Author

# David Ruelle

Other affiliations: IHS Inc., Carnegie Mellon University, Rutgers University ...read more

Bio: David Ruelle is an academic researcher from Institut des Hautes Études Scientifiques. The author has contributed to research in topic(s): Statistical mechanics & Dynamical systems theory. The author has an hindex of 72, co-authored 230 publication(s) receiving 31960 citation(s). Previous affiliations of David Ruelle include IHS Inc. & Carnegie Mellon University.

Topics: Statistical mechanics, Dynamical systems theory, Ergodic theory, Attractor, Differentiable function

##### Papers

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Abstract: Let z be the activity of point particles described by classical equilibrium statistical mechanics in
$$\mathbf{R}^
u $$
. The correlation functions
$$\rho ^z(x_1,\dots ,x_k)$$
denote the probability densities of finding k particles at
$$x_1,\dots ,x_k$$
. Letting
$$\phi ^z(x_1,\dots ,x_k)$$
be the cluster functions corresponding to the
$$\rho ^z(x_1,\dots ,x_k)/z^k$$
we prove identities of the type
$$\begin{aligned}&\phi ^{z_0+z'}(x_1,\dots ,x_k)\\&\quad =\sum _{n=0}^\infty {z'^n\over n!}\int dx_{k+1}\dots \int dx_{k+n}\,\phi ^{z_0}(x_1,\dots ,x_{k+n}) \end{aligned}$$
It is then non-rigorously argued that, assuming a suitable cluster property (decay of correlations) for a crystal state, the pressure and the translation invariant correlation functions
$$\rho ^z(x_1,\dots ,x_k)$$
are real analytic functions of z.

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Abstract: Let $z$ be the activity of point particles described by classical equilibrium statistical mechanics in ${\bf R}^
u$. The correlation functions $\rho^z(x_1,\dots,x_k)$ denote the probability densities of finding $k$ particles at $x_1,\dots,x_k$. Letting $\phi^z(x_1,\dots,x_k)$ be the cluster functions corresponding to the $\rho^z(x_1,\dots,x_k)/z^k$ we prove identities of the type $$ \phi^{z_0+z'}(x_1,\dots,x_k) $$ $$ =\sum_{n=0}^\infty{z'^n\over n!}\int dx_{k+1}\dots\int dx_{k+n}\,\phi^{z_0}(x_1,\dots,x_{k+n}) $$ It is then non-rigorously argued that, assuming a suitable cluster property (decay of correlations) for a crystal state, the pressure and the translation invariant correlation functions \- $\rho^z(x_1,\dots,x_k)$ are real analytic functions of $z$.

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Abstract: This note presents a non-rigorous study of the linear response for an SRB (or 'natural physical') measure ρ of a diffeomorphism f in the presence of tangencies of the stable and unstable manifolds of ρ. We propose that generically, if ρ has no zero Lyapunov exponent, if its stable dimension is sufficiently large (greater than 1/2 or perhaps 3/2) and if it is exponentially mixing in a suitable sense, then the following formal expression for the first derivative of with respect to f along X is convergent: This suggests that an SRB measure may exist for small perturbations of f, with weak differentiability.

5 citations

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Abstract: If $$\mathcal{F}$$
is a set of subgraphs F of a finite graph E we define a graph-counting polynomial $$p_\mathcal{F}(z)=\sum _{F\in \mathcal{F}}z^{|F|}$$
In the present note we consider oriented graphs and discuss some cases where $$\mathcal{F}$$
consists of unbranched subgraphs E. We find several situations where something can be said about the location of the zeros of $$p_\mathcal{F}$$
.

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Abstract: If ${\cal F}$ is a set of subgraphs $F$ of a finite graph $E$ we define a graph-counting polynomial $$ p_{\cal F}(z)=\sum_{F\in{\cal F}}z^{|F|} $$ In the present note we consider oriented graphs and discuss some cases where ${\cal F}$ consists of unbranched subgraphs $E$. We find several situations where something can be said about the location of the zeros of $p_{\cal F}$.

