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The Sparse Identification of Nonlinear Dynamics (SINDy) algorithm can be applied to stochastic differential equations to estimate the drift and the diffusion function using data from a realization of the SDE. The SINDy algorithm requires sample data from each of these functions, which is typically estimated numerically from the data of the state. We analyze the performance of the previously proposed estimates for the drift and diffusion function to give bounds on the error for finite data. However, since this algorithm only converges as both the sampling frequency and the length of trajectory go to infinity, obtaining approximations within a certain tolerance may be infeasible. To combat this, we develop estimates with higher orders of accuracy for use in the SINDy framework. For a given sampling frequency, these estimates give more accurate approximations of the drift and diffusion functions, making SINDy a far more feasible system identification method., 25 pages, 6 figures and 1 table

At the core of many machine learning methods resides an iterative optimization algorithm for their training. Such optimization algorithms often come with a plethora of choices regarding their implementation. In the case of deep neural networks, choices of optimizer, learning rate, batch size, etc. must be made. Despite the fundamental way in which these choices impact the training of deep neural networks, there exists no general method for identifying when they lead to equivalent, or non-equivalent, optimization trajectories. By viewing iterative optimization as a discrete-time dynamical system, we are able to leverage Koopman operator theory, where it is known that conjugate dynamics can have identical spectral objects. We find highly overlapping Koopman spectra associated with the application of online mirror and gradient descent to specific problems, illustrating that such a data-driven approach can corroborate the recently discovered analytical equivalence between the two optimizers. We extend our analysis to feedforward, fully connected neural networks, providing the first general characterization of when choices of learning rate, batch size, layer width, data set, and activation function lead to equivalent, and non-equivalent, evolution of network parameters during training. Among our main results, we find that learning rate to batch size ratio, layer width, nature of data set (handwritten vs. synthetic), and activation function affect the nature of conjugacy. Our data-driven approach is general and can be utilized broadly to compare the optimization of machine learning methods., Comment: 15 pages, 6 figures, 1 appendix

To achieve the desired biological activity of synthetically produced peptides, several properties are important, such as peptide hydrophobicity, net charge, amphipathic nature, and secondary structure. [1] The secondary structure of ferrocene-derived peptidomimetics can be tuned by the number and type of constitutive amino acids, modification of their chirality, and bulkiness of C- and N-protecting groups, starting from the choice of ferrocene backbone: ferrocene amino acid (Fca) or symmetrically disubstituted ferrocene diacid (−OC−Fn−CO−) and diamine (−NH−Fn−NH−). Our study of ferrocene diamine derived peptides started with Y-Ala-NH-Fn-NH-Ala-Y (Ia, Y=Ac, Boc) and was extended by the preparation of their homologues Ac-AA-NH-Fn-NH-AA- Boc (Ib, AA=Val, Leu, Phe). [2-5] To find out how the side chain of amino acids affects the pattern of hydrogen bonding, we decided to replace an Ala residue with hydrophobic L- and D-amino acids (Leu, Val and Phe) to generate peptidomimetics II (Fig. 1). To investigate whether the dipeptides II adopt ordered conformations supported by IHBs in solution, we performed IR, NMR and CD spectroscopic studies.

Compared to traditional, rigid robots, soft robots offer inherent compliance that gives these robots the ability to work safely in close proximity with humans. However, the flexibility of soft robots induces highly nonlinear dynamical behavior which makes them very difficult to model and control. Considering also the infinite number degrees-of-freedom present in such systems, obtaining mathematical models grounded on first-principles and knowledge of material properties is very difficult and generally enables their operating only in linear regime (low deflection and low velocity). To address the above listed issues, a data-driven learning approach based on the Koopman operator framework is used to identify the model of pneumatically driven soft arm. The arm is equipped with a motion capture system to measure its position in workspace. An Extended Dynamic Mode Decomposition (EDMD) with time delay observables is applied to obtain a finite dimensional approximation of control Koopman operator. Such approach results in a globally linear model of the nonlinear soft robot dynamics using only a couple of minutes of training data generated by random step inputs. Based on the obtained linear model, a simple linear state feedback controller is developed and precise, fast, high-deflection displacements to arbitrary reference positions commanded in real-time are achieved.

