Elementaria mathematicae

Diese Lecture Note beinhaltet drei Beiträge zur lehramts-orientierten elementaren Mathematik. Ein 1. Beitrag liefert Module der elementaren Mengenlehre und Ordnungstheorie, Algebra und Informatik, Geometrie und Stochastik, ein 2.Beitrag liefert Zitate großer Meister und großer Geister und ein 3.Beitrag ist den Elementen der Analysis gewidmet.

Magister-Aspekte der Mathematik

In dieser Lecture Note sind Inhalte der Linearen Algebra, der Algebra, der Ringtheorie, der Ordnungs- und der Graphentheorie gebündelt, so wie sie der Autor - verteilt auf Vorlesungen, Übungen, Seminare und Staatsexamens- bzw. Diplom-Anteile während seiner aktiven Zeit als Hochschullehrer wiederholt angeboten bzw. eingefordert hat.

Identifikation spezieller Funktionen, die durch Rodriguesformeln gegeben sind

Es ist allgemein bekannt, dass sich zwei gegebene Systeme spezieller Funktionen durch Angabe einer Rekursionsgleichung und entsprechend vieler Anfangswerte identifizieren lassen, denn computeralgebraisch betrachtet hat man damit eine Normalform vorliegen. Daher hat sich die interessante Forschungsfrage ergeben, Funktionensysteme zu identifizieren, die über ihre Rodriguesformel gegeben sind. Zieht man den in den 1990er Jahren gefundenen Zeilberger-Algorithmus für holonome Funktionenfamilien hinzu, kann die Rodriguesformel algorithmisch in eine Rekursionsgleichung überführt werden. Falls die Funktionenfamilie überdies hypergeometrisch ist, sogar laufzeiteffizient. Um den Zeilberger-Algorithmus überhaupt anwenden zu können, muss es gelingen, die Rodriguesformel in eine Summe umzuwandeln. Die vorliegende Arbeit beschreibt die Umwandlung einer Rodriguesformel in die genannte Normalform für den kontinuierlichen, den diskreten sowie den q-diskreten Fall vollständig. Das in Almkvist und Zeilberger (1990) angegebene Vorgehen im kontinuierlichen Fall, wo die in der Rodriguesformel auftauchende n-te Ableitung über die Cauchysche Integralformel in ein komplexes Integral überführt wird, zeigt sich im diskreten Fall nun dergestalt, dass die n-te Potenz des Vorwärtsdifferenzenoperators in eine Summenschreibweise überführt wird. Die Rekursionsgleichung aus dieser Summe zu generieren, ist dann mit dem diskreten Zeilberger-Algorithmus einfach. Im q-Fall wird dargestellt, wie Rekursionsgleichungen aus vier verschiedenen q-Rodriguesformeln gewonnen werden können, wobei zunächst die n-te Potenz der jeweiligen q-Operatoren in eine Summe überführt wird. Drei der vier Summenformeln waren bislang unbekannt. Sie wurden experimentell gefunden und per vollständiger Induktion bewiesen. Der q-Zeilberger-Algorithmus erzeugt anschließend aus diesen Summen die gewünschte Rekursionsgleichung. In der Praxis ist es sinnvoll, den schnellen Zeilberger-Algorithmus anzuwenden, der Rekursionsgleichungen für bestimmte Summen über hypergeometrische Terme ausgibt. Auf dieser Fassung des Algorithmus basierend wurden die Überlegungen in Maple realisiert. Es ist daher sinnvoll, dass alle hier aufgeführten Prozeduren, die aus kontinuierlichen, diskreten sowie q-diskreten Rodriguesformeln jeweils Rekursionsgleichungen erzeugen, an den hypergeometrischen Funktionenfamilien der klassischen orthogonalen Polynome, der klassischen diskreten orthogonalen Polynome und an der q-Hahn-Klasse des Askey-Wilson-Schemas vollständig getestet werden. Die Testergebnisse liegen tabellarisch vor. Ein bedeutendes Forschungsergebnis ist, dass mit der im q-Fall implementierten Prozedur zur Erzeugung einer Rekursionsgleichung aus der Rodriguesformel bewiesen werden konnte, dass die im Standardwerk von Koekoek/Lesky/Swarttouw(2010) angegebene Rodriguesformel der Stieltjes-Wigert-Polynome nicht korrekt ist. Die richtige Rodriguesformel wurde experimentell gefunden und mit den bereitgestellten Methoden bewiesen. Hervorzuheben bleibt, dass an Stelle von Rekursionsgleichungen analog Differential- bzw. Differenzengleichungen für die Identifikation erzeugt wurden. Wie gesagt gehört zu einer Normalform für eine holonome Funktionenfamilie die Angabe der Anfangswerte. Für den kontinuierlichen Fall wurden umfangreiche, in dieser Gestalt in der Literatur noch nie aufgeführte Anfangswertberechnungen vorgenommen. Im diskreten Fall musste für die Anfangswertberechnung zur Differenzengleichung der Petkovsek-van-Hoeij-Algorithmus hinzugezogen werden, um die hypergeometrischen Lösungen der resultierenden Rekursionsgleichungen zu bestimmen. Die Arbeit stellt zu Beginn den schnellen Zeilberger-Algorithmus in seiner kontinuierlichen, diskreten und q-diskreten Variante vor, der das Fundament für die weiteren Betrachtungen bildet. Dabei wird gebührend auf die Unterschiede zwischen q-Zeilberger-Algorithmus und diskretem Zeilberger-Algorithmus eingegangen. Bei der praktischen Umsetzung wird Bezug auf die in Maple umgesetzten Zeilberger-Implementationen aus Koepf(1998/2014) genommen. Die meisten der umgesetzten Prozeduren werden im Text dokumentiert. Somit wird ein vollständiges Paket an Algorithmen bereitgestellt, mit denen beispielsweise Formelsammlungen für hypergeometrische Funktionenfamilien überprüft werden können, deren Rodriguesformeln bekannt sind. Gleichzeitig kann in Zukunft für noch nicht erforschte hypergeometrische Funktionenklassen die beschreibende Rekursionsgleichung erzeugt werden, wenn die Rodriguesformel bekannt ist.

