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Statistical Properties of Time-dependent Systems
Diego Fregolente Mendes De Oliveira, 2012, doktorska disertacija

Opis: In the dissertation I have dealt with time-dependent (nonautonomous) systems, the conservative (Hamiltonian) as well as dissipative, and investigated their dynamical and statistical properties. In conservative (Hamiltonian) time-dependent systems the energy is not conserved, whilst the Liouville theorem about the conservation of the phase space volume still applies. We are interested to know, whether the system can gain energy, and whether this energy can grow unbounded, up to infinity, and we are interested in the system's behaviour in the mean, as well as its statistical properties. An example of such a system goes back to the 1940s, when Fermi proposed the acceleration of cosmic rays (in the first place protons) upon the collisions with moving magnetic domains in the interstellar medium of our Galaxy, and in other galaxies. He then proposed a simple mechanical one-dimensional model, the so-called Fermi-Ulam Model (FUM), where a point particle is moving between two rigid walls, one being at rest and the other one oscillating. If the oscillation is periodic and smooth, it turned out in a nontrivial way, which is, in the modern era of understanding the chaotic dynamical systems, well understood, namely that the unbounded increasing of the energy (the so-called Fermi acceleration) is not possible, due to the barriers in form of invariant tori, which partition the phase space into regions, between which the transitions are not possible. The research has then been extended to other simple dyanamical systems, which have complex dynamics. The first was so-called bouncer model, in which a point particle bounces off the oscillating platform in a gravitational field. In this simple system the Fermi acceleration is possible. Later the research was directed towards two-dimensional billiard systems. It turned out that the Fermi acceleration is possible in all such systems, which are at least partially chaotic (of the mixed type), or even in a system that is integrable as static, namely in case of the elliptic billiard. (The circle billiard is an exception, because it is always integrable, as the angular momentum is conserved even in time-dependent case.) The study of time-dependent systems has developed strongly worldwide around the 1990s, in particular in 2000s, and became one of the central topics in nonlinear dynamics. It turned out, quite generally, but formal and implicit, in the sense of mathematical existence theorems, that in nonautonomous Hamilton systems the energy can grow unbounded, meaning that the system ``pumps" the energy from the environment with which it interacts. There are many open questions: how does the energy increase with time, in particular in the mean of some representative ensemble of initial conditions (typically the phase space of two-dimensional time-dependent billiards is four-dimensional.) It turned out that almost everywhere the power laws apply, empirically, based on the numerical calculations, but with various acceleration exponents. If the Fermi acceleration is not posssible, like e.g. in the FUM, due to the invariant tori, then after a certain time of acceleration stage the crossover into the regime of saturation takes place, whose characteristics also follow the power laws. One of the central themes in the dissertation is the study of these power laws, their critical exponents, analytical relationships among them, using the scaling analysis (Leonel, McClintock and Silva, Phys. Rev. Lett. 2004). Furthermore, the central theme is the question, what happens, if, in a nonautonomous Hamilton system which exhibits Fermi acceleration, we introduce dissipation, either at the collisions with the walls (collisional dissipation) or during the free motion (in-flight dissipation, due to the viscosity of the fluid or the drag force etc.). Dissipation typically transforms the periodic points into point attractors and chaotic components into chaotic attractors. The Fermi acceleration is always suppressed. We are interested in the phase portraits of
Ključne besede: nonlinear dynamics, dynamical systems, conservative and dissipative systems, time-dependent systems, Fermi acceleration, billiards, kicked systems, chaos, chaotic and periodic attractors, bifurcations, boundary crisis
Objavljeno: 19.09.2012; Ogledov: 1424; Prenosov: 49
.pdf Polno besedilo (16,09 MB)

Exact analysis of adiabatic invariants in time dependent harmonic oscillator
Marko Robnik, Valery Romanovski, 2009, objavljeni znanstveni prispevek na konferenci

Objavljeno: 07.06.2012; Ogledov: 511; Prenosov: 1
URL Polno besedilo (0,00 KB)

Expanded boundary integral method and chaotic time-reversal in quatum billiards
Gregor Veble, Tomaž Prosen, Marko Robnik, 2007, izvirni znanstveni članek

Opis: We present the expanded boundary integral method for solving the planar Helmholtz problem, which combines the ideas of the boundary integral method and the scaling method and is applicable to arbitrary shapes. We apply the method to a chaotic billiard with unidirectional transport, where we demonstrate the existence of doublets of chaotic eigenstates, which are quasi-degenerate due to time-reversal symmetry, and a very particular level spacing distribution that attains a chaotic Shnirelman peak at short energy ranges and exhibits Gaussian Unitary Ensemble (GUE) like statistics for large energy ranges. We show that, as a consequence of such particular level statistics or algebraic tunnelling between disjoint chaotic components connected by time-reversal operation, the system exhibits quantum current reversals.
Ključne besede: 2D Helmholtz equation, energy spectrum, quantum billiard
Objavljeno: 03.07.2017; Ogledov: 50; Prenosov: 0
.pdf Polno besedilo (2,90 MB)

