Epigenetic landscapes, represented by patterns of chemical modifications on histone tails, are essential for maintaining cell identity and tissue homeostasis. These landscapes are shaped by multiple factors, including local biochemical signals and the three-dimensional organisation of chromatin. However, their response to genomic stress, such as DNA double-strand breaks (DSBs), remains incompletely understood. Here, we use a stochastic model of histone modification dynamics integrated with chromatin architecture to investigate how local depletion of sirtuins, histone deacetylases involved in DSB repair, destabilises epigenetic patterns. Our simulations recapitulate experimental findings in which sirtuin relocalisation to DSB sites leads to the epigenetic erosion and suggest that the resulting landscape depends on enzyme levels and chromatin geometry. Importantly, chromatin regions with large domains of long-range contacts are more resilient to epigenetic destabilisation. These findings suggest that chromatin folding can buffer against relocation of histone-modifying enzymes, highlighting a structural mechanism for preserving epigenetic integrity under stress.

Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality.

We prove that any class of bipartite graphs with linear neighborhood complexity has bounded odd chromatic number. As a result, if $\mathcal{G}$ is the class of all circle graphs, or if $\mathcal{G}$ is any class with bounded twin-width, bounded merge-width, or a forbidden vertex-minor, then $\mathcal{G}$ is $\chi_{odd}$-bounded.

Understanding the dynamics of large-scale brain models remains a central challenge due to the inherent complexity of these systems. In this work, we explore the emergence of complex spatiotemporal patterns in a large scale-brain model composed of 90 interconnected brain regions coupled through empirically derived anatomical connectivity. An important aspect of our formulation is that the local dynamics of each brain region are described by a next-generation neural mass model, which explicitly captures the macroscopic gamma activity of coupled excitatory and inhibitory neural populations (PING mechanism). We first identify the system's homogeneous states-both resting and oscillatory-and analyze their stability under uniform perturbations. Then, we determine the stability against non-uniform perturbations by obtaining dispersion relations for the perturbation growth rate. This analysis enables us to link unstable directions of the homogeneous solutions to the emergence of rich spatiotemporal patterns, that we characterize by means of Lyapunov exponents and frequency spectrum analysis. Our results show that, compared to previous studies with classical neural mass models, next-generation neural mass models provide a broader dynamical repertoire, both within homogeneous states and in the heterogeneous regime. Additionally, we identify a key role for anatomical connectivity in cross-frequency coupling, allowing for the emergence of gamma oscillations with amplitude modulated by slower rhythms. These findings suggest that such models are not only more biophysically grounded but also particularly well-suited to capture the full complexity of large-scale brain dynamics. Overall, our study advances the analytical understanding of emerging spatiotemporal patterns in whole-brain models.

The Lonely Runner Conjecture originated in Diophantine approximation will turn 60 in 2028. Even if the conjecture is still widely open, the flow of partial results, innovative tools and connections to different problems and applications has been steady on its long life. This survey attempts to give a panoramic view of the status of the problem, trying to highlight the contributions of the many papers that it has originated.

This chapter deals with the notion of the resolvent of a self-adjoint operator. We pay special attention to the convergence of unbounded self-adjoint operators in several resolvent senses, and how they are related to the convergence of their spectra. We also explore the relations that these notions of convergence have with the so-called strong graph limit, $G$-convergence, and $\Gamma$-convergence.

An important body of quantitative linguistics is constituted by a series of statistical laws about language usage. Despite the importance of these linguistic laws, some of them are poorly formulated, and, more importantly, there is no unified framework that encompasses all them. This paper presents a new perspective to establish a connection between different statistical linguistic laws. Characterizing each word type by two random variables, length (in number of characters) and absolute frequency, we show that the corresponding bivariate joint probability distribution shows a rich and precise phenomenology, with the type-length and the type-frequency distributions as its two marginals, and the conditional distribution of frequency at fixed length providing a clear formulation for the brevity-frequency phenomenon. The type-length distribution turns out to be well fitted by a gamma distribution (much better than with the previously proposed lognormal), and the conditional frequency distributions at fixed length display power-law-decay behavior with a fixed exponent $\alpha\simeq 1.4$ and a characteristic-frequency crossover that scales as an inverse power $\delta\simeq 2.8$ of length, which implies the fulfilment of a scaling law analogous to those found in the thermodynamics of critical phenomena. As a by-product, we find a possible model-free explanation for the origin of Zipf's law, which should arise as a mixture of conditional frequency distributions governed by the crossover length-dependent frequency.