Iterative algorithms are of utmost importance in decision and control. With an ever growing number of algorithms being developed, distributed, and proprietarized, there is a similarly growing need for methods that can provide classification and comparison. By viewing iterative algorithms as discrete-time dynamical systems, we leverage Koopman operator theory to identify (semi-)conjugacies between algorithms using their spectral properties. This provides a general framework with which to classify and compare algorithms., 6 pages, 5 figures, accepted to IEEE CDC 2022

The TomKat UC Carbon Neutrality Project was established in 2016 to develop and deploy solutions to mitigate climate change by capitalizing on the vast resources and researchers within the University of California (UC) system. The goal of the Project is to support innovative multidisciplinary research projects that will substantially accelerate progress of the University of California Carbon Neutrality Initiative. The Natural Gas Exit Strategies Working Group makes available its final report below. The TomKat UC Carbon Neutrality Project was launched by UC Santa Barbara’s Institute for Energy Efficiency (IEE) in partnership with the National Center for Ecological Analysis and Synthesis (NCEAS) to help advance the University of California's system-wide goal to achieve zero net greenhouse gas emissions from its buildings and vehicle fleet by 2025. Funding for this project was made possible by a gift from the TomKat Foundation together with supplemental funding from the University of California Office of the President.

The original intent of the Koopman–von Neumann formalism was to put classical and quantum mechanics on the same footing by introducing an operator formalism into classical mechanics. Here we pursue their path the opposite way and examine what transfer operators can say about quantum mechanical evolution. To that end, we introduce a physically motivated scalar wavefunction formalism for a velocity field on a 4-dimensional pseudo-Riemannian manifold, and obtain an evolution equation for the associated wavefunction, a generator for an associated weighted transfer operator. The generator of the scalar evolution is of first order in space and time. The probability interpretation of the formalism leads to recovery of the Schrödinger equation in the non-relativistic limit. In the special relativity limit, we show that the scalar wavefunction of Dirac spinors satisfies the new equation. A connection with string theoretic considerations for mass is provided.

Frictional characteristics are an important factor in the design and exploitation of precision positioning systems and components. In fact, friction is a complex stochastic phenomenon depending on an intricate interplay between materials’ physical and chemical properties, as well as on the contact conditions influenced by the exerted normal forces, sliding velocities, contact area, temperature etc. Experimental measurements of the nanoscale friction force of an X39CrMo17-1 high-alloy steel thin film, deposited on a silicon wafer, are obtained in this work via elaborated lateral force microscopy (LFM) measurements performed on a scanning probe microscope using silicon-nitride microcantilever probes. The thin-film samples are synthesised by using pulsed laser deposition, which enables obtaining precise stochiometric elemental properties. The LFM measurements allow obtaining single-asperity contact conditions in the studied tribo-pair, which is studied experimentally considering the mentioned external influences as the variable process parameters. In order to obtain an in-depth insight into the dependencies of nanoscale friction on these parameters, the obtained experimental data is used to train various machine learning (ML) algorithms. The resulting predicted values of the nanoscale friction force are therefore obtained by using random forest, multilayer perceptron and support vector regression ML methods. In order to assess their predictive performances, all the used algorithms are validated on a separately measured test dataset. The best predictive performances are obtained by using the support vector regression algorithm, resulting in the highest achieved coefficient of determination (R2), allowing the prediction of 90 % of the variance of the experimental measurements.

Dynamic Mode Decomposition (DMD) is a data driven spectral analysis technique for a time series. For a sequence of snapshot vectors $\mathbf{; ; f}; ; _1, \mathbf{; ; f}; ; _2, \dots, \mathbf{; ; f}; ; _{; ; m+1}; ; $ in $\mathbb{; ; C}; ; ^{; ; n}; ; $, assumed driven by a linear operator $\mathbb{; ; A}; ; $, $\mathbf{; ; f}; ; _{; ; i+1}; ; =\mathbb{; ; A}; ; \mathbf{; ; f}; ; _i$, the goal is to represent the snapshots in terms of the computed eigenvectors and eigenvalues of $\mathbb{; ; A}; ; $. We can think of $\mathbb{; ; A}; ; $ as a discretization of the underlying physics that drives the measured $\mathbf{; ; f}; ; _i$'s. In a pure data driven setting we have no access to $\mathbb{; ; A}; ; $. Instead, the $\mathbf{; ; f}; ; _i$'s are the results of measurements, e.g. computed from pixel values from a high speed camera recorded video. No other information on the action of $\mathbb{; ; A}; ; $ is available. (In another scenario of data acquisition, $\mathbb{; ; A}; ; $ represents PDE/ODE solver (software toolbox) that generates solution in high resolution, with given initial condition $\mathbf{; ; f}; ; _1$.) Such a representation of the data sequence provides an insight into the evolution of the underlying dynamics, in particular on dynamically relevant spatial structures (eigenvectors) and amplitudes and frequencies of their evolution (encoded in the corresponding eigenvalues) -- it can be considered a finite dimensional realization of the Koopman spectral analysis, corresponding to the Koopman operator associated with the nonlinear dynamics under study \cite{; ; zd:arbabi-mezic-siam-2017}; ; . This important theoretical connection with the Koopman operator and the ergodic theory, and the availability of numerical algorithm \cite{; ; zd:schmid2010}; ; make the DMD a tool of trade in computational study of complex phenomena in fluid dynamics, see e.g. \cite{; ; zd:Williams2015}; ; . Its exceptional performance motivated developments of several modifications that make the DMD an attractive method for analysis and model reduction of nonlinear systems in data driven settings. A peculiarity of the data driven setting is that direct access to the operator is not available, thus an approximate representation of $\mathbb{; ; A}; ; $ is achieved using solely the snapshot vectors $\mathbf{; ; f}; ; _i$ whose number $m+1$ is usually much smaller than the dimension $n$ of the domain of $\mathbb{; ; A}; ; $. In this talk, we present recent development of numerically robust computational tools for dynamic mode decomposition and data driven Koopman spectral analysis. First, we consider computations of the Ritz pairs $(z_j, \lambda_j)$ of $\mathbb{; ; A}; ; $, using both the SVD based Schmid's DMD \cite{; ; zd:schmid2010}; ; and the natural formulation via the Krylov decomposition with the Frobenius companion matrix. We show how to use the eigenvectors of the companion matrix explicitly -- these are the columns of the inverse of the notoriously ill-conditioned Vandermonde matrix. The key step to curb ill-conditioning is the discrete Fourier transform of the snapshots ; in the new representation, the Vandermonde matrix is transformed into a generalized Cauchy matrix, which then allows accurate computation by specially tailored algorithms of numerical linear algebra. Numerical experiments show robustness in extremely ill-conditioned cases. More details can be found in \cite{; ; zd:P1}; ; , \cite{; ; zd:P2}; ; . Secondly, the goal is to identify the most important coherent structures in the dynamic process under study, i.e. after computing the Ritz pairs $(z_j, \lambda_j)$ of $\mathbb{; ; A}; ; $ the task is to determine $\ell

