A Computer Algebra Application to Determination of Lie Symmetries of Partial Differential Equations

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006

The Conception of Creation of Integrated Development Environment for Computer Training Systems

The various questions of creation of integrated development environment for computer training systems are considered in the given paper. The information technologies that can be used for creation of the integrated development environment are described. The different didactic aspects of realization of such systems are analyzed. The ways to improve the efficiency and quality of learning process with computer training systems for distance education are pointed.

Operational Methods in the Environment of a Computer Algebra System

This article presents the principal results of the doctoral thesis “Direct Operational Methods in the Environment of a Computer Algebra System” by Margarita Spiridonova (Institute of mathematics and Informatics, BAS), successfully defended before the Specialised Academic Council for Informatics and Mathematical Modelling on 23 March, 2009.

Computer Mathematics with MuPAD in School Years 6. To 9

Report published in the Proceedings of the National Conference on "Education and Research in the Information Society", Plovdiv, May, 2014

Adaptive Brain-Computer Interface Systems For Communication in People with Severe Neuromuscular Disabilities

Brain-computer interfaces (BCI) have the potential to restore communication or control abilities in individuals with severe neuromuscular limitations, such as those with amyotrophic lateral sclerosis (ALS). The role of a BCI is to extract and decode relevant information that conveys a user's intent directly from brain electro-physiological signals and translate this information into executable commands to control external devices. However, the BCI decision-making process is error-prone due to noisy electro-physiological data, representing the classic problem of efficiently transmitting and receiving information via a noisy communication channel.

This research focuses on P300-based BCIs which rely predominantly on event-related potentials (ERP) that are elicited as a function of a user's uncertainty regarding stimulus events, in either an acoustic or a visual oddball recognition task. The P300-based BCI system enables users to communicate messages from a set of choices by selecting a target character or icon that conveys a desired intent or action. P300-based BCIs have been widely researched as a communication alternative, especially in individuals with ALS who represent a target BCI user population. For the P300-based BCI, repeated data measurements are required to enhance the low signal-to-noise ratio of the elicited ERPs embedded in electroencephalography (EEG) data, in order to improve the accuracy of the target character estimation process. As a result, BCIs have relatively slower speeds when compared to other commercial assistive communication devices, and this limits BCI adoption by their target user population. The goal of this research is to develop algorithms that take into account the physical limitations of the target BCI population to improve the efficiency of ERP-based spellers for real-world communication.

In this work, it is hypothesised that building adaptive capabilities into the BCI framework can potentially give the BCI system the flexibility to improve performance by adjusting system parameters in response to changing user inputs. The research in this work addresses three potential areas for improvement within the P300 speller framework: information optimisation, target character estimation and error correction. The visual interface and its operation control the method by which the ERPs are elicited through the presentation of stimulus events. The parameters of the stimulus presentation paradigm can be modified to modulate and enhance the elicited ERPs. A new stimulus presentation paradigm is developed in order to maximise the information content that is presented to the user by tuning stimulus paradigm parameters to positively affect performance. Internally, the BCI system determines the amount of data to collect and the method by which these data are processed to estimate the user's target character. Algorithms that exploit language information are developed to enhance the target character estimation process and to correct erroneous BCI selections. In addition, a new model-based method to predict BCI performance is developed, an approach which is independent of stimulus presentation paradigm and accounts for dynamic data collection. The studies presented in this work provide evidence that the proposed methods for incorporating adaptive strategies in the three areas have the potential to significantly improve BCI communication rates, and the proposed method for predicting BCI performance provides a reliable means to pre-assess BCI performance without extensive online testing.

Field investigation of contactless displacement measurement using computer vision systems for civil engineering applications

Much of the bridge stock on major transport links in North America and Europe was constructed in the 1950s and 1960s and has since deteriorated or is carrying loads far in excess of the original design loads. Structural Health Monitoring Systems (SHM) can provide valuable information on the bridge capacity but the application of such systems is currently limited by access and bridge type. This paper investigates the use of computer vision systems for SHM. A series of field tests have been carried out to test the accuracy of displacement measurements using contactless methods. A video image of each test was processed using a modified version of the optical flow tracking method to track displacement. These results have been validated with an established measurement method using linear variable differential transformers (LVDTs). The results obtained from the algorithm provided an accurate comparison with the validation measurements. The calculated displacements agree within 2% of the verified LVDT measurements, a number of post processing methods were then applied to attempt to reduce this error.

