present new numerical simulations of this weakly nonlinear approximation using the version with exact linear dispersion. We compare them against computations from the full water wave system and find very good agreement.

The Hall–Littlewood polynomials $\mathbf{H}_\lambda = Q'_\lambda[X;q]$ are an important symmetric function basis that appears in many contexts. In this work, we give an accessible combinatorial formula for expanding the related symmetric functions $\mathbf{H}_\alpha$ for any composition $\alpha$, in terms of the complete homogeneous basis. We do this by analyzing Jing's operators, which extend to give nice expansions for the related symmetric functions $\mathbf{C}_\alpha$ and $\mathbf{B}_\alpha$ which appear in the formulation of the Compositional Shuffle Theorem. We end with some consequences related to eigenoperators of the modified Macdonald basis.

The MacWilliams identities for the homogeneous weight enumerator over $\mathbb{Z}/m\mathbb{Z}$ do not hold for composite $m \geq 6$. For such $m$, there exist two linear codes over $\mathbb{Z}/m\mathbb{Z}$ that have the same homogeneous weight enumerator, yet whose dual codes have different homogeneous weight enumerators.

D. B. Cohen, C. Goodman-Strauss, and the author [Ergodic Theory Dynam. Systems 42 (2022), no. 9, 2740–2783] proved that a hyperbolic group admits an SA SFT if and only if it has at most one end. This paper has two distinct parts: the first is a conversation explaining what an SA SFT is and how it may be of use, while in the second part I attempt to explain both old and new ideas that go into that proof. References to specific claims in the work cited above are given, with the hope that any interested reader may be able to find the details there more accesible after reading this exposition.

Radial basis functions (RBFs) have been gaining popularity recently in the development of methods for solving partial differential equations (PDEs) numerically. These functions have become an extremely effective tool for interpolation on scattered node sets in several dimensions. One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter $\varepsilon$, which controls the flatness of the function. It is observed that best accuracy is often achieved when $\varepsilon$ tends to zero. However, the system of discrete equations from interpolation matrices becomes ill-conditioned. A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit, such as the RBF-QR method and the RBF-GA method. We present these techniques in the context of boundary integral methods to improve the solution of PDEs with RBFs. These stable calculations open up new opportunities for applications and developments of local integral methods based on local RBF approximations. Numerical results for a small shape parameter that stabilizes the error are presented. Accuracy and comparisons are also shown for elliptic PDEs.

The oscillatory nature of basal ganglia activity is highly tied with some movement disorders. We study, through bifurcation analysis and computer simulations, the appearance of abnormal oscillations in an already proposed reduced neural circuit model of the subthalamic nucleus and globus pallidus loop. The results show that the model exhibits stable steady states associated to normal activity and oscillatory activity corresponding to the often termed “tremor frequency” oscillations due to their coherence with parkinsonian tremor observed in patients with Parkinson's disease.

This is a study of weakly integral braided fusion categories with elementary fusion rules to determine which possess nondegenerately braided extensions of theoretically minimal dimension, or equivalently in this case, which satisfy the minimal modular extension conjecture. We classify near-group braided fusion categories satisfying the minimal modular extension conjecture; the remaining Tambara–Yamagami braided fusion categories provide arbitrarily large families of braided fusion categories with identical fusion rules violating the minimal modular extension conjecture. These examples generalize to braided fusion categories with the fusion rules of the representation categories of extraspecial $p$-groups for any prime $p$, which possess a minimal modular extension only if they arise as the adjoint subcategory of a twisted double of an extraspecial $p$-group.

We present a survey of results concerning arithmetic properties of generalized Fibonacci sequences and certain Diophantine equations involving terms from that family of numbers. Most of these results have been recently obtained by the research groups in number theory at the Universities of Cauca (in Popayán) and of Valle (in Cali), Colombia, lead by the first two authors.

This survey builds on the two surveys by Wang and Lauret, written 10–15 years ago, to give the current state of affairs regarding homogeneous Einstein spaces.

Mosquito-borne diseases like dengue, Zika, and chikungunya, carried by Aedes aegypti mosquitoes, pose significant health threats. Controlling these diseases primarily involves reducing mosquito populations. Current approaches rely on simplistic measures to decide on actions like mosquito fogging or larval control. Adult mosquito numbers are often estimated from aquatic populations due to easier counting. While these methods have provided some assistance, there's a need for improvement. Existing risk and population models overlook the various developmental stages of Aedes aegypti and their complex interactions. In this work, several mathematical models for the life cycle of the Aedes aegypti in its aquatic phase are proposed. They consider the different aquatic developmental stages and differ in how the competition between the stages occurs. Then, all the models are discriminated against experimental data to select the one with the best predictive power. The chosen model will help estimate the adult mosquito population with a greater degree of precision as well as when they will emerge. It will also help design better control strategies and better risk indices.