As a major source of uncertainties in micro- and nanopositioning devices, friction, characterised by complex stochastic phenomena, is one of the major challenges in developing reliable predictive models od systems’ behaviour. Based on recently performed lateral force microscopy experimental measurements of thin films’ nanoscale friction, fundamental frictional mechanisms in atomic-scale single- asperity contacts are investigated in this work vs. the concurrent influence of multivariate process parameters. The hence proposed nanoscale friction models are developed using innovative artificial intelligence-based methods. In fact, although in a previous study the employment of conventional (black box) machine learning methods provided rather good predictive performances, the intrinsic nature of these models prevents their usage in most practical applications. The novel methodology proposed in this work allows, in turn, attaining an extremely simple mathematical formulation (i.e., a white box model) providing an immediate insight and a unique scientific perspective into the multidimensional dependence of nanoscale friction on the studied variable influencing parameters. What is more, this artificial intelligence-based approach allows achieving a high predictive performance with R2 values in the range of 0.75, while the simplicity of the obtained expressions makes future studies and possible practical applications (e.g. in the corresponding control algorithms) rather straightforward.

Usporedno s povećanjem svjetske populacije povećava se i potražnja za kvalitetnijom i funkcionalnom hranom. Cilj rada je definirati pojam funkcionalne hrane te ukazati na važnost makro elemenata u uzgoju bilja, te njihov utjecaj na proizvodnju sekundarnih metabolita. Za normalan rast i razvoj biljke, je potrebno devet makro elemenata.. Nedostatak pojedinih makro elemenata uzrokuje značajne promjene i poremećaje u metabolizmu biljke, te njezinoj kvaliteti Poznavanjem utjecaja makro elemenata na biljku moguće je provesti ciljanu gnojidbu sa svrhom postizanja proizvodnje kvalitetnije i funkcionalne hrane.

Soft robots promise improved safety and capability over rigid robots when deployed in complex, delicate, and dynamic environments. However, the infinite degrees of freedom and highly nonlinear dynamics of these systems severely complicate their modeling and control. As a step toward addressing this open challenge, we apply the data-driven, Hankel Dynamic Mode Decomposition (HDMD) with time delay observables to the model identification of a highly inertial, helical soft robotic arm with a high number of underactuated degrees of freedom. The resulting model is linear and hence amenable to control via a Linear Quadratic Regulator (LQR). Using our test bed device, a dynamic, lightweight pneumatic fabric arm with an inertial mass at the tip, we show that the combination of HDMD and LQR allows us to command our robot to achieve arbitrary poses using only open loop control. We further show that Koopman spectral analysis gives us a dimensionally reduced basis of modes which decreases computational complexity without sacrificing predictive power., Comment: Submitted to IEEE International Conference on Robotics and Automation, 2021

Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel-Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an $N\times N$ Hermitian positive-definite Toeplitz matrix, where $N$ is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite-dimensional approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.

