Дисертації з теми "Condition numbers of matrices"

Random matrix theory has found many applications in various fields such as physics, statistics, number theory and so on. One important approach of studying random matrices is based on their spectral properties. In this thesis, we investigate the limiting behaviors of condition numbers of suitable random matrices in terms of large deviations. The thesis is divided into two parts. Part I, provides to the readers an short introduction on the theory of large deviations, some spectral properties of random matrices, and a summary of the results we derived, and in Part II, two papers are appended. In the first paper, we study the limiting behaviors of the 2-norm condition number of p x n random matrix in terms of large deviations for large n and p being fixed or p = p(n) → ∞ with p(n) = o(n). The entries of the random matrix are assumed to be i.i.d. whose distribution is quite general (namely sub- Gaussian distribution). When the entries are i.i.d. normal random variables, we even obtain an application in statistical inference. The second paper deals with the β-Laguerre (or Wishart) ensembles with a general parameter β > 0. There are three special cases β = 1, β = 2 and β = 4 which are called, separately, as real, complex and quaternion Wishart matrices. In the paper, large deviations of the condition number are achieved as n → ∞, while p is either fixed or p = p(n) → ∞ with p(n) = o(n/ln(n)).

This paper is devoted to asymptotic estimates for the condition numbers $\kappa(T_n(a))=||T_n(a)|| ||T_n^(-1)(a)||$ of large $n\cross n$ Toeplitz matrices $T_N(a)$ in the case where $\alpha \element L^\infinity$ and $Re \alpha \ge 0$ . We describe several classes of symbols $\alpha$ for which $\kappa(T_n(a))$ increases like $(log n)^\alpha, n^\alpha$ , or even $e^(\alpha n)$ . The consequences of the results for singular values, eigenvalues, and the finite section method are discussed. We also consider Wiener-Hopf integral operators and multidimensional Toeplitz operators.

The growing interest in computing structured matrix functions stems from the fact that preserving and exploiting the structure of matrices can help us gain physically meaningful solutions with less computational cost and memory requirement. The work presented here is divided into two parts. The first part deals with the computation of functions of structured matrices. The second part is concerned with the structured error analysis in the computation of matrix functions. We present algorithms applying the inverse scaling and squaring method and using the Schur-like form of the symplectic matrices as an alternative to the algorithms using the Schur decomposition to compute the logarithm of symplectic matrices. There are two main calculations in the inverse scaling and squaring method: taking a square root and evaluating the Padé approximants. Numerical experiments suggest that using the Schur-like form with the structure preserving iterations for the square root helps us to exploit the Hamiltonian structure of the logarithm of symplectic matrices. Some type of matrices are nearly structured. We discuss the conditions for using the nearest structured matrix to the nearly structured one by analysing the forward error bounds. Since the structure preserving algorithms for computing the functions of matrices provide advantages in terms of accuracy and data storage we suggest to compute the function of the nearest structured matrix. The analysis is applied to the nearly unitary, nearly Hermitian and nearly positive semi-definite matrices for the matrix logarithm, square root, exponential, cosine and sine functions. It is significant to investigate the effect of the structured perturbations in the sensitivity analysis of matrix functions. We study the structured condition number of matrix functions defined between smooth square matrix manifolds. We develop algorithms computing and estimating the structured condition number. We also present the lower and upper bounds on the structured condition number, which are cheaper to compute than the "exact" structured condition number. We observe that the lower bounds give a good estimation for the structured condition numbers. Comparing the structured and unstructured condition number reveals that they can differ by several orders of magnitude. Having discussed how to compute the structured condition number of matrix functions defined between smooth square matrix manifolds we apply the theory of structured condition numbers to the structured matrix factorizations. We measure the sensitivity of matrix factors to the structured perturbations for the structured polar decomposition, structured sign factorization and the generalized polar decomposition. Finally, we consider the unstructured perturbation analysis for the canonical generalized polar decomposition by using three different methods. Apart from theoretical aspect of the perturbation analysis, perturbation bounds obtained from these methods are compared numerically and our findings show an improvement on the sharpness of the perturbation bounds in the literature.

原著論文リスト[A1]: “The final publication is available at Springer via http://dx.doi.org/10.1007/s11075-012-9607-5.”. [A2]: “The final publication is available at Springer via http://dx.doi.org/10.1007/s10092-013-0085-5.”, [A3]: DOI“10.1016/j.camwa.2015.11.022”
Kyoto University (京都大学)
0048
新制・課程博士
博士(情報学)
甲第19859号
情博第610号
新制||情||106(附属図書館)
32895
京都大学大学院情報学研究科数理工学専攻
(主査)教授 中村 佳正, 教授 矢ｹ崎 一幸, 教授 山下 信雄
学位規則第4条第1項該当

Cette thèse porte sur certains aspects symboliques-numériques du problème des paires invariantes pour les polynômes de matrices. Les paires invariantes généralisent la définition de valeur propre / vecteur propre et correspondent à la notion de sous-espaces invariants pour le cas nonlinéaire. Elles trouvent leurs applications dans le calcul numérique de plusieurs valeurs propres d’un polynôme de matrices; elles présentent aussi un intérêt dans le contexte des systèmes différentiels. En utilisant une approche basée sur les intégrales de contour, nous déterminons des expressions du nombre de conditionnement et de l’erreur rétrograde pour le problème du calcul des paires invariantes. Ensuite, nous adaptons la méthode des moments de Sakurai-Sugiura au calcul des paires invariantes et nous étudions le comportement de la version scalaire et par blocs de la méthode en présence de valeurs propres multiples. Le résultats obtenus à l’aide des approches directes peuvent éventuellement être améliorés numériquement grâce à une méthode itérative: nous proposons ici une comparaison de deux variantes de la méthode de Newton appliquée aux paires invariantes. Le problème des solvants de matrices est très proche de celui des paires invariants. Le résultats présentés ci-dessus sont donc appliqués au cas des solvants pour obtenir des expressions du nombre de conditionnement et de l’erreur, et un algorithme de calcul basé sur la méthode des moments. De plus, nous étudions le lien entre le problème des solvants et la transformation des polynômes de matrices en forme triangulaire
In this thesis, we study some symbolic-numeric aspects of the invariant pair problem for matrix polynomials. Invariant pairs extend the notion of eigenvalue-eigenvector pairs, providing a counterpart of invariant subspaces for the nonlinear case. They have applications in the numeric computation of several eigenvalues of a matrix polynomial; they also present an interest in the context of differential systems. Here, a contour integral formulation is applied to compute condition numbers and backward errors for invariant pairs. We then adapt the Sakurai-Sugiura moment method to the computation of invariant pairs, including some classes of problems that have multiple eigenvalues, and we analyze the behavior of the scalar and block versions of the method in presence of different multiplicity patterns. Results obtained via direct approaches may need to be refined numerically using an iterative method: here we study and compare two variants of Newton’s method applied to the invariant pair problem. The matrix solvent problem is closely related to invariant pairs. Therefore, we specialize our results on invariant pairs to the case of matrix solvents, thus obtaining formulations for the condition number and backward errors, and a moment-based computational approach. Furthermore, we investigate the relation between the matrix solvent problem and the triangularization of matrix polynomials

