Inertialess gyrating engines

A typical model for a gyrating engine consists of an inertial wheel powered by an energy source that generates an angle-dependent torque. Examples of such engines include a pendulum with an externally applied torque, Stirling engines, and the Brownian gyrating engine. Variations in the torque are averaged out by the inertia of the system to produce limit cycle oscillations. While torque generating mechanisms are also ubiquitous in the biological world, where they typically feed on chemical gradients, inertia is not a property that one naturally associates with such processes. In the present work, seeking ways to dispense of the need for inertial effects, we study an inertia-less concept where the combined effect of coupled torque-producing components averages out variations in the ambient potential and helps overcome dissipative forces to allow sustained operation for vanishingly small inertia. We exemplify this inertia-less concept through analysis of two of the aforementioned engines, the Stirling engine, and the Brownian gyrating engine. An analogous principle may be sought in biomolecular processes as well as in modern-day technological engines, where for the latter, the coupled torque-producing components reduce vibrations that stem from the variability of the generated torque.

Energy harvesting from anisotropic fluctuations

We consider a rudimentary model for a heat engine, known as the Brownian gyrator, that consists of an overdamped system with two degrees of freedom in an anisotropic temperature field. Whereas the hallmark of the gyrator is a nonequilibrium steady-state curl-carrying probability current that can generate torque, we explore the coupling of this natural gyrating motion with a periodic actuation potential for the purpose of extracting work. We show that path lengths traversed in the manifold of thermodynamic states, measured in a suitable Riemannian metric, represent dissipative losses, while area integrals of a work density quantify work being extracted. Thus, the maximal amount of work that can be extracted relates to an isoperimetric problem, trading off area against length of an encircling path. We derive an isoperimetric inequality that provides a universal bound on the efficiency of all cyclic operating protocols, and a bound on how fast a closed path can be traversed before it becomes impossible to extract positive work. The analysis presented provides guiding principles for building autonomous engines that extract work from anisotropic fluctuations.

Underdamped stochastic thermodynamic engines in contact with a heat bath with arbitrary temperature profile

We study thermodynamic processes in contact with a heat bath that may have an arbitrary time-varying periodic temperature profile. Within the framework of stochastic thermodynamics, and for models of thermodynamic engines in the idealized case of underdamped particles in the low-friction regime subject to a harmonic potential, we derive explicit bounds as well as optimal control protocols that draw maximum power and achieve maximum efficiency at any specified level of power.

Rotated Spectral Principal Component Analysis (rsPCA) for Identifying Dynamical Modes of Variability in Climate Systems

Spectral PCA (sPCA), in contrast to classical PCA, offers the advantage of identifying organized spatiotemporal patterns within specific frequency bands and extracting dynamical modes. However, the unavoidable trade-off between frequency resolution and robustness of the PCs leads to high sensitivity to noise and overfitting, which limits the interpretation of the sPCA results. We propose herein a simple nonparametric implementation of sPCA using the continuous analytic Morlet wavelet as a robust estimator of the cross-spectral matrices with good frequency resolution. To improve the interpretability of the results, especially when several modes of similar amplitude exist within the same frequency band, we propose a rotation of the complex-valued eigenvectors to optimize their spatial regularity (smoothness). The developed method, called rotated spectral PCA (rsPCA), is tested on synthetic data simulating propagating waves and shows impressive performance even with high levels of noise in the data. Applied to global historical geopotential height (GPH) and sea surface temperature (SST) daily time series, the method accurately captures patterns of atmospheric Rossby waves at high frequencies (3-60-day periods) in both GPH and SST and El Niño-Southern Oscillation (ENSO) at low frequencies (2-7-yr periodicity) in SST. At high frequencies the rsPCA successfully unmixes the identified waves, revealing spatially coherent patterns with robust propagation dynamics.

