In this study, we develop epidemic reaction-diffusion models by incorporating the dependency of the diffusion rate of susceptible individuals on new infection cases, employing both Fickian and Fokker–Planck-type diffusion laws. As the first part of a two-part series, we focus on epidemics driven by frequency-dependent incidence. We explore linear, exponential and algebraic relationships between diffusion rate of the susceptible population and new infection cases to provide deeper biological insights. Our analysis establishes the global existence of solutions and characterizes the threshold dynamics using basic reproduction numbers. We find that in quasilinear parabolic systems, the Fokker–Planck-type diffusion law tends to induce spatial segregation of susceptible and infected individuals, while the Fickian law favours spatial homogenization of susceptible individuals. Additionally, the Fokker–Planck-type model, where the diffusion rate of infected individuals depends on new recovery cases, more accurately captures the cognitive diffusion behaviour of individuals.

Using the CR structure of the Heisenberg group, we construct several different Kähler structures in the Siegel domain. However, all these are PCR-Kähler equivalent, that is, essentially the same when restricted to the CR structure.

We study the Ambrozio–Carlotto–Sharp (ACS) criterion on minimal isoparametric hypersurfaces $N^{n+1}\subset S^{n+2}$ with positive Ricci curvature, motivated by the Schoen–Marques–Neves conjecture on Morse index. For $g=4$ distinct principal curvatures with multiplicities $m_1,m_2$, we prove that the pointwise ACS inequality holds if and only if $\min \{m_1,m_2\}\ge 4$. Sufficiency is obtained via a moment-relaxation technique yielding the sharp bound $4a^2$ on the quadratic part of the integrand; necessity follows from an explicit extremal configuration in the top eigenspace of the shape operator. We also verify the ACS condition for $g=3$ with $m_1=m_2\in \{4,8\}$. As a consequence, for any closed embedded minimal hypersurface $M^n$ in such an ambient manifold, $\operatorname {index}(M)\ge \tfrac {2}{d(d-1)}\, b_1(M)$ with $d=n+3$.

We study analytic torsion and eta-like invariants on contact manifolds admitting a CR structure invariant under a transverse circle action, and equipped with a unitary representation. We show that, when defined using the spectrum of relevant operators arising in this geometry, the spectral series involved can been interpreted in their whole, both from a topological viewpoint, and as purely dynamical functions of the Reeb flow.

We prove that every oriented tree on $n$ vertices with bounded maximum degree appears as a spanning subdigraph of every directed graph on $n$ vertices with minimum semidegree at least $n/2+{\mathrm{o}}(n)$. This can be seen as a directed graph analogue of a well-known theorem of Komlós, Sárközy, and Szemerédi. Our result for trees follows from a more general result, allowing the embedding of arbitrary orientations of a much wider class of spanning ‘tree-like’ structures, such as collections of at most $O(n^{0.99})$ pairwise vertex-disjoint cycles and subdivisions of graphs $H$ with $|H|\lt \exp \bigl (\sqrt {\textrm {O}(\log n)}\,\bigr )$ in which each edge is subdivided at least once.

We prove a quantitative isoperimetric inequality for nearly spherical domains in the Bergman ball in $\mathbb{C}^2$. We prove a Fuglede-type theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic unit ball, and it provides the first result on the isoperimetric phenomenon in the Bergman ball.

This paper addresses the simultaneous homogenization and dimension reduction of thin composite plates reinforced with rigid substructures, within the framework of non-linear elasticity. The reference (undeformed) configuration consists of a periodically perforated elastic matrix, where the holes are occupied by rigid inclusions. We first establish a decomposition of the deformation for such composite structures. Under the assumption that the thickness parameter $\delta$ is asymptotically smaller than the periodicity $\varepsilon$, we derive a reduced asymptotic model as both parameters tend to zero. Using rescaled unfolding operators, we characterize the limiting behaviour of the Green–St. Venant strain tensor. Through $\Gamma$-convergence, we obtain the homogenized limit energy and establish the existence of a minimizer. The resulting limit model is of constrained Kármán type, where the limiting displacement satisfies a first-order infinitesimal isometry constraint.

We introduce and analyse a variant of the Becker–Döring equations that models the growth of clusters through the gain or loss of monomers. Motivated by enzymatic reactions in biology, this model incorporates irreversible fragmentation and monomers injection. We establish the well-posedness of the equations under suitable conditions on the kinetic rates. Then, as in the Becker–Döring equations, we distinguish two cases for the long-time behaviour of our solution; however, the distinction is made from the constant rate injection of monomers. While under strong fragmentation rate the system may exhibit infinite steady states, we prove that for low injection rate and moderate fragmentation, the solution converges locally exponentially fast to the steady state. Finally, we present an efficient scheme that preserves the asymptotic and allows fast computation by sub-sampling the clusters.

We further develop the abstract representation theory of affine Hecke algebras with arbitrary positive parameters. We establish analogues of several results that are known for reductive p-adic groups. These include: the relation between parabolic induction/restriction and Hermitian duals, Bernstein’s second adjointness and generalizations of the Langlands classification. We check that, in the known cases of equivalences between module categories of affine Hecke algebras and Bernstein blocks for reductive p-adic groups, such equivalences preserve Hermitian duality.

We also initiate the study of generic representations of affine Hecke algebras. Based on an analysis of the Hecke algebras associated to generic Bernstein blocks for quasi-split reductive p-adic groups, we propose a fitting definition of genericity for modules over affine Hecke algebras. With that notion we prove special cases of the generalized injectivity conjecture, about generic subquotients of standard modules for affine Hecke algebras.