Dans cette thèse, nous calculons les formules asymptotiques pour n grand, des coefficients de Fourier de la puissance n ième d'un facteur de Blaschke, permettant de prolonger et d'affiner les estimations déjà existantes. Pour cela, nous utilisons des outils classiques d'analyse asymptotique: la méthode de la phase stationnaire et celle de la descente la plus raide. Puis en application, nous construisons des fonctions fortement annulaires dont les coefficients de Taylor satisfont des propriétés de sommation nous permettant de généraliser et d'affiner les résultats de D.D. Bonar, F.W. Carroll et G. Piranian (1977). En utilisant des polynômes plats, nous élaborons aussi une autre construction de telles fonctions à partir d'un théorème de E. Bombieri et J. Bourgain (2009). Par ailleurs, nous obtenons une majoration asymptotiquement exacte, pour n grand, de la suite (\widehat{B^n} (k))_{k \geq 0} des coefficients de Fourier de la puissance n ième d'un produit de Blaschke fini quelconque B, que nous appliquerons dans la dernière partie de la thèse à une question d'analyse matricielle/théorie des opérateurs, énoncée par J. J. Schäffer en 1970. Nous élaborons aussi des exemples constructifs de produits de Blaschke finis qui atteignent nos majorants. Enfin nous étudions le conditionnement de matrices T \in \mathcal{M}_n(\mathbb{C}) pour n grand, matrices dont le spectre est donné et qui agissent sur un espace de Hilbert ou de Banach, en particulier pour les matrices de Kreiss. Dans le cas banachique, nous utilisons notre majoration des \widehat{B^n}(k) pour construire des matrices de spectres donnés arbitraires réfutant la conjecture de Schäffer
In this thesis we first compute asymptotic formulas for Fourier coefficients of the n th-power of a Blaschke factor as n gets large which extend and sharpen known estimates on those coefficients. To perform this study we use standard tools of asymptotic analysis: the so-called method of the stationary phase and the method of the steepest descent. Next as an application of our asymptotic formulas we construct strongly annular functions with Taylor coefficients satisfying sharp summation properties. This allows us to generalize and sharpen results by D.D. Bonar, F.W. Carroll and G. Piranian (1977). Making use of properties of flat polynomials, we also present another construction of such functions built on a theorem by E. Bombieri and J. Bourgain (2009). In another part of the thesis we obtain sharp upper bounds as n gets large, on the sequence (\widehat{B^{n}}(k))_{k\geq0} of the Fourier coefficients of the n th-power of an arbitrary finite Blaschke product B, which we apply in the last part of the thesis to a question raised by J.J. Schäffer (1970) in matrix analysis/operator theory. We also provide constructive examples of finite Blaschke products that achieve our upper bounds. The last chapter is dedicated to the study of the condition numbers of large matrices T\in\mathcal{M}_{n}(\mathbb{C}) with given spectrum acting on a Hilbert space or on a Banach space, espacially for some specific classes of matrices, the so-called Kreiss matrices. In the Banach case, we use our upper bound on (\widehat{B^{n}}(k))_{k\geq0} where B is arbitrary to exhibit matrices with arbitrary given spectrum refuting Schäffer's conjecture

In this paper we deal with the `strength' of ill-posedness for ill-posed linear operator equations Ax = y in Hilbert spaces, where we distinguish according_to_M. Z. Nashed [15] the ill-posedness of type I if A is not compact, but we have R(A) 6= R(A) for the range R(A) of A; and the ill-posedness of type II for compact operators A: From our considerations it seems to follow that the problems with noncompact operators A are not in general `less' ill-posed than the problems with compact operators. We motivate this statement by comparing the approximation and stability behaviour of discrete least-squares solutions and the growth rate of Galerkin matrices in both cases. Ill-posedness measures for compact operators A as discussed in [10] are derived from the decay rate of the nonincreasing sequence of singular values of A. Since singular values do not exist for noncompact operators A; we introduce stability rates in order to have a common measure for the compact and noncompact cases. Properties of these rates are illustrated by means of convolution equations in the compact case and by means of equations with multiplication operators in the noncompact case. Moreover using increasing rearrangements of the multiplier functions specific measures of ill-posedness called ill-posedness rates are considered for the multiplication operators. In this context, the character of sufficient conditions providing convergence rates of Tikhonov regularization are compared for compact operators and multiplication operators.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, 2002.
Includes bibliographical references (p. 213-216).
The modern theory of condition numbers for convex optimization problems was developed for convex problems in conic format: ... The condition number C(d) for (CPd) has been shown in theory to provide upper and/or lower bounds on many behavioral and computational characteristics of (CPd), from sizes of feasible and optimal solutions to the complexity of algorithms for solving (CPd). However, it is not known to what extent these bounds might be reasonably close to their actual measures of interest. One difficulty in testing the practical relevance of such theoretical bounds is that most practical problems are not presented in conic format. While it is usually easy to transform convex optimization problems into conic format, such transformations are not unique and do not maintain the original data, making this strategy somewhat irrelevant for computational testing of the theory. The purpose of this thesis is to overcome the obstacles stated above. We introduce an extension of condition number theory to include convex optimization problems not in conic form, and is thus more amenable to computational evaluation. This extension considers problems of the form: ... where P is a closed convex set, no longer required to be a cone. We extend many results of condition number theory to problems of form (GPd), including bounds on optimal solution sizes, optimal objective function values, interior-point algorithm complexity, etc.
(cont.) We also test the practical relevance of condition number bounds on quantities of interest for linear optimization problems. We use the NETLIB suite of linear optimization problems as a test-bed for condition number computation and analysis. Our computational results indicate that: (i) most of the NETLIB suite problems have infinite condition number (prior to pre-processing heuristics) (ii) there exists a positive linear relationship between the IPM iterations and log C(d) for the post-processed problem instances, which accounts for 42% of the variation in IPM iterations, (iii) condition numbers provide fairly tight upper bounds on the sizes of minimum-norm feasible solutions, and (iv) condition numbers provide fairly poor upper bounds on the sizes of optimal solutions and optimal objective function values.
by Fernando Ordóñez.
Ph.D.

This thesis is about high–dimensional problems considered under the so{called Kolmogorov condition. Hence, we consider research questions related to random matrices with p rows (corresponding to the parameters) and n columns (corresponding to the sample size), where p > n, assuming that the ratio  converges when the number of parameters and the sample size increase. We focus on the eigenvalue distribution of the considered matrices, since it is a well–known information–carrying object. The spectral distribution with compact support is fully characterized by its moments, i.e., by the normalized expectation of the trace of powers of the matrices. Moreover, such an expectation can be seen as a free moment in the non–commutative space of random matrices of size p x p equipped with the functional . Here, the connections with free probability theory arise. In the relation to that eld we investigate the closed form of the asymptotic spectral distribution for the sum of the quadratic forms. Moreover, we put a free cumulant–moment relation formula that is based on the summation over partitions of the number. This formula is an alternative to the free cumulant{moment relation given through non{crossing partitions ofthe set. Furthermore, we investigate the normalized  and derive, using the dierentiation with respect to some symmetric matrix, a recursive formula for that expectation. That allows us to re–establish moments of the Marcenko–Pastur distribution, and hence the recursive relation for the Catalan numbers. In this thesis we also prove that the , where , is a consistent estimator of the . We consider , where , which is proven to be normally distributed. Moreover, we propose, based on these random variables, a test for the identity of the covariance matrix using a goodness{of{t approach. The test performs very well regarding the power of the test compared to some presented alternatives for both the high–dimensional data (p > n) and the multivariate data (p ≤ n).