Atenci??n a la diversidad en el aula de matem??ticas

Recopilaci??n de materiales cuyo objetivo es el tratamiento de la diversidad en el aula de Matem??ticas. El primer nivel a estudiar es el correspondiente a las capacidades num??ricas que tienen los alumnos que comienzan la ESO, cuyos resultados pueden ser recogidos en unas fichas elaboradas con el fin de efectuar el seguimiento individual de cada alumno a lo largo de los dos ciclos de la ense??anza obligatoria. Los bloques tem??ticos elegidos para esta prueba inicial son: n??meros, algebra, estad??stica, y azar y probabilidad, cada uno de ellos dividido en bloques y subbloques con tres cuestiones de respuesta abierta o cerrada. La ficha de diagn??stico y seguimiento sirve para conocer la situaci??n de partida, en cada uno de los bloques estudiados, con la finalidad de dar respuesta diversa a las necesidades educativas de cada alumno. El material se completa con otra serie de fichas de recuperaci??n con una propuesta de ejercicios de dificultad creciente. Tanto las pruebas iniciales como las fichas de recuperaci??n pueden utilizarse como material de aula para aplicar de forma individual a cada alumno en situaciones diversas a lo largo del segundo ciclo de la ESo, principalmente.

Predicate Dispatching in the Common Lisp Object System

I have added support for predicate dispatching, a powerful generalization of other dispatching mechanisms, to the Common Lisp Object System (CLOS). To demonstrate its utility, I used predicate dispatching to enhance Weyl, a computer algebra system which doubles as a CLOS library. My result is Dispatching-Enhanced Weyl (DEW), a computer algebra system that I have demonstrated to be well suited for both users and programmers.

Programmable Self-Assembly: Constructing Global Shape using Biologically-inspire

In this thesis I present a language for instructing a sheet of identically-programmed, flexible, autonomous agents (``cells'') to assemble themselves into a predetermined global shape, using local interactions. The global shape is described as a folding construction on a continuous sheet, using a set of axioms from paper-folding (origami). I provide a means of automatically deriving the cell program, executed by all cells, from the global shape description. With this language, a wide variety of global shapes and patterns can be synthesized, using only local interactions between identically-programmed cells. Examples include flat layered shapes, all plane Euclidean constructions, and a variety of tessellation patterns. In contrast to approaches based on cellular automata or evolution, the cell program is directly derived from the global shape description and is composed from a small number of biologically-inspired primitives: gradients, neighborhood query, polarity inversion, cell-to-cell contact and flexible folding. The cell programs are robust, without relying on regular cell placement, global coordinates, or synchronous operation and can tolerate a small amount of random cell death. I show that an average cell neighborhood of 15 is sufficient to reliably self-assemble complex shapes and geometric patterns on randomly distributed cells. The language provides many insights into the relationship between local and global descriptions of behavior, such as the advantage of constructive languages, mechanisms for achieving global robustness, and mechanisms for achieving scale-independent shapes from a single cell program. The language suggests a mechanism by which many related shapes can be created by the same cell program, in the manner of D'Arcy Thompson's famous coordinate transformations. The thesis illuminates how complex morphology and pattern can emerge from local interactions, and how one can engineer robust self-assembly.