We analyze the performance of Dynamic Mode Decomposition (DMD)-based approximations of the stochastic Koopman operator for random dynamical systems where either the dynamics or observables are affected by noise. For many DMD algorithms, the presence of noise can introduce a bias in the DMD operator, leading to poor approximations of the dynamics. In particular, methods using time delayed observables, such as Hankel DMD, are biased when the dynamics are random. We introduce a new, robust DMD algorithm that can approximate the stochastic Koopman operator despite the presence of noise. We then demonstrate how this algorithm can be applied to time delayed observables, which allows us to generate a Krylov subspace from a single observable. This allows us to compute a realization of the stochastic Koopman operator using a single observable measured over a single trajectory. We test the performance of the algorithms over several examples., Comment: 20 pages, six figures

Friction is a ubiquitous phenomenon of great research interest in engineering practice. Fundamental frictional features of two solids in contact and in relative motion are governed by microscopic single asperity contacts at their interface. A structured multidisciplinary approach to the experimental determination of friction in the nanometric domain is presented in this work. The dependence of nanoscale friction on process parameters comprising the materials in relative motion, normal forces, sliding velocities and the temperature conditions is studied experimentally by employing scanning probe microscopy. The data hence attained from multidimensional experimental measurements on thin-film samples is used for the development of machine learning-based models. In fact, due to the stochastic nature of the considered phenomena, conventional regression methods yield poor predictive performances, prompting thus the usage of the machine learning numerical paradigm. Such an approach enables obtaining an insight into the concurrent influence of the process parameters on nanoscale friction. A comparative study allows thus showing that, while the best typical regression models result in coefficients of determination (R2) of the order of 0.3, the predictive performances of the used machine learning models, depending on the considered sample, yield R2 in the range from 0.54 to 0.9. The proposed method, aimed at accomplishing an in-depth insight into the physical phenomena influencing nanoscale frictional interactions, will be complemented next with advanced studies based on genetic programming-based artificial intelligence methods. These could, in fact, allow obtaining a functional description of the dependence of nanoscale friction on the studied variable parameters, thus enabling not only true nanoscale friction prediction but also an important tool for control purposes.

The Paris Agreement highlights the need for local climate leadership. The University Of California’s approach to deep decarbonization offers lessons in efficiency, alternative fuels and electrification. Bending the emissions curve globally requires efforts that blend academic insights with practical solutions.

The problem of prediction of behavior of dynamical systems has undergone a change in the second half of the 20th century with the discovery of the possibility of chaotic dynamics in simple dynamical systems. However, that approach does not account for another type of unpredictability: the ``black swan" event. In our framework, the black-swan-type dynamics occurs when an underlying dynamical system becomes coupled to, or decoupled from, another one. Here we explore the problem of prediction in systems that exhibit such behavior. The mathematical theory and algorithms we use are based on an operator-theoretic approach in which the dynamics of the system are embedded into an infinite-dimensional space. We show that the framework correctly identifies a black swan event. Moreover, we show that the algorithms we developed enabled a successful prediction of the flu season, and prediction in other complex dynamics datasets such as physiology models.

The Dynamic Mode Decomposition (DMD, introduced by P. Schmid) has become a tool of trade in computational data driven analysis of complex dynamical systems, e.g. fluid flows, where it can be used to decompose the flow field into component fluid structures, called DMD modes, that describe the evolution of the flow. The DMD is deeply connected with the Koopman spectral analysis of nonlinear dynamical systems, and it can be considered as a computational device in the Koopman analysis framework. Its exceptional performance motivated developments of several modifications that make the DMD an attractive method for analysis and model reduction of nonlinear systems in data driven settings. In this talk, we will present our recent results on the numerical aspects of the DMD/Koopman analysis. We show how the state of the art numerical linear algebra can be deployed to improve the numerical performances in the cases that are usually considered notoriously ill-conditioned. Further, we show how even in the data driven setting, we can work with residual bounds, which allows error estimates for the computed modes.

© 2018 The Author(s). The Paris Agreement highlights the need for local climate leadership. The University Of California's approach to deep decarbonization offers lessons in efficiency, alternative fuels and electrification. Bending the emissions curve globally requires efforts that blend academic insights with practical solutions.

A generalization of the Koopman operator framework, originally developed for de- terministic dynamical systems, to discrete and continuous time random dynamical systems results with the stochastic Koopman operators. The study of the spectral objects of these operators (Koopman eigenvalues, eigenfunctions and modes) could be useful in analysing the behaviour of the considered random dynamical system. We provide the results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operators for the particular classes of random and stochastic dynamical systems. The numerical approximations of the eigenvalues and eigen- functions of the stochastic Koopman operator are computed by using DMD RRR algorithm. Its behaviour in the stochastic framework is explored on several test examples. Furthermore, we introduce the isostables and isochrones associated to the random dynamical systems and compute their numerical approximations on the chosen example.