Dans cette thèse, j’étudie trois sujets reliés : les modèles de tenseurs, les nombres de Hurwitz et les polynômes de Macdonald-Koornwinder. Les modèles de tenseurs généralisent les modèles de matrices en tant qu’une approche à la gravité quantique en dimension arbitraire (les modèles de matrices donnent une version bidimensionnelle). J’étudie un modèle particulier qui s’appelle le modèle quartique mélonique. Sa spécialité est qu’il s’écrit en termes d’un modèle de matrices qui est lui-même aussi intéressant. En utilisant les outils bien établis, je calcule les deux premiers ordres de leur 1=N expansion. Parmi plusieurs interprétations, les nombres de Hurwitz comptent le nombre de revêtements ramifiés de surfaces de Riemann. Ils sont connectés avec de nombreux sujets en mathématiques contemporaines telles que les modèles de matrices, les équations intégrables et les espaces de modules. Ma contribution principale est une formule explicite pour les nombres doubles avec 3-cycles complétées d’une part. Cette formule me permet de prouver plusieurs propriétés intéressantes de ces nombres. Le dernier sujet de mon étude est les polynôme de Macdonald et Koornwinder, plus précisément les identités de Littlewood. Ces polynômes forment les bases importantes de l’algèbre des polynômes symétriques. Un des problèmes intrinsèques dans la théorie des fonctions symétriques est la décomposition d’un polynôme symétrique dans la base de Macdonald. La décomposition obtenue (notamment si les coefficients sont raisonnablement explicites et compacts) est nommée une identité de Littlewood. Dans cette thèse, j’étudie les identités démontrées récemment par Rains et Warnaar. Mes contributions incluent une preuve d’une extension d’une telle identité et quelques progrès partiels vers la généralisation d’une autre
In this thesis, I study three related subjects: tensor models, Hurwitz numbers and Macdonald-Koornwinder polynomials. Tensor models are generalizations of matrix models as an approach to quantum gravity in arbitrary dimensions (matrix models give a 2D version). I study a specific model called the quartic melonic tensor model. Its specialty is that it can be transformed into a multi-matrix model which is very interesting by itself. With the help of well-established tools, I am able to compute the first two leading orders of their 1=N expansion. Among many interpretations, Hurwitz numbers count the number of weighted ramified coverings of Riemann surfaces. They are connected to many subjects of contemporary mathematics such as matrix models, integrable equations and moduli spaces of complex curves. My main contribution is an explicit formula for one-part double Hurwitz numbers with completed 3-cycles. This explicit formula also allows me to prove many interesting properties of these numbers. The final subject of my study is Macdonald-Koornwinder polynomials, in particular their Littlewood identities. These polynomials form important bases of the algebra of symmetric polynomials. One of the most important problems in symmetric function theory is to decompose a symmetric polynomial into the Macdonald basis. The obtained decomposition (in particular, if the coefficients are explicit and reasonably compact) is called a Littlewood identity. In this thesis, I study many recent Littlewood identities of Rains and Warnaar. My own contributions include a proof of an extension of one of their identities and partial progress towards generalization of one another

We present bounds on various quantities of interest regarding the central trajectory of a semi-definite program (SDP), where the bounds are functions of Renegar's condition number C(d) and other naturally-occurring quantities such as the dimensions n and m. The condition number C(d) is defined in terms of the data instance d = (A, b, C) for SDP; it is the inverse of a relative measure of the distance of the data instance to the set of ill-posed data instances, that is, data instances for which arbitrary perturbations would make the corresponding SDP either feasible or infeasible. We provide upper and lower bounds on the solutions along the central trajectory, and upper bounds on changes in solutions and objective function values along the central trajectory when the data instance is perturbed and/or when the path parameter defining the central trajectory is changed. Based on these bounds, we prove that the solutions along the central trajectory grow at most linearly and at a rate proportional to the inverse of the distance to ill-posedness, and grow at least linearly and at a rate proportional to the inverse of C(d)2 , as the trajectory approaches an optimal solution to the SDP. Furthermore, the change in solutions and in objective function values along the central trajectory is at most linear in the size of the changes in the data. All such bounds involve polynomial functions of C(d), the size of the data, the distance to ill-posedness of the data, and the dimensions n and m of the SDP.