We study the joint image of lines incident to points, meaning the set of image tuples obtained from fixed cameras observing a varying 3D point-line incidence. We prove a formula for the number of complex critical points of the triangulation problem that aims to compute a 3D point-line incidence from noisy images. Our formula works for an arbitrary number of images and measures the intrinsic difficulty of this triangulation. Additionally, we conduct numerical experiments using homotopy continuation methods, comparing different approaches of triangulation of such incidences. In our setup, exploiting the incidence relations gives both a faster point reconstruction and in three views more accurate.

Extending constructions by Gabriel and Zisman, we develop a functorial framework for the cohomology and homology of simplicial sets with very general coefficient systems given by functors on simplex categories into abelian categories. Furthermore we construct Leray type spectral sequences for any map of simplicial sets. We also show that these constructions generalise and unify the various existing versions of cohomology and homology of small categories and as a bonus provide new insight into their functoriality.

Experimental and empirical observations on cell metabolism cannot be understood as a whole without their integration into a consistent systematic framework. However, the characterization of metabolic flux phenotypes is typically reduced to the study of a single optimal state, like maximum biomass yield that is by far the most common assumption. Here we confront optimal growth solutions to the whole set of feasible flux phenotypes (FFP), which provides a benchmark to assess the likelihood of optimal and high-growth states and their agreement with experimental results. In addition, FFP maps are able to uncover metabolic behaviors, such as aerobic fermentation accompanying exponential growth on sugars at nutrient excess conditions, that are unreachable using standard models based on optimality principles. The information content of the full FFP space provides us with a map to explore and evaluate metabolic behavior and capabilities, and so it opens new avenues for biotechnological and biomedical applications.

We consider semi-discrete discontinuous Galerkin approximations of a general elastodynamics problem, in both {\it displacement} and {\it displacement-stress} formulations. We present the stability analysis of all the methods in the natural energy norm and derive optimal a-priori error estimates. For the displacement-stress formulation, schemes preserving the total energy of the system are introduced and discussed. We include some numerical experiments in three dimensions to verify the theory.

Complex networks grow subject to structural constraints which affect their measurable properties. Assessing the effect that such constraints impose on their observables is thus a crucial aspect to be taken into account in their analysis. To this end,we examine the effect of fixing the strength sequence in multi-edge networks on several network observables such as degrees, disparity, average neighbor properties and weight distribution using an ensemble approach. We provide a general method to calculate any desired weighted network metric and we show that several features detected in real data could be explained solely by structural constraints. We thus justify the need of analytical null models to be used as basis to assess the relevance of features found in real data represented in weighted network form.

Power-law distributions contain precious information about a large variety of processes in geoscience and elsewhere. Although there are sound theoretical grounds for these distributions, the empirical evidence in favor of power laws has been traditionally weak. Recently, Clauset et al. have proposed a systematic method to find over which range (if any) a certain distribution behaves as a power law. However, their method has been found to fail, in the sense that true (simulated) power-law tails are not recognized as such in some instances, and then the power-law hypothesis is rejected. Moreover, the method does not work well when extended to power-law distributions with an upper truncation. We explain in detail a similar but alternative procedure, valid for truncated as well as for non-truncated power-law distributions, based in maximum likelihood estimation, the Kolmogorov-Smirnov goodness-of-fit test, and Monte Carlo simulations. An overview of the main concepts as well as a recipe for their practical implementation is provided. The performance of our method is put to test on several empirical data which were previously analyzed with less systematic approaches. The databases presented here include the half-lives of the radionuclides, the seismic moment of earthquakes in the whole world and in Southern California, a proxy for the energy dissipated by tropical cyclones elsewhere, the area burned by forest fires in Italy, and the waiting times calculated over different spatial subdivisions of Southern California. We find the functioning of the method very satisfactory.

