Limited accuracy of conduction band effective mass equations for semiconductor quantum dots

Effective mass equations are the simplest models of carrier states in a semiconductor structures that reduce the complexity of a solid-state system to Schr\"odinger- or Pauli-like equations resempling those well known from quantum mechanics textbooks. Here we present a systematic derivation of a conduction-band effective mass equation for a self-assembled semiconductor quantum dot in a magnetic field from the 8-band kp theory. The derivation allows us to classify various forms of the effective mass equations in terms of a hierarchy of approximations. We assess the accuracy of the approximations in calculating selected spectral and spin-related characteristics. We indicate the importance of preserving the off-diagonal terms of the valence band Hamiltonian and argue that an effective mass theory cannot reach satisfactory accuracy without self-consistently including non-parabolicity corrections and renormalization of kp parameters. Quantitative comparison with the 8-band kp results supports the phenomenological Roth-Lax-Zwerdling formula for the g-factor in a nanostructure.Comment: Final versio

Generalising the matter coupling in massive gravity: a search for new interactions

Massive gravity theory introduced by de Rham, Gabadadze, Tolley (dRGT) is restricted by several uniqueness theorems that protect the form of the potential and kinetic terms, as well as the matter coupling. These restrictions arise from the requirement that the degrees of freedom match the expectation from Poincar\'e representations of a spin--2 field. Any modification beyond the dRGT form is known to invalidate a constraint that the theory enjoys and revive a dangerous sixth mode. One loophole is to exploit the effective nature of the theory by pushing the sixth mode beyond the strong coupling scale without completely removing it. In this paper, we search for modifications to dRGT action by coupling the matter sector to an arbitrary metric constructed out of the already existing degrees of freedom in the dRGT action. We formulate the conditions that such an extension should satisfy in order to prevent the sixth mode from contaminating the effective theory. Our approach provides a new perspective for the "composite coupling" which emerges as the unique extension up to four-point interactions.Comment: 19 pages; v2: new references added, accepted for publication in PR

Binary Darboux Transformations in Bidifferential Calculus and Integrable Reductions of Vacuum Einstein Equations

We present a general solution-generating result within the bidifferential calculus approach to integrable partial differential and difference equations, based on a binary Darboux-type transformation. This is then applied to the non-autonomous chiral model, a certain reduction of which is known to appear in the case of the D-dimensional vacuum Einstein equations with D-2 commuting Killing vector fields. A large class of exact solutions is obtained, and the aforementioned reduction is implemented. This results in an alternative to the well-known Belinski-Zakharov formalism. We recover relevant examples of space-times in dimensions four (Kerr-NUT, Tomimatsu-Sato) and five (single and double Myers-Perry black holes, black saturn, bicycling black rings)

Similarity of structures based on matrix similarity

Rad prikazuje numerički postupak za uspostavljanje odnosa između ponašanja dvije različite konstrukcije, odnosno određivanje mjerila (faktora skaliranja) između dvije konstrukcije. Ovo novo rješenje zasnovano je na ideji sličnosti matrica i linearnim transformacijama, uz ograničenja da se mjerilo između konstrukcija određuje tek nakon provođenja diskretizacije te da obje konstrukcije moraju biti u elastičnom području. Mjerilo konstrukcije može se odrediti u polju opterećenja ili pomaka (ovisno o tome dovode li se u vezu sile ili pomaci konstrukcije) gdje se skaliranje statičkog modela konstrukcije zasniva na principu ekvivalentnosti matrica, dok je skaliranje dinamičkog modela konstrukcije bazirano na Smith normalnoj formi. Skaliranje konstrukcije u operatorskom prostoru (matrice krutosti ili fleksije stavljaju se u međuodnos) trebalo bi biti bazirano na Sylvester matričnoj jednadžbi. Međutim, takav pristup nije praktičan te je zamijenjen Levenberg-Marquardt metodom za dobivanje približno ekvivalentnih matrica krutosti. Numerički primjeri ilustriraju predloženi drugačiji pristup.The paper presents a numerical procedure for relating the behaviour of two different structures, i.e. determining a scale between two structures. This novel solution is based on the notion of matrix similarity and linear transformations, with the restriction that the scale between structures is determined only after structural discretization, and that both structures have to be in the elastic regime. The structure scale can be determined in loading space or displacement space (i.e. structure forces or displacements are put into relation) where the scaling of the static structure model is based on the matrix equivalence principle, and scaling of the dynamic structure model is based on the Smith normal form. The structure scale in operator space (structure stiffness or flexibility matrices are put into relation) should be based on the Sylvester matrix equation. However, that approach is not practical and is replaced with the Levenberg-Marquardt method for obtaining only approximately equivalent stiffness matrices. Numerical examples illustrate the proposed novel approach

