The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by taking graph complements, it becomes isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincare polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs.

We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

From a modern perspective cosmology is a historical science in so far that it deals with the development of the universe since its origin some 14 billion years ago. The origin itself may not be subject to scientific analysis and explanation. Nonetheless, there are theories that claim to explain the ultimate origin or “creation” of the universe. As shown by the history of cosmological thought, the very concept of “origin” is problematic and can be understood in different ways. While it is normally understood as a temporal concept, cosmic origin is not temporal by necessity.

This text was published in the book “Penser les mathematiques: seminaire de philosophie et mathematiques de l’Ecole normale superieure (J. Dieudonne, M. Loi, R. Thom)” edited by F. Guenard and G. Lelievre, Paris, editions du Seuil, 1982, pp. 58-72. It is reproduced with the kind authorisation of Francois Apery.

El fin de siglo invita siempre a hacer un recuento de la centuria pasada en todas sus manifestaciones: políticas, militares, artísticas, científicas… La filosofía se hace también su pequeño hueco y cuenta el siglo –lo está haciendo ya en multitud de congresos y publicaciones– desde múltiples perspectivas:

The article aims at providing an introduction to Russell’s and Whitehead’s neglected mature theory of magnitude, presented in the last published part of Principia Mathematica. I intend to show that Principia, VI, is the culmination of a line of thought whose beginning goes back to the time of Russell’s first works on the theory of relations, in 1900. But I insist as well on Whitehead’s own important contribution. At the end, I address a more general problem: how to articulate this quantitative doctrine of numbers with Russell’s and Whitehead’s logicist stance?

If empirical evidence has been so far lacking as regards the bombastic claim by US President Donald Trump that China – and its leader President Xi Jinping personally – has been «helping» the United States to address the North Korea crisis, a terrible beauty was born this week, which provides a clue. On Wednesday, Pyongyang opened heavy artillery against Beijing, with the North’s state news agency admonishing China by name for exacerbating tensions on the Korean Peninsula. If Trump’s team has been looking for the proverbial needle in the haystack to figure out the true Chinese intentions on North Korea, here it is in the North’s fury toward Beijing.

We start with reviewing the origin of the idea that entropy and the Second Law are associated with the Arrow of Time. We then introduced a new definition of entropy based on Shannons Measure of Information, SMI. The SMI may be defined on any probability distribution, and therefore it is a very general concept. On the other hand entropy is defined on a very special set of probability distributions. More specifically the entropy of a thermodynamic system is related the probability distribution of locations and velocities or momenta of all the particles, which maximized the Shannon Measure of Information. As such, entropy is not a function of time. We also show that the H function, as defined by Boltzmann is an SMI but not entropy. Therefore, while the H-function, as an SMI may change with time, Entropy, as a limit of the SMI does not change with time.

Let G be the group Z d or the monoid N d where d is a positive integer. Let X be a subshift over G , i.e., a closed and shift-invariant subset of A G where A is a finite alphabet. We prove that the topological entropy of X is equal to the Hausdorff dimension of X and has a sharp characterization in terms of the Kolmogorov complexity of finite pieces of the orbits of X . In the version of this paper that has been published in Theory of Computing Systems, the proof of Lemma 4.3 contains a confusing typographical error. This version of the paper corrects that error.

A framework integrating information theory and network science is proposed, giving rise to a potentially new area of network information science. By incorporating and integrating concepts such as complexity, coding, topological projections and network dynamics, the proposed network-based framework paves the way not only to extending traditional information science, but also to modeling, characterizing and analyzing a broad class of real-world problems, from language communication to DNA coding. Basically, an original network is supposed to be transmitted, with our without compaction, through a time-series obtained by sampling its topology by some network dynamics, such as random walks. We show that the degree of compression is ultimately related to the ability to predict the frequency of symbols based on the topology of the original network and the adopted dynamics.