[ES] La calidad superficial en las piezas mecanizadas depende del acabado superficial, resultado de las marcas dejadas por la herramienta durante el proceso de corte. Las aproximaciones teóricas tradicionales indican que estas marcas están relacionadas con los parámetros de corte (velocidad de corte, avance, profundidad de corte...), el tipo de máquina, el material de la pieza, la geometría de la herramienta, etc. Pero no todos los tipos de mecanizado y selección de materiales pueden dar un resultado ambiguo. Hoy en día, de manera progresiva, se están utilizando las técnicas de fresado de Alta Velocidad sobre materiales de difícil mecanizado cada vez más. El fresado de Alta Velocidad implica a un considerable número de parámetros del proceso que pueden afectar a la formación topográfica 3D de la superficie. La hipótesis de que los parámetros de rugosidad superficial dependen de las huellas dejadas por la herramienta, determinadas por las condiciones de trabajo y las propiedades del entorno, condujo al desarrollo de una metodología de investigación personalizada. Este trabajo de investigación muestra como la combinación de los parámetros, inclinación del eje de la herramienta, deflexión geométrica de la herramienta y comportamiento vibracional del entorno, influencian sobre el parámetro de rugosidad superficial 3D, Sz. El modelo general fue dividido en varias partes, donde se ha descrito la influencia de parámetros del proceso adicionales, siendo incluidos en el modelo general propuesto. El proceso incremental seguido permite al autor desarrollar un modelo matemático general, paso a paso, testeando y añadiendo los componentes que más afectan a la formación de la topografía de la superficie. En la primera parte de la investigación se seleccionó un proceso de fresado con herramientas de punta plana. Primero, se analiza la geometría de la herramienta, combinada con múltiples avances, para distinguir los principales parámetros que afectan a la rugosidad superficial. Se introduce un modelo de predicción con un componente básico para la altura de la rugosidad, obtenida por la geometría de la herramienta de corte. A continuación, se llevan a cabo experimentos más específicamente diseñados, variando parámetros tecnológicos. Esto empieza con el análisis de la inclinación del eje de la herramienta contra la mesa de fresado. Los especímenes de análisis son muestras con cuatro recorridos de corte rectos con corte en sentido contrario. Las trayectorias lineales con diferentes direcciones dan la oportunidad de analizar la inclinación del husillo de fresado en la máquina. Un análisis visual reveló diferencias entre direcciones de corte opuestas, así como marcas dejadas por el filo posterior de la herramienta. Considerando las desviaciones de las marcas de corte observadas en las imágenes de rugosidad superficial obtenidas a partir de las medidas, se introdujo un análisis sobre el comportamiento dinámico del equipo y de la herramienta de corte. Las vibraciones producen desviaciones en la mesa de fresado y en la herramienta de corte. Estas desviaciones fueron detectadas e incluidas en el modelo matemático para completar la precisión en la predicción del modelo. Finalmente, el modelo de predicción del parámetro de rugosidad Sz fue comprobado con un mayor número de parámetros del proceso. Los valores de Sz medidos y predichos, fueron comparados y analizados estadísticamente. Los resultados revelaron una mayor desviación de la rugosidad predicha en las muestras fabricadas con diferentes máquinas y con diferentes avances. Importantes conclusiones sobre la precisión del equipo de fabricación han sido extraídas y de ellas se desprende que la huella de la herramienta de corte está directamente relacionada con los parámetros de la topografía de la superficie. Además, la influencia de la huella está afectada por la geometría de la herramienta de corte, la rigidez de la herramienta y la precisión del equipo. La geometría de la herramienta conforma la base del parámetro Sz, desviación de la altura de la superficie. Las conclusiones alcanzadas son la base para recomendaciones prácticas, aplicables en la industria.
[CA] La qualitat superficial en les peces mecanitzades depèn de l'acabat superficial, resultat de les marques deixades per l'eina durant el procés de tall. Les aproximacions teòriques tradicionals indiquen que aquestes marques estan relacionades amb els paràmetres de tall (velocitat de tall, avanç, profunditat de tall...), el tipus de màquina, el material de la peça, la geometria de l'eina, etc. Però no tots els tipus de mecanitzat i selecció de materials poden donar un resultat ambigu. Avui en dia, de manera progressiva, s'estan utilitzant les tècniques de fresat d'Alta Velocitat sobre materials de difícil mecanització cada vegada més. El fresat d'Alta Velocitat implica un considerable nombre de paràmetres del procés que poden afectar la formació topogràfica 3D de la superfície. La hipòtesi que els paràmetres de rugositat superficial depenen de les empremtes deixades per l'eina, determinades per les condicions de treball i les propietats de l'entorn, va conduir al desenvolupament d'una metodologia d'investigació personalitzada. Aquest treball de recerca mostra com la combinació dels paràmetres, inclinació de l'eix de l'eina, deflexió geomètrica de l'eina i comportament vibracional de l'entorn, influencien sobre el paràmetre de rugositat superficial 3D, Sz. El model general va ser dividit en diverses parts, on s'ha descrit la influència de paràmetres addicionals del procés, sent inclosos en el model general proposat. El procés incremental seguit permet a l'autor desenvolupar un model matemàtic general, pas a pas, testejant i afegint els components que més afecten a la formació de la topografia de la superfície. En la primera part de la investigació es va seleccionar un procés de fresat amb eines de punta plana. Primer, s'analitza la geometria de l'eina, combinada amb múltiples avanços, per distingir els principals paràmetres que afecten la rugositat superficial. S'introdueix un model de predicció amb un component bàsic per a l'altura de la rugositat, obtinguda a través de la geometria de l'eina de tall. A continuació, es duen a terme experiments més específicament dissenyats, variant paràmetres tecnològics. Això comença amb l'anàlisi de la inclinació de l'eix de l'eina contra la taula de fresat. Els espècimens d'anàlisi són mostres amb quatre recorreguts de tall rectes amb tall en sentit contrari. Les trajectòries lineals amb diferents direccions donen l'oportunitat d'analitzar la inclinació del fus de fresat en la màquina. Una anàlisi visual revelà diferències entre direccions de tall oposades, així com marques deixades pel tall posterior de l'eina. Considerant les desviacions de les marques de tall observades en les imatges de rugositat superficial obtingudes a partir de les mesures, es va introduir una anàlisi sobre el comportament dinàmic de l'equip i de l'eina de tall. Les vibracions produeixen desviacions en la taula de fresat i en l'eina de tall. Aquestes desviacions van ser detectades i incloses en el model matemàtic per completar la precisió en la predicció de el model. Finalment, el model de predicció de el paràmetre de rugositat Sz va ser comprovat amb un major nombre de paràmetres del procés. Els valors de Sz mesurats i predits, van ser comparats i analitzats estadísticament. Els resultats van revelar una major desviació de la rugositat predita en les mostres fabricades amb diferents màquines i amb diferents avanços. Importants conclusions sobre la precisió de l'equip de fabricació han estat extretes i d'elles es desprèn que l'empremta de l'eina de tall està directament relacionada amb els paràmetres de la topografia de la superfície. A més, la influència de la empremta està afectada per la geometria de l'eina de tall, la rigidesa de l'eina i la precisió de l'equip. La geometria de l'eina conforma la base del paràmetre Sz, desviació de l'altura de la superfície. Les conclusions assolides són la base per recomanacions pràctiques, aplicables en la indústria.
[EN] Surface quality of machined parts highly depends on the surface texture that reflects the marks, left by the tool during the cutting process. The traditional theoretical approaches indicate that these marks are related to the cutting parameters (cutting speed, feed, depths of cut...), the machining type, the part material, the tool geometry, etc. But, different machining type and material selection can give a variable result. In nowadays, more progressively, High Speed milling techniques have been applied on hard-to-cut materials more and more extensively. High-speed milling involves a considerable number of process parameters that may affect the 3D surface topography formation. The hypothesis that surface topography parameters depends on the traces left by the tool, determined by working conditions and environmental properties, led to the development of a custom research methodology. This research work shows how the parameters combination, tool axis inclination, tool geometric deflection, cutting tool geometry and environment vibrational behavior, influence on 3D surface topography parameter Sz. The general model was divided in multiple parts, where additional process parameters influence has been described and included in general model proposed. The incremental process followed allows the author to develop a general mathematical model, step by step, testing and adding the components that affect surface topography formation the most. In the first part of the research a milling procedure with flat end milling tools was selected. First, tool geometry, combined with multiple cutting feed rates, is analyzed to distinguish the main parameters that affect surface topography. A prediction model is introduced with a basic topography height component, performed by cutting tool geometry. Next, specifically designed experiments were conducted, varying technological parameters. That starts with cutting tool axis inclination against the milling table analysis. The specimens of analysis are samples with 4 contrary aimed straight cutting paths. Linear paths in different directions give a chance to analyze milling machine spindle axis topography, as well as marks left from cutting tool back cutting edge. Considering the deviations of cutting marks observed in the images of the surface topography obtained through the measurements, the milling equipment and cutting tool dynamical behavior analysis were introduced. Vibrations produce deviations in the milling table and cutting tool. These deviations were detected and included in the mathematical model to complete the prediction model accuracy. Finally, the prediction model of the topography parameter SZ was tested with increased number of process parameters. Measured and predicted SZ values were compared and analyzed statistically. Results revealed high predicted topography deviation on samples manufactured with different machines and with different feed rates. Relevant conclusions about the manufacturing equipment accuracy have been drawn and they state that cutting tool's footprint is directly related with surface topography parameters. Besides, footprint influence is affected by cutting tool geometry, tool stiffness and equipment accuracy.
Logins, A. (2021). High speed milling technological regimes, process condition and technological equipment condition influence on surface quality parameters of difficult to cut materials [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/164122
TESIS

The core of stock portfolio diversification is to pick stocks from different correlation clusters when forming portfolios. The result is that the chosen stocks will be only weakly correlated with each other. However, since correlation matrices are high dimensional, it is close to impossible to determine correlation clusters by simply looking at a correlation matrix. It is therefore common to regard industry groups as correlation clusters. In this thesis, we used three visualization methods namely Hierarchical Cluster Trees, Minimum Spanning Trees and neighbor-Net splits graphs to “collapse” correlation matrices’ high dimensional structures onto two-dimensional planes, and then assign stocks into different clusters to create the correlation clusters. We then simulated sets of portfolios where each set contains 1000 portfolios, and stocks in each of the portfolio were picked from the correlation clusters suggested by each of the three visualization methods and industry groups (another way of determine correlation clusters). The mean and variance distribution of each set of 1000 simulated portfolios gives us an indication of how well those clusters were determined. The examinations were conducted on two sets of financial data. The first one is the 30 stocks in the Dow Jones Industrial average which contains relatively small number of stocks and the second one is the ASX 200 which contains relatively larger number of stocks. We found none of the methods studied consistently defined correlation clusters more efficiently than others in out-of-sample testing. The thesis does contribute the finance literature in two ways. Firstly, it introduces the neighbor-Net method as an alternative way to visualize financial data’s underlying structures. Secondly, it used a novel “visualization

