Perron Frobenius Research Articles

Tight-Binding Modeling of Nucleic Acid Sequences: Interplay between Various Types of Order or Disorder and Charge Transport

This review is devoted to tight-binding (TB) modeling of nucleic acid sequences like DNA and RNA. It addresses how various types of order (periodic, quasiperiodic, fractal) or disorder (diagonal, non-diagonal, random, methylation et cetera) affect charge transport. We include an introduction to TB and a discussion of its various submodels [wire, ladder, extended ladder, fishbone (wire), fishbone ladder] and of the process of renormalization. We proceed to a discussion of aperiodicity, quasicrystals and the mathematics of aperiodic substitutional sequences: primitive substitutions, Perron–Frobenius eigenvalue, induced substitutions, and Pisot property. We discuss the energy structure of nucleic acid wires, the coupling to the leads, the transmission coefficients and the current–voltage curves. We also summarize efforts aiming to examine the potentiality to utilize the charge transport characteristics of nucleic acids as a tool to probe several diseases or disorders.

Mean field systems on networks, with singular interaction through hitting times

Building on the line of work (Ann. Appl. Probab. 25 (2015) 2096–2133; Stochastic Process. Appl. 125 (2015) 2451–2492; Ann. Appl. Probab. 29 (2019) 89–129; Arch. Ration. Mech. Anal. 233 (2019) 643–699; Ann. Appl. Probab. 29 (2019) 2338–2373; Finance Stoch. 23 (2019) 535–594), we continue the study of particle systems with singular interaction through hitting times. In contrast to the previous research, we (i) consider very general driving processes and interaction functions, (ii) allow for inhomogeneous connection structures and (iii) analyze a game in which the particles determine their connections strategically. Hereby, we uncover two completely new phenomena. First, we characterize the “times of fragility” of such systems (e.g., the times when a macroscopic part of the population defaults or gets infected simultaneously, or when the neuron cells “synchronize”) explicitly in terms of the dynamics of the driving processes, the current distribution of the particles’ values and the topology of the underlying network (represented by its Perron–Frobenius eigenvalue). Second, we use such systems to describe a dynamic credit-network game and show that, in equilibrium, the system regularizes, that is, the times of fragility never occur, as the particles avoid them by adjusting their connections strategically. Two auxiliary mathematical results, useful in their own right, are uncovered during our investigation: a generalization of Schauder’s fixed-point theorem for the Skorokhod space with the M1 topology, and the application of max-plus algebra to the equilibrium version of the network flow problem.

Nonnegative Eigenvectors of Symmetric Matrices

For matrices with all nonnegative entries, the Perron–Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different class of matrices, namely real symmetric matrices with exactly two eigenvalues. We also prove a partial converse, that among real symmetric matrices with any more than two eigenvalues there exist some having no nonnegative eigenvector.

Robust verification algorithm for stabilizing solutions of discrete-time algebraic Riccati equations

A robust algorithm is proposed for numerically computing an interval matrix containing the stabilizing solution of a discrete-time algebraic Riccati equation. This algorithm is based on estimating an upper bound for the spectral radius of a matrix power utilizing the Perron–Frobenius theory. The algorithm moreover verifies the uniqueness of the contained solution. Numerical results show that the algorithm is more successful than the previous algorithms.

Estimating contaminant distribution from finite sensor data: Perron Frobenious operator and ensemble Kalman Filtering

Accurate and rapid monitoring of indoor air quality is critical to ensure occupant safety in the built environment. This is especially important in events where hazardous substances are released, and prompt estimation of contaminant distribution will facilitate quick evacuation and control response. The built environment is usually equipped with a finite set of sensors that measure local concentration of contaminants. The goal is to use this streaming dataset to estimate the contaminant concentration distribution in the complete domain. We accomplish this by integrating two powerful concepts. We utilize an operator theoretic approach – specifically the Perron-Frobenius (P-F) operator – to model the contaminant transport. Previous work has shown that the PF approach is a fast, effective, and accurate paradigm for sensor placement and contaminant transport prediction. The PF approach is integrated with an Ensemble Kalman Filter to rapidly estimate contaminant distribution under unknown release scenarios, given minimal sensor data. The framework is illustrated for two scenarios: a 2D problem involving an office space, and a 3D problem involving a furnished hotel room. Both examples show that the contaminant distribution is accurately predicted within a few sensor measurement cycles. The general applicability of the framework is illustrated by testing the framework for multiple, unknown release locations. This approach provides a unified, extendable framework for rapid contaminant estimation.

A linear model for ranking soccer teams

In [1] an interesting in-depth analysis on possible ranking methods, mainly concentrated on football teams ranking, may be found. Within certain hypotheses the ranking problem may be easily formulated as a linear eigenvalue problem. In this paper we present the application of the linear model to the recent editions of the Italian national soccer championship, showing that the results from the model give some significant differences with respect to the final official ordering.We recall first some important mathematical definitions and results, which are relevant in the linear model. Among these, the concepts of eigenvalues and eigenvectors of a square matrix, together with the fundamental properties of the eigenvalues of positive matrices, namely Perron–Frobenius theorem on the so called dominant eigenvalue. Then we collect the data, mainly the final results of the matches played by the teams in the Italian championship, in order to have the matrix of the model. The computation of the dominant eigenvalue and the corresponding eigenvector gives us the solution of the model, that is a final ordering for the teams in the championship. Finally we try to interpret the result, by comparing this to the official final ordering.