The main limitation in modelling nanoscale friction is the variety of physicochemical processes and their interactions in nanoscale contacts that depend on the materials and on the conditions relating to sliding velocity, normal forces, temperature etc. To assess the value of the nanoscale friction coefficients, in this work an experimental methodology using a scanning probe microscope (SPM) in the lateral force microscopy (LFM) mode is hence used on various thin film samples. The measurements are designed by employing the centroidal Voronoi tessellation technique with 50 sample points. By using Support Vector Machine-based machine learning, the trends of the values of the friction coefficients can thus be established despite the marked stochastic contributions, creating therefore the preconditions for obtaining a multi-dimensional model relating the value of nanometric friction to its influencing parameters.

We propose a fast iterative algorithm for computation of isostables and isochronsfor dynamical systems with stable limit cycles or fixed points in high dimensions.We formulate the problem as a solution to a first order static Hamilton–Jacobiequation with a constant source term and we solve the boundary problem using aEulerian Fast Sweeping Method developed for this type of problems. We apply thismethod on several illustrative examples of dynamical systems and show that this isan efficient, accurate method which can be used in a data-driven setting.

Heart rate variability biofeedback intervention involves slow breathing at a rate of ∼6 breaths/min (resonance breathing) to maximize respiratory and baroreflex effects on heart period oscillations. This intervention has wide-ranging clinical benefits and is gaining empirical support as an adjunct therapy for biobehavioral disorders, including asthma and depression. Yet, little is known about the system-level cardiovascular changes that occur during resonance breathing or the extent to which individuals differ in cardiovascular benefit. This study used a computational physiology approach to dynamically model the human cardiovascular system at rest and during resonance breathing. Noninvasive measurements of heart period, beat-to-beat systolic and diastolic blood pressure, and respiration period were obtained from 24 healthy young men and women. A model with respiration as input was parameterized to better understand how the cardiovascular processes that control variability in heart period and blood pressure change from rest to resonance breathing. The cost function used in model calibration corresponded to the difference between the experimental data and model outputs. A good match was observed between the data and model outputs (heart period, blood pressure, and corresponding power spectral densities). Significant improvements in several modeled cardiovascular functions (e.g., blood flow to internal organs, sensitivity of the sympathetic component of the baroreflex, ventricular elastance) were observed during resonance breathing. Individual differences in the magnitude and nature of these dynamic responses suggest that computational physiology may be clinically useful for tailoring heart rate variability biofeedback interventions for the needs of individual patients.

We propose new methods for the evaluation of eigenvalues λ(t, t0)of the Koopman operator family K(t, t0), t>t0 of the non-autonomous dynamic systems. The first step in the development is a new data-driven method for very accurate evaluation of eigenvalues λ(t, t0), t>t0 in the hybrid linear non-autonomous case. Then, the approach is extended to continuous linear non-autonomous systems and non-autonomous systems in general. We also propose a relationship between eigenvalues λ(t, t0), t>t0 and eigenvalues computed by Arnoldi-like methods on large sets of snapshots. We apply the new approach to baroreflex physiology, i.e. to the resonant breathing which is used in PTSD treatment. For the resonant breathing we have both data and parameterized mathematical model. The model incorporates a delay and thus is infinite dimensional, hybrid, with a stochastic input. Its asymptotic dynamics is close to quasi-periodic. The applied new methods give an improved insight to the eigenvalues of the related Koopman operator family K(t, t0).

This paper considers the evolution of Koopman principal eigenfunctions of cascaded dynamical systems. If each component subsystem is asymptotically stable, the matrix norms of the linear parts of the component subsystems are strictly increasing, and the component subsystems have disjoint spectrums, there exist perturbation functions for the initial conditions of each component subsystem such that the orbits of the cascaded system and the decoupled component subsystems have zero asymptotic relative error. This implies that the evolutions are asymptotically equivalent; cascaded compositions of stable systems are stable. These results hold for both cascaded systems with linear component subsystem dynamics and linear coupling terms and nonlinear cascades topologically conjugate to the linear case. We further show that the Koopman principal eigenvalues of each component subsystem are also Koopman eigenvalues of the cascaded system. The corresponding Koopman eigenfunctions of the cascaded system are formed by extending the domain of definition of the component systems' principal eigenfunctions and then composing them with the perturbation function.