We develop an agent-based model of vasculogenesis, the de novo formation of blood vessels. Endothelial cells in the vessel network are viewed as linearly elastic spheres and are of two types: vessel elements are contained within the network; tip cells are located at endpoints. Tip cells move in response to forces due to interactions with neighbouring vessel elements, the local tissue environment, chemotaxis and a persistence force modeling their tendency to continue moving in the same direction. Vessel elements experience similar forces but not chemotaxis. An angular persistence force representing local tissue interactions stabilises buckling instabilities due to proliferation. Vessel elements proliferate, at rates that depend on their degree of stretch: elongated elements proliferate more rapidly than compressed elements. Following division, new cells are more likely to form new sprouts if the parent vessel is highly compressed and to be incorporated into the parent vessel if it is stretched. Model simulations reproduce key features of vasculogenesis. Parameter sensitivity analyses reveal significant changes in network size and morphology on varying the chemotactic sensitivity of tip cells, and the sensitivities of the proliferation rate and sprouting probability to mechanical stretch. Varying chemotactic sensitivity also affects network directionality. Branching and network density are influenced by the sprouting probability. Glyphs depicting multiple network properties show how network quantities change over time and as model parameters vary. We also show how glyphs constructed from in vivo data could be used to discriminate between normal and tumour vasculature and, ultimately, for model validation. We conclude that our biomechanical hybrid model generates vascular networks similar to those generated from in vitro and in vivo experiments.

We show that the model category of diagrams of spaces generated by a proper class of orbits is not cofibrantly generated. In particular the category of maps between spaces may be given a non-cofibrantly generated model structure.

The theory of finite-size scaling explains how the singular behavior of thermodynamic quantities in the critical point of a phase transition emerges when the size of the system becomes infinite. Usually, this theory is presented in a phenomenological way. Here, we exactly demonstrate the existence of a finite-size scaling law for the Galton-Watson branching processes when the number of offsprings of each individual follows either a geometric distribution or a generalized geometric distribution. We also derive the corrections to scaling and the limits of validity of the finite-size scaling law away the critical point. A mapping between branching processes and random walks allows us to establish that these results also hold for the latter case, for which the order parameter turns out to be the probability of hitting a distant boundary.

This manuscript aims to provide a self-contained introduction to the regularity theory for elliptic PDE, focusing on the main ideas rather than proving all results in their greatest generality. It can be seen as a bridge between an elementary PDE course and more advanced textbooks. This is a draft of the book "Regularity Theory for Elliptic PDE". The final version has been published in Zurich Lectures in Advanced Mathematics, EMS Press, 2022.

Building upon the pioneering work [Merle, Rapha\"el, Rodnianski, and Szeftel, Ann. of Math., 196(2):567-778, 2022, Ann. of Math., 196(2):779-889, 2022, Invent. Math., 227(1):247-413, 2022] we construct exact, smooth self-similar imploding solutions to the 3D isentropic compressible Euler equations for ideal gases for all adiabatic exponents $\gamma>1$. For the particular case $\gamma=\frac75$ (corresponding to a diatomic gas, e.g. oxygen, hydrogen, nitrogen), akin to the previous result, we show the existence of a sequence of smooth, self-similar imploding solutions. In addition, we provide simplified proofs of linear stability and non-linear stability, which allow us to construct asymptotically self-similar imploding solutions to the compressible Navier-Stokes equations with density independent viscosity for the case $\gamma=\frac75$. Moreover, the solutions constructed have density bounded away from zero and converge to a constant at infinity, representing the first example of singularity formation in such a setting.

Despite being a paradigm of quantitative linguistics, Zipf's law for words suffers from three main problems: its formulation is ambiguous, its validity has not been tested rigorously from a statistical point of view, and it has not been confronted to a representatively large number of texts. So, we can summarize the current support of Zipf's law in texts as anecdotic. We try to solve these issues by studying three different versions of Zipf's law and fitting them to all available English texts in the Project Gutenberg database (consisting of more than 30000 texts). To do so we use state-of-the art tools in fitting and goodness-of-fit tests, carefully tailored to the peculiarities of text statistics. Remarkably, one of the three versions of Zipf's law, consisting of a pure power-law form in the complementary cumulative distribution function of word frequencies, is able to fit more than 40% of the texts in the database (at the 0.05 significance level), for the whole domain of frequencies (from 1 to the maximum value) and with only one free parameter (the exponent).

It has been proposed that the number of tropical cyclones as a function of the energy they release is a decreasing power-law function, up to a characteristic energy cutoff determined by the spatial size of the ocean basin in which the storm occurs. This means that no characteristic scale exists for the energy of tropical cyclones, except for the finite-size effects induced by the boundaries of the basins. This has important implications for the physics of tropical cyclones. We discuss up to what point tropical cyclones are related to critical phenomena (in the same way as earthquakes, rainfall, etc.), providing a consistent picture of the energy balance in the system. Moreover, this perspective allows one to visualize more clearly the effects of global warming on tropical-cyclone occurrence.