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The objective of this work is to give a more geometric approach in the introduction of complex numbers in order to make them more compreenssíveis and eliminating the idea of strange numbers and difficult to understand. To achieve this far will be a study of the properties of matrices 2x2 operative type [ a −b b a ] ,with a, b ϵR, reaching the result that these matrices form one body. Then associated with such matrices to points on the plane R2. From the result of this association gets to multiply a vector by a matrix of this type corresponds to a spin efeturar and multiply it by a scalar. From then makes two-way matching between the matrices and complex numbers so that all properties studied in the previous section remain true. As a result of this correspondence we obtain that multiplying by i2 corresponds to a spin 180o , I.e., keep the direction and reverse direction which corresponds to multiplying by (−1), I.e., i2 = −1 . Thus one arrives at a result which is usually presented to students in the introduction of complex numbers but with a meaning that once lacked. Then did a study of compliance and deformation of transformations of variables through functions complexas.Com this approach is facilitated understanding by students of their same concepts and the same function, to conclude we present a practical situation in which it uses the complexs numbers.
O objetivo deste trabalho é dar um enfoque mais geométrico na introdução dos números complexos, de forma a torná-los mais compreensíveis e eliminando a ideia de números estranhos e de difícil compreensão.Para alcançar tal objetivo far-se-á um estudo das propriedades operatórias das matrizes 2x2 do tipo [ a −b b a ] , com a, b ϵR, chegando ao resultado de que tais matrizes formam um corpo. Em seguida associa-se tais matrizes a pontos do plano R2. A partir desta associação obtém o resultado que multiplicar um vetor por uma matriz deste tipo corresponde a efeturar um giro e multiplicá-lo por um escalar. A partir daí faz a correspondência biunívoca entre as matrizes e os números complexos de forma que todas as propriedades estudadas no item anterior permanecem verdadeiras. Como resultado desta correspondência obtemos que multiplicar por i2 corresponde a um giro de 180o , isto é, manter a direção e inverter o sentido o que corresponde a multiplicar por (−1), ou seja que i2 = −1. Desta forma chega-se ao resultado que normalmente é apresentado aos alunos na introdução dos números complexos porém com um significado que outrora não possuía. A seguir fez um estudo da conformidade e deformação das transformações através de funçõeoes de variáveis complexas.Com esta abordagem fica facilitada a compreensão por parte dos alunos dos seus conceitos e mesmo a função dos mesmos, para concluir apresentamos uma situação prática em que se utiliza os números complexos.

La géométrie complexe est un outil puissant pour étudier les systèmes intégrables classiques, la physique statistique sur réseau aléatoire, les problèmes de matrices aléatoires, la théorie topologique des cordes, …Tous ces problèmes ont en commun la présence de relations, appelées équations de boucle ou contraintes de Virasoro. Dans le cas le plus simple, leur solution complète a été trouvée récemment, et se formule naturellement en termes de géométrie différentielle sur une surface de Riemann : la "courbe spectrale", qui dépend du problème. Cette thèse est une contribution au développement de ces techniques et de leurs applications.Pour commencer, nous abordons les questions de développement asymptotique à tous les ordres lorsque N tend vers l’infini, des intégrales N-dimensionnelles venant de la théorie des matrices aléatoires de taille N par N, ou plus généralement des gaz de Coulomb. Nous expliquons comment établir, dans les modèles de matrice beta et dans un régime à une coupure, le développement asymptotique à tous les ordres en puissances de N. Nous appliquons ces résultats à l'étude des grandes déviations du maximum des valeurs propres dans les modèles beta, et en déduisons de façon heuristique des informations sur l'asymptotique à tous les ordres de la loi de Tracy-Widom beta, pour tout beta positif. Ensuite, nous examinons le lien entre intégrabilité et équations de boucle. En corolaire, nous pouvons démontrer l'heuristique précédente concernant l'asymptotique de la loi de Tracy-Widom pour les matrices hermitiennes.Nous terminons avec la résolution de problèmes combinatoires en toute topologie. En théorie topologique des cordes, une conjecture de Bouchard, Klemm, Mariño et Pasquetti affirme que des séries génératrices bien choisies d'invariants de Gromov-Witten dans les espaces de Calabi-Yau toriques, sont solution d'équations de boucle. Nous l'avons démontré dans le cas le plus simple, où ces invariants coïncident avec les nombres de Hurwitz simples. Nous expliquons les progrès récents vers la conjecture générale, en relation avec nos travaux. En physique statistique sur réseau aléatoire, nous avons résolu le modèle O(n) trivalent sur réseau aléatoire introduit par Kostov, et expliquons la démarche à suivre pour résoudre des modèles plus généraux.Tous ces travaux soulignent l'importance de certaines "intégrales de matrices généralisées" pour les applications futures. Nous indiquons quelques éléments appelant à une théorie générale, encore basée sur des "équations de boucles", pour les calculer
Complex analysis is a powerful tool to study classical integrable systems, statistical physics on the random lattice, random matrix theory, topological string theory, … All these topics share certain relations, called "loop equations" or "Virasoro constraints". In the simplest case, the complete solution of those equations was found recently : it can be expressed in the framework of differential geometry over a certain Riemann surface which depends on the problem : the "spectral curve". This thesis is a contribution to the development of these techniques, and to their applications.First, we consider all order large N asymptotics in some N-dimensional integrals coming from random matrix theory, or more generally from "log gases" problems. We shall explain how to use loop equations to establish those asymptotics in beta matrix models within a one cut regime. This can be applied in the study of large fluctuations of the maximum eigenvalue in beta matrix models, and lead us to heuristic predictions about the asymptotics of Tracy-Widom beta law to all order, and for all positive beta. Second, we study the interplay between integrability and loop equations. As a corollary, we are able to prove the previous prediction about the asymptotics to all order of Tracy-Widom law for hermitian matrices.We move on with the solution of some combinatorial problems in all topologies. In topological string theory, a conjecture from Bouchard, Klemm, Mariño and Pasquetti states that certain generating series of Gromov-Witten invariants in toric Calabi-Yau threefolds, are solutions of loop equations. We have proved this conjecture in the simplest case, where those invariants coincide with the "simple Hurwitz numbers". We also explain recent progress towards the general conjecture, in relation with our work. In statistical physics on the random lattice, we have solved the trivalent O(n) model introduced by Kostov, and we explain the method to solve more general statistical models.Throughout the thesis, the computation of some "generalized matrices integrals" appears to be increasingly important for future applications, and this appeals for a general theory of loop equations

The aim of this work is the study of arbitrary mobius transformations by means of simple complex transformations, namely: the Translation, the Rotation, the Homotetia (Contraction and Dilatation) and Inversion. The results obtained were applied in circles and straight line. At the end, we give the the alternative of studying mobius transformations via matrices.
O objetivo deste trabalho é estudar transformações de Möbius arbitrárias por meio de transformações complexas mais simples, a saber: a Translação, a Rotação, a Homotetia (Contração e Dilatação) e a Inversão. Os resultados obtidos foram aplicados em círculos e retas. No final, damos a alternativa de estudar transformações de Möbius via matrizes.

