Related papers: La notion d'involution dans le Brouillon Project d…
We investigate the initial-value problem of the non-linear Liouville hierarchy. For the general form of the interaction potential we construct an explicit solution in terms of an expansion over particle clusters whose evolution is described…
A full accounting of biological robustness remains elusive; both in terms of the mechanisms by which robustness is achieved and the forces that have caused robustness to grow over evolutionary time. Although its importance to topics such as…
The concept of evolutionary development of structures constituted a \emph{real} revolution in biology: it was possible to understand how the very complex structures of life can arise in an out-of-equilibrium system. The investigation of…
We define a trivolution on a complex algebra $A$ as a non-zero conjugate-linear, anti-homomorphism $\tau$ on $A$, which is a generalized inverse of itself, that is, $\tau^3=\tau$. We give several characterizations of trivolutions and show…
The evolution of complexity has been a central theme for Biology [2] and Artificial Life research [1]. It is generally agreed that complexity has increased in our universe, giving way to life, multi-cellularity, societies, and systems of…
Classifications and phylogenetic inferences of organismal groups change in light of new insights. Over time these changes can result in an imperfect tracking of taxonomic perspectives through the re-/use of Code-compliant or informal names.…
We apply the theory of learning to physically renormalizable systems in an attempt to develop a theory of biological evolution, including the origin of life, as multilevel learning. We formulate seven fundamental principles of evolution…
We develop a new bijective framework for the enumeration of bipartite planar maps with control on the degree distribution of black and white vertices. Our approach builds on the blossoming-tree paradigm, introducing a family of orientations…
In this note, we give item-by-item responses to the criticisms raised in [TZ] by Tehrani ad Zinger on our paper [LR]. We illuminate the main ideas and contributions in [LR] in section 2, itemize the responses to issues raised in [TZ] and…
Diffusion models have recently shown considerable potential in solving Bayesian inverse problems when used as priors. However, sampling from the resulting denoising posterior distributions remains a challenge as it involves intractable…
The driving force behind convolutional networks - the most successful deep learning architecture to date, is their expressive power. Despite its wide acceptance and vast empirical evidence, formal analyses supporting this belief are scarce.…
From 1837, when he returned to England aboard the $\textit{HMS Beagle}$, to 1860, just after publication of $\textit{The Origin of Species}$, Charles Darwin kept detailed notes of each book he read or wanted to read. His notes and…
The purpose of this expository note is to give the proof of a theorem of Bourgain with some additional details and updated notation. The theorem first appeared as an appendix to the breakthrough paper by Friedgut, \emph{Sharp Thresholds of…
In recent years, optimization problems have become increasingly more prevalent due to the need for more powerful computational methods. With the more recent advent of technology such as artificial intelligence, new metaheuristics are needed…
We give a motivated introduction to the theory of perverse sheaves, culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical…
We will discuss our experiences and design decisions obtained from building a formal library for the convolution of two functions. Convolution is a fundamental concept with applications throughout mathematics. We will focus on the design…
Incidence theorems concern configurations of points, lines, and, more generally, higher-dimensional subspaces in projective space. Broadly speaking, such theorems fall into two classes: those that hold over an arbitrary division ring, such…
The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. We describe in each case the corresponding notions of degeneration and rigidity. We illustrate these notions with…
Blur is an image degradation that is difficult to remove. Invariants with respect to blur offer an alternative way of a~description and recognition of blurred images without any deblurring. In this paper, we present an original unified…
The Theorem on Invariance of Domain due to L.E.J. Brouwer states that one connected, compact (Hausdorff) m-dimensional manifold embedded into another actually realizes a homeomorphism. This fundamental result is relevant to Functional…