Exploring Bit-Difference for Approximate KNN Search in High-dimensional Databases

In this paper, we develop a novel index structure to support efficient approximate k-nearest neighbor (KNN) query in high-dimensional databases. In high-dimensional spaces, the computational cost of the distance (e.g., Euclidean distance) between two points contributes a dominant portion of the overall query response time for memory processing. To reduce the distance computation, we first propose a structure (BID) using BIt-Difference to answer approximate KNN query. The BID employs one bit to represent each feature vector of point and the number of bit-difference is used to prune the further points. To facilitate real dataset which is typically skewed, we enhance the BID mechanism with clustering, cluster adapted bitcoder and dimensional weight, named the BID⁺. Extensive experiments are conducted to show that our proposed method yields significant performance advantages over the existing index structures on both real life and synthetic high-dimensional datasets.

Statistical modeling on (α, +∞)

Observations in daily practice are sometimes registered as positive values larger then a given threshold α. The sample space is in this case the interval (α,+∞), α > 0, which can be structured as a real Euclidean space in different ways. This fact opens the door to alternative statistical models depending not only on the assumed distribution function, but also on the metric which is considered as appropriate, i.e. the way differences are measured, and thus variability

Hilbert space on probability density functions with Aitchison geometry

Compositional data analysis motivated the introduction of a complete Euclidean structure in the simplex of D parts. This was based on the early work of J. Aitchison (1986) and completed recently when Aitchinson distance in the simplex was associated with an inner product and orthonormal bases were identified (Aitchison and others, 2002; Egozcue and others, 2003). A partition of the support of a random variable generates a composition by assigning the probability of each interval to a part of the composition. One can imagine that the partition can be refined and the probability density would represent a kind of continuous composition of probabilities in a simplex of infinitely many parts. This intuitive idea would lead to a Hilbert-space of probability densities by generalizing the Aitchison geometry for compositions in the simplex into the set probability densities

Distributions on the simplex

The simplex, the sample space of compositional data, can be structured as a real Euclidean space. This fact allows to work with the coefficients with respect to an orthonormal basis. Over these coefficients we apply standard real analysis, inparticular, we define two different laws of probability trought the density function and we study their main properties

Compositional Data Analysis with R

R from http://www.r-project.org/ is ‘GNU S’ – a language and environment for statistical computing and graphics. The environment in which many classical and modern statistical techniques have been implemented, but many are supplied as packages. There are 8 standard packages and many more are available through the cran family of Internet sites http://cran.r-project.org . We started to develop a library of functions in R to support the analysis of mixtures and our goal is a MixeR package for compositional data analysis that provides support for operations on compositions: perturbation and power multiplication, subcomposition with or without residuals, centering of the data, computing Aitchison’s, Euclidean, Bhattacharyya distances, compositional Kullback-Leibler divergence etc. graphical presentation of compositions in ternary diagrams and tetrahedrons with additional features: barycenter, geometric mean of the data set, the percentiles lines, marking and coloring of subsets of the data set, theirs geometric means, notation of individual data in the set . . . dealing with zeros and missing values in compositional data sets with R procedures for simple and multiplicative replacement strategy, the time series analysis of compositional data. We’ll present the current status of MixeR development and illustrate its use on selected data sets

