Search

Background. Monte Carlo simulation of radiation transport for medical linear accelerators (linacs) requires accurate knowledge of the geometrical description of the linac head. Since the geometry of Varian TrueBeam machines has not been disclosed, the manufacturer distributes phase-space files of the linac patient-independent part to allow researchers to compute absorbed dose distributions using the Monte Carlo method. This approach limits the possibility of achieving an arbitrarily small statistical uncertainty. This work investigates the use of the geometry of the Varian Clinac 2100, which is included in the Monte Carlo system PRIMO, as a surrogate. Methods. Energy, radial and angular distributions extracted from the TrueBeam phase space files published by the manufacturer and from phase spaces tallied with PRIMO for the Clinac 2100 were compared for the 6, 8, 10 and 15 MV flattened-filtered beams. Dose distributions in water computed for the two sets of PSFs were compared with the Varian Representative Beam Data (RBD) for square fields with sides ranging from 3 to 30 cm. Output factors were calculated for square fields with sides ranging from 2 to 40 cm. Results. Excellent agreement with the RBD was obtained for the simulations that employed the phase spaces distributed by Varian as well as for those that used the surrogate geometry, reaching in both cases Gamma ( , 2 mm) pass rates larger than , except for the 15 MV surrogate. This result supports previous investigations that suggest a change in the material composition of the TrueBeam 15 MV flattening filter. In order to get the said agreement, PRIMO simulations were run using enlarged transport parameters to compensate the discrepancies between the actual and surrogate geometries. Conclusions. This work sustains the claim that the simulation of the 6, 8 and 10 MV flattening-filtered beams of the TrueBeam linac can be performed using the Clinac 2100 model of PRIMO without significant loss of accuracy., Open Access funding enabled and organized by Projekt DEAL. The authors acknowledge support by the Open Access Publication Fund of the University of Duisburg-Essen. MR acknowledges funding by the Sistema Nacional de Investigación de Panamá (SNI). JS thanks the Spanish Ministerio de Ciencia e Innovación, project PID2019-104714GB-C22. LB acknowledges funding by the HARMONIC project. The HARMONIC project (Health effects of cArdiac fluoRoscopy and MOderN radIotherapy in paediatriCs) has received funding from the Euratom research and training programme 2014-2018 under grant agreement number 847707., Postprint (published version)

El texto completo de este trabajo no se encuentra disponible por no haber sido facilitado aún por su autor, por restricciones de copyright, o por no existir una versión digital, A procedure to characterize beams of a medical linear accelerator for their use in Monte Carlo (MC) dose calculations for intraoperative electron radiation therapy (IOERT) is presented. The procedure relies on dose measurements in homogeneous media as input, avoiding the need for detailed simulations of the accelerator head. An iterative algorithm (EM-ML) has been employed to extract the relevant details of the phase space (PHSP) of the particles coming from the accelerator, such as energy spectra, spatial distribution and angle of emission of particles. The algorithm can use pre-computed dose volumes in water and/or air, so that the machine-specific tuning with actual data can be performed in a few minutes. To test the procedure, MC simulations of a linear accelerator with typical IOERT applicators and energies, have been performed and taken as reference. A solution PHSP derived from the dose produced by the simulated accelerator has been compared to the reference PHSP. Further, dose delivered by the simulated accelerator for setups not included in the fit of the PHSP were compared to the ones derived from the solution PHSP. The results show that it is possible to derive from dose measurements PHSP accurate for IOERT MC dose estimations., Comunidad de Madrid, Ministerio de Ciencia e Innovación (España), European Regional Development Fund, Cluster de Calculo de Alta Capacidad para Tecnicas Fisicas, Depto. de Estructura de la Materia, Física Térmica y Electrónica, Fac. de Ciencias Físicas, TRUE, pub

Memristors were proposed in the early 1970s by Leon Chua as a new electrical element linking charge to flux. Since that first introduction, these devices have positioned themselves to be considered as possible fundamental ones for the generations of electronic devices to come. In this paper, we propose a new way to investigate the effects of the electrical variables on the memristance of a device, and we successfully apply this technique to model the behavior of a TiN/Ti/HfO2/W ReRAM structure. To do so, we initially apply the Dynamic Route Map technique in the general case to obtain an approximation to the differential equation that determines the behaviour of the device. This is performed by choosing a variable of interest and observing the evolution of its own temporal derivative versus both its value and the applied voltage. Then, according to this technique, it is possible to obtain an approach to the governing equations with no need to make any assumption about the underlying physical mechanisms, by fitting a function to this. We have used a polynomial function, which allows accurate reproduction of the observed electrical behavior of the measured devices, by integrating the resulting differential equation system.

[EN] Statistical mechanics is a physics theory that deals with ensembles of microstates of a system compatible with environmental constraints and that on average define a thermodynamic state. The evolution of a small system is normally subjected to changing constraints, as set by a protocol, and involves a stochastic dependence on previous events. Here, we generalize the dynamic trajectories described by a realization of a physical system without dissipation to include those in which the history of previous events is necessary to understand its future. This framework is then used to characterize the processes experienced by the stochastic system, as derived from ensemble averages over the available pathways. We find that the pathways that the system traces in the presence of a protocol entail different statistics from those in its absence and prove that both types of pathways are equivalent in the limit of independent events. Such equivalence implies that a thermodynamic system cannot evolve away from equilibrium in the absence of memory. These results are useful to interpret single-molecule experiments in biophysics and other fields in nanoscience, as well as an adequate platform to describe non-equilibrium processes.