For every non-autonomous system, there is the related family of Koopman operators $\mathcal{K}^{(t,t_0)}$, parameterized by the time pair $(t,t_0)$. In this paper we are investigating the time dependency of the spectral properties of the Koopman operator family in the linear non-autonomous case and we propose an algorithm for computation of its spectrum from observed data only. To build this algorithm we use the concept of the fundamental matrix of linear non-autonomous systems and some specific aspects of Arnoldi-like methods. In particular, we use Arnoldi-like methods on local data stencils, we exploit the information contained in the Krylov subspace projection error, and discover limitations in the application of Arnoldi-like methods to cases with continous time dependency. We present results of this data-driven algorithm on various linear non-autonomous systems, hybrid as well as continuous in time. In all the examples comparison with exact eigenvalues and eigenfunctions shows excellent performance of the proposed algorithm., Comment: 30 pages, 30 figures

This paper considers a nonlinear dynamical system on a complex, finite dimensional Banach space which has an asymptotically stable, hyperbolic fixed point. We investigate the connection between the so-called principle eigenfunctions of the Koopman operator and the existence of a topological conjugacy between the nonlinear dynamics and its linearization in the neighborhood of the fixed point. The principle eigenfunctions generate an algebra of observables for the linear dynamics --- called the principle algebra --- which can be used to generate a sequence of approximate conjugacy maps in the same manner as it is done in normal form theory. Each element of the principle algebra has an expansion into eigenfunctions of the Koopman operator and composing an eigenfunction with one of the approximate topological conjugacies gives an approximate eigenfunction of the Koopman operator associated with the nonlinear dynamical system. When the limit of the approximate conjugacies exists and attention is restricted to real Banach spaces, a simple application of the Stone-Weierstrass theorem shows that both the principle algebra and the pull-back algebra --- defined by composing the principle algebra with the topological conjugacy --- are uniformly dense in either the space of continuous functions or the maximal ideal of continuous functions vanishing at the fixed point. The point is that, a priori, it is difficult to know which space of observables to use for dissipative nonlinear dynamical systems whose elements have spectral expansions into eigenfunctions. These results say that any continuous observable is arbitrarily close to one that has such an expansion.

This paper extends Yosida's mean ergodic theorem in order to compute projections onto non-unitary eigenspaces for spectral operators of scalar-type on locally convex linear topological spaces. For spectral operators with dominating point spectrum, the projections take the form of Laplace averages, which are a generalization of the Fourier averages used when the spectrum is unitary. Inverse iteration and Laplace averages project onto eigenspaces of spectral operators with minimal point spectrum. Two classes of dynamical systems --- attracting fixed points in $\mathbb{C}^{d}$ and attracting limit cycles in $\mathbb{R}^{2}$ --- and their respective spaces of observables are given for which the associated composition operator is spectral. It is shown that the natural spaces of observables are completions with an $\ell^{2}$ polynomial norm of a space of polynomials over a normed unital commutative ring. These spaces are generalizations of the Hardy spaces $H^{2}(\mathbb{D})$ and $H^{2}(\mathbb{D}^{d})$. Elements of the ring are observables defined on the attractor --- the fixed point or the limit cycle, in our examples. Furthermore, we are able to provide a (semi)global spectral theorem for the composition operators associated with a large class of dissipative nonlinear dynamical systems; any sufficiently smooth dynamical system topologically conjugate to either of the two cases above admits an observable space on which the associated Koopman operator is spectral. It is conjectured that this is generically true for systems where the basin of attraction can be properly "coordinatized"., Comment: 25 pages, 1 figure. Output generated via PDFLatex

Predstavljeni su rezultati međunarodnog istraživačkog projekta pod naslovom Defining Dispositional Employability in Croatian Society koji je nastao u suradnji Odsjeka za psihologiju Filozofskog fakulteta u Zagrebu i Odsjeka za psihologiju, Državnog Sveučilišta u Coloradu (SAD). Pokrovitelji projekta su Vlada SAD-a i Ministarstvo rada i mirovinskog sustava RH. Ciljevi projekta bili su: (1) utvrditi ponašanja, stavove i osobine ljudi koje, bez obzira na formalno obrazovanje i druge demografske značajke, utječu na uspješno zapošljavanje ostvareno kroz radnu fleksibilnost i mobilnost, skraćuju razdoblja nezaposlenosti i potiču samostalno upravljanje karijerom ; (2) istražiti individualnu zapošljivost radnih ljudi različitih demografskih i radnih karakteristika ; (3) razviti instrument za mjerenje dispozicijske zapošljivosti definirane kao višedimenzionalni konstrukt poželjnog stanja radne motivacije. Projekt je obuhvaćao kvalitativne i kvantitativne studije radne motivacije osoba različitih demografskih i radnih karakteristika te empirijsku validaciju konstrukta zapošljivosti u hrvatskoj kulturi. Rezultati se temelje na kvalitativnim podacima prikupljenima metodom fokusnih grupa te kvantitativnim istraživanjima provedenima na preko 1200 zaposlenika različitih demografskih i radnih karakteristika. Utvrđeni su stavovi prema radu i radna ponašanja koja opisuju zapošljivost na individualnoj razini. To su zaokupljenost radom, radna samoefikasnost i proaktivnost, otvorenost prema promjenama u poslu, radna otpornost i socijalni kapital. Predstavljanje projekta će obuhvaćati prezentaciju prikupljenih rezultata i konstruiranog instrument te raspravu o praktičnim implikacijama dobivenih nalaza i smjernicama daljnje suradnje u ovom području. Na predstavljanje projekta bit će pozvani predstavnici Ministarstva rada i mirovinskog sustava i Veleposlanstva SAD-a u Zagrebu. Novokonstruirani instrument i pripadajući validacijski podaci bit će ponuđeni na besplatno korištenje članovima Sekcije za Organizacijsku psihologiju HPD-a.

