A proof of Mader's conjecture on large clique subdivisions in C4C_4-free graphs

Given any integers $s,t\geq 2$, we show there exists some $c=c(s,t)>0$ such that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $cd^{\frac{1}{2}\frac{s}{s-1}}$ vertices. In particular, when $s=2$ this resolves in a strong sense the conjecture of Mader in 1999 that every $C_4$-free graph has a subdivision of a clique with order linear in the average degree of the original graph. In general, the widely conjectured asymptotic behaviour of the extremal density of $K_{s,t}$-free graphs suggests our result is tight up to the constant $c(s,t)$.Comment: 25 pages, 1 figur

The C_\ell-free process

The C_\ell-free process starts with the empty graph on n vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of C_\ell is created. For every $\ell \geq 4$ we show that, with high probability as $n \to \infty$, the maximum degree is $O((n \log n)^{1/(\ell-1)})$, which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the C_\ell-free process typically terminates with $\Theta(n^{\ell/(\ell-1)}(\log n)^{1/(\ell-1)})$ edges, which answers a question of Erd\H{o}s, Suen and Winkler. This is the first result that determines the final number of edges of the more general H-free process for a non-trivial \emph{class} of graphs H. We also verify a conjecture of Osthus and Taraz concerning the average degree, and obtain a new lower bound on the independence number. Our proof combines the differential equation method with a tool that might be of independent interest: we establish a rigorous way to `transfer' certain decreasing properties from the binomial random graph to the H-free process.Comment: 34 pages, 5 figures. Minor revisions and addition

Quantum-classical transition in Scale Relativity

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrodinger and Klein-Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrodinger and Klein-Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds - ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx^mu - dx^mu) and by requiring invariance under parity and time inversion. The Pauli equation is recovered as a non-relativistic-motion approximation of the Dirac equation.Comment: 28 pages, no figur

How is the Reionization Epoch Defined?

We study the effect of a prolonged epoch of reionization on the angular power spectrum of the Cosmic Microwave Background. Typically reionization studies assume a sudden phase transition, with the intergalactic gas moving from a fully neutral to a fully ionized state at a fixed redshift. Such models are at odds, however, with detailed investigations of reionization, which favor a more extended transition. We have modified the code CMBFAST to allow the treatment of more realistic reionization histories and applied it to data obtained from numerical simulations of reionization. We show that the prompt reionization assumed by CMBFAST in its original form heavily contaminates any constraint derived on the reionization redshift. We find, however, that prompt reionization models give a reasonable estimate of the epoch at which the mean cosmic ionization fraction was ~50%, and provide a very good measure of the overall Thomson optical depth. The overall differences in the temperature (polarization) angular power spectra between prompt and extended models with equal optical depths are less than 1% (10%).Comment: 5 pages, 3 figures; accepted on MNRA

Mathematics of CLIFFORD - A Maple package for Clifford and Grassmann algebras

CLIFFORD performs various computations in Grassmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for Clifford product are implemented: 'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.Comment: 24 pages, update contains new material included in published versio

On-line Ramsey numbers of paths and cycles

Consider a game played on the edge set of the infinite clique by two players, Builder and Painter. In each round, Builder chooses an edge and Painter colours it red or blue. Builder wins by creating either a red copy of $G$ or a blue copy of $H$ for some fixed graphs $G$ and $H$. The minimum number of rounds within which Builder can win, assuming both players play perfectly, is the on-line Ramsey number $\tilde{r}(G,H)$. In this paper, we consider the case where $G$ is a path $P_k$. We prove that $\tilde{r}(P_3, P_{\ell+1}) = \lceil 5\ell/4 \rceil = \tilde{r}(P_3, C_\ell)$ for all $\ell \ge 5$, and determine $\tilde{r}(P_4, P_{\ell+1}$) up to an additive constant for all $\ell \ge 3$. We also prove some general lower bounds for on-line Ramsey numbers of the form $\tilde{r}(P_{k+1},H)$.Comment: Preprin

Multiple scattering of polarized light in disordered media exhibiting short-range structural correlations

We develop a model based on a multiple scattering theory to describe the diffusion of polarized light in disordered media exhibiting short-range structural correlations. Starting from exact expressions of the average field and the field spatial correlation function, we derive a radiative transfer equation for the polarization-resolved specific intensity that is valid for weak disorder and we solve it analytically in the diffusion limit. A decomposition of the specific intensity in terms of polarization eigenmodes reveals how structural correlations, represented via the standard anisotropic scattering parameter $g$, affect the diffusion of polarized light. More specifically, we find that propagation through each polarization eigenchannel is described by its own transport mean free path that depends on $g$ in a specific and non-trivial way