[EN] Statistical mechanics is a physics theory that deals with ensembles of microstates of a system compatible with environmental constraints and that on average define a thermodynamic state. The evolution of a small system is normally subjected to changing constraints, as set by a protocol, and involves a stochastic dependence on previous events. Here, we generalize the dynamic trajectories described by a realization of a physical system without dissipation to include those in which the history of previous events is necessary to understand its future. This framework is then used to characterize the processes experienced by the stochastic system, as derived from ensemble averages over the available pathways. We find that the pathways that the system traces in the presence of a protocol entail different statistics from those in its absence and prove that both types of pathways are equivalent in the limit of independent events. Such equivalence implies that a thermodynamic system cannot evolve away from equilibrium in the absence of memory. These results are useful to interpret single-molecule experiments in biophysics and other fields in nanoscience, as well as an adequate platform to describe non-equilibrium processes.

[EN] Statistical mechanics is a physics theory that deals with ensembles of microstates of a system compatible with environmental constraints and that on average define a thermodynamic state. The evolution of a small system is normally subjected to changing constraints, as set by a protocol, and involves a stochastic dependence on previous events. Here, we generalize the dynamic trajectories described by a realization of a physical system without dissipation to include those in which the history of previous events is necessary to understand its future. This framework is then used to characterize the processes experienced by the stochastic system, as derived from ensemble averages over the available pathways. We find that the pathways that the system traces in the presence of a protocol entail different statistics from those in its absence and prove that both types of pathways are equivalent in the limit of independent events. Such equivalence implies that a thermodynamic system cannot evolve away from equilibrium in the absence of memory. These results are useful to interpret single-molecule experiments in biophysics and other fields in nanoscience, as well as an adequate platform to describe non-equilibrium processes.

We consider the system of N two-level atoms, of which N0 atoms are unexcited and N1 are excited. This system of N two-level atoms, which forms a linear quantum amplifier, interacts with a single-mode electromagnetic field. The problem of amplification of the L-photon states using such an amplifier is studied. The evolution of the electromagnetic field density matrix is described by the master equation for the field under amplification. The dynamics of this process is such that it can be described as the transformation of the scale of the phase space. The exact solution of the master equation is expressed using the transformed Husimi function of the L-quantum state of the harmonic oscillator. The properties of this function are studied and using it the average photon number and its fluctuations in the amplified state are found.

Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of 10 - 6.

We consider the system of N two-level atoms, of which N0 atoms are unexcited and N1 are excited. This system of N two-level atoms, which forms a linear quantum amplifier, interacts with a single-mode electromagnetic field. The problem of amplification of the L-photon states using such an amplifier is studied. The evolution of the electromagnetic field density matrix is described by the master equation for the field under amplification. The dynamics of this process is such that it can be described as the transformation of the scale of the phase space. The exact solution of the master equation is expressed using the transformed Husimi function of the L-quantum state of the harmonic oscillator. The properties of this function are studied and using it the average photon number and its fluctuations in the amplified state are found. © 2021, Editura Academiei Romane. All rights reserved.

Radiation shielding is a central part of the design of treatment rooms for radiation therapy systems. The dose levels that medical staff and members of the public can be exposed to outside the treatment rooms are regulated by authorities and influence the required wall thicknesses and possible locations for the systems. Several standard methods exist for performing shielding calculations, but they are not well adapted to the stereotactic radiosurgery system Leksell Gamma Knife because of its self-shielding properties. The built-in shielding makes the leakage radiation anisotropic and generally have lower energy than the primary radiation from the Gamma Knife's cobalt sources. Oversimplifications made in the standard shielding calculation methods regarding the field can lead to excessively thick shielding and limit the number of suitable locations for the system. In this thesis project, a simulation-based dose calculation algorithm was developed, that uses Monte Carlo-generated data in two steps. The algorithm uses a phase space to accurately describe the radiation field around the Gamma Knife. Information about individual photons in the field is then combined with a generated library of data describing the resulting dose outside a wall depending on the wall thickness and the photon energy. The dose calculation algorithm is fast enough to be integrated into optimization processes, in which the algorithm is used iteratively while varying room design parameters. Demonstrated in this report is a case with a room of fixed size, in which the Gamma Knife's position and the walls' thicknesses are varied, with the aim to find the room design resulting in the minimum wall thicknesses needed to achieve acceptable dose levels outside. The results in this thesis indicate that the dose calculation algorithm performs well and could likely be used in more complex optimizations with more design variables and more advanced design goals., Strålsäkerhet är en viktig aspekt vid uppförandet av behandlingsrum för strål-terapisystem. Strålningsnivåerna som sjukvårdspersonal och allmänheten kan exponeras för utanför behandlingsrummet regleras av myndigheter och påverkar vilken väggtjocklek som behövs och vilka platser som är lämpliga att placera systemen på. Flertalet metoder för strålskyddsberäkning existerar, men de är inte väl anpassade till det stereotaktiska radiokirurgiska systemet Leksell Gamma Knife, eftersom det har ett inbyggt strålskydd. Det inbyggda strålskyddet gör att strålfältet runt Gamma Knife är anisotropt och generellt har lägre energi än primärstrålningen från systemets koboltkällor. Förenklingar som görs rörande strålfältet i flera existerande metoder för strålskyddsberäkning kan leda till att överdrivet tjocka strålskydd används eller begränsa antalet lämpliga platser att placera systemet på. I detta projekt utvecklades en dosberäkningsalgoritm, som i två steg använder data genererad genom Monte Carlo-simuleringar. Algoritmen använder ett fasrum för att detaljerat beskriva strålfältet runt Gamma Knife. Information om enskilda fotoner i fältet används sen i kombination med ett genererat bibliotek av data som beskriver det dosbidrag som en foton bidrar med utanför behandlingsrummet, baserat på fotonens energi och väggarnas tjocklek. Dosberäkningsalgoritmen är snabb nog att integreras i optimeringsprocesser där den används iterativt samtidigt som rumsdesignparametrar varieras. I denna rapport demonstreras ett fall med ett rum av bestämd storlek, där positionen av Gamma Knife i rummet och väggarnas tjocklekar varieras. Optimeringens syfte i exemplet är att hitta den rumsdesign som med de minsta väggtjocklekarna resulterar i acceptabla strålningsnivåer utanför rummet. Resultaten tyder på att dosberäkningsalgoritmen sannolikt kan användas i mer komplexa optimeringar med fler designvariabler och mer avancerade designmål.