Dynamical and statistical properties of time-dependent one-dimensional nonlinear Hamilton systems
Dimitrios Andresas, 2015, doktorska disertacija

Opis: We study the one-dimensional time-dependent Hamiltonian systems and their statistical behaviour, assuming the microcanonical ensemble of initial conditions and describing the evolution of the energy distribution in three characteristic cases: 1) parametric kick, which by definition means a discontinuous jump of a control parameter of the system, 2) linear driving, and 3) periodic driving. For the first case we specifically analyze the change of the adiabatic invariant (the canonical action) of the system under a parametric kick: A conjecture has been put forward by Papamikos and Robnik (2011) that the action at the mean energy always increases, which means, for the given statistical ensemble, that the Gibbs entropy in the mean increases (PR property). By means of a detailed rigorous analysis of a great number of case studies we show that the conjecture largely is satisfied, except if either the potential is not smooth enough (e.g. has discontinuous first derivative), or if the energy is too close to a stationary point of the potential (separatrix in the phase space). We formulate the conjecture in full generality, and perform the local theoretical analysis by introducing the ABR property. For the linear driving we study first 1D Hamilton systems with homogeneous power law potential and their statistical behaviour under monotonically increasing time-dependent function A(t) (prefactor of the potential). We used the nonlinear WKB-like method by Papamikos and Robnik J. Phys. A: Math. Theor., 44:315102, (2012) and following a previous work by Papamikos G and Robnik M J. Phys. A: Math. Theor., 45:015206, (2011) we specifically analyze the mean energy, the variance and the adiabatic invariant (action) of the system for large time t→∞. We also show analytically that the mean energy and the variance increase as powers of A(t), while the action oscillates and finally remains constant. By means of a number of detailed case studies we show that the theoretical prediction is correct. For the periodic driving cases we study the 1D periodic quartic oscillator and its statistical behaviour under periodic time-dependent function A(t) (prefactor of the potential). We compare the results for three different drivings, the periodic parametrically kicked case (discontinuous jumps of $A(t)$), the piecewise linear case (sawtooth), and the smooth case (harmonic). Considering the Floquet map and the energy distribution we perform careful numerical analysis using the 8th order symplectic integrator and present the phase portraits for each case, the evolution of the average energy and the distribution function of the final energies. In the case where we see a large region of chaos connected to infinity, we indeed find escape orbits going to infinity, meaning that the energy growth can be unbounded, and is typically exponential in time. The main results are published in two papers: Andresas, Batistić and Robnik Phys. Rev. E, 89:062927, (2014) and Andresas and Robnik J. Phys. A: Math. Theor., 47:355102, (2014).
Ključne besede: one-dimensional nonlinear Hamiltonian systems, adiabatic invariant, parametric kick, periodic driving, linear driving, energy distribution, WKB method, action
Objavljeno: 02.03.2015; Ogledov: 686; Prenosov: 12
.pdf Polno besedilo (11,07 MB)

Model napovedovanja prodajnih priložnosti v medorganizacijskem poslovanju z uporabo metod strojnega učenja
Marko Bohanec, 2017, doktorsko delo/naloga

