In this thesis, we study several nonlocal models to obtain their numerical solutions accurately and efficiently. In contrast to the classical (local) partial differential equation models, these nonlocal models are integro-differential equations that do not contain spatial derivatives. As a result, these nonlocal models allow their solutions to have discontinuities. Hence, they can be widely used for fracture problems and anisotropic problems. This thesis mainly includes two parts. The first part focuses on presenting accurate and efficient numerical methods. In this part, we first introduce three meshless methods including two global schemes, namely the radial basis functions collocation method (RBFCM) and the radial ba- sis functions-based pseudo-spectral method (RBF-PSM) and a localized scheme, namely the localized radial basis functions-based pseudo-spectral method (LRBF-PSM), which also gives the development process of the RBF methods from global to local. The comparison of these methods shows that LRBF-PSM not only avoids the Runge phenomenon but also has similar accuracy to the global scheme. Since the LRBF-PSM uses only a small subset of points, the calculation consumes less CPU time. Afterwards, we improve this scheme by adding enrichment functions so that it can be effectively applied to discontinuity problems. This thesis abbreviates this enriched method as LERBF-PSM (Localized enriched radial basis functions-based pseudo-spectral method). In the second part, we focus on applying the derived methods from the first part to nonlocal topics of current research, including nonlocal diffusion models, linear peridynamic models, parabolic/hyperbolic nonlocal phase field models, and nonlocal nonlinear Schrödinger equations arising in quantum mechanics. The first point worth noting is that in order to verify the meshless nature of LRBF-PSM, we apply this method to solve a two-dimensional steady-state continuous peridynamic model in regular, irregular (L-shaped and Y-shaped) domains with uniform and non-uniform discretizations and even extend this method to three dimensions. It is also worth noting that before solving nonlinear nonlocal Schrödinger equations, according to the property of the convolution, these partial integro-differential equations are transformed into equivalent or approximate partial differential equations (PDEs) in the whole space and then the LRBF-PSM is used for the spatial discretization in a finite domain with suitable boundary conditions. Therefore, the solutions can be quickly approximated.

We compute the expected numbers of both formal concepts and proper premises in a formal context that is chosen uniformly at random among all formal contexts of given dimensions.

Constructing description logic ontologies is a difficult task that is normally conducted by experts. Recent results show that parts of ontologies can be constructed from description logic interpretations. However, these results assume the interpretations to be free of errors, which may not be the case for real-world data. To provide some mechanism to handle these errors, the notion of confidence from data mining is introduced into description logics, yielding confident general concept inclusions (confident GCIs) of finite interpretations. The main focus of this work is to prove the existence of finite bases of confident GCIs and to describe some of theses bases explicitly.

In the recent past, the field of scientific computing has become of more and more importance for scientific as well as for industrial research, playing a comparable role as experiment and theory do. This success of computational methods in scientific and engineering research is next to the enormous improvement of computer hardware to a large extend due to contributions from applied mathematicians, who have developed algorithms which make real life applications feasible. Examples are adaptive methods, high order discretization, fast linear and non-linear solvers and multi-level methods. The application of these methods in a large class of problems demands for suitable and robust tools for a flexible and efficient implementation. In order to play a crucial role in scientific and engineering research, besides efficiency in the numerical solution, also efficiency in problem setup and interpretation of simulation results is of utmost importance. As modeling and computing comes closer together, efficient computational methods need to be applied to new sets of equations. The problems to be addressed by simulation methods become more and more complicated, ranging over different scales, interacting on different dimensions and combining different physics. Such problems need to be implemented in a short period of time, solved on complicated domains and visualized with respect to the demand of the user. %Only a modular abstract simulation environment will fulfill these requirements and allow to setup, solve and visualize real-world problems appropriately. In this work, the concepts and the design of the C++ finite element toolbox AMDiS (adaptive multidimensional simulations) are described. It is shown, how abstract data structures and modern software concepts can help to design user-friendly finite element software, which provides large flexibility in problem definition while on the other hand efficiently solves these problems. Also systems of coupled problems can be solved in an intuitive way. In order to demonstrate its possibilities, AMDiS has been applied to several non-standard problems. The most time-consuming part in most simulations is the solution of linear systems of equations. Multi-level methods use discretization hierarchies to solve these systems in a very efficient way. In AMDiS, such multi-level techniques are implemented in the context of adaptive finite elements. Several numerical results are given which compare this multigrid solver with classical iterative methods. Besides the development of more efficient algorithms also the growing hardware capabilities lead to an improvement of simulation possibilities. Modern computing clusters contain more and more processors and also personal computers today are often equipped with multi-core processors. In this work, a new parallelization approach has been developed which allows the parallelization of sequential code in a very easy way and reduces the communication overhead compared to classical parallelization concepts.

