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We introduce an algorithm that constructs a random uniform graph with prescribed degree sequence together with a depth first exploration of it. In the so-called supercritical regime where the graph contains a giant component, we prove that…
We sharpen and generalize the dimension growth bounds for the number of points of bounded height lying on an irreducible algebraic variety of degree $d$, over any global field. In particular, we focus on the affine hypersurface situation by…
We investigate the bifurcation structure of equilibria in a class of non-autonomous ordinary differential equations governed by a season length parameter, $\tau$, which determines the alternation between growth and decline dynamics. This…
The nonsingular real plane algebraic curves of given degree $d$ are considered either up to isotopy or up to deformation. The asymptotic behavior of the number $I_d$ of isotopy classes and the number $D_d$ of deformation classes are…
Any affine map on the (n+1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by…
We provide a framework for modeling social network formation through conditional multinomial logit models from discrete choice and random utility theory, in which each new edge is viewed as a "choice" made by a node to connect to another…
The various phase spaces involved in the dynamics of parametrized nonrelativistic Hamiltonian systems are displayed by using Crnkovic and Witten's covariant canonical formalism. It is also pointed out that in Dirac's canonical formalism…
This paper considers the solution structure of non-trivial, non-constant stationary states of 1D spatial parabolic equations with nonlinear self-diffusion and logistic growth terms. A two-dimensional ordinary differential equation…
We consider the discrete surface growth process with relaxation to the minimum [F. Family, J. Phys. A {\bf 19} L441, (1986).] as a possible synchronization mechanism on scale-free networks, characterized by a degree distribution $P(k) \sim…
We prove that there exists an open subset of the set of real-analytic Hamiltonian diffeomorphisms of a closed surface in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense. We also prove that there…
We show that two automorphisms of an affine surface with dynamical degree strictly larger than 1 share a Zariski dense set of periodic points if and only if they have the same periodic points. We construct canonical heights for these…
We define a set of boundary conditions that ensure the presence of a null hypersurface with the essential characteristics of a horizon, using the formalism of weakly isolated horizons as a guide. We then determine the diffeomorphisms that…
The dynamical degrees of a rational map $f:X\dashrightarrow X$ are fundamental invariants describing the rate of growth of the action of iterates of $f$ on the cohomology of $X$. When $f$ has nonempty indeterminacy set, these quantities can…
The Chapter is devoted to reviewing the main features of aging in non disordered systems relaxing via domain growth, after an istantaneous temperature quench. Using the autocorrelation and autoresponse functions to gauge the deviation from…
The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann…
This work is devoted to the study of the relationships between graph theory and the qualitative analysis of ordinary differential equations, with a special focus on two-dimensional systems. In particular, we reinterpret classical results…
We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space…
A discretisation of differential geometry using the Whitney forms of algebraic topology is consistently extended via the introduction of a pairing on the space of chains. This pairing of chains enables us to give a definition of the…
Gradient dynamics play a central role in determining the stability and generalization of deep neural networks. In this work, we provide an empirical analysis of how variance and standard deviation of gradients evolve during training,…
A broad class of nonlinear acoustic wave models possess a Hamiltonian structure in their dissipation-free limit and a gradient flow structure for their dissipative dynamics. This structure may be exploited to design numerical methods which…