Fractional SIR epidemiological models

The purpose of this work is to make a case for epidemiological models with fractional exponent in the contribution of sub-populations to the incidence rate. More specifically, we question the standard assumption in the literature on epidemiological models, where the incidence rate dictating propagation of infections is taken to be proportional to the product between the infected and susceptible sub-populations; a model that relies on strong mixing between the two groups and widespread contact between members of the groups. We contend, that contact between infected and susceptible individuals, especially during the early phases of an epidemic, takes place over a (possibly diffused) boundary between the respective sub-populations. As a result, the rate of transmission depends on the product of fractional powers instead. The intuition relies on the fact that infection grows in geographically concentrated cells, in contrast to the standard product model that relies on complete mixing of the susceptible to infected sub-populations. We validate the hypothesis of fractional exponents (1) by numerical simulation for disease propagation in graphs imposing a local structure to allowed disease transmissions and (2) by fitting the model to the JHU CSSE COVID-19 Data for the period Jan-22-20 to April-30-20, for the countries of Italy, Germany, France, and Spain.

Relaxed Schrödinger bridges and robust network routing

We seek network routing towards a desired final distribution that can mediate possible random link failures. In other words, we seek a routing plan that utilizes alternative routes so as to be relatively robust to link failures. To this end, we provide a mathematical formulation of a relaxed transport problem where the final distribution only needs to be close to the desired one. The problem is cast as a maximum entropy problem for probability distributions on paths with an added terminal cost. The entropic regularizing penalty aims at distributing the choice of paths amongst possible alternatives. We prove that the unique solution may be obtained by solving a generalized Schrödinger system of equations. An iterative algorithm to compute the solution is provided. Each iteration of the algorithm contracts the distance (in the Hilbert metric) to the optimal solution by more than 1/2, leading to extremely fast convergence.

Fractional SIR Epidemiological Models

The purpose of this work is to make a case for epidemiological models with fractional exponent in the contribution of sub-populations to the transmission rate. More specifically, we question the standard assumption in the literature on epidemiological models, where the transmission rate dictating propagation of infections is taken to be proportional to the product between the infected and susceptible sub-populations; a model that relies on strong mixing between the two groups and widespread contact between members of the groups. We content, that contact between infected and susceptible individuals, especially during the early phases of an epidemic, takes place over a (possibly diffused) boundary between the respective sub-populations. As a result, the rate of transmission depends on the product of fractional powers instead. The intuition relies on the fact that infection grows in geographically concentrated cells, in contrast to the standard product model that relies on complete mixing of the susceptible to infected sub-populations. We validate the hypothesis of fractional exponents i) by numerical simulation for disease propagation in graphs imposing a local structure to allowed disease transmissions and ii) by fitting the model to a COVID-19 data set provided by John Hopkins University (JHUCSSE) for the period Jan-31–20 to Mar-24–20, for the countries of Italy, Germany, Iran, and France.

Network curvature as a hallmark of brain structural connectivity

Although brain functionality is often remarkably robust to lesions and other insults, it may be fragile when these take place in specific locations. Previous attempts to quantify robustness and fragility sought to understand how the functional connectivity of brain networks is affected by structural changes, using either model-based predictions or empirical studies of the effects of lesions. We advance a geometric viewpoint relying on a notion of network curvature, the so-called Ollivier-Ricci curvature. This approach has been proposed to assess financial market robustness and to differentiate biological networks of cancer cells from healthy ones. Here, we apply curvature-based measures to brain structural networks to identify robust and fragile brain regions in healthy subjects. We show that curvature can also be used to track changes in brain connectivity related to age and autism spectrum disorder (ASD), and we obtain results that are in agreement with previous MRI studies.

The brain can often continue to function despite lesions in many areas, but damage to particular locations may have serious effects. Here, the authors use the concept of Ollivier-Ricci curvature to investigate the robustness of brain networks.

Optimal transport for Gaussian mixture models

We introduce an optimal mass transport framework on the space of Gaussian mixture models. These models are widely used in statistical inference. Specifically, we treat Gaussian mixture models as a submanifold of probability densities equipped with the Wasserstein metric. The topology induced by optimal transport is highly desirable and natural because, in contrast to total variation and other metrics, the Wasserstein metric is weakly continuous (i.e., convergence is equivalent to convergence of moments). Thus, our approach provides natural ways to compare, interpolate and average Gaussian mixture models. Moreover, the approach has low computational complexity. Different aspects of the framework are discussed and examples are presented for illustration purposes.