Recently, Formal Concept Analysis has proven to be an efficient method for the analysis and representation of information. However, the possibility to visualize concept hierarchies is being affected by the difficulty of drawing attractive diagrams automatically. Reducing the number of edge crossings seems to increase the readability of those drawings. This dissertation concerns with a mandatory prerequisite of this constraint, namely the characterization and visual representation of planar lattices. The manifold existing approaches and algorithms are thereby considered under a different point of view. It is well known that exactly the planar lattices (or planar posets) possess an additional order ``from left to right''. Our aim in this work is to define left-relations and left-orders more precisely and to describe several aspects of planar lattices with their help. The three approaches employed structure the work in as many parts: Left-relations on lattices allow a more efficient consideration of conjugate orders since they are uniquely determined by the sorting of the meet-irreducibles. Additionally, the restriction on the meet-irreducibles enables us to achieve an intuitive description of standard contexts of planar lattices similar to the consecutive-one property. With the help of left-relations on diagrams, planar lattices can indeed be drawn without edge crossings in the plane. Thereby, lattice-theoretically found left-orders can be detected in the graphical representation again. Furthermore, we modify the left-right-numbering algorithm in order to obtain attribute-additive and plane drawings of planar lattices. Finally, we will consider left-relations on contexts. They turn out to be fairly similar structures to the Ferrers-graphs. Planar lattices can be characterized by a property of these graphs, namely the bipartiteness. We will constructively prove this result. Subsequently, we can design an efficient algorithm that finds all non-similar plane diagrams of a lattice. / Die Formale Begriffsanalyse hat sich in den letzten Jahren als effizientes Werkzeug zur Datenanalyse und -repräsentation bewährt. Die Möglichkeit der visuellen Darstellung von Begriffshierarchien wird allerdings durch die Schwierigkeit, ansprechende Diagramme automatisch generieren zu können, beeinträchtigt. Offenbar sind Diagramme mit möglichst wenig Kantenkreuzungen für den menschlichen Anwender leichter lesbar. Diese Arbeit beschäftigt sich mit mit einer diesem Kriterium zugrunde liegenden Vorleistung, nämlich der Charakterisierung und Darstellung planarer Verbände. Die schon existierenden vielfältigen Ansätze und Methoden werden dabei unter einem neuen Gesichtspunkt betrachtet. Bekannterweise besitzen genau die planaren Verbände (bzw. planare geordnete Mengen) eine zusätzliche Ordnung "von links nach rechts". Unser Ziel in dieser Arbeit ist es, solche Links-Relationen bzw. Links-Ordnungen genauer zu definieren und verschiedene Aspekte planarer Verbände mit ihrer Hilfe zu beschreiben. Die insgesamt drei auftretenden Sichtweisen gliedern die Arbeit in ebensoviele Teile: Links-Relationen auf Verbänden erlauben eine effizientere Behandlung konjugierter Ordnungen, da sie durch die Anordnung der Schnitt-Irreduziblen schon eindeutig festgelegt sind. Außerdem erlaubt die Beschränkung auf die Schnitt-Irreduziblen eine anschauliche Beschreibung von Standardkontexten planarer Verbände ähnlich der consecutive-one property. Mit Hilfe der Links-Relationen auf Diagrammen können planare Verbände tatsächlich eben gezeichnet werden. Dabei lassen sich verbandstheoretisch ermittelte Links-Ordnungen in der graphischen Darstellung wieder finden. Weiterhin geben wir in eine Modifikation des left-right-numbering an, mit der planare Verbände merkmaladditiv und eben gezeichnet werden können. Schließlich werden wir Links-Relationen auf Kontexten betrachten. Diese stellen sich als sehr ähnlich zu Ferrers-Graphen heraus. Planare Verbände lassen sich durch eine Eigenschaft dieser Graphen, nämlich die Bipartitheit, charakterisieren. Wir werden dieses Ergebnis konstruktiv beweisen und darauf aufbauend einen effizienten Algorithmus angeben, mit dem alle nicht-ähnlichen ebenen Diagramme eines Verbandes bestimmt werden können.