[Date] November 28 (Mon) - December 2 (Fri), 2011: [Place] Kyoto University Clock Tower Centennial Hall, Kyoto, JAPAN

Using an elementary application of Birkhoff's ergodic theorem, necessary and sufficient conditions are given for the existence of asymptotically t^2 dispersion of a distribution of nondiffusive passive tracer in a class of incompressible laminar flows. Nonergodicity is shown to be the dynamical mechanism giving rise to this behavior.

Human adaptability involves interconnected biological and psychological control processes that determine how successful we are in meeting internal and environmental challenges. Heart rate variability (HRV), the variability in consecutive R-wave to R-wave intervals (RRI) of the electrocardiogram, captures synergy between the brain and cardiovascular control systems that modulate adaptive responding. Here we introduce a qualitatively new dimension of adaptive change in HRV quantified as a redistribution of spectral power by applying the Wasserstein distance with exponent 1 metric (W(1)) to RRI spectral data. We further derived a new index, D, to specify the direction of spectral redistribution and clarify physiological interpretation. We examined gender differences in real time RRI spectral power response to alcohol, placebo and visual cue challenges. Adaptive changes were observed as changes in power of the various spectral frequency bands (i.e., standard frequency domain HRV indices) and, during both placebo and alcohol intoxication challenges, as changes in the structure (shape) of the RRI spectrum, with a redistribution towards lower frequency oscillations. The overall conclusions from the present study are that the RRI spectrum is capable of a fluid and highly flexible response, even when oscillations (and thus activity at the sinoatrial node) are pharmacologically suppressed, and that low frequency oscillations serve a crucial but less studied role in physical and mental health.

Modelling of nanoscale friction presents a long-lasting challenge. In fact, while there are several generalised models that provide good results for macro- and micro-scale friction, due to the complex concurrent physicochemical interactions in nanoscale contacts, when modelling nanoscale friction there is a clear lack of reliable predicting tools. The modelling methodology proposed in this work is based on the recently performed multidimensional experimental measurements of thin-films’ nanoscale friction, where the concurrent effects of several process parameters are considered. Due to the stochastic nature of the considered phenomena, conventional regression methods yield poor predictive performances. A machine learning (ML) numerical paradigm is hence proposed. Via a comparative study it is hence shown that, while the best typical regression models result in coefficients of determination (R2) of the order of 0.3, the predictive performances of the used ML models, depending on the considered sample, yield R2 in the range from 0.54 to 0.9. The developed models provide also new insights into the functional dependence of the variable process parameters, but also sound basis for future extensions of existing friction models to the nanometric range.

