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

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

We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection method in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2.6 million customers and by analyzing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad-hoc modular networks. .

Fast unfolding of communities in large networks

Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

/pdf/linear-matrix-inequalities-in-system-and-control-theory-2q5cf0pqm2.pdf

Linear Matrix Inequalities in System and Control Theory

We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection methods in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2 million customers and by analysing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad hoc modular networks.

/pdf/fast-unfolding-of-communities-in-large-networks-3hz1lz8b9n.pdf

In this paper, we analyze statistical properties of a communication network constructed from the records of a mobile phone company. The network consists of 2.5 million customers that have placed 810 million communications (phone calls and text messages) over a period of 6 months and for whom we have geographical home localization information. It is shown that the degree distribution in this network has a power-law degree distribution k(-5) and that the probability that two customers are connected by a link follows a gravity model, i.e. decreases as d(-1). where d is the distance between the customers. We also consider the geographical extension of communication triangles and we show that communication triangles are not only composed of geographically adjacent nodes but that they may extend over large distances. This last property is not captured by the existing models of geographical networks and in a last section we propose a new model that reproduces the observed property. Our model, which is based on the migration and on the local adaptation of agents, is then studied analytically and the resulting predictions are confirmed by computer simulations. (C) 2008 Elsevier B.V. All rights reserved.

/pdf/geographical-dispersal-of-mobile-communication-networks-4n7ku3t6rr.pdf

Geographical dispersal of mobile communication networks

We introduce a concept of {similarity} between vertices of directed graphs. Let GA and GB be two directed graphs with, respectively, nA and nB vertices. We define an nB \times nA similarity matrix S whose real entry sij expresses how similar vertex j (in GA) is to vertex i (in GB): we say that sij is their similarity score. The similarity matrix can be obtained as the limit of the normalized even iterates of Sk+1 = BSkAT + BTSkA, where A and B are adjacency matrices of the graphs and S0 is a matrix whose entries are all equal to 1. In the special case where GA = GB = G, the matrix S is square and the score sij is the similarity score between the vertices i and j of G. We point out that Kleinberg's "hub and authority" method to identify web-pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them. In analogy to Kleinberg, we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix. Potential applications of our similarity concept are numerous. We illustrate an application for the automatic extraction of synonyms in a monolingual dictionary.

/pdf/a-measure-of-similarity-between-graph-vertices-applications-9fpcggwjo2.pdf

A Measure of Similarity between Graph Vertices: Applications to Synonym Extraction and Web Searching

A numerically stable algorithm is derived to compute orthonormal bases for any deflating subspace of a regular pencil $\lambda B - A$. The method is based on an update of the $QZ$-algorithm, in order to obtain any desired ordering of eigenvalues in the quasitriangular forms constructed by this algorithm. As applications we discuss a new approach to solve Riccati equations arising in linear system theory. The computation of deflating subspaces with specified spectrum is shown to be of crucial importance here.

A Generalized Eigenvalue Approach for Solving Riccati Equations

In order to test the numerical methods for model reduction we present here a benchmark collection, which contain some useful real world examples reflecting current problems in applications. All simulations were obtained via Matlab and some SLICOT programs of Niconet.

/pdf/a-collection-of-benchmark-examples-for-model-reduction-of-1ils6rz9cw.pdf

A collection of benchmark examples for model reduction of linear time invariant dynamical systems.

Support vector machines (SVM's) have been introduced in literature as a method for pattern recognition and function estimation, within the framework of statistical learning theory and structural risk minimization. A least squares version (LSSVM) has been recently reported which expresses the training in terms of solving a set of linear equations instead of quadratic programming as for the standard SVM case. In this paper we present an iterative training algorithm for LS-SVM's which is based on a conjugate gradient method. This enables solving large scale classification problems which is illustrated on a multi two-spiral benchmark problem. Keywords. Support vector machines, classification, neural networks, RBF kernels, conjugate gradient method.

Paul Van Dooren

Papers

Geographical dispersal of mobile communication networks

A Measure of Similarity between Graph Vertices: Applications to Synonym Extraction and Web Searching

A Generalized Eigenvalue Approach for Solving Riccati Equations

A collection of benchmark examples for model reduction of linear time invariant dynamical systems.

Least squares support vector machine classifiers: a large scale algorithm