Analyse et commande des systèmes paramétrés, par la fonction signe matricielle

This thesis focuses on some new methods for analysis and control problems of parameter-dependent systems. These problems are reformulated as a parameter-dependent Riccati, Lyapunov and/or Sylvester equation.As an alternative to the LMI framework, we show initially that these problems can be solved thanks to the matrix sign function, or direct inversion methods of parameter-dependent matrices.Several approaches are proposed along the manuscript. First the use of non-iterative methods, based either on a direct inversion method, or on the integral definition of the matrix sign function is proposed.A second way mixes the matrix sign function together with the spectral decomposition of a matrix in an original method for the solution of non-standard Sylvester equations.Then, several methods, called iterative methods, are also proposed based on different mathematic tools such as the Laurent polynomial expansion, the inverse via Discrete Fourier Transform, etc.All along the manuscript, examples are shown, simultaneously in the constant and parameter-dependent cases, in order to show to the reader the applicability and the limits of the proposed methods.La thèse présentée ici a pour objet de proposer des méthodes de résolution de problèmes d’analyse et de commande de systèmes paramétrés. Ces problèmes sont ramenés à la résolution d’équations de Riccati, de Lyapunov et/ou de Sylvester paramétrées.S’inscrivant comme une alternative au formalisme LMI nous montrons dans un premier temps que ces équations peuvent être résolues à l’aide de la fonction signe matricielle ou des méthodes d’inversion directe de matrices paramétrées.Différentes approches sont proposées tout au long du manuscrit. On notera dans un premier temps l’utilisation de méthodes non itératives, se basant soit sur une méthode d’inversion directe, soit sur la définition dite intégrale de la fonction signe matricielle. Une seconde voie explorée combine la fonction signe matricielle avec la séparation de spectre d’une matrice pour la résolution d’équations de Sylvester non standards. Ensuite, différentes méthodes, dites itératives, sont proposées également se basant sur des outils mathématiques tels que l’expansion en polynômes de Laurent, les inversions de Fourier, etc. Tout au long du manuscrit, des exemples sont proposés, tant sur le plan constant que paramétré afin d’estimer l’applicabilité, l’efficacité ainsi que les limites des différentes méthodes proposées

Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases

Many real problems in control systems are related to the solvability of the generalized Sylvester matrix equation either using analytical or numerical methods. However, in many applications, the classical generalized Sylvester matrix equation are not well equipped to handle uncertainty in real-life problems such as conflicting requirements during the system process, the distraction of any elements and noise. Thus, crisp number in this matrix equation is replaced by fuzzy numbers and called generalized fully fuzzy Sylvester matrix equation when all parameters are in fuzzy form. The existing fuzzy analytical methods have four main drawbacks, the avoidance of using near-zero fuzzy numbers, the lack of accurate solutions, the limitation of the size of the systems, and the positive sign restriction of the fuzzy matrix coefficients and fuzzy solutions. Meanwhile, the convergence, feasibility, existence and uniqueness of the fuzzy solution are not examined in many fuzzy numerical methods. In addition, many studies are limited to positive fuzzy systems only due to the limitation of fuzzy arithmetic operation, especially for multiplication between trapezoidal fuzzy numbers.Therefore, this study aims to construct new analytical and numerical methods, namely fuzzy matrix vectorization, fuzzy absolute value, fuzzy Bartle’s Stewart, fuzzy gradient iterative and fuzzy least-squares iterative for solving arbitrary generalized Sylvester matrix equation for special cases and couple Sylvester matrix equations. In constructing these methods, new fuzzy arithmetic multiplication operators for trapezoidal fuzzy numbers are developed. The constructed methods overcome the positive restriction by allowing the negative, near-zero fuzzy numbers as the coefficients and fuzzy solutions. The necessary and sufficient conditions for the existence, uniqueness, and convergence of the fuzzy solutions are discussed, and a complete analysis of the fuzzy solution is provided. Some numerical examples and the verification of the solutions are presented to demonstrate the constructed methods. As a result, the constructed methods have successfully demonstrated the solutions for the arbitrary generalized Sylvester matrix equation for special and general cases based on the new fuzzy arithmetic operations, with minimum complexity fuzzy operations. The constructed methods are applicable to either square or non-square coefficient matrices up to 100 × 100. In conclusion, the constructed methods have significant contribution to the application of control system theory without any restriction on the system