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For several decades, there have been a varying number of books on Calculus following the classic line of mathematical thought, where Mathematics is taught for everybody by means of rigorous definitions, theorems, and carefully detailed and extensive demonstrations. For mathematical education into the XXI Century the students need to achieve ability in handling of present mathematical tools and concepts from the beginning of their courses. These needs can be achieved today by means of a paradigmatic change in the focus of mathematics teaching: to learn to develop ideas and to experiment and test those ideas in such way that students can verify their own inferences. In this paper we report an educational research in teaching and learning functions models according to a new paradigm in hands-on experimental mathematics, with applications in the real world, i.e. sciences and engineering by using Computer Algebra Systems. The study of functions is presented, focused into the framing of Exploratory Learning Systems, where students learn by means of the action and their participation in it. It is designed for teachers working together with students in a computer laboratory like hands-on workshops-type activities on other sciences. In this way students have a more “alive”, “realistic” and “accessible” touch in Calculus.

In recent years, the idea of language influencing the cognitive development of an understanding of place value has received increasing attention. This study explored the influence of using explicit number names on prekindergarten and kindergarten students’ ability to rote count, read two-digit numerals, model two-digit numbers, and identify the place value of individual digits in two-digit numerals. Through individual student interviews, preand post-assessments were administered to evaluate rote counting, reading five two-digit numerals, modeling five two-digit numbers, and identifying place value in two two-digit numerals. Chi-square tests for independence showed two significant relations: (1) the relationship between the control and treatment group membership on the postassessment of modeling two-digit numbers and (2) the relationship between place value identifications and group membership. Analysis of the children’s performance and error patterns revealed interesting differences between children taught with explicit number names and children taught with traditional number names. The improvement of the treatment group overall exceeded the improvement of the control group. This study indicates that teaching children to use explicit number names can, indeed, have a positive influence on their understanding of place value.

The study of matrices have been of interest to mathematicians for some time. Recently the use of matrices has assumed greater importance also in the fields of management, social science, and natural science because they are very useful in the organization and presentation of data and in the solution of linear equations. The theory of matrices is yet another type of mathematical model which we can use to solve many problems that arise in these fields. The aim of this paper is to show how we can use matrices and their mathematical model to solve some problems in the process of routing. First we will introduce the term of routing and the new approach in the process of selecting paths. We will show some simple examples. We will also pint out how we can learn about matrices in the classroom. At the end we will discuss about advantages and potential disadvantages that may occur in the described technique.

The purpose of this research was to compare American and Albanian students’ achievement in Algebra 1. The study compared algebraic solving abilities of 219 students in a city of Albania and 242 ninth-grade American students, residents of an American region. Albanian sample did not use calculators on the test. Of the American sample, 97 students used calculators on the test, whereas 145 did not use them. The three research questions addressed: (1) students’ mastering of the overall algebraic achievement, (2) students’ mastering of specific domains of algebraic understanding: knowing, applying, and reasoning, and (3) students’ preference of algebraic strategies for solving word-problems. The study found that Albanian students outperformed American students on the overall achievement. However, American students who used calculators on the test significantly outperformed not only the American group who did not use calculators on the test, but also the entire Albanian sample. In addition, Albanian students scored significantly higher than their American peers both on 2 out of 3 cognitive domains and on using algebraic strategies.