La thèse se compose de 6 articles. (1) Nous calculons les générateurs des SA₃(ℝ)-invariants pour les surfaces paraboliques. (2) Nous calculons les invariants rigides relatifs pour les hypersurfaces rigides 2-non-dégénérées de rang de Levi constant 1 dans ℂ³: V₀, I₀, Q₀ ayant 11, 52, 824 monômes au numérateur. (3) Nous organisons tous les modèles affinement homogènes non-dégénérés dans ℂ³ en branches inéquivalentes. (4) Pour un courant harmonique dirigé autour d'une singularité linéarisée non-hyperbolique qui ne charge pas les séparatrices triviales dont l'extension triviale à travers 0 est ddc-fermée, nous démontrons que le nombre de Lelong en 0 est : 4.1) strictement positif si λ>0 ; 4.2) nul si λ est rationnel et négatif ; 4.3) nul si λ est négatif et si T est invariant sous l'action d'un sous-groupe cofini du groupe de monodromie. (5) Nous construisons des fibrés holomorphes en droites en toute dimension n>=2 non-prolongeables au sens de Hartogs. (6) Nous montrons que les matrices circulantes ayant k entrées 1 et k+1 entrées 0 dans leur première rangée sont toujours non singulières lorsque 2k+1 est soit une puissance d'un nombre premier, soit un produit de deux nombres premiers distincts. Pour tout autre entier 2k+1, nous exhibons une matrice circulante singulière
The thesis consists of 6 papers. (1) We calculate the generators of SA₃(ℝ)-invariants for parabolic surfaces. (2) We calculate rigid relative invariants for rigid constant Levi-rank 1 and 2-non-degenerate hypersurfaces in ℂ³: V₀, I₀, Q₀ having 11, 52, 824 monomials in their numerators. (3) We organize all affinely homogeneous nondegenerate surfaces in ℂ³ in inequivalent branches. (4) For a directed harmonic current near a non-hyperbolic linearized singularity which does not give mass to any of the trivial separatrices and whose trivial extension across 0 is ddc-closed, we show that the Lelong number at 0 is: 4.1) strictly positive if the eigenvalue λ>0; 4.2) zero if λ is a negative rational number; 4.3) zero if λ<0 and if T is invariant under the action of some cofinite subgroup of the monodromy group. (5) We construct non-extendable, in the sense of Hartogs, holomorphic line bundles in any dimension n>=2. (6) We show that circulant matrices having k ones and k+1 zeros in the first row are always nonsingular when 2k+1 is either a power of a prime, or a product of two distinct primes. For any other integer 2k+1 we exhibit a singular circulant matrix

Em uma detalhada revisão nós obtemos a lei do semi-círculo para a densidade de estados no ensemble gaussiano de Wigner. Também falamos sobre a analogia eletrostática de Dyson, enxergando os autovalores como cargas que se repelem no círculo unitário, mostrando que nesse caso a densidade de estados é uniforme. Em um contexto mais geral nós obtemos a lei do semicírculo, provando o teorema de Glivenko-Cantelli para variáveis fortemente correlacionadas usando um método combinatorial de contagem de trajetos, o que nos dá subsídios para falar em estabilidade da lei do semi-círculo. Também, nesta dissertação nós estudamos as funções de correlação nos ensembles gaussiano e circular, mostrando que sob um adequado reescalamento elas são idênticas. Outros ensembles nesta dissertação foram investigados usando o Método de Gram para o caso em que os autovalores são limitados em um intervalo. Computamos a densidade de estados para cada um desses ensembles. Mais precisamente no ensemble de Chebychev, os resultados foram obtidos analiticamente e nesse ensemble além da densidade de estados, também traçamos grá…cos da função de correlação truncada.
In a detailed review we obtain a semi-circle law for the density of states in theWigner’s Gaussian Ensemble. Also we talk about Dyson’s Analogy, seeing the eigenvalues like charges that repulse themselves in the unitary circle, showing that this case the density of states is uniform. In a more general context we obtain the semi-circle law, proving the Glivenko-Cantelli Theorem to strongly correlated variables, using a combinatorial method of Paths' Counting. Thus we are showing the stability of the semi-circle Law. Also, in this dissertation we study the correlation functions in the Gaussian and Circular ensembles showing that using the Gram's Method in the case that eigenvalues are limited in a interval. In these ensembles we computed the density of states. More precisely, in a Chebychev ensemble the results were obtained analytically. In this ensemble, we also obtain graphics of the truncated correlation function.

Les Systèmes à Evénements Discrets (SED) peuvent être définis comme des systèmes dans lesquels les variables d'état changent sous l'occurrence d'évènements au fil du temps. Les SED mettant en jeu des phénomènes de synchronisation peuvent être modélisés par des équations linéaires dans les algèbres de type (max,+). L'analyse d'atteignabilité est une problématique majeure pour les systèmes dynamiques. L'objectif est de calculer l'ensemble des états atteignables d'un système dynamique pour toutes les valeurs admissibles d'un ensemble d'états initiaux. Le problème de l'analyse d'atteignabilité pour les systèmes Max-Plus Linéaire (MPL) a été, proprement, résolu en décomposant le système MPL en une combinaison de systèmes affines par morceaux où les composantes affines du système sont représentées par des matrices de différences bornées (Difference Bound Matrix, DBM). La contribution principale de cette thèse est de présenter une procédure similaire pour résoudre le problème de l'atteignabilité pour des systèmes MPL incertains (uMPL), c'est-à-dire des systèmes MPL soumis à des bruits bornés, des perturbations et/ou des erreurs de modélisation. Tout d'abord, nous présentons une procédure permettant de partionner l'espace d'état d'un système uMPL en parties représentables par des DBM. Ensuite, nous étendons l'analyse d'atteignabilité des systèmes MPL aux systèmes uMPL. Enfin, les résultats sur l'analyse d'atteignabilité sont mis en oeuvre pour résoudre le problème d'atteignabilité conditionnelle, qui est étroitement lié au calcul du support de la densité de probabilité impliquée dans le problème de filtage stochastique
Discrete Event Dynamic Systems (DEDS) are discrete-state systems whose dynamics areentirely driven by the occurrence of asynchronous events over time. Linear equations in themax-plus algebra can be used to describe DEDS subjected to synchronization and time delayphenomena. The reachability analysis concerns the computation of all states that can bereached by a dynamical system from an initial set of states. The reachability analysis problemof Max Plus Linear (MPL) systems has been properly solved by characterizing the MPLsystems as a combination of Piece-Wise Affine (PWA) systems and then representing eachcomponent of the PWA system as Difference-Bound Matrices (DBM). The main contributionof this thesis is to present a similar procedure to solve the reachability analysis problemof MPL systems subjected to bounded noise, disturbances and/or modeling errors, calleduncertain MPL (uMPL) systems. First, we present a procedure to partition the state spaceof an uMPL system into components that can be completely represented by DBM. Then weextend the reachability analysis of MPL systems to uMPL systems. Moreover, the results onreachability analysis of uMPL systems are used to solve the conditional reachability problem,which is closely related to the support calculation of the probability density function involvedin the stochastic filtering problem
Os Sistemas a Eventos Discretos (SEDs) constituem uma classe de sistemas caracterizada por apresentar espaço de estados discreto e dinâmica dirigida única e exclusivamente pela ocorrência de eventos. SEDs sujeitos aos problemas de sincronização e de temporização podem ser descritos em termos de equações lineares usando a álgebra max-plus. A análise de alcançabilidade visa o cálculo do conjunto de todos os estados que podem ser alcançados a partir de um conjunto de estados iniciais através do modelo do sistema. A análise de alcançabilidade de sistemas Max Plus Lineares (MPL) pode ser tratada por meio da decomposição do sistema MPL em sistemas PWA (Piece-Wise Affine) e de sua correspondente representação por DBM (Difference-Bound Matrices). A principal contribuição desta tese é a proposta de uma metodologia similar para resolver o problema de análise de alcançabilidade em sistemas MPL sujeitos a ruídos limitados, chamados de sistemas MPL incertos ou sistemas uMPL (uncertain Max Plus Linear Systems). Primeiramente, apresentamos uma metodologia para particionar o espaço de estados de um sistema uMPL em componentes que podem ser completamente representados por DBM. Em seguida, estendemos a análise de alcançabilidade de sistemas MPL para sistemas uMPL. Além disso, a metodologia desenvolvida é usada para resolver o problema de análise de alcançabilidade condicional, o qual esta estritamente relacionado ao cálculo do suporte da função de probabilidade de densidade envolvida o problema de filtragem estocástica

This thesis consists of two main parts. First of all, the condition number of rectangular Vandermonde matrices with nodes on the complex unit circle is studied. The first time quantitative bounds for the extreme singular values are proven in the multivariate setting and when nodes of the Vandermonde matrix form clusters. In the second part, an optimized presentation of the deterministic stability analysis of the subspace method ESPRIT is given and results from the first part are applied.