Opis: Področje medorganizacijske (B2B) prodaje je zahtevno. Običajno ne predvideva le prodaje končnih izdelkov, temveč so predmet prodaje kompleksnejše rešitve, prilagojene kupcu. Kupci pri tem sledijo svojim internim procesom, želijo doseči prilagoditve elementov pogodbe, se pogajajo in podobno. To zahteva od prodajalcev dobro poznavanje pričakovanj strank, njihovih želja in potreb. Proces B2B prodaje je zato daljši in kompleksnejši. V disertaciji se osredotočamo na napovedovanje prodajnega izida v medorganizacijski prodaji, ki v praksi večinoma temelji na subjektivni presoji prodajalcev. Glede na napovedi prodajalcev se podjetja odločajo o virih in aktivnostih, zato netočne napovedi lahko vodijo v nepopravljive posledice. Raziskave so pokazale, da podjetja, ki svoje odločitve temeljijo na podatkih (angl. ``data driven decison-making''), izkazujejo boljše poslovne rezultate. Vendar pa raziskave kažejo tudi, da je uporaba metod in orodij, ki temeljijo na podatkih, v praksi še vedno šibka. To lahko pripišemo slabemu razumevanju metod in orodij za podporo odločanju ter tudi nezaupanju v tehnologijo. Motivacija za raziskavo izhaja iz zaznanega problemskega stanja v medorganizacijski prodaji in vrzeli, ki smo jo zaznali v akademski literaturi. Izhajamo iz teze, da je mogoče z uporabo modelov stojnega učenja pomagati prodajalcu in podjetju tako, da pri napovedovanju poslovnega izida delajo manj napak. V ta namen smo po metodologiji akcijske raziskave in razvoja (angl. ``action design research'', s kratico ADR) razvili model napovedovanja prodajnih priložnosti v medorganizacijskem poslovanju z uporabo metod strojnega učenja. Pri tem sta nastala dva artefakta: informacijsko-tehnološki (IT) artefakt, ki temelji na modelih strojnega učenja, podkrepljenih s transparentnimi razlagami, ter organizacijski artefakt, ki spodbuja vključevanje spoznanj iz IT-artefakta v proces napovedovanja in organizacijsko učenje. Prednost ADR je v tem, da vključuje uporabnike v razvoj in evalvacijo modela že na začetku raziskave. Na ta način uporabniki lahko izrazijo svoja pričakovanja, sproti vrednotijo model ter tudi sproti predlagajo spremembe. To krepi zaupanje v razvit model in povečuje zavezanost h kasnejši uporabi v praksi. Jedro disertacije predstavlja model napovedovanja prodajnih priložnosti v medorganizacijskem poslovanju z uporabo metod strojnega učenja. Metode strojnega učenja se iz prodajne zgodovine naučijo prepoznati značilnosti prodaje. Ko se pojavijo nove priložnosti, metode ocenijo njihovo zrelost in ponudijo odločevalcem razlago vplivnih dejavnikov, omogočajo pa tudi analizo vpliva različnih prodajnih aktivnosti s pomočjo ``kaj-če'' analize. Pri tem uporabljamo poenoten format napovedi in njihovih razlag, ki podpirajo različne modele. Tako omogočamo uporabo visoko zmogljivih metod strojnega učenja (npr. naključni gozd), ki so običajno zaradi svoje zapletenosti netransparentne in jim uporabniki stežka zaupajo. Da smo lahko razvili model, smo opravili dodatne raziskave za oblikovanje nabora atributov, ki opisujejo proces B2B prodaje in razvili optimizacijske postopke za detekcijo šuma in redundance atributov. Za učinkovito detekcijo kvalitete učne množice smo razvili vizualno metodo. Potrdili smo domnevo, da je želen organizacijsko-informacijski model mogoče zgraditi, saj je večina uporabljenih metod dosegla klasifikacijsko točnost nad 70\%. Za podrobnejšo analizo vpliva atributov smo razvili simulacijske in optimizacijske algoritme. Praksa potrjuje koristnost razvitega modela, saj se je v realnem podjetju z uporabo modela točnost napovedi bistveno izboljšala. S kombinacijo uporabe modelov, znanja in prakse ekspertov, smo tako prispevali k preseganju pomanjkljivosti posameznih pristopov. Uporabljene metode predstavljajo novost na področju organizacijskih znanosti in tako prispevajo k znanstveni literaturi na področju organizacijskega učenja in uporabe metod strojnega učenja.
Ključne besede: strojno učenje, medorganizacijska prodaja, organizacijsko učenje, napovedovanje izida, razlaga modelov, analiza kaj-če
Objavljeno: 24.04.2017; Ogledov: 158; Prenosov: 12
.pdf Polno besedilo (3,49 MB)

Organizational learning supported by machine learning models coupled with general explanation methods
Marko Bohanec, Marko Robnik Šikonja, Mirjana Kljajić Borštnar, 2017, izvirni znanstveni članek

Opis: Background and Purpose: The process of business to business (B2B) sales forecasting is a complex decision-making process. There are many approaches to support this process, but mainly it is still based on the subjective judgment of a decision-maker. The problem of B2B sales forecasting can be modeled as a classification problem. However, top performing machine learning (ML) models are black boxes and do not support transparent reasoning. The purpose of this research is to develop an organizational model using ML model coupled with general explanation methods. The goal is to support the decision-maker in the process of B2B sales forecasting. Design/Methodology/Approach: Participatory approach of action design research was used to promote acceptance of the model among users. ML model was built following CRISP-DM methodology and utilizes R software environment. Results: ML model was developed in several design cycles involving users. It was evaluated in the company for several months. Results suggest that based on the explanations of the ML model predictions the users’ forecasts improved. Furthermore, when the users embrace the proposed ML model and its explanations, they change their initial beliefs, make more accurate B2B sales predictions and detect other features of the process, not included in the ML model. Conclusions: The proposed model promotes understanding, foster debate and validation of existing beliefs, and thus contributes to single and double-loop learning. Active participation of the users in the process of development, validation, and implementation has shown to be beneficial in creating trust and promotes acceptance in practice.
Ključne besede: decision support, organizational learning, machine learning, explanations
Objavljeno: 01.09.2017; Ogledov: 50; Prenosov: 0
.pdf Polno besedilo (1,31 MB)

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