1 citations

##### Cited by

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Abstract: Consider a compact metric space $$(M, d_M)$$
and $$X = M^{{\mathbb {N}}}$$
. We prove a Ruelle’s Perron Frobenius Theorem for a class of compact subshifts with Markovian structure introduced in da Silva et al. (Bull Braz Math Soc 45:53–72, 2014) which are defined from a continuous function $$A : M \times M \rightarrow {\mathbb {R}}$$
that determines the set of admissible sequences. In particular, this class of subshifts includes the finite Markov shifts and models where the alphabet is given by the unit circle $$S^1$$
. Using the involution Kernel, we characterize the normalized eigenfunction of the Ruelle operator associated to its maximal eigenvalue and present an extension of its corresponding Gibbs state to the bilateral approach. From these results, we prove existence of equilibrium states and accumulation points at zero temperature in a particular class of countable Markov shifts.

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Abstract: Cardiac arrhythmia refers to irregularities in heartbeats. Left undiagnosed arrhythmias can cause severe and potentially fatal complications. As a result, early finding of such abnormalities is critical. Electrocardiogram (ECG) is regularly used by medical professionals to diagnose and differentiate cardiac arrhythmias. As a result, there have been many deep learning methods over the years in an attempt to automate this process. But traditional deep learning methods require big training data which often clearly do not reflect the age, weight and gender spectrum of patients and are prone to misclassification when data from different demographics is shown. Hence, temporal features extracted from these datasets are demographically biased. Consequently, in this paper, we intend to introduce Optimum Recurrence Plot based Classifier (OptRPC); a dynamical systems-based method of classifying ECG beats by embedding them in higher dimensions and devising an optimized recurrence plot. A Convolutional Neural Network architecture is then used to classify these recurrence plots. The proposed scheme accomplished an overall accuracy of 98.67% and 98.48% on two benchmark databases and delivered better performance than the previous state-of-the-art methods.

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Anastasia Pentari

^{1}, Anastasia Pentari^{2}, George Tzagkarakis^{2}, Panagiotis Tsakalides^{1}+11 more•Institutions (3)Abstract: There is growing interest in dynamic approaches to functional brain connectivity (FC), and their potential applications in understanding atypical brain function. In this study, we assess the relative sensitivity of cross recurrence quantification analysis CRQA) to identify aberrant FC in patients with neuropsychiatric systemic lupus erythematosus (NPSLE) in comparison with conventional static and dynamic bivariate FC measures, as well as univariate (nodal) RQA. This technique was applied to resting-state fMRI data obtained from 45 NPSLE patients and 35 healthy volunteers (HC). Cross recurrence plots were computed for all pairs of 16 frontoparietal brain regions known to be critically involved in visuomotor control and suspected to show hemodynamic disturbance in NPSLE. Multivariate group comparisons revealed that the combination of six CRQA measures differentiated the two groups with large effect sizes ( . 214 > η 2 > . 061 ) in 40 out of the 120 region pairs. The majority of brain regions forming these pairs also showed group differences on nodal RQA indices ( . 146 > η 2 > . 09 ) Overall, larger values were found in NPSLE patients vs. HC with the exception of FC formed by the paracentral lobule. Determinism within five pairs of right-hemisphere sensorimotor regions (paracentral lobule, primary somatosensory, primary motor, and supplementary motor areas), correlated positively with visuomotor performance among NPSLE patients ( p . 001 ). By comparison, group differences on static FC displayed large effect sizes in only 4 of the 120 region pairs ( . 126 > η 2 > . 061 ), none of which correlated significantly with visuomotor performance. Indices derived from dynamic, temporal-based FC analyses displayed large effect sizes in 11 / 120 region pairs ( . 11 > η 2 > . 063 ). These findings further support the importance of feature-based dFC in advancing current knowledge on correlates of cognitive dysfunction in a clinically challenging disorder, such as NPSLE.

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José M. Amigó

^{1}, Roberto Dale^{1}, Piergiulio Tempesta^{2}, Piergiulio Tempesta^{3}•Institutions (3)Abstract: This is a paper in the intersection of time series analysis and complexity theory that presents new results on permutation complexity in general and permutation entropy in particular. In this context, permutation complexity refers to the characterization of time series by means of ordinal patterns (permutations), entropic measures, decay rates of missing ordinal patterns, and more. Since the inception of this “ordinal” methodology, its practical application to any type of scalar time series and real-valued processes have proven to be simple and useful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of allowed permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of allowed permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. The next step is to adapt the concept of Z-entropy to each of those classes, which we call permutation entropy because it coincides with the conventional permutation entropy on the exponential class. Z-entropies are a family of group entropies, each of them extensive on a given complexity class. The result is a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise, with new concepts and tools. Numerical simulations show that permutation entropy discriminates time series from all complexity classes.