博士
國立交通大學
應用數學系所
102
ABSTRACT For any n-"by" -n matrix" " A, let k(A) stand for the maximal number of orthonormal vectors x_j such that the scalar products ⟨Ax_j,├ x_j ⟩┤ lie in the boundary of the numerical range W(A). This number k(A) is called the Gau-Wu number of the matrix A. If A is a normal or a quadratic matrix, then the exact value of k(A) can be computed. For a matrix A of the form B⊕C, we show that k(A)=2 if and only if the numerical range of one summand, say, B, is contained in the interior of the numerical range of the other summand C and k(C)=2. For an irreducible matrix A, we can determine exactly when the value of k(A) equals the size of A. These are then applied to determine k(A) for a reducible matrix A of size 4 in terms of the shape of W(A). Moreover, if A is an n-"by" -n (n ≥2) nonnegative matrix of the form [■(0&;A_1&;&;0@&;0&;⋱&;@&;&;⋱&;A_(m-1)@0&;&;&;0)], where m ≥3 and the diagonal zeros are zero square matrices, with irreducible real part, then k(A) has an upper bound m-1. In addition, we also obtain necessary and sufficient conditions for k(A)=m-1 for such a matrix A. The other class of nonnegative matrices we study is the doubly stochastic ones. We prove that the value of k(A) is equal to 3 for any 3-by-3 doubly stochastic matrix A. Next, for any 4-by-4 doubly stochastic matrix, we also determine its numerical range. This result can be applied to find the value of k(A) for any doubly stochastic matrix A of size 4 in terms of the shape of W(A). Furthermore, the lower bound of k(A) is also found for a general n-"by" -n (n ≥5) doubly stochastic matrix A via the possible shapes of W(A).

We consider Jacobi matrices J with real b_n on the diagonal, positive a_n on the next two diagonals, and with u'(x) the density of the absolutely continuous part of the spectral measure. In particular, we are interested in compact perturbations of the free matrix J_0, that is, such that the a_n go to 1 and b_n go to 0. We study the Case sum rules for such matrices. These are trace formulae relating sums involving the a_n's and b_n's on one side and certain quantities in terms of the spectral measure on the other. We establish situations where the sum rules are valid, extending results of Case and Killip-Simon.

The matrix J is said to satisfy the Szego condition whenever the integral

int_{0}^{pi} log [u'(2 cos t)] dt,

which appears in the sum rules, is finite. Applications of our results include an extension of Shohat's classification of certain Jacobi matrices satisfying the Szego condition to cases with an infinite point spectrum, and a proof that if n(a_n - 1) go to a, nb_n go to b, and 2a < |b|, then the Szego condition fails. Related to this, we resolve a conjecture by Askey on the Szego condition for Jacobi matrices which are Coulomb perturbations of J_0. More generally, we prove that if

a_n = 1 + a/n^c + O(n^{-1-eps}) and b_n = b/n^c + O(n^{-1-eps})

with 0 < γ ≤ 1 and eps > 0, then the Szego condition is satisfied if and only if 2a ≥|b|

The goal of this paper is to develop some computational experience and test the practical relevance of the theory of condition numbers C(d) for linear optimization, as applied to problem instances that one might encounter in practice. We used the NETLIB suite of linear optimization problems as a test bed for condition number computation and analysis. Our computational results indicate that 72% of the NETLIB suite problem instances are ill-conditioned. However, after pre-processing heuristics are applied, only 19% of the post-processed problem instances are ill-conditioned, and log C(d) of the finitely-conditioned post-processed problems is fairly nicely distributed. We also show that the number of IPM iterations needed to solve the problems in the NETLIB suite varies roughly linearly (and monotonically) with log C(d) of the post-processed problem instances. Empirical evidence yields a positive linear relationship between IPM iterations and log C(d) for the post-processed problem instances, significant at the 95% confidence level. Furthermore, 42% of the variation in IPM iterations among the NETLIB suite problem instances is accounted for by log C(d) of the problem instances after pre-processin

The goal of this paper is to develop some computational experience and test the practical relevance of the theory of condition numbers C(d) for linear optimization, as applied to problem instances that one might encounter in practice. We used the NETLIB suite of linear optimization problems as a test bed for condition number computation and analysis. Our computational results indicate that 72% of the NETLIB suite problem instances are ill-conditioned. However, after pre-processing heuristics are applied, only 19% of the post-processed problem instances are ill-conditioned, and log C(d) of the finitely-conditioned post-processed problems is fairly nicely distributed. We also show that the number of IPM iterations needed to solve the problems in the NETLIB suite varies roughly linearly (and monotonically) with log C(d) of the post-processed problem instances. Empirical evidence yields a positive linear relationship between IPM iterations and log C(d) for the post-processed problem instances, significant at the 95% confidence level. Furthermore, 42% of the variation in IPM iterations among the NETLIB suite problem instances is accounted for by log C(d) of the problem instances after pre-processing. Keywords: Convex Optimization, Complexity, Interior-Point Method, Barrier Method.
Fernando Ordonez [and] Robert M. Freund.
Abstract in HTML and working paper for download in PDF available via World Wide Web at the Social Science Research Network.
Title from cover. "January 2002."
Includes bibliographical references (leaves 32-34).

Un nombre naturel n est dit congruent si il est l’aire d’un triangle rectangle dont tous les cotés sont de longueur rationnelle. Le problème des nombres congruents consiste à déterminer quels nombres sont congruents. Cette question, connue depuis plus de 1000 ans, est toujours ouverte. Elle est liée à la théorie des courbes elliptiques, car le naturel n est congruent si et seulement si la courbe elliptique y²=x³-n²x possède un point rationnel d’ordre infini. Ce lien entre les nombres congruents et les courbes elliptiques permet d’accéder à des techniques venant de la géométrie algébrique. Une de ces méthodes est le concept des matrices de Monsky qui peuvent être utilisées pour calculer la taille du groupe de 2-Selmer de la courbe elliptique y²=x³-n²x. On peut utiliser ces matrices afin de trouver de nouvelles familles infinies de nombres non-congruents. Cette relation introduit aussi des généralisations possibles au problème des nombres congruents. Par exemple, nous pouvons considérer le problème des nombres θ-congruent qui étudie des triangles avec un avec un angle fixé de taille θ au lieu de seulement des triangles rectangles. Ce problème est aussi lié aux courbes elliptiques et le concept des matrices de Monsky peut être étendu à ce cas. En fait, les matrices de Monsky peuvent être généralisées à n’importe quelle courbe elliptique qui possède une forme de Legendre sur les rationnels. Le but de ce mémoire est de construire une telle généralisation puis de l’appliquer à des problèmes de géométrie arithmétique afin de reprouver efficacement de vieux résultats ainsi que d’en trouver de nouveaux.
A positive integer n is said to be congruent if it is the area of a right triangle whose sides are all of rational length. The task of finding which integers are congruent is an old and famous yet still open question in arithmetic geometry called the congruent number problem. It is linked to the theory of elliptic curves as the integer n is congruent if and only if the elliptic curve y²=x³-n²x has a rational point of infinite order. The link between congruent numbers and elliptic curves enables the application of techniques from algebraic geometry to study the problem. One of these methods is the concept of Monsky matrices that can be used to calculate the size of the 2-Selmer group of the elliptic curve y²=x³-n²x. One can use these matrices in order to find new infinite families of non-congruent numbers. The connection to elliptic curves also introduces generalizations to the congruent number problem. For example, one may consider the θ-congruent number problem which studies triangles with a fixed angle of θ instead of only right triangles. This problem is also related to elliptic curves and the concept of Monsky matrices can be generalized to it. In fact, Monsky matrices can be generalized to any elliptic curve that has a Legendre form over the rationals. The goal of this thesis is to construct such a generalization and then to apply it to relevant problems in arithmetic geometry to efficiently reprove old results and find new ones.