We consider the system of N two-level atoms, of which N0 atoms are unexcited and N1 are excited. This system of N two-level atoms, which forms a linear quantum amplifier, interacts with a single-mode electromagnetic field. The problem of amplification of the L-photon states using such an amplifier is studied. The evolution of the electromagnetic field density matrix is described by the master equation for the field under amplification. The dynamics of this process is such that it can be described as the transformation of the scale of the phase space. The exact solution of the master equation is expressed using the transformed Husimi function of the L-quantum state of the harmonic oscillator. The properties of this function are studied and using it the average photon number and its fluctuations in the amplified state are found. © 2021, Editura Academiei Romane. All rights reserved.

Liouville’s equation on phase space in geometrical optics describes the evolution of an energy distribution through an optical system, which is discontinuous across optical interfaces. The discontinuous Galerkin spectral element method is conservative and can achieve higher order of convergence locally, making it a suitable method for this equation. When dealing with optical interfaces in phase space, non-local boundary conditions arise. Besides being a difficulty in itself, these non-local boundary conditions must also satisfy energy conservation constraints. To this end, we introduce an energy conservative treatment of optical interfaces. Numerical experiments are performed to prove that the method obeys energy conservation. Furthermore, the method is compared to the industry standard ray tracing. The numerical experiments show that the discontinuous Galerkin spectral element method outperforms ray tracing by reducing the computation time by up to three orders of magnitude for an error of 10 - 6.

We consider the system of N two-level atoms, of which N0 atoms are unexcited and N1 are excited. This system of N two-level atoms, which forms a linear quantum amplifier, interacts with a single-mode electromagnetic field. The problem of amplification of the L-photon states using such an amplifier is studied. The evolution of the electromagnetic field density matrix is described by the master equation for the field under amplification. The dynamics of this process is such that it can be described as the transformation of the scale of the phase space. The exact solution of the master equation is expressed using the transformed Husimi function of the L-quantum state of the harmonic oscillator. The properties of this function are studied and using it the average photon number and its fluctuations in the amplified state are found.

Radiation shielding is a central part of the design of treatment rooms for radiation therapy systems. The dose levels that medical staff and members of the public can be exposed to outside the treatment rooms are regulated by authorities and influence the required wall thicknesses and possible locations for the systems. Several standard methods exist for performing shielding calculations, but they are not well adapted to the stereotactic radiosurgery system Leksell Gamma Knife because of its self-shielding properties. The built-in shielding makes the leakage radiation anisotropic and generally have lower energy than the primary radiation from the Gamma Knife's cobalt sources. Oversimplifications made in the standard shielding calculation methods regarding the field can lead to excessively thick shielding and limit the number of suitable locations for the system. In this thesis project, a simulation-based dose calculation algorithm was developed, that uses Monte Carlo-generated data in two steps. The algorithm uses a phase space to accurately describe the radiation field around the Gamma Knife. Information about individual photons in the field is then combined with a generated library of data describing the resulting dose outside a wall depending on the wall thickness and the photon energy. The dose calculation algorithm is fast enough to be integrated into optimization processes, in which the algorithm is used iteratively while varying room design parameters. Demonstrated in this report is a case with a room of fixed size, in which the Gamma Knife's position and the walls' thicknesses are varied, with the aim to find the room design resulting in the minimum wall thicknesses needed to achieve acceptable dose levels outside. The results in this thesis indicate that the dose calculation algorithm performs well and could likely be used in more complex optimizations with more design variables and more advanced design goals., Strålsäkerhet är en viktig aspekt vid uppförandet av behandlingsrum för strål-terapisystem. Strålningsnivåerna som sjukvårdspersonal och allmänheten kan exponeras för utanför behandlingsrummet regleras av myndigheter och påverkar vilken väggtjocklek som behövs och vilka platser som är lämpliga att placera systemen på. Flertalet metoder för strålskyddsberäkning existerar, men de är inte väl anpassade till det stereotaktiska radiokirurgiska systemet Leksell Gamma Knife, eftersom det har ett inbyggt strålskydd. Det inbyggda strålskyddet gör att strålfältet runt Gamma Knife är anisotropt och generellt har lägre energi än primärstrålningen från systemets koboltkällor. Förenklingar som görs rörande strålfältet i flera existerande metoder för strålskyddsberäkning kan leda till att överdrivet tjocka strålskydd används eller begränsa antalet lämpliga platser att placera systemet på. I detta projekt utvecklades en dosberäkningsalgoritm, som i två steg använder data genererad genom Monte Carlo-simuleringar. Algoritmen använder ett fasrum för att detaljerat beskriva strålfältet runt Gamma Knife. Information om enskilda fotoner i fältet används sen i kombination med ett genererat bibliotek av data som beskriver det dosbidrag som en foton bidrar med utanför behandlingsrummet, baserat på fotonens energi och väggarnas tjocklek. Dosberäkningsalgoritmen är snabb nog att integreras i optimeringsprocesser där den används iterativt samtidigt som rumsdesignparametrar varieras. I denna rapport demonstreras ett fall med ett rum av bestämd storlek, där positionen av Gamma Knife i rummet och väggarnas tjocklekar varieras. Optimeringens syfte i exemplet är att hitta den rumsdesign som med de minsta väggtjocklekarna resulterar i acceptabla strålningsnivåer utanför rummet. Resultaten tyder på att dosberäkningsalgoritmen sannolikt kan användas i mer komplexa optimeringar med fler designvariabler och mer avancerade designmål.

