Properties of the density for a three dimensional stochastic wave equation

We consider a stochastic wave equation in space dimension three driven by a noise white in time and with an absolutely continuous correlation measure given by the product of a smooth function and a Riesz kernel. Let $p_{t,x}(y)$ be the density of the law of the solution $u(t,x)$ of such an equation at points (t,x)\in]0,T]\times \IR^3. We prove that the mapping $(t,x)\mapsto p_{t,x}(y)$ owns the same regularity as the sample paths of the process \{u(t,x), (t,x)\in]0,T]\times \mathbbR^3\} established Dalang and Sanz-Sol\'e [Memoirs of the AMS, to appear]. The proof relies on Malliavin calculus and more explicitely, Watanabe's integration by parts formula and estimates derived form it.Comment: 29 page

Cosmopolitan speakers and their cultural cartographies

Language learners' increased mobility and the ubiquity of virtual intercultural encounters has challenged traditional ideas of ‘cultures’. Moreover, representations of cultures as consumable life-choices has meant that learners are no longer locked into standard and static cultural identities. Language learners are better defined as cosmopolitan individuals with subjective and complex socio-political and historical identities. Such models push the boundaries of current concepts in language pedagogy to new understandings of who the language learner is and a refashioning of the cultural maps they inhabit. This article presents a model for cultural understanding that draws on the theoretical framework of Beck's Cosmopolitan Vision and its related concepts of ‘Banal Cosmopolitanism’ and ‘Cosmopolitan Empathy’. Narrative accounts are used to illustrate the experience of a group of students of Arabic and Serbian/Croatian and their use of the cultural resources at their disposal to construct their own subjective cosmopolitan life-worlds. Through the analysis of learners' everyday cultural practices inside and outside the educational environment, the scope of the intercultural experience is revisited and a new paradigm for the language learner is presented. The Cosmopolitan Speaker (CS) described in this article is a subject who adopts a flâneur-like disposition to reflect on and scrutinise the target culture. Armed with this highly personal interpretation of reality, CSs will be able to take part in their own cultural trajectories and imagine and ‘figure’ their own cartography of the world

Nonequilibrium entropic bounds for Darwinian replicators

Life evolved on our planet by means of a combination of Darwinian selection and innovations leading to higher levels of complexity. The emergence and selection of replicating entities is a central problem in prebiotic evolution. Theoretical models have shown how populations of different types of replicating entities exclude or coexist with other classes of replicators. Models are typically kinetic, based on standard replicator equations. On the other hand, the presence of thermodynamical constrains for these systems remain an open question. This is largely due to the lack of a general theory of out of statistical methods for systems far from equilibrium. Nonetheless, a first approach to this problem has been put forward in a series of novel developements in non-equilibrium physics, under the rubric of the extended second law of thermodynamics. The work presented here is twofold: firstly, we review this theoretical framework and provide a brief description of the three fundamental replicator types in prebiotic evolution: parabolic, malthusian and hyperbolic. Finally, we employ these previously mentioned techinques to explore how replicators are constrained by thermodynamics.Comment: 12 Pages, 5 Figure

Efficient prime counting and the Chebyshev primes

The function \epsilon(x)=\mbox{li}(x)-\pi(x) is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions \epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x) and \epsilon_{\psi}(x)=\mbox{li}[\psi(x)]-\pi(x) are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $\theta(x)=\sum_{p \le x} \log p$ and $\psi(x)=\sum_{n=1}^x \Lambda(n)$, respectively, \mbox{li}(x) is the logarithmic integral, $\mu(n)$ and $\Lambda(n)$ are the M\"obius and the Von Mangoldt functions). Negative jumps in the above functions $\epsilon$, $\epsilon_{\theta}$ and $\epsilon_{\psi}$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes j_p=\mbox{li}(p)-\mbox{li}(p-1) and one investigates the jumps $j_p$, $j_{\theta(p)}$ and $j_{\psi(p)}$. In particular, $j_p<1$, and $j_{\theta(p)}>1$ for $p<10^{11}$. Besides, $j_{\psi(p)}<1$ for any odd p \in \mathcal{\mbox{Ch}}, an infinite set of so-called {\it Chebyshev primes } with partial list $\{109, 113, 139, 181, 197, 199, 241, 271, 281, 283, 293, 313, 317, 443, 449, 461, 463, \ldots\}$. We establish a few properties of the set \mathcal{\mbox{Ch}}, give accurate approximations of the jump $j_{\psi(p)}$ and relate the derivation of \mbox{Ch} to the explicit Mangoldt formula for $\psi(x)$. In the context of RH, we introduce the so-called {\it Riemann primes} as champions of the function $\psi(p_n^l)-p_n^l$ (or of the function $\theta(p_n^l)-p_n^l$ ). Finally, we find a {\it good} prime counting function S_N(x)=\sum_{n=1}^N \frac{\mu(n)}{n}\mbox{li}[\psi(x)^{1/n}], that is found to be much better than the standard Riemann prime counting function.Comment: 15 pages section 2.2 added, new sequences added, Fig. 2 and 3 are ne

Extreme values of the Dedekind Ψ\Psi function

Let $\Psi(n):=n\prod_{p | n}(1+\frac{1}{p})$ denote the Dedekind $\Psi$ function. Define, for $n\ge 3,$ the ratio $R(n):=\frac{\Psi(n)}{n\log\log n}.$ We prove unconditionally that $R(n)< e^\gamma$ for $n\ge 31.$ Let $N_n=2...p_n$ be the primorial of order $n.$ We prove that the statement $R(N_n)>\frac{e^\gamma}{\zeta(2)}$ for $n\ge 3$ is equivalent to the Riemann Hypothesis.Comment: 5 pages, to appear in Journal of Combinatorics and Number theor

How Turing parasites expand the computational landscape of digital life

Why are living systems complex? Why does the biosphere contain living beings with complexity features beyond those of the simplest replicators? What kind of evolutionary pressures result in more complex life forms? These are key questions that pervade the problem of how complexity arises in evolution. One particular way of tackling this is grounded in an algorithmic description of life: living organisms can be seen as systems that extract and process information from their surroundings in order to reduce uncertainty. Here we take this computational approach using a simple bit string model of coevolving agents and their parasites. While agents try to predict their worlds, parasites do the same with their hosts. The result of this process is that, in order to escape their parasites, the host agents expand their computational complexity despite the cost of maintaining it. This, in turn, is followed by increasingly complex parasitic counterparts. Such arms races display several qualitative phases, from monotonous to punctuated evolution or even ecological collapse. Our minimal model illustrates the relevance of parasites in providing an active mechanism for expanding living complexity beyond simple replicators, suggesting that parasitic agents are likely to be a major evolutionary driver for biological complexity.Comment: 13 pages, 8 main figures, 1 appendix with 5 extra figure

Large deviations for rough paths of the fractional Brownian motion

Starting from the construction of a geometric rough path associated with a fractional Brownian motion with Hurst parameter $H\in]{1/4}, {1/2}[$ given by Coutin and Qian (2002), we prove a large deviation principle in the space of geometric rough paths, extending classical results on Gaussian processes. As a by-product, geometric rough paths associated to elements of the reproducing kernel Hilbert space of the fractional Brownian motion are obtained and an explicit integral representation is given.Comment: 32 page