Given a positive, non-increasing sequence $a$ with finite sum equal to $1$, we consider the family of all closed subsets of $[0,1]$ whose complementary open intervals have lengths given by a rearrangement of the sequence $a$. We study the full range of possible $\theta$-intermediate dimensions of these sets and, under suitable assumptions on the sequence, we show that this range forms a closed interval, whose endpoints we compute explicitly. This paper fills a gap in the literature concerning the dimensional properties of complementary sets.

We prove that the positive-dimensional part of the torsion locus of the Ceresa normal function in $\mathcal {M}_g$ is not Zariski dense when $g\geq 3$. Moreover, it has only finitely many components with generic Mumford-Tate group equal to $\mathrm {GSp}_{2g}$; these components are defined over $\overline {\mathbb Q}$, and their union is closed under the action of $\mathrm {Gal}(\overline {\mathbb Q}/\mathbb Q)$. More generally, we study the distribution of the torsion locus of arbitrary admissible normal functions.

We find upper and lower bounds on the number of rational points with bounded denominators that are contained in a rectangular neighbourhood of some $n$-dimensional $p$-adic integer. To find the upper bound, we use lattice point counting techniques on $p$-adic approximation lattices, and for the corresponding lower-bound statement, a classical pigeonhole principle-style argument is used. We apply this counting result to prove a statement in the setting of weighted simultaneous $p$-adic Diophantine approximation on coordinate hyperplanes. For the lower-bound Hausdorff dimension result, we construct a local ubiquitous system of rectangles and then apply the recent Mass Transference Principle result of Wang and Wu (Math. Ann., 2021).

A known condition for the integrability of the one-dimensional Fourier transform, in which the derivative is assumed to belong to the real Hardy space, is extended to the $n$-dimensional case, in terms of the $n$-th Riesz derivative. We obtain results for the Riesz derivatives of other orders. The function spaces involved in this study are compared and analyzed.

Two-sort species yield differential equations for functional digraphs of Cayley permutations. From these, we obtain an explicit formula for fixed-point-free Cayley permutations and prove that their proportion tends to $1/e$, as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.

We give a new proof of a result by Fathi, which states that, to any homeomorphism of a closed surface which is isotopic to a pseudo-Anosov homeomorphism, we can associate a stable and an unstable invariant partition of the surface with properties which are similar to the unstable and the stable foliation of a pseudo-Anosov homeomorphism.

Malaria remains a significant global health challenge, with sub-Saharan Africa bearing the majority of the burden. While vector control measures such as pyrethroid-based insecticidal nets and indoor residual spraying have significantly reduced malaria incidence, the emergence of insecticide resistance in Anopheles mosquito populations threatens these gains. Resistance develops through genetic mutations under prolonged selection pressure, complicating control efforts and necessitating a deeper understanding of its evolutionary dynamics. This study introduces a novel mathematical framework to investigate the emergence and spread of insecticide resistance in mosquito populations. By modelling insecticide resistance as a continuous (quantitative) trait influenced by multiple genes, we capture its variability and evolutionary transient dynamics. We propose an age-structured mosquito population model using integro-differential equations, where the resistance trait influences life-history parameters such as mortality and reproduction. Our approach provides new insights into how resistance emerges and spreads within mosquito populations over time. We analyse the model’s properties, including the existence of a unique maximal bounded semiflow, and derive conditions for the existence and stability of steady states. Through parameterization and simulations, we explore the transient and long-term dynamics of resistance evolution under different scenarios. The results offer valuable insights into the evolutionary mechanisms driving insecticide resistance and inform the design of sustainable vector control strategies.

Let $r_s(n)$ denote the number of representations of $n$ as a sum of $s$ squares. Hurwitz established eleven identities expressing the generating function of $r_3(an+b)$ as a simple infinite product. Cooper and Hirschhorn (Discrete Math 274 (1-3):9–24, 2004) proved that for any $k\geq0$, the generating functions $\sum_{n=0}^\infty r_3\big(3^{2k}n\big)q^n$ and $\sum_{n=0}^\infty r_3\big(3^{2k+1}n\big)q^n$ can be written as linear combinations of two specified generalized eta-quotients. In this paper, we substantially extend these results to high dimensions. Specifically, we prove that for any $k\geq0$ and $3\leq s\leq 100$, the generating functions $\sum_{n=0}^\infty r_s\big(3^{2k+1}n\big)q^n$ and $\sum_{n=0}^\infty r_s\big(3^{2k+2}n\big)q^n$ can also be expressed as linear combinations of certain generalized eta-quotients. Motivated by these results, we conjecture that this phenomenon holds for $r_s(n)$ for all $s\geq3$, and further that $r_s(n)$ satisfies an infinite family of internal congruences modulo high powers of $3$.

Cohomological equation is of special interest because it concerns the study of time change for flows, topological stability and topological conjugacy in dynamical systems, which is also an auxiliary equation to study the problem of linearization. In this paper, we consider a general form of cohomological equation for planar contractions. By using the ideas of invariant manifold and estimations in [W. Zhang and W. Zhang, $C^1$ linearization for planar contractions, J. Funct. Anal. 260 (2011), 2043–2063.], we present new criteria on eigenvalues of the linear parts for the existence of $C^1$ solutions in the Poincaré domain. Our results are a generalization of $C^1$ linearization for contractions.