Метою роботи є розробка спеціалізованого математичного апарату та алгоритмічного забезпечення вирішення задачі мінімізації енергетичних витрат БПЛА за рахунок оптимальної системи автоматичної стабілізації програмного руху БПЛА. Для розв'язання даної задачі розроблено процедуру оцінювання параметрів лінійних динамічних моделей БПЛА, яка використовує сплайн-функції та функції Уолша. Запропоновано оригінальний метод перетворення одного класу лінійних нестаціонарних моделей БПЛА в еквівалентні їм лінійні стаціонарні моделі, для яких можна застосувати добре відомі та апробовані методи оптимізації. На основі запропонованої в роботі процедури синтезу, що використовує принцип мінімуму в поєднанні з математичним апаратом функцій Уолша, вирішена задача синтезу розімкнутих оптимальних по витраті палива законів стабілізації програмного руху одного класу нестаціонарних БПЛА, а також для нестаціонарних БПЛА з монотонними і знакопостійними параметрами у випадку узагальненого квадратичного критерію синтезовано оптимальні закони стабілізації в аналітичній формі, зручній для їх практичної реалізації. Вперше запропоновано метод невизначених коефіцієнтів для реалізації процедури модального синтезу лінійних динамічних систем, також процедуру оптимізації процесів стабілізації лінійних динамічних систем з параметричною невизначеністю на основі модального синтезу і принципу гарантованою динаміки, що дає можливість синтезу робастного стабілізатора програмного руху БПЛА.

Метою роботи є розробка спеціалізованого математичного апарату та алгоритмічного забезпечення вирішення задачі мінімізації енергетичних витрат БПЛА за рахунок оптимальної системи автоматичної стабілізації програмного руху БПЛА. Для розв'язання даної задачі розроблено процедуру оцінювання параметрів лінійних динамічних моделей БПЛА, яка використовує сплайн-функції та функції Уолша. Запропоновано оригінальний метод перетворення одного класу лінійних нестаціонарних моделей БПЛА в еквівалентні їм лінійні стаціонарні моделі, для яких можна застосувати добре відомі та апробовані методи оптимізації. На основі запропонованої в роботі процедури синтезу, що використовує принцип мінімуму в поєднанні з математичним апаратом функцій Уолша, вирішена задача синтезу розімкнутих оптимальних по витраті палива законів стабілізації програмного руху одного класу нестаціонарних БПЛА, а також для нестаціонарних БПЛА з монотонними і знакопостійними параметрами у випадку узагальненого квадратичного критерію синтезовано оптимальні закони стабілізації в аналітичній формі, зручній для їх практичної реалізації. Вперше запропоновано метод невизначених коефіцієнтів для реалізації процедури модального синтезу лінійних динамічних систем, також процедуру оптимізації процесів стабілізації лінійних динамічних систем з параметричною невизначеністю на основі модального синтезу і принципу гарантованою динаміки, що дає можливість синтезу робастного стабілізатора програмного руху БПЛА., The theoretical and practical significance of the tasks of analysis and synthesis of optimal systems of stabilization of the software movement of the UAV, with the relative complexity of its mathematical and computer apparatus, clearly determine the need for widespread use of modern means of computer technology and modern computer technologies for the implementation of appropriate methods and computational algorithms. However, known approaches and specific universal methods for the optimal synthesis of automatic control laws are not aimed at their wide use for solving specific problems arising in the practice of developing research on control systems of UAVs. In this regard, it is urgent to carry out research aimed at the development of mathematical methods for optimizing the transitional processes of stabilizing the UAV software movement in different flight conditions. The purpose of the work is to develop a specialized mathematical apparatus and algorithmic support for solving the problems of minimizing energy costs of UAV at the expense of an optimal system of automatic stabilization of the software movement of the UAV. The use of optimal stabilization systems for UAV software movement is justified Linearized models for which a generalized procedure for estimating the parameters of UAV motion models based on splice functions and Walsh functions is developed, which is practically implemented for a number of UAV propulsion models. To select the step of the identification process an adaptive algorithm is proposed, which allows a significant increase in the accuracy of the estimates obtained. The synthesis of the optimal laws of stabilizing the software motion of non-stationary UAVs in an analytical form has some difficulties. This is due to the need to solve the nonlinear Riccati differential equation, as well as the practical implementation of the obtained control law. For the first time, a method for converting a class of linear non-stationary models of UAV motion, Целью работы является разработка специализированного математического аппарата и алгоритмического обеспечения решения задачи минимизации энергетических затрат БПЛА за счет оптимальной системы автоматической стабилизации программного движения БПЛА. Для решения данной задачи разработана процедура оценивания параметров линейных динамических моделей БПЛА, которая использует сплайн-функции и функции Уолша. Предложен оригинальный метод преобразования одного класса линейных нестационарных моделей БПЛА в эквивалентные им линейные стационарные модели, для которых можно применить хорошо известные и апробированные методы оптимизации. На основе предложенной в работе процедуры синтеза, использующей принцип минимума в сочетании с математическим аппаратом функций Уолша, решена задача синтеза разомкнутых оптимальных по расходу топлива законов стабилизации программного движения одного класса нестационарных БПЛА, а также для нестационарных БПЛА с монотонными и знакопостоянными параметрами в случае обобщенного квадратичного критерия синтезированы оптимальные законы стабилизации в аналитической форме, удобной для их практической реализации. Впервые предложен метод неопределенных коэффициентов для реализации процедуры модального синтеза оптимальных законов стабилизации программного движения стационарных БПЛА с заданными показателями качества переходных процессов, в том числе и при наличии запаздывания в контуре управления. Кроме того, предложена процедура оптимизации процессов стабилизации линейных динамических систем с параметрической неопределенностью на основе модального синтеза и принципа гарантируемой динамики, что дает возможность синтеза робастного стабилизатора программного движения БПЛА.