Ray pattern matrices requiring Perron–Frobenius properties

A ray pattern matrix is a matrix whose nonzero entries are all unimodular. An n×n ray pattern matrix A is said to require the weak Perron–Frobenius property if for any n×n entrywise positive matrix K, the spectral radius ρ(K∘A) is one of the eigenvalues of K∘A; to require the strong Perron–Frobenius property if ρ(K∘A) is an algebraically simple eigenvalue of K∘A and the left and right eigenvectors associated to ρ(K∘A) are strictly nonzero. In this paper, necessary and sufficient conditions for ray pattern matrices requiring the weak/strong Perron–Frobenius property are given.

The persistent statistical structure of the US input–output coefficient matrices: 1963–2007

ABSTRACTThe paper finds evidence for the existence of a statistical structure in the US input–output coefficient matrices for 1963–2007 and characterizes the identified statistical regularities. For various aspects of matrices, we find smooth and unimodal empirical distributions (EDs) with a remarkable stability in their functional form for most of the samples. The EDs of all entries, row sums, and the entries of the (left- and right-hand) Perron–Frobenius eigenvectors are well described by fat-tailed distributions, while the EDs of column sums and eigenvalues' moduli are explained by the normal and the beta distribution. The paper provides several economic interpretations of these statistical results as well as some implications and potential uses for structural and stochastic input–output analysis.

Galton–Watson and branching process representations of the normalized Perron–Frobenius eigenvector

Let A be a primitive matrix and let λ be its Perron–Frobenius eigenvalue. We give formulas expressing the associated normalized Perron–Frobenius eigenvector as a simple functional of a multitype Galton–Watson process whose mean matrix is A, as well as of a multitype branching process with mean matrix e(A−I)t. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.

New criteria for mean square exponential stability of stochastic systems with variable and distributed delays

In this study, the authors are interested in the mean square exponential stability of stochastic systems with variable and distributed delays. Different from the traditional methods, based on the well-known Perron–Frobenius theorem and Ito formula, a proof by contradiction to explore some new criteria for the mean square exponential stability of stochastic delay systems is introduced. In particular, the proposed novel stability criteria reduce the traditional restrictions imposed on variable delays and also provide an optimal upper bound for delays. Two examples are given as applications to verify the effectiveness of the obtained results.

Looking for alternatives to the superspace description of icosahedral quasicrystals.

A multiple-cell approach is discussed as a possible alternative to the higher dimensional crystallography of icosahedral quasicrystals. It is based on the Socolar-Steinhardt tiling combined with the quasi-unit cell model. Quasi-unit cells fill the space without gaps and overlappings similar to those in periodic crystals. Similarly, the atoms can occupy general and special positions. The alloy stoichiometry and the packing density can be calculated through the relative tile frequencies, which in turn are determined as the components of the Perron-Frobenius eigenvector of the corresponding substitution matrix. The calculation of the diffraction pattern reduces to the Perron projection of a special matrix, the entries of which reflect the contribution of each type of quasi-unit cell to the coherent scattering.

Sign‐Consensus of Linear Multiagent Systems under a State Observer Protocol

The sign‐consensus problem for linear time‐invariant systems under signed digraph is considered. The information of the agents’ states is reconstructed, and then, a state observer‐type sign‐consensus protocol is proposed, whose performance is analyzed using matrix analysis and ordinary differential equation theory. Sufficient conditions for ensuring sign‐consensus are given. It is proven that if the adjacency matrix of the signed digraph has strong Perron–Frobenius property or is eventually positive, sign‐consensus can be achieved under the proposed protocol. In particular, conventional consensus is a special case of sign‐consensus under mild conditions.

The weak Bernoulli property for matrix Gibbs states

We study the ergodic properties of a class of measures on$\unicode[STIX]{x1D6F4}^{\mathbb{Z}}$for which$\unicode[STIX]{x1D707}_{{\mathcal{A}},t}[x_{0}\cdots x_{n-1}]\approx e^{-nP}\Vert A_{x_{0}}\cdots A_{x_{n-1}}\Vert ^{t}$, where${\mathcal{A}}=(A_{0},\ldots ,A_{M-1})$is a collection of matrices. The measure$\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$is called a matrix Gibbs state. In particular, we give a sufficient condition for a matrix Gibbs state to have the weak Bernoulli property. We employ a number of techniques to understand these measures, including a novel approach based on Perron–Frobenius theory. We find that when$t$is an even integer the ergodic properties of$\unicode[STIX]{x1D707}_{{\mathcal{A}},t}$are readily deduced from finite-dimensional Perron–Frobenius theory. We then consider an extension of this method to$t>0$using operators on an infinite- dimensional space. Finally, we use a general result of Bradley to prove the main theorem.