Compositional analysis of bivariate discrete probabilities

A joint distribution of two discrete random variables with finite support can be displayed as a two way table of probabilities adding to one. Assume that this table has n rows and m columns and all probabilities are non-null. This kind of table can be seen as an element in the simplex of n · m parts. In this context, the marginals are identified as compositional amalgams, conditionals (rows or columns) as subcompositions. Also, simplicial perturbation appears as Bayes theorem. However, the Euclidean elements of the Aitchison geometry of the simplex can also be translated into the table of probabilities: subspaces, orthogonal projections, distances. Two important questions are addressed: a) given a table of probabilities, which is the nearest independent table to the initial one? b) which is the largest orthogonal projection of a row onto a column? or, equivalently, which is the information in a row explained by a column, thus explaining the interaction? To answer these questions three orthogonal decompositions are presented: (1) by columns and a row-wise geometric marginal, (2) by rows and a columnwise geometric marginal, (3) by independent two-way tables and fully dependent tables representing row-column interaction. An important result is that the nearest independent table is the product of the two (row and column)-wise geometric marginal tables. A corollary is that, in an independent table, the geometric marginals conform with the traditional (arithmetic) marginals. These decompositions can be compared with standard log-linear models. Key words: balance, compositional data, simplex, Aitchison geometry, composition, orthonormal basis, arithmetic and geometric marginals, amalgam, dependence measure, contingency table

Mixing compositions and other scales

Theory of compositional data analysis is often focused on the composition only. However in practical applications we often treat a composition together with covariables with some other scale. This contribution systematically gathers and develop statistical tools for this situation. For instance, for the graphical display of the dependence of a composition with a categorical variable, a colored set of ternary diagrams might be a good idea for a first look at the data, but it will fast hide important aspects if the composition has many parts, or it takes extreme values. On the other hand colored scatterplots of ilr components could not be very instructive for the analyst, if the conventional, black-box ilr is used. Thinking on terms of the Euclidean structure of the simplex, we suggest to set up appropriate projections, which on one side show the compositional geometry and on the other side are still comprehensible by a non-expert analyst, readable for all locations and scales of the data. This is e.g. done by defining special balance displays with carefully- selected axes. Following this idea, we need to systematically ask how to display, explore, describe, and test the relation to complementary or explanatory data of categorical, real, ratio or again compositional scales. This contribution shows that it is sufficient to use some basic concepts and very few advanced tools from multivariate statistics (principal covariances, multivariate linear models, trellis or parallel plots, etc.) to build appropriate procedures for all these combinations of scales. This has some fundamental implications in their software implementation, and how might they be taught to analysts not already experts in multivariate analysis

Using self organizing maps on compositional data

Self-organizing maps (Kohonen 1997) is a type of artificial neural network developed to explore patterns in high-dimensional multivariate data. The conventional version of the algorithm involves the use of Euclidean metric in the process of adaptation of the model vectors, thus rendering in theory a whole methodology incompatible with non-Euclidean geometries. In this contribution we explore the two main aspects of the problem: 1. Whether the conventional approach using Euclidean metric can shed valid results with compositional data. 2. If a modification of the conventional approach replacing vectorial sum and scalar multiplication by the canonical operators in the simplex (i.e. perturbation and powering) can converge to an adequate solution. Preliminary tests showed that both methodologies can be used on compositional data. However, the modified version of the algorithm performs poorer than the conventional version, in particular, when the data is pathological. Moreover, the conventional ap- proach converges faster to a solution, when data is \well-behaved". Key words: Self Organizing Map; Artificial Neural networks; Compositional data

Clustering compositional data trajectories

Our essay aims at studying suitable statistical methods for the clustering of compositional data in situations where observations are constituted by trajectories of compositional data, that is, by sequences of composition measurements along a domain. Observed trajectories are known as “functional data” and several methods have been proposed for their analysis. In particular, methods for clustering functional data, known as Functional Cluster Analysis (FCA), have been applied by practitioners and scientists in many fields. To our knowledge, FCA techniques have not been extended to cope with the problem of clustering compositional data trajectories. In order to extend FCA techniques to the analysis of compositional data, FCA clustering techniques have to be adapted by using a suitable compositional algebra. The present work centres on the following question: given a sample of compositional data trajectories, how can we formulate a segmentation procedure giving homogeneous classes? To address this problem we follow the steps described below. First of all we adapt the well-known spline smoothing techniques in order to cope with the smoothing of compositional data trajectories. In fact, an observed curve can be thought of as the sum of a smooth part plus some noise due to measurement errors. Spline smoothing techniques are used to isolate the smooth part of the trajectory: clustering algorithms are then applied to these smooth curves. The second step consists in building suitable metrics for measuring the dissimilarity between trajectories: we propose a metric that accounts for difference in both shape and level, and a metric accounting for differences in shape only. A simulation study is performed in order to evaluate the proposed methodologies, using both hierarchical and partitional clustering algorithm. The quality of the obtained results is assessed by means of several indices