The global importance of and need for sustainable development demand an informed decision-making process from the built environment to ensure optimum service life, which depends on the ability to quantify changes in condition of building materials over time. The objective of this thesis is to develop a model, which translates expert knowledge and reasoning into probability values through the application of Fuzzy Logic Artificial Intelligence to supplement limited historical performance data on degradation of building materials for the development of Markov Chain transitional probability matrices to predict service life, condition changes over time, and consequences of maintenance levels on service life of buildings. The Markov Chain methodology, a stochastic approach used for simulating the transition from one condition to another over time, has been identified as the preferred method for service life prediction by a number of studies. Limited availability of historic performance data on degradation and durability of building materials, required to populate the Markovian transition probability matrices, however restricts the application of the Markov Chain methodology. The durability and degradation factors, defined as design and maintenance levels, material and workmanship quality, external and internal climate, and operational environment, similar to the factors identified in the state-of-the–art ‘Factor Method’ for service life prediction, and current condition are rated on a uniform colour-coded five-point rating system and used to develop “IF-THEN” rules based on expert knowledge and reasoning. Fuzzy logic artificial intelligence is then used to translate these rules into crisp probability values to populate the Markovian transitional probability matrices. Historic performance data from previous condition assessments of six academic hospitals are used to calibrate and test the model. There is good correlation between the transitional probability matrices developed for the proposed model and other Markov applications in concrete bridge deck deterioration and roof maintenance models, based on historic performance data collected over extended periods, which makes the correlation more significant. Proof is presented that the Markov Chain can be used to calculate the estimated service life of a building or component, quantify changes in condition over time and determine the effect of maintenance levels on service life. It is also illustrated that the limited availability of historic performance data on degradation of building materials can be supplemented with expert knowledge, translated into probability values through the application of Fuzzy Logic Artificial Intelligence, to develop transition probability matrices for the Markov Chain. The proposed model can also be used to determine the estimated loss of or gain in service life of a building or component for various levels of maintenance.
Thesis (PhD(Civil Engineering))--University of Pretoria, 2007.
Civil Engineering
unrestricted

Les Systèmes à Evénements Discrets (SED) peuvent être définis comme des systèmes dans lesquels les variables d'état changent sous l'occurrence d'évènements au fil du temps. Les SED mettant en jeu des phénomènes de synchronisation peuvent être modélisés par des équations linéaires dans les algèbres de type (max,+). L'analyse d'atteignabilité est une problématique majeure pour les systèmes dynamiques. L'objectif est de calculer l'ensemble des états atteignables d'un système dynamique pour toutes les valeurs admissibles d'un ensemble d'états initiaux. Le problème de l'analyse d'atteignabilité pour les systèmes Max-Plus Linéaire (MPL) a été, proprement, résolu en décomposant le système MPL en une combinaison de systèmes affines par morceaux où les composantes affines du système sont représentées par des matrices de différences bornées (Difference Bound Matrix, DBM). La contribution principale de cette thèse est de présenter une procédure similaire pour résoudre le problème de l'atteignabilité pour des systèmes MPL incertains (uMPL), c'est-à-dire des systèmes MPL soumis à des bruits bornés, des perturbations et/ou des erreurs de modélisation. Tout d'abord, nous présentons une procédure permettant de partionner l'espace d'état d'un système uMPL en parties représentables par des DBM. Ensuite, nous étendons l'analyse d'atteignabilité des systèmes MPL aux systèmes uMPL. Enfin, les résultats sur l'analyse d'atteignabilité sont mis en oeuvre pour résoudre le problème d'atteignabilité conditionnelle, qui est étroitement lié au calcul du support de la densité de probabilité impliquée dans le problème de filtage stochastique
Discrete Event Dynamic Systems (DEDS) are discrete-state systems whose dynamics areentirely driven by the occurrence of asynchronous events over time. Linear equations in themax-plus algebra can be used to describe DEDS subjected to synchronization and time delayphenomena. The reachability analysis concerns the computation of all states that can bereached by a dynamical system from an initial set of states. The reachability analysis problemof Max Plus Linear (MPL) systems has been properly solved by characterizing the MPLsystems as a combination of Piece-Wise Affine (PWA) systems and then representing eachcomponent of the PWA system as Difference-Bound Matrices (DBM). The main contributionof this thesis is to present a similar procedure to solve the reachability analysis problemof MPL systems subjected to bounded noise, disturbances and/or modeling errors, calleduncertain MPL (uMPL) systems. First, we present a procedure to partition the state spaceof an uMPL system into components that can be completely represented by DBM. Then weextend the reachability analysis of MPL systems to uMPL systems. Moreover, the results onreachability analysis of uMPL systems are used to solve the conditional reachability problem,which is closely related to the support calculation of the probability density function involvedin the stochastic filtering problem
Os Sistemas a Eventos Discretos (SEDs) constituem uma classe de sistemas caracterizada por apresentar espaço de estados discreto e dinâmica dirigida única e exclusivamente pela ocorrência de eventos. SEDs sujeitos aos problemas de sincronização e de temporização podem ser descritos em termos de equações lineares usando a álgebra max-plus. A análise de alcançabilidade visa o cálculo do conjunto de todos os estados que podem ser alcançados a partir de um conjunto de estados iniciais através do modelo do sistema. A análise de alcançabilidade de sistemas Max Plus Lineares (MPL) pode ser tratada por meio da decomposição do sistema MPL em sistemas PWA (Piece-Wise Affine) e de sua correspondente representação por DBM (Difference-Bound Matrices). A principal contribuição desta tese é a proposta de uma metodologia similar para resolver o problema de análise de alcançabilidade em sistemas MPL sujeitos a ruídos limitados, chamados de sistemas MPL incertos ou sistemas uMPL (uncertain Max Plus Linear Systems). Primeiramente, apresentamos uma metodologia para particionar o espaço de estados de um sistema uMPL em componentes que podem ser completamente representados por DBM. Em seguida, estendemos a análise de alcançabilidade de sistemas MPL para sistemas uMPL. Além disso, a metodologia desenvolvida é usada para resolver o problema de análise de alcançabilidade condicional, o qual esta estritamente relacionado ao cálculo do suporte da função de probabilidade de densidade envolvida o problema de filtragem estocástica