Friction and wear are recognized as one of the most puzzling problems, not only in many engineering and manufacturing applications, but also in a fundamental scientific sense. In fact, friction is a nonlinear stochastic effect with a distinct time, position and temperature variability. While frictional phenomena on the macro- and meso-scales can be considered well described, and their characteristic features can be simulated via suitable models, as well as generally efficiently compensated by using proper control typologies, the study of friction, the parameters that influence its value and the respective models in the nanometric domain are still in an early stage, due to various experimental and modelling complexities. The research performed in the framework of the doctoral thesis provides a scientific contribution to the study of dry (unlubricated) friction by characterising the parameters influencing its value at the nanometric scale, and especially the dependence of friction on material properties, loading conditions, the velocity of motion, as well as temperature. The characterisation of the dependence of friction on the listed parameters is based on experimental measurements performed by employing a Scanning Probe Microscope (SPM). Due to the number and variety of the monitored influences, the number and type of measurements is determined by a state-of-the-art Design of Experiment (DoE) methodology by employing Voronoi tessellations. To obtain predictive models linking the process variables to the value of nanometric friction, the obtained measurement results are then validated numerically via a thorough comparative analysis of state-of-the-art machine learning methods. Despite the marked complexity of the analysed phenomena and the inherent dispersion of the measurements, the developed symbolic regression models, show, depending on the type of the sample, an excellent prediction accuracy between 72 and 91%., Trenje i trošenje su jedan od najizazovnijih problema u mnogih inženjerskim i proizvodnim primjenama. Doista, trenje je nelinearna stohastička pojava s izraženom vremenskom, prostornom i temperaturnom varijabilnošću. Dok je trenje u makro- i mezodomeni dobro objašnjeno te je njegove učinke, primjerenim modelima, moguće modelirati i, primjerenim sustavima regulacije, najčešće i uspješno kompenzirati, u nanometarskom području je proučavanje trenja, parametara koji utječu na trenje, te nalaženje odgovarajućeg modela tih pojavnosti još u zametku. Karakterizacijom utjecajnih parametara u nanometarskom području, a posebice ovisnosti trenja o svojstvima materijala, opterećenju, brzini relativnog gibanja te temperaturi tribološkog para, istraživanje provedeno u sklopu doktorske disertacije daje znanstveni doprinos izučavanju trenja klizanja bez podmazivanja. Eksperimentalno su analizirani tanki filmovi pet različitih materijala: aluminijevog oksida (Al2O3), aluminija, molibden disulfida (MoS2), titanijevog dioksida (TiO2) te martenzitnog nehrđajućeg čelika (X39CrMo17-1). Doista, zbog povoljnih svojstava otpornosti na trošenje i njihove tvrdoće, prevlake od Al2O3 i TiO2 su često korišteni materijali za primjene u mikro- i nanoelektromehaničkim sustavima (M(N)EMS) kao i, općenito, u preciznim konstrukcijama. Ti su uzorci sintetizirani u obliku tankog filma metodom taloženja atomskih slojeva (Atomic Layer Deposition – ALD) na silicijev (Si) supstrat. Ostali proučavani uzorci su odabrani za istraživanje zbog njihove široke primjene u općem strojarstvu te u preciznim konstrukcijama. Zbog povoljnih kliznih svojstava sulfida, MoS2 se, tako, često koristi kao kruto mazivo. Čisti Al je jedan od najčešće korištenih materijala za lake i precizne konstrukcije, dok je martenzitni nehrđaju čelik X39CrMo17-1 odabran kao predstavnik visokokvalitetne grupe čelika za strojne elemente. Ovi uzorci su sintetizirani metodom taloženja pulsirajućim laserom (Pulsed Laser Deposition – PLD), što je omogućilo i da se prvi puta postigne sintetiziranje tankog filma visokolegiranog čelika. Eksperimentalno mjerenje pretražnim mikroskopom atomskih sila u modalitetu rada mjerenja poprečne sile (Lateral Force Microscopy – LFM) vršeno je na svim uzorcima strukturiranim načinom u eksperimentalnim točkama definiranim trima promjenjivim tehnološkim parametrima: normalnom silom FN = 10...150 nN, brzinom klizanja v = 5...500 nm/s i temperaturom  = 20...80 °C. Pedeset mjernih točaka je pritom definirano Voronoi teselacijskom Design of Experiment (DoE) metodom podjele domene mjerenja u granicama promjene navedenih utjecajnih parametara, te je mjerenje u svakoj točki ponovljeno pet puta, Marko Perčić: Characterization of Parameters Influencing Friction in the Nanometric Domain pa je tako u analizi ukupno izvršeno 1,250 mjerenja. Eksperimentalna metodologija je u tom kontekstu strukturirana na način da u kalibracijskom postupku uzima u obzir i promjenjive učinke adhezije ali i vodeći računa o potrebi kompenzacije temperaturnih rastezanja. Razvijenom metodologijom je po prvi puta uopće postignuto mjerenje trenja na nanometarskoj razini s tri promjenjive veličine. Dobiveni rezultati mjerenja omogućavaju ne samo uvid u ponašanje pojedinog analiziranog materijala u danim promjenjivim uvjetima, već i određivanje korelacijskih funkcija koje povezuju parametre procesa s vrijednošću sile trenja na nanometarskoj razini. Temeljita komparativna analiza primjene različitih naprednih metoda strojnog učenja na mjerne podatke je omogućila određivanje korelacijskih funkcija, odnosno prediktivnog modela trenja. Usprkos kompleksnosti analiziranih fizikalnih pojava te značajnom rasipanju mjernih rezultata, provedena je analiza omogućila da se, ovisno o uzorku, primjenom razvijenog matematičkog modela metodom simboličke regresije, dobije točnost predviđanja sile trenja, u odnosu na radne parametre, na razini od 72 do 91%. Takva izvanredna točnost predikcije omogućava ne samo uvid u funkcijsku ovisnost trenja na nanometarskoj razini o promatranim varijablama, nego i stvara preduvjete za proširenje postojećih modela trenja, čime bi se njihova praktična primjenjivost proširila i na nanometarsku razinu.