The paper proposes a nonlocal method of additive noise suppression in digital image based on presenting the image in matrix phase space and using nonconventional methods of multivariate statistical analysis, namely, the surrogate data technology that makes it possible to generate a pseudo-ensemble of surrogate images with their subsequent averaging using a single snapshot. This approach is based on properties of the coherent accumulation of signal component of the observation and noncoherent accumulation of its noise component as the size of observation ensemble increases that allows us to partially solve the contradiction between the denoising level and the distortion or loss of small-sized details of image, i.e. reduction of spatial resolution. The simulation modeling of the proposed method of generalized SDT-filtering of noise was conducted using the application software package of MathCad and Matlab. A comparative analysis of the spatial resolution of the proposed and several known methods of denoising has been carried out using the resolution–measurement criterion and the modified Rayleigh criterion. As is shown, the proposed method demonstrates a better spatial resolution as compared to the most common methods of denoising that is confirmed by the results of simulation modeling.

We consider a linear quantum amplifier consisting of NA two-level atoms and study the problem of amplification of N-photon states. The N-photon states are associated with N-quantum states of the harmonic oscillator. We show that the process of interaction of the electromagnetic field with atoms can be associated with some transformation of the phase space and functions defined on this phase space. We consider the Husimi functions QN(q, p) of N-quantum states of the harmonic oscillator, which are defined on the phase space, investigate transformation of these functions, and find an explicit form of the density matrix of the amplified N-photon state. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

We consider a linear quantum amplifier consisting of NA two-level atoms and study the problem of amplification of N-photon states. The N-photon states are associated with N-quantum states of the harmonic oscillator. We show that the process of interaction of the electromagnetic field with atoms can be associated with some transformation of the phase space and functions defined on this phase space. We consider the Husimi functions QN (q, p) of N-quantum states of the harmonic oscillator, which are defined on the phase space, investigate transformation of these functions, and find an explicit form of the density matrix of the amplified N-photon state.

We consider a linear quantum amplifier consisting of NA two-level atoms and study the problem of amplification of N-photon states. The N-photon states are associated with N-quantum states of the harmonic oscillator. We show that the process of interaction of the electromagnetic field with atoms can be associated with some transformation of the phase space and functions defined on this phase space. We consider the Husimi functions QN(q, p) of N-quantum states of the harmonic oscillator, which are defined on the phase space, investigate transformation of these functions, and find an explicit form of the density matrix of the amplified N-photon state. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.

4D phase space descriptions of light capture wave-optical and angular information, enabling digitally refocusing, 3D reconstructions and aberration removal. The wave-optical theory includes diffraction and interference effects, making phase space applicable to scales near the wavelength of light (e.g. in microscopy); however, at the cost of making phase space functions more complicated than their ray optics counterparts (light fields). In this thesis, we aim at bridging the gap between the abstract high-dimensional phase space and actual experiments upon which the reconstruction of unknown objects relies. We achieve the aim by 1) providing practical methods of measuring phase-space functions with good resolution in all 4D and 2) developing phase-space theories that we use to computationally mitigate scattering in experimental situations. We extend phase-space measurement schemes from from lenslet arrays to a scanning-based coded aperture method in order to improve information throughput. Theory and experiment for designed coded apertures is proposed that can efficiently capture the entire 4D phase space. Next, we develop a phase-space theory for imaging through scattering and apply it to experimentally imaging point sources through scattering and tracking neural activity in a scattering environment, such as mouse brain tissue. The method utilizes the dimension mismatch between 3D object and 4D phase-space measurements, along with a sparsity prior, to ensure robustness and allow 3D localization of point sources relatively deep into scattering tissue. We develop theory and verify the mathematical phase-space scattering operator, then study how light interacts with scatterers and propose a fast wave-equation solver. This method uses an accelerated gradient descent solver and expands the solution to the wave equation as a series of the gradient solver updates. The method outperforms the first Born approximation and the Rytov approximation in predicting the scattered fie

4D phase space descriptions of light capture wave-optical and angular information, enabling digitally refocusing, 3D reconstructions and aberration removal. The wave-optical theory includes diffraction and interference effects, making phase space applicable to scales near the wavelength of light (e.g. in microscopy); however, at the cost of making phase space functions more complicated than their ray optics counterparts (light fields). In this thesis, we aim at bridging the gap between the abstract high-dimensional phase space and actual experiments upon which the reconstruction of unknown objects relies. We achieve the aim by 1) providing practical methods of measuring phase-space functions with good resolution in all 4D and 2) developing phase-space theories that we use to computationally mitigate scattering in experimental situations. We extend phase-space measurement schemes from from lenslet arrays to a scanning-based coded aperture method in order to improve information throughput. Theory and experiment for designed coded apertures is proposed that can efficiently capture the entire 4D phase space. Next, we develop a phase-space theory for imaging through scattering and apply it to experimentally imaging point sources through scattering and tracking neural activity in a scattering environment, such as mouse brain tissue. The method utilizes the dimension mismatch between 3D object and 4D phase-space measurements, along with a sparsity prior, to ensure robustness and allow 3D localization of point sources relatively deep into scattering tissue. We develop theory and verify the mathematical phase-space scattering operator, then study how light interacts with scatterers and propose a fast wave-equation solver. This method uses an accelerated gradient descent solver and expands the solution to the wave equation as a series of the gradient solver updates. The method outperforms the first Born approximation and the Rytov approximation in predicting the scattered fie

A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states

A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states

This paper discusses comparative analysis of trajectories in the development of participating countries of the Eurasian Economic Union (EAEC) in a two-dimensional phase space. The coordinates in the space is represented by the value of a dynamic variable that is a key indicator of the country's development, and the rate of its relative growth. This allows for construction of a ternary classification diagram describing competitive behavior strategies of countries in question. The comparative analysis was run for two primary factors: the size of investment in the main capital and R&D spendings. The authors carried out analysis and identification of competitive strategies for the behavior of the EAEC countries, as well as he proposed conclusions and recommendations on improving the policy of economic development. © 2017 Author(s).