Matricial Wasserstein-1 Distance

We propose an extension of the Wasserstein 1-metric (W₁) for density matrices, matrix-valued density measures, and an unbalanced interpretation of mass transport. We use duality theory and, in particular, a "dual of the dual" formulation of W₁. This matrix analogue of the Earth Mover's Distance has several attractive features including ease of computation.

Ricci curvature: An economic indicator for market fragility and systemic risk

Quantifying the systemic risk and fragility of financial systems is of vital importance in analyzing market efficiency, deciding on portfolio allocation, and containing financial contagions. At a high level, financial systems may be represented as weighted graphs that characterize the complex web of interacting agents and information flow (for example, debt, stock returns, and shareholder ownership). Such a representation often turns out to provide keen insights. We show that fragility is a system-level characteristic of "business-as-usual" market behavior and that financial crashes are invariably preceded by system-level changes in robustness. This was done by leveraging previous work, which suggests that Ricci curvature, a key geometric feature of a given network, is negatively correlated to increases in network fragility. To illustrate this insight, we examine daily returns from a set of stocks comprising the Standard and Poor's 500 (S&P 500) over a 15-year span to highlight the fact that corresponding changes in Ricci curvature constitute a financial "crash hallmark." This work lays the foundation of understanding how to design (banking) systems and policy regulations in a manner that can combat financial instabilities exposed during the 2007-2008 crisis.

Linear Models Based on Noisy Data and the Frisch Scheme

We address the problem of identifying linear relations among variables based on noisy measurements. This is a central question in the search for structure in large data sets. Often a key assumption is that measurement errors in each variable are independent. This basic formulation has its roots in the work of Charles Spearman in 1904 and of Ragnar Frisch in the 1930s. Various topics such as errors-in-variables, factor analysis, and instrumental variables all refer to alternative viewpoints on this problem and on ways to account for the anticipated way that noise enters the data. In the present paper we begin by describing certain fundamental contributions by the founders of the field and provide alternative modern proofs to certain key results. We then go on to consider a modern viewpoint and novel numerical techniques to the problem. The central theme is expressed by the Frisch-Kalman dictum, which calls for identifying a noise contribution that allows a maximal number of simultaneous linear relations among the noise-free variables-a rank minimization problem. In the years since Frisch's original formulation, there have been several insights, including trace minimization as a convenient heuristic to replace rank minimization. We discuss convex relaxations and theoretical bounds on the rank that, when met, provide guarantees for global optimality. A complementary point of view to this minimum-rank dictum is presented in which models are sought leading to a uniformly optimal quadratic estimation error for the error-free variables. Points of contact between these formalisms are discussed, and alternative regularization schemes are presented.

On Matrix-Valued Monge-Kantorovich Optimal Mass Transport

We present a particular formulation of optimal transport for matrix-valued density functions. Our aim is to devise a geometry which is suitable for comparing power spectral densities of multivariable time series. More specifically, the value of a power spectral density at a given frequency, which in the matricial case encodes power as well as directionality, is thought of as a proxy for a "matrix-valued mass density." Optimal transport aims at establishing a natural metric in the space of such matrix-valued densities which takes into account differences between power across frequencies as well as misalignment of the corresponding principle axes. Thus, our transportation cost includes a cost of transference of power between frequencies together with a cost of rotating the principle directions of matrix densities. The two endpoint matrix-valued densities can be thought of as marginals of a joint matrix-valued density on a tensor product space. This joint density, very much as in the classical Monge-Kantorovich setting, can be thought to specify the transportation plan. Contrary to the classical setting, the optimal transport plan for matrices is no longer supported on a thin zero-measure set.

Noninvasive estimation of tissue temperature via high-resolution spectral analysis techniques

We address the noninvasive temperature estimation from pulse-echo radio frequency signals from standard diagnostic ultrasound imaging equipment. In particular, we investigate the use of a high-resolution spectral estimation method for tracking frequency shifts at two or more harmonic frequencies associated with temperature change. The new approach, employing generalized second-order statistics, is shown to produce superior frequency shift estimates when compared to conventional high-resolution spectral estimation methods Seip and Ebbini (1995). Furthermore, temperature estimates from the new algorithm are compared with results from the more commonly used echo shift method described in Simon et al. (1998).