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 study the interplay of topology and dynamics in a neural network connected with spike-timing-dependent plasticity (STDP) synapses. Stimulated with periodic spike trains, the STDP-driven network undergoes a synaptic pruning process and evolves to a residual network. We examine the variation of topological and dynamical properties of the residual network by varying two key parameters of STDP: Synaptic delay and the ratio between potentiation and depression. Our extensive numerical simulations of the Leaky Integrate-and-Fire model show that there exists two regions in the parameter space. The first corresponds to fixed point configurations, where the distribution of peak synaptic conductances and the firing rate of neurons remain constant over time. The second corresponds to oscillating configurations, where both topological and dynamical properties vary periodically which is a result of a fixed point becoming a limit cycle via a Hopf bifurcation. This leads to interesting questions regarding the implications of these rhythms in the topology and dynamics of the network for learning and cognitive processing.

/pdf/hopf-bifurcation-in-the-evolution-of-stdp-driven-networks-3xqz83d1su.pdf

Hopf bifurcation in the evolution of STDP-driven networks

Autism Spectrum Disorder (ASD) is a set of neurodevelopmental disorders that pose a significant global health burden. Measures from graph theory have been used to characterise ASD-related changes in resting-state fMRI functional connectivity networks (FCNs), but recently developed geometry-inspired measures have not been applied so far. In this study, we applied geometry-inspired graph Ricci curvatures to investigate ASD-related changes in resting-state fMRI FCNs. To do this, we applied Forman-Ricci and Ollivier-Ricci curvatures to compare networks of ASD and healthy controls (N = 1112) from the Autism Brain Imaging Data Exchange I (ABIDE-I) dataset. We performed these comparisons at the brain-wide level as well as at the level of individual brain regions, and further, determined the behavioral relevance of region-specific differences with Neurosynth meta-analysis decoding. We found brain-wide ASD-related differences for both Forman-Ricci and Ollivier-Ricci curvatures. For Forman-Ricci curvature, these differences were distributed across 83 of the 200 brain regions studied, and concentrated within the Default Mode, Somatomotor and Ventral Attention Network. Meta-analysis decoding identified the brain regions showing curvature differences as involved in social cognition, memory, language and movement. Notably, comparison with results from previous non-invasive stimulation (TMS/tDCS) experiments revealed that the set of brain regions showing curvature differences overlapped with the set of brain regions whose stimulation resulted in positive cognitive or behavioural outcomes in ASD patients. These results underscore the utility of geometry-inspired graph Ricci curvatures in characterising disease-related changes in ASD, and possibly, other neurodevelopmental disorders.

https://www.biorxiv.org/content/biorxiv/early/2021/11/28/2021.11.28.470231.full.pdf

Graph Ricci Curvatures Reveal Disease-related Changes in Autism Spectrum Disorder

Let \(u : \Omega \rightarrow \mathbb{R}\) be integrable, \(\alpha := ({\alpha }_{1},\ldots ,{\alpha }_{d})\)

Sobolev Spaces and L 2 Regularity Theory

This chapter treats Kahler manifolds and symmetric spaces as important examples of Riemannian manifolds in detail.

Chapter 7 Symmetric Spaces and Kähler Manifolds

Extending work of Caffarelli-Yang and Tarantello, we present a variational existence proof for two-vortex solutions of the periodic Chern-Simons Higgs model and analyze the asymptotic behavior of these solutions as the parameter coupling the gauge field with the scalar field tends to 0.

An analysis of the two-vortex case in the Chern-Simons Higgs model

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