We consider scale transformations (q, p) - gt (lambda q, lambda p) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson-Schrodinger uncertainty relations for stretched states.

We consider scale transformations (q, p) - GT (lambda q, lambda p) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson-Schrodinger uncertainty relations for stretched states.

We consider scale transformations (q, p) - GT (lambda q, lambda p) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson-Schrodinger uncertainty relations for stretched states.

We consider scale transformations (q, p) - gt (lambda q, lambda p) in phase space. They induce transformations of the Husimi functions H(q, p) defined in this space. We consider the Husimi functions for states that are arbitrary superpositions of n-particle states of a harmonic oscillator. We develop a method that allows finding so-called stretched states to which these superpositions transform under such a scale transformation. We study the properties of the stretched states and calculate their density matrices in explicit form. We establish that the density matrix structure can be described using negative binomial distributions. We find expressions for the energy and entropy of stretched states and calculate the means of the number-ofstates operator. We give the form of the Heisenberg and Robertson-Schrodinger uncertainty relations for stretched states.

We describe a setup for obtaining uncertainty relations for arbitrary pairs of observables related by a Fourier transform. The physical examples discussed here are the standard position and momentum, number and angle, finite qudit systems, and strings of qubits for quantum information applications. The uncertainty relations allow for an arbitrary choice of metric for the outcome distance, and the choice of an exponent distinguishing, e.g., absolute and root mean square deviations. The emphasis of this article is on developing a unified treatment, in which one observable takes on values in an arbitrary locally compact Abelian group and the other in the dual group. In all cases, the phase space symmetry implies the equality of measurement and preparation uncertainty bounds. There is also a straightforward method for determining the optimal bounds.

A method of constructing a system of multi-moment balance equations, based on the statistical description of a production system is proposed. The need for research is determined by current trends in the development of production systems. The system of equations, which simulates the behavior of the production line for the transient conditions is obtained. It is shown that the resulting balance equations are not closed. The methods of closure of the self-linking chain of balance equations: the small-parameter method and the method of setting the equations of states for higher-order moments are considered. The known models using various methods of closure of the system of equations are analyzed. The model of the production line for the assembly-line production method is considered. The limitations and constraint equations, which enable the transition to single-moment PDE model of description of the assembly line and two-moment PDE model of the production line using the Burgers' equation are shown. The model of the production line for the company with the flow production method is considered. One-, two- and three-moment systems of equations for the two-level model of the production line are obtained. A general system of balance equations for the flow parameters of the production line is constructed., Запропонована методика побудови системи багатомоментних рівнянь дворівневого опису виробничої потокової лінії. Отримані незамкнені балансові рівняння. Розглянуто відомі моделі, в яких використані різні способи замикання системи рівнянь. Показані обмеження, що дозволяють отримати перехід до одномоментної PDE-моделі опису конвеєрної лінії і двухмоментної PDE-моделі з використанням рівняння Бюргерса, Предложена методика построения системы многомоментных балансовых уравнений для двухуровневого описания производственной поточной линии. Полученные незамкнуты балансовые уравнения. Рассмотрены известные модели, в которых использованы разные способы замыкания системы уравнений. Показаны ограничения и уравнения связей, позволяющие осуществить переход к одномоментной PDE-модели описания конвейерной линии и двухмоментной PDE-модели поточной линии с использованием уравнения Бюргерса

Práce je věnována problematice modelování dynamiky epileptického záchvatu. Nejprve jsou popsány některé zjednodušené modely neuronových populací používané v oblasti neurověd. Poté je analyzováno jejich chování ve fázovém prostoru a pomocí bifurkační analýzy, přičemž jsou diskutovány fyziologické souvislosti. Na modelu části hipokampu je provedena identifikace parametrů s využitím spektrálních vlastností normální a epileptické aktivity. Dosažené výsledky jsou porovnány s jinými výzkumy. Zároveň je naznačen další postup, který by mohl vést k ukončení epileptického záchvatu. V práci je dále popsán program vytvořený v prostředí MATLAB, který může být využit pro roztřídění signálů generovaných modelem., The work is devoted to the issue of modelling seizure dynamics. At first, some basic neural population models used in computational neuroscience are described. Then, their behavior is analyzed in phase space. Bifurcation analysis is performed and the link between physiology and the model is discussed. Model of hippocampal slice is used in the process of parameter identification that is based on spectral properties of normal and epileptiform activity. Finally, results are compared with other studies. Further approach that could possibly result in seizure termination is suggested as well. Computer program created in MATLAB environment that can be used for classification of signals generated by the model is described., Fakulta elektrotechniky a informatiky, Dokončená práce s úspěšnou obhajobou

A DGFEM discretization has been developed for the Hamiltonian dynamics of stratified incompressible linear fluid flow. The developed discretization can handle the numerical challenges posed by wave attractors: the three dimensionality of the domain, the focusing of wave energy and the incompressibility of the flow. The discretization is unconditionally stable. The conservation of phase space and energy ensure that the numerical error is physically more correct since an unphysical numerical error that changes the total energy is not possible., Numerical analysis, Electrical Engineering, Mathematics and Computer Science

