Mathematical Model of Aedes Outbreaks in Padang City Considering Climate Parameter

Basic reproduction number is a threshold number that frequently found in the analysis of determining the spreads of population. Many methods have been derived to calculate basic reproduction number including eigen values or radius spectral and statistical approximation. However, the calculation of the threshold number is still not easy and tends to use a lot of computation. Another approach has been proposed using graph reduction. This approach has been applied to determine threshold number of Aedes spreads in Padang city depending on climate parameter. The daily rainfall data is used as an empirical study.     Keywords: basic reproduction number, population projection, graph reduction, climate parameter

Joint Spectral Radius and Path-Complete Graph Lyapunov Functions

We introduce the framework of path-complete graph Lyapunov functions for approximation of the joint spectral radius. The approach is based on the analysis of the underlying switched system via inequalities imposed among multiple Lyapunov functions associated to a labeled directed graph. Inspired by concepts in automata theory and symbolic dynamics, we define a class of graphs called path-complete graphs, and show that any such graph gives rise to a method for proving stability of the switched system. This enables us to derive several asymptotically tight hierarchies of semidefinite programming relaxations that unify and generalize many existing techniques such as common quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov functions. We compare the quality of approximation obtained by certain classes of path-complete graphs including a family of dual graphs and all path-complete graphs with two nodes on an alphabet of two matrices. We provide approximation guarantees for several families of path-complete graphs, such as the De Bruijn graphs, establishing as a byproduct a constructive converse Lyapunov theorem for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has gone through two major rounds of revision. In particular, a section on the performance of our algorithm on application-motivated problems has been added and a more comprehensive literature review is presente

Asynchronous Approximation of a Single Component of the Solution to a Linear System

We present a distributed asynchronous algorithm for approximating a single component of the solution to a system of linear equations $Ax = b$, where $A$ is a positive definite real matrix, and $b \in \mathbb{R}^n$. This is equivalent to solving for $x_i$ in $x = Gx + z$ for some $G$ and $z$ such that the spectral radius of $G$ is less than 1. Our algorithm relies on the Neumann series characterization of the component $x_i$, and is based on residual updates. We analyze our algorithm within the context of a cloud computation model, in which the computation is split into small update tasks performed by small processors with shared access to a distributed file system. We prove a robust asymptotic convergence result when the spectral radius $\rho(|G|) < 1$, regardless of the precise order and frequency in which the update tasks are performed. We provide convergence rate bounds which depend on the order of update tasks performed, analyzing both deterministic update rules via counting weighted random walks, as well as probabilistic update rules via concentration bounds. The probabilistic analysis requires analyzing the product of random matrices which are drawn from distributions that are time and path dependent. We specifically consider the setting where $n$ is large, yet $G$ is sparse, e.g., each row has at most $d$ nonzero entries. This is motivated by applications in which $G$ is derived from the edge structure of an underlying graph. Our results prove that if the local neighborhood of the graph does not grow too quickly as a function of $n$, our algorithm can provide significant reduction in computation cost as opposed to any algorithm which computes the global solution vector $x$. Our algorithm obtains an $\epsilon \|x\|_2$ additive approximation for $x_i$ in constant time with respect to the size of the matrix when the maximum row sparsity $d = O(1)$ and $1/(1-\|G\|_2) = O(1)$

On the complexity of computing the capacity of codes that avoid forbidden difference patterns

We consider questions related to the computation of the capacity of codes that avoid forbidden difference patterns. The maximal number of $n$-bit sequences whose pairwise differences do not contain some given forbidden difference patterns increases exponentially with $n$. The exponent is the capacity of the forbidden patterns, which is given by the logarithm of the joint spectral radius of a set of matrices constructed from the forbidden difference patterns. We provide a new family of bounds that allows for the approximation, in exponential time, of the capacity with arbitrary high degree of accuracy. We also provide a polynomial time algorithm for the problem of determining if the capacity of a set is positive, but we prove that the same problem becomes NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, we prove the existence of extremal norms for the sets of matrices arising in the capacity computation. This result makes it possible to apply a specific (even though non polynomial) approximation algorithm. We illustrate this fact by computing exactly the capacity of codes that were only known approximately.Comment: 7 pages. Submitted to IEEE Trans. on Information Theor

Tropical Kraus maps for optimal control of switched systems

Kraus maps (completely positive trace preserving maps) arise classically in quantum information, as they describe the evolution of noncommutative probability measures. We introduce tropical analogues of Kraus maps, obtained by replacing the addition of positive semidefinite matrices by a multivalued supremum with respect to the L\"owner order. We show that non-linear eigenvectors of tropical Kraus maps determine piecewise quadratic approximations of the value functions of switched optimal control problems. This leads to a new approximation method, which we illustrate by two applications: 1) approximating the joint spectral radius, 2) computing approximate solutions of Hamilton-Jacobi PDE arising from a class of switched linear quadratic problems studied previously by McEneaney. We report numerical experiments, indicating a major improvement in terms of scalability by comparison with earlier numerical schemes, owing to the "LMI-free" nature of our method.Comment: 15 page

The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201