Automatic computation of quantum-mechanical bound states and wavefunctions

We discuss the automatic solution of the multichannel Schr\"odinger equation. The proposed approach is based on the use of a CP method for which the step size is not restricted by the oscillations in the solution. Moreover, this CP method turns out to form a natural scheme for the integration of the Riccati differential equation which arises when introducing the (inverse) logarithmic derivative. A new Pr\"ufer type mechanism which derives all the required information from the propagation of the inverse of the log-derivative, is introduced. It improves and refines the eigenvalue shooting process and implies that the user may specify the required eigenvalue by its index

The Spectral Web of stationary plasma equilibria. I. General theory

A new approach to computing the complex spectrum of magnetohydrodynamic waves and instabilities of moving plasmas is presented. It is based on the concept of the Spectral Web, exploiting the self-adjointness of the generalized Frieman–Rotenberg force operator, G, and the Doppler–Coriolis gradient operator parallel to the velocity, U. The problem is solved with an open boundary, where the complementary energy Wcom represents the amount of energy to be delivered to or extracted from the system to maintain a harmonic time-dependence. The eigenvalues are connected by a system of curves in the complex ω-plane, the solution path and the conjugate path (where Wcom is real or imaginary) which together constitute the Spectral Web, having a characteristic geometry that has to be clarified yet, but that has a deep physical significance. It is obtained by straightforward contour plotting of the two paths. The complex eigenvalues, within a specified rectangle of the complex ω-plane, are found by fast, reliable, and accurate iterations. Real and complex oscillation theorems, replacing the familiar tool of counting nodes of eigenfunctions, provide an associated mechanism of mode tracking along the two paths. The Spectral Web method is generalized to toroidal systems and extended to include a resistive wall by accounting for the dissipation in such a wall. It is applied in an accompanying Paper II [J. P. Goedbloed, Phys. Plasmas 25, 032110 (2018).] to a multitude of the basic fundamental instabilities operating in cylindrical plasmas.</p

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

Canonical quantization of superconducting circuits

226 p.Los circuitos superconductores han surgido como una de las implementaciones físicas más prometedorasen tecnologías cuánticas, fusionando la física, la ingeniería y las matemáticas. Esta tesis expone modeloshamiltonianos matemáticamente consistentes y precisos para describir redes superconductoras idealesformadas por un número arbitrario de elementos concentrados y distribuidos como condensadores,inductores, uniones de Josephson, giradores, y líneas de transmisión. Aunque son ideales, hemosdemostrado que estos modelos que están basados en las leyes de Kirchhoff, son finitos y no presentanproblemas de divergencias, disipando malentendidos de la literatura previa. Finalmente se describe unaextensión de la teoría estándar para cuantizar circuitos que incluyen elementos ideales no recíprocos deforma sistemática, y se allana el camino para su extensión a giradores y circuladores dependientes defrecuencia

Fluid-Structure Interaction between Structural Components of Hydraulic Turbine and Fluid Flow

Tato dizertační práce se zabývá dvěma případy interakce tělesa s tekutinou (FSI). První z nich se zabývá analýzou vzájemné interakce mezi rotorem čerpadla a kapalinou uvnitř těsnící spáry. Vliv těsnící spáry na dynamiku celého stoje je popsán pomocí dynamických parametrů, které jsou také označovaný jako přídavné účinky. V současnosti používané modely těsnících spár používají pro stanovení dynamických parametrů řadu zjednodušujících předpokladů. V této práci je prezentováno pět různých analýz dynamických parametrů těsnící spáry čerpadla na okysličovadlo. Každá z těchto pěti analýz používá jinou míru zjednodušení výpočetního modelu. V případě největšího zjednodušení je modelován pouze objem kapaliny uvnitř těsnící spáry. Nejkomplexnější analýza pro stanovení dynamických parametrů těsnící spáry používá pro výpočet model celého čerpadla s excentrickou polohou rotoru. Druhá část této dizertační práce definuje novou metodu pro řešení interakce kapaliny s pružným tělesem. Tato metoda využívá řešení inverzního problému kmitání. Přímý problém kmitání, který je také označován jako problém vlastních hodnot, používá jako vstupy pro řešení matice hmotnosti, tuhosti a tlumení, které jsou dohromady označovány jako koeficientové matice, na základě kterých je v nejobecnějším případě stanovena Jordanovská matice a také modální matice pravostranných a levostranných vlastních vektorů. Při řešení inverzního problému kmitání jsou stanoveny koeficientové matice na základě Jordanovské matice a modálních matic pravostranných a levostranných vlastních vektorů. Existují dva případy inverzního problému kmitání. V případě, že jsou známy všechny vstupní vlastní čísla a vlastní vektory, pak se jedná o tzv. plný problém. Naopak v případě, že alespoň 1 mód kmitání soustavy není znám, tak se jedná o tzv. částečný problém. V této práci je prezentováno 5 algoritmů pro řešení inverzního problému v kmitání. Nicméně pro každý typ inverzního problému kmitání je prezentován jeden univerzální algoritmus. Algoritmus pro řešení plných problémů byl poprvé prezentován v roce 1979 Otakarem Daňkem. Algoritmy pro řešení částečných problémů, které jsou prezentovány v této práci, jsou vůbec prvními algoritmy pro řešení tohoto typu inverzního problému kmitání. Univerzální algoritmus pro řešení částečných problémů je označován jako algoritmus pro řešení částečných problémů s volbou doplňkových vlastních hodnot. Aplikace těchto dvou univerzálních algoritmů pro řešení inverzního problému kmitání pro případ plných i částečných problémů je ukázána na řešení dvou případů interakce pružného tělesa s kapalinou.This doctoral thesis deals with two cases of fluid-structure interaction (FSI). The concern of the first part is to investigate the mutual interaction between the rotor of rotating machinery and fluid within the annular seals. The effect of the annular seals on the dynamic behaviour of the whole machine is described by the rotordynamic coefficients. The current models for the determination of the rotordynamic coefficients of the annular seal use many simplifications. This thesis presents five different analyses of rotordynamic coefficients of the plain annular seal of the oxidizer pump. Each of those five analyses uses a different level of simplification. The most simple analysis models only the volume of fluid within the annular seal. And the most sophisticated analysis models fluid flow within the entire pump with the eccentric rotor. The second part of this thesis defines a new method for the solution of interaction between the fluid and flexible body. This method is based on the solution of the inverse vibration problem. The direct vibration problem, which is as well known as the eigenvalue problem, uses the mass, damping and stiffness matrices, which are collectively called ''the structural matrices'', and determines in the most general case the Jordan matrix and modal matrices of right and left eigenvectors. The inverse vibration problem is used for the definition of the structural matrices based on the Jordan matrix and modal matrices of right and left eigenvectors. The inverse vibration problems can be divided into two types. If all eigenvalues and eigenvectors are known, then it is called the full problem. On contrary, if at least one mode of vibration is unknown, then it is called the partial problem. Five algorithms for the solution of the inverse vibration problem are defined in this thesis. However, two of these five algorithms are versatile, each one for one type of inverse vibration problem. The algorithm for the solution of the full problems was presented in 1979 by Otakar Daněk. The algorithms for the solution of the partial problem, which are presented in this thesis, are the very first algorithms for the solution of this type of inverse vibration problem. And the versatile algorithm for partial problems is called the algorithm for the partial problems with the selection of additional eigenvalues. The application of these two algorithms for the solution of the inverse vibration problem for the full problems and the partial problems are demonstrated on the solution of two cases of interaction between the fluid and flexible body.

POD-Galerkin modelling of the Martian atmosphere

The aim of this thesis is to seek a low-dimensional description of baroclinic instability in general, and of the Martian atmosphere in particular, where both forcing and spatial resonance are relevant to the dynamics of the system being analysed. The Proper Orthogonal Decomposition (POD) is used to determine a basis for the modal decomposition of climatic simulations of Mars, obtained by using two General Circulation Models (GCMs): (a) a simple GCM, which is an idealised model in which the meteorological primitive equations are solved on a sphere with simplified physical parameters and (b) the Martian GCM, a more realistic model in which a comprehensive range of the relevant Martian physical parameters and topography are represented. Results of these analyses are presented for a range of Martian seasons and climatic conditions. The effects of using different forms of energy norm in performing the analysis is considered, with the objective of providing analyses which represents the physically most significant components of the circulation, with optimal efficiency. Reduced low-dimensional models that replicated the full simple GCM streamfunction simulations are formulated by projecting the spherical quasi-geostrophic equations onto the PODs of the large-scale calculations. The resulting models are analysed by using a combination of solution continuation and numerical integration methods. A thorough analysis of the models reveals that a 6-D POD model is capable of reproducing the amplitude, frequency and behaviour of the leading oscillatory structures of the simple GCM, to within a 1% error. Such an excellent reproduction of the original system is shown to be due to (1) an accurate vertical formulation scheme, (2) the use of the correct norm, (3) a sufficiently high level of truncation and (4) the fact that the original system is a steady wave flow. The behaviour of the various regimes observed in the low-order models are comparable with observations from studies of large-scale waves and instabilities in planetary atmospheres, including a range of hydrodynamical experiments on baroclinic wave interactions of a stratified fluid in cylindrical containers