A DGFEM discretization has been developed for the Hamiltonian dynamics of stratified incompressible linear fluid flow. The developed discretization can handle the numerical challenges posed by wave attractors: the three dimensionality of the domain, the focusing of wave energy and the incompressibility of the flow. The discretization is unconditionally stable. The conservation of phase space and energy ensure that the numerical error is physically more correct since an unphysical numerical error that changes the total energy is not possible., Numerical analysis, Electrical Engineering, Mathematics and Computer Science

The basic effects and patterns that characterize the model of coexistence of two species with weak sinusoidal external effect on the reproduction rate is considered. Solving Lotka-Volterra differential equations describes the behavior of the elementary ecosystem. Numerical solutions for exposure frequencies, close to the frequency of the unperturbed system cycle are found. The stability of such a non-autonomous system is investigated.It is determined that the sinusoidal effect on the population, e.g., by changing the reproduction rate of one or both species because of seasonal changes in nutrition or hunting leads to a nonperiodic system dynamics, having the type of degenerate 2-dimensional non-resonant torus. Various forms of irregular behavior of “predators” and “victims” appear in the phase portraits for similar perturbations. All this confirms that even relatively simple models of ecosystems reveal their instability, i. e., sensitivity to small external perturbations, Исследованы основные эффекты и закономерности, характеризующие модель сосуществования двух видов при слабых синусоидальных внешних воздействиях на скорость размножения, описываемую решением системы дифференциальных уравнений типа Лотки-Вольтерра. Найдены численные решения при частотах воздействия, близких к частоте цикла невозмущенной системы; исследована устойчивость неавтономной системы., Досліджено основні ефекти і закономірності, що характеризують модель співіснування двох видів при слабких синусоїдальних зовнішніх впливах на швидкість розмноження, описувану рішенням системи диференціальних рівнянь типу Лотки-Вольтерра. Знайдено чисельні рішення при частотах впливу, близьких до частоти циклу незбуреної системи; досліджено стійкість неавтономної системи.

The information technology of forecasting non-stationary time series data, which cannot be reduced to stationary is proposed in the paper. Today, this time series class is often found in various fields, including economics, sociology, and is characterized by nonlinear trend, presence of several periodic components with variable frequency and amplitude, high noise level. Identification of non-stationary time series components is achieved using the method of singular spectrum analysis (SSA), which does not require a priori information about the time series structure. It is proposed to use several phase spaces, which can be constructed using different parameter of time window length in the SSA method, for building the models of predicting and identifying the most stable time series components. It is assumed that the time series is described by linear recurrence formulas, the coefficients of which are calculated in various phase spaces. Forecasting results are characterized by stability and efficiency as the non-stationary time series data analysis is performed in various states of the system and the most significant components are considered. The proposed information technology allows to select the amount of considered phase spaces in the forecasting model and their dimensions, as well as to make an effective short-term forecast of non-stationary time series data., В работе предложена информационная технология прогнозирования нестационарных временных рядов, которые не приводятся к стационарным, характеризуются нелинейным трендом и завуалированными периодическими компонентами. Для построения модели прогнозирования определяется поведение компонент временного ряда в нескольких фазовых пространствах, построенных с использованием метода сингулярного спектрального анализа (SSA)., В роботі запропонована інформаційна технологія прогнозування нестаціонарних часових рядів, не зводяться до стаціонарних, характерізуються нелінійним трендом та завуальованимі періодичними компонентами. З метою побудови моделі прогнозування визначається поведінка компонент часового ряду у декілької фазових просторах, побудованних з використанням методу сингулярного спектрального аналізу (SSA).

We have studied time series of a grout technology in making beer wort on the basis of grout temperature time series, using non-linear dynamics methods. We have developed algorithms for studying complex dynamic management systems as well as reconstructed attractors due to historical data on a brewing unit operation. Ultimately, we have estimated the Hurst index, the fractal volume, the delay time, the maximum phase space, and the correlation format. Our analysis of the indices has determined that the technological process of making beer wort has a complex non-linear behavior. This requires adequate methods and systems of synergetic management, which would be compatible with the physical nature of the object. This approach allows using natural mechanisms of the beer wort technology at full capacity, which saves energy and expenses., Исследовано и проведен анализ влияния временных рядов с использованием методов нелинейной динамики. Использовались такие алгоритмы, как: показатель Херста, корреляционная размерность, восстановление фазового пространства. Проведена разработка алгоритмов исследования сложных динамических систем управления и реконструкцию аттракторов на основе исторических данных работы варочного отделения пивзавода., Досліджено і проаналізовано вплив часових рядів з використанням методів нелінійної динаміки. Використовувались такі алгоритми як: показник Херста, кореляційна розмірність, відновлення фазового простору. Проведено розробку алгоритмів дослідження складних динамічних систем управління та реконструкцію атракторів на основі історичних даних роботи варочного відділення пивзаводу.

A variational procedure is given for finding the pulses for which the initial temporal rms width and the rate of increase of this width are jointly minimized for propagation in non-absorbing media with arbitrary dispersive properties. We show that, while in linearly dispersive media the optimal pulses are Gaussian, in other situations such as a hollow metallic waveguide or for purely cubic dispersion departures from Gaussian behavior become evident. An interpretation of the results in terms of suitable phase-space representations is also given.

We present an overview of the setup for all-optical cooling and beam diagnostics for relativistic C3+ ion beams at the Experimental Storage Ring (ESR) at GSI. With new optical diagnostics it is foreseen to improve the measurement of the longitudinal momentum spread of the beam by at least an order of magnitude. The new optical diagnostics together with the new Schottky diagnosis and beam profile monitor available at ESR will allow to access the complete phase space evolution of the beam inside the storage ring. With new laser systems developed for cooling beams with an initially large energy spread it will be possible to replace the electron cooler that was used to reduce the initial momentum spread of the ion beam.

We present an overview of the setup for all-optical cooling and beam diagnostics for relativistic C3+ ion beams at the Experimental Storage Ring (ESR) at GSI. With new optical diagnostics it is foreseen to improve the measurement of the longitudinal momentum spread of the beam by at least an order of magnitude. The new optical diagnostics together with the new Schottky diagnosis and beam profile monitor available at ESR will allow to access the complete phase space evolution of the beam inside the storage ring. With new laser systems developed for cooling beams with an initially large energy spread it will be possible to replace the electron cooler that was used to reduce the initial momentum spread of the ion beam.

Emulsions are widely utilised in commercial environments, such as in the food and cosmetic industries. In their simplest form, emulsions are a system consisting of two immiscible liquids in the presence of emulsifiers. To form an emulsion, an input of energy is required. In this thesis, Na-caseinate was used as the emulsifier and three systems were studied: soybean oil/Na-caseinate/water, palm oil/Na-caseinate/water and tetradecane/Nacaseinate/ water. Four main techniques were used to characterise the stabilised emulsions: laser diffraction particle sizing, PGSTE-NMR, rheology and cryo-SEM. Emulsion systems are extremely complex making control and predictability over their phase behaviour practically difficult. This is because the required overall characteristics of these colloids are strongly dependent on both the energy of formulation and the choice of an appropriate combination of emulsifier, dispersed phase and continuous phase. A full understanding of the microstructure, stability and physicochemical properties of caseinatestabilised emulsions has as yet not been achieved. For example, how does caseinate selfassembly control emulsion stability? How do concentrated caseinate-based emulsions differ from dilute ones and how do the different oils (food grade oils vs. straight chain hydrocarbon) affect the formation of emulsions? The aim of this PhD programme was to obtain data to allow a better fundamental understanding of the underlying parameters defining emulsion behaviour to be obtained ...

Emulsions are widely utilised in commercial environments, such as in the food and cosmetic industries. In their simplest form, emulsions are a system consisting of two immiscible liquids in the presence of emulsifiers. To form an emulsion, an input of energy is required. In this thesis, Na-caseinate was used as the emulsifier and three systems were studied: soybean oil/Na-caseinate/water, palm oil/Na-caseinate/water and tetradecane/Nacaseinate/ water. Four main techniques were used to characterise the stabilised emulsions: laser diffraction particle sizing, PGSTE-NMR, rheology and cryo-SEM. Emulsion systems are extremely complex making control and predictability over their phase behaviour practically difficult. This is because the required overall characteristics of these colloids are strongly dependent on both the energy of formulation and the choice of an appropriate combination of emulsifier, dispersed phase and continuous phase. A full understanding of the microstructure, stability and physicochemical properties of caseinatestabilised emulsions has as yet not been achieved. For example, how does caseinate selfassembly control emulsion stability? How do concentrated caseinate-based emulsions differ from dilute ones and how do the different oils (food grade oils vs. straight chain hydrocarbon) affect the formation of emulsions? The aim of this PhD programme was to obtain data to allow a better fundamental understanding of the underlying parameters defining emulsion behaviour to be obtained ...

We present evidence that the simplest particle-physics scalar-field models of dynamical dark energy can be separated into distinct behaviors based on the acceleration or deceleration of the field as it evolves down its potential towards a zero minimum. We show that these models occupy narrow regions in the phase-plane of w and w', the dark energy equation-of-state and its time-derivative in units of the Hubble time. Restricting an energy scale of the dark energy microphysics limits how closely a scalar field can resemble a cosmological constant. These results, indicating a desired measurement resolution of order \sigma(w')\approx (1+w), define firm targets for observational tests of the physics of dark energy.

We present evidence that the simplest particle-physics scalar-field models of dynamical dark energy can be separated into distinct behaviors based on the acceleration or deceleration of the field as it evolves down its potential towards a zero minimum. We show that these models occupy narrow regions in the phase-plane of w and w', the dark energy equation-of-state and its time-derivative in units of the Hubble time. Restricting an energy scale of the dark energy microphysics limits how closely a scalar field can resemble a cosmological constant. These results, indicating a desired measurement resolution of order \sigma(w')\approx (1+w), define firm targets for observational tests of the physics of dark energy.

Phase-space methods combined with metal oxide gas sensors are introduced for the first time. The usefulness of the method is assessed by analysing the transient response of metal oxide gas sensors either to a step change in gas concentration or to thermal modulation. The first application considered consists in sorting samples from the headspace of crisps according to their rancidity state. It is shown that using the new dynamic features results in an improvement of rancidity classification: 85.4% instead of the 68.7% reached when standard features are used (AG). In the second step, the method is applied to analyse the response of temperature-modulated micro-hotplate sensors. In this case, phase-space methods give the opportunity to use novel features that describe sensor trajectories while adsorption and reaction kinetics are altered by the temperature modulation. Tungsten oxide-based sensors are employed to detect ammonia, nitrogen dioxide and their binary mixtures at different concentrations. Dynamic features are used to build and validate PLS-DA classifiers and good results are obtained, not only ameliorating the gas identification rate (95.6%), but also dramatically reducing the time needed to perform measurements (less than 2% of the response transient is needed to perform the analysis). (c) 2006 Elsevier B.V. All rights reserved.

An Abelian gerbe is constructed over classical phase space. The two-cocycles defining the gerbe are given by Feynman path integrals whose integrands contain the exponential of the Poincare-Cartan form. The U(1) gauge group on the gerbe has a natural interpretation as the invariance group of the Schrödinger equation on phase space.

This course provides an introduction to the theory and phenomenology of nonlinear dynamics and chaos in dissipative systems. The content is structured to be of general interest to undergraduates in science and engineering.