There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.

/pdf/the-structure-and-function-of-complex-networks-34deompc9l.pdf

The Structure and Function of Complex Networks

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

http://www.maths.qmul.ac.uk/%7Elatora/report_06.pdf

Complex networks: Structure and dynamics

We show that if a solution of the Dirichlet problem for harmonic maps lies in a convex ball and its boundary values are contained in some smaller ball, then the whole solution itself is contained in this smaller ball.

A maximum principle for harmonic mappings which solve a Dirichlet problem

Coupled map lattices (CMLs) are prototypical dynamical systems on networks/graphs They exhibit complex patterns generated via the interplay of diffusive/Laplacian coupling and nonlinear reactions modelled by a single iterated map at each node; the maps are often taken as unimodal, eg, logistic or tent maps In this letter, we propose a class of higher-order coupled dynamical systems involving the hypergraph Laplacian, which we call coupled hypergraph maps (CHMs) By combining linearized (in-)stability analysis of synchronized states, hypergraph spectral theory, and numerical methods, we detect robust regions of chaotic cluster synchronization occurring in parameter space upon varying coupling strength and the main bifurcation parameter of the unimodal map Furthermore, we find key differences between Laplacian and hypergraph Laplacian coupling and detect various other classes of periodic and quasi-periodic patterns The results show the high complexity of coupled graph maps and indicate that they might be an excellent universal model class to understand the similarities and differences between dynamics on classical graphs and dynamics on hypergraphs

Coupled Hypergraph Maps and Chaotic Cluster Synchronization

We consider harmonic functions of polynomial growth of some order $$d$$
 on Cayley graphs of discrete groups of polynomial volume growth of order $$D$$
 w.r.t. the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions. More precisely, the dimension of this space of harmonic functions is at most of order $$d^{D-1}$$
 . As in the already known Riemannian case, this estimate is polynomial in the growth degree. More generally, our techniques also apply to graphs roughly isometric to Cayley graphs of groups of polynomial volume growth.

pdf/polynomial-growth-harmonic-functions-on-groups-of-polynomial-2a15fmlrlr.pdf

Polynomial growth harmonic functions on groups of polynomial volume growth

Suppose we have a pair of information channels, κ 1 , κ 2 , with a common input. The Blackwell order is a partial order over channels that compares κ 1 and κ 2 by the maximal expected utility an agent can obtain when decisions are based on the channel outputs. Equivalently, κ 1 is said to be Blackwell-inferior to κ 2 if and only if κ 1 can be constructed by garbling the output of κ 2 . A related partial order stipulates that κ 2 is more capable than κ 1 if the mutual information between the input and output is larger for κ 2 than for κ 1 for any distribution over inputs. A Blackwell-inferior channel is necessarily less capable. However, examples are known where κ 1 is less capable than κ 2 but not Blackwell-inferior. We show that this may even happen when κ 1 is constructed by coarse-graining the inputs of κ 2 . Such a coarse-graining is a special kind of “pre-garbling” of the channel inputs. This example directly establishes that the expected value of the shared utility function for the coarse-grained channel is larger than it is for the non-coarse-grained channel. This contradicts the intuition that coarse-graining can only destroy information and lead to inferior channels. We also discuss our results in the context of information decompositions.

/pdf/coarse-graining-and-the-blackwell-order-k17qg7eydh.pdf

Coarse-graining and the Blackwell Order

We introduce and study an evolutionary complementarity game where in each round a player of population 1 is paired with a member of population 2 The game is symmetric, and each player tries to obtain an advantageous deal, but when one of them pushes too hard, no deal at all can be concluded, and they both lose The game has many equilibria, and which of them is reached depends on the history of the interactions as the players evolve according to a fitness function that measures their gains across the interactions We can then break the symmetry by assigning different strategy spaces to the populations, varying in particular with respect to the information available to the agents The agents can, for example, adapt to the behavior of their opponents met in previous rounds, or they can try to copy the strategies of their successful friends It turns out that, in general, the more restricted strategy spaces, that is, those which utilize less information, are more advantageous for a population as a whole as their adoption drives the equilibrium in a direction advantageous to that population One reason is that a simpler strategy can be learned faster in an evolutionary setting, another is that it is good for a population to have some individuals that are unfit in the sense that they make offers that are individually unsuccessful, but have a systematic effect on the strategies of their opponents All these effects are demonstrated through systematic simulations

Individual strategies in complementarity games and population dynamics

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