Laboratory investigation of contactless displacement measurement using computer vision systems

Much of the bridge stock on major transport links in North America and Europe was constructed in the 1950’s and 1960’s and has since deteriorated or is carrying loads far in excess of the original design loads. Structural Health Monitoring Systems (SHM) can provide valuable information on the bridge capacity but the application of such systems is currently limited by access and system cost. This paper investigates the development of a low cost portable SHM system using commercially available cameras and computer vision techniques. A series of laboratory tests have been carried out to test the accuracy of displacement measurements using contactless methods. The results from each of the tests have been validated with established measurement methods, such as linear variable differential transformers (LVDTs). A video image of each test was processed using two different digital image correlation programs. The results obtained from the digital image correlation methods provided an accurate comparison with the validation measurements. The calculated displacements agree within 4% of the verified measurements LVDT measurements in most cases confirming the suitability full camera based SHM systems

Modified gravity with tensor computer algebra: a case study

Il modello ΛCDM è il modello cosmologico più semplice, ma finora più efficace, per descrivere l'evoluzione dell'universo. Esso si basa sulla teoria della Relatività Generale di Einstein e fornisce una spiegazione dell'espansione accelerata dell'universo introducendo la costante cosmologica Λ, che rappresenta il contributo della cosiddetta energia oscura, un'entità di cui ben poco si sa con certezza. Sono stati tuttavia proposti modelli teorici alternativi che descrivono gli effetti di questa quantità misteriosa, introducendo ad esempio gradi di libertà aggiuntivi, come nella teoria di Horndeski. L'obiettivo principale di questa testi è quello di studiare questi modelli tramite il tensor computer algebra xAct. In particolare, il nostro scopo sarà quello di implementare una procedura universale che permette di derivare, a partire dall'azione, le equazioni del moto e l'evoluzione temporale di qualunque modello generico.

Faktorisierung in Schief-Polynomringen

In der Arbeit werden zunächst die wesentlichsten Fakten über Schiefpolynome wiederholt, der Fokus liegt dabei auf Shift- und q-Shift-Operatoren in Charakteristik Null. Alle für die Arithmetik mit diesen Objekten notwendigen Konzepte und Algorithmen finden sich im ersten Kapitel. Einige der zur Bestimmung von Lösungen notwendigen Daten können aus dem Newtonpolygon, einer den Operatoren zugeordneten geometrischen Figur, abgelesen werden. Die Herleitung dieser Zusammenhänge ist das Thema des zweiten Kapitels der Arbeit, wobei dies insbesondere im q-Shift-Fall in dieser Form neu ist. Das dritte Kapitel beschäftigt sich mit der Bestimmung polynomieller und rationaler Lösungen dieser Operatoren, dabei folgt es im Wesentlichen der Darstellung von Mark van Hoeij. Der für die Faktorisierung von (q-)Shift Operatoren interessanteste Fall sind die sogenannten (q-)hypergeometrischen Lösungen, die direkt zu Rechtsfaktoren erster Ordnung korrespondieren. Im vierten Kapitel wird der van Hoeij-Algorithmus vom Shift- auf den q-Shift-Fall übertragen. Außerdem wird eine deutliche Verbesserung des q-Petkovsek-Algorithmus mit Hilfe der Daten des Newtonpolygons hergeleitet. Das fünfte Kapitel widmet sich der Berechnung allgemeiner Faktoren, wozu zunächst der adjungierte Operator eingeführt wird, der die Berechnung von Linksfaktoren erlaubt. Dann wird ein Algorithmus zur Berechnung von Rechtsfaktoren beliebiger Ordnung dargestellt. Für die praktische Benutzung ist dies allerdings für höhere Ordnungen unpraktikabel. Bei fast allen vorgestellten Algorithmen tritt das Lösen linearer Gleichungssysteme über rationalen Funktionenkörpern als Zwischenschritt auf. Dies ist in den meisten Computeralgebrasystemen nicht befriedigend gelöst. Aus diesem Grund wird im letzten Kapitel ein auf Evaluation und Interpolation basierender Algorithmus zur Lösung dieses Problems vorgestellt, der in allen getesteten Systemen den Standard-Algorithmen deutlich überlegen ist. Alle Algorithmen der Arbeit sind in einem MuPAD-Package implementiert, das der Arbeit beiliegt und eine komfortable Handhabung der auftretenden Objekte erlaubt. Mit diesem Paket können in MuPAD nun viele Probleme gelöst werden, für die es vorher keine Funktionen gab.

Algorithmen für q-holonome Funktionen und q-hypergeometrische Reihen

Die q-Analysis ist eine spezielle Diskretisierung der Analysis auf einem Gitter, welches eine geometrische Folge darstellt, