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.

Aspects of Black Holes in Higher Dimensions

This thesis is divided into three Parts. In Part I the general theory of black holes in higher dimensions is discussed. In addition to an introductory essay, two studies of linear perturbations of Myers-Perry black holes are presented. These black holes are the higher dimensional generalization of the Kerr black hole, and their analysis reveals numerous instabilities. Threshold unstable modes provide the connection between the Myers-Perry black holes and novel stationary black hole solutions such as black rings or black Saturns, as well as other non-stationary solutions known as single Killing vector field black holes.In Part II gauge/gravity duality is briefly reviewed and two aspects are studied in detail. First, the problem of finding a holographic dual to a superconductor with d-wave order parameter is investigated, and second, we examine the problem of holographic thermalization in field theories dual to rotating black holes.Lastly, in Part III the role of de Sitter solutions in string theory is discussed. A recent puzzle surrounding the fate of the de Sitter landscape is reviewed, and it is shown how the study of black holes in certain flux backgrounds can provide insight into this puzzle. We then present a theorem ruling out the addition of black holes to a certain class of flux backgrounds. Finally, a study is presented which shows that black holes can be added to the flux backgrounds relevant for the de Sitter landscape in string theory, thereby providing strong evidence for the resolution of the puzzle

Dielectrophoresis of colloids and polyelectrolytes

This PhD dissertation describes experimental and theoretical investigations on the dielectrophoretic movement of colloidal particles and polyelectrolytes suspended in aqueous solution. Dielectrophoresis (DEP) is the movement of polarisable particles in non-uniform electric fields according to their induced, or effective, polarisability. The colloidal particles used in experiments were fluorescently labelled 216 nm diameter carboxyl-modified polystyrene micro-spheres (beads) and the polyelectrolyte particles were fluorescently labelled 12 kilobase pair DNA plasmids with approximately 1 gm planar diameter. The dielectrophoretic force was generated by applying electrical alternating current (AC) potentials of varying frequency to micro-fabricated electrodes covered with low conductivity aqueous suspending media. The electrodes used for quantitative particle measurements were interdigitated Ti/Pd/Au electrode arrays (10 μm width and 10 μm gap) microfabricated on glass microscope slides using standard photolithography techniques. The frequency dependent effective particle polarisability, ap, is a key parameter in governing the dielectrophoretic force. Time domain dielectric spectroscopic measurements of solutions of DNA gave values of ap at 2 to 80x 10"31 (F m2), in the frequency range 12 MHz - 140 kHz. For latex micro-spheres, the DEP cross-over technique was used to predict ap. Since the diameters of micro-spheres and plasmid DNA were up to a micron in size, their movement in an aqueous medium at room temperature was influenced by random, thermal Brownian motion. One and two-dimensional Fokker-Planck equation (FPE) models were constructed to predict DEP-driven collection of particles onto electrodes. The model comprised DEP-induced particle flux and thermally driven diffusion flux. The FPE computer model also predicted the diffusion of particles away from the electrode surfaces after the DEP force was switched off, called particle relaxation. Using the values of ap, the FPE model was used to simulate particle collections and relaxations under the action of DEP onto a planar interdigitated electrode surface for a range of applied frequencies and voltages. The collection of particles (beads and plasmid DNA) onto interdigitated electrodes was observed using epi-fluorescence microscopy together with video-recording of images. The images were processed using software written in MATLAB 5.0. The processed images yielded timedependent particle collection profiles representing particle accumulation on the electrodes, and particle relaxation profiles after the DEP potential was switched off. Theoretical predictions were used to compare DEP collection experiments of 216 nm diameter beads and DNA plasmids. Collection and relaxation profiles were measured for AC frequencies from 100 kHz to 20 MHz and applied voltages from 1 to 4.5 V (peak). The data was in broad agreement with theoretical predictions, but there were significant quantitative differences. There are a number of reasons for these discrepancies between theory and experiment. These include electrohydrodynamically induced fluid motion that can disturb particle movement, and distortion of the electric field generated by the interdigitated electrodes due to the presence of charges associated with colloidal particles and DNA. As a first approximation, these factors were not included in the FPE model

PT-symmetric quantum mechanics

It is generally assumed that a Hamiltonian for a physically acceptable quantum system (one that has a positive-definite spectrum and obeys the requirement of unitarity) must be Hermitian. However, a PT-symmetric Hamiltonian can also define a physically acceptable quantum-mechanical system even if the Hamiltonian is not Hermitian. The study of PT-symmetric quantum systems is a young and extremely active research area in both theoretical and experimental physics. The purpose of this Review is to provide established scientists as well as graduate students with a compact, easy-to-read introduction to this field that will enable them to understand more advanced publications and to begin their own theoretical or experimental research activity. The ideas and techniques of PT symmetry have been applied in the context of many different branches of physics. This Review introduces the concepts of PT symmetry by focusing on elementary one-dimensional PT-symmetric quantum and classical mechanics and relies in particular on oscillator models to illustrate and explain the basic properties of PT-symmetric quantum theory.Comment: 58 pages, 35 figures, submitted to Reviews of Modern Physic

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

Fire Effects on Suspension Bridge Main Cables: Methods for Determining Both Temperature and Strain Distributions Within an Exposed Cable

Fire resistance design and analysis is an under-studied and under-codified area of bridge engineering. With the lessening of conservatism in bridge design, the aging or our bridge infrastructure, and the increase in the ground transport of highly-flammable and -combustible materials, it is essential that the bridge engineering community better understand and incorporate methods for modeling the effects of fire on bridges. Typical fire resistance analysis looks at the response of individual structural components. Analysis for the component of a bridge is nowhere more important than for that of the main cables of suspension bridges. As such, we will survey and introduce the necessary analysis techniques to provide the bridge engineering community with the knowledge and tools to understand fire modeling and both rapidly and accurately assess their effects on suspension bridge main cables. The work of this dissertation is twofold. In the first portion, we address proper fire modeling techniques for bridge conditions and apply them in a sequential thermal-mechanical analysis of a three-dimensional model main cable with thermally-dependent material and mechanical properties. Although fire modeling has been addressed in a variety of scenarios, including extensive studies for building design and analysis as well as tunnel design and analysis, the types of fires, fire geometries, and air conditions associated with bridge fires vary drastically. Our work identifies the time to failure for our particular main cable example and subsequently compares both the temperature and strain distributions for temperature-dependent and temperature-independent models. Although the three-dimensional analysis is complete, we hope to emulate and expand on the work done in the building fire engineering community and bring to the literature methods to produce significant two-dimensional temperature distributions for when a main cable component is either partially or fully-exposed to fire. As such, the main fire modeling analyses mentioned in the previous paragraph lay the groundwork for our pursuit of closed-form analytical solutions necessary to rapidly and accurately assess the time-dependent temperature distribution within a cable cross-section exposed to fire. These solutions are formed with different approaches depending on the fire scenario in question. They include a separation of variables (eigenfunction) approach, sinusoidal transforms, Laplace transforms, Green's function solutions, and a semi-analytical hybrid method. We validate each of the approaches numerically using three different fire models