A NOTE ON RÉNYI'S ENTROPY RATE FOR TIME-INHOMOGENEOUS MARKOV CHAINS

AbstractIn this note, we use the Perron–Frobenius theorem to obtain the Rényi's entropy rate for a time-inhomogeneous Markov chain whose transition matrices converge to a primitive matrix. As direct corollaries, we also obtain the Rényi's entropy rate for asymptotic circular Markov chain and the Rényi's divergence rate between two time-inhomogeneous Markov chains.

Perron–Frobenius theory recovers more than you might think: The example of limited participation

In their seminal article, Hansen and Scheinkman (2009) proved that Perron–Frobenius theory helps to recover a probability measure that can be used to price long-term claims. In this paper, we show that the recovered probability also contains information about market structure. More precisely, we provide an example in which Perron–Frobenius theory can be used to measure the degree of limited market participation.

The steady-state lifting bifurcation problem associated with the valency on networks

In this paper we consider homogeneous coupled cell networks with asymmetric inputs — networks where each cell receives exactly one input of each edge type. The coupled cell systems associated with a network are the dynamical systems that respect the network structure. There are subspaces, determined solely by the network structure, that are flow-invariant under any such coupled cell system — the synchrony subspaces. For a homogeneous network with asymmetric inputs, one of the eigenvalues of the Jacobian matrix of any coupled cell system at an equilibrium in the full-synchrony subspace corresponds to the valency of the network. In this work, we study the codimension-one steady-state bifurcations of coupled cell systems with a bifurcation condition associated with the valency. We start by giving an adaptation of the Perron–Frobenius Theorem for the eigenspace associated with the valency showing that the dimension of that eigenspace equals the number of the network source components. A network source component is a strongly connected component of the network whose cells receive inputs only from cells in the component. Each synchrony subspace determines a smaller network called quotient network. The lifting bifurcation problem addresses the issue of understanding when the bifurcation branches of a network can be lifted from one of its quotient networks. We consider the lifting bifurcation problem when the bifurcation condition is associated with the valency. We give sufficient conditions on the number of source components for the answer to the lifting bifurcation problem to be positive and prove that those conditions are necessary and sufficient for a class of networks.

Simulating realistic correlation matrices for financial applications: correlation matrices with the Perron–Frobenius property

ABSTRACTThis article is concerned with the simulation of correlation matrices with realistic properties for financial applications. Previous studies found that the major part of observed correlation matrices in financial applications exhibits the Perron-Frobenius property, namely a dominant eigenvector with only positive entries. We present a simulation algorithm for random correlation matrices satisfying this property, which can be augmented to take into account a realistic eigenvalue structure. From the construction principle applied in our algorithm, and the fact that it is able to generate all such correlation matrices, we are further able to prove that the proportion of Perron-Frobenius correlation matrices in the set of all correlation matrices is in dimension d.

A Perron–Frobenius type result for integer maps and applications

It is shown that for certain maps, including concave maps, on the d-dimensional lattice of positive integer points, ‘approximate’ eigenvectors can be found. Applications in epidemiology as well as distributed resource allocation are discussed as examples.

Perron–Frobenius theorem for hypermatrices in the max algebra

Recently, there have been many intriguing new developments in the study of hypermatrices and their associated eigenvalue problems. In particular, results coming from the matrix setting when studying the max algebra have shown especially attractive combinatorial features. We now extend this max algebra setting into the realm of hypermatrices. Considering that the max algebra has shown particular significance in optimization problems for the matrix setting, we look to examine and extend these results in the higher order conditions. Furthermore, we establish some algebraic properties for hypermatrices and then proceed to extend the Perron–Frobenius Theorem for this setting and prove the existence of a unique eigenvalue. We continue by stating a result from Nussbaum, that the Min–Max theorem holds, and provide a proof for completeness. For strongly increasing hypermatrices, an iterative algorithm which converges to our unique eigenvalue is given. Finally, we conclude with an analysis of our results in the hypergraph setting.

Constrained Ulam Dynamic Mode Decomposition: Approximation of the Perron-Frobenius Operator for Deterministic and Stochastic Systems

Dynamical systems described by ordinary and stochastic differential equations can be analyzed through the eigen-decomposition of the Perron–Frobenius (PF) and Koopman transfer operators. While the Koopman operator may be approximated by data-driven techniques, e.g., extended dynamic mode decomposition (EDMD), the approximation of the PF operator uses a single-pass Monte Carlo approach in Ulam’s method, which requires a sufficiently long time step. This letter proposes a finite-dimensional approximation technique for the PF operator that uses multi-pass Monte Carlo data to pose and solve a constrained EDMD-like least-squares problem to approximate the PF operator on a finite-dimensional basis. The basis functions used to project the PF operator are the characteristic functions of the state-space partitions. The results are analyzed theoretically and illustrated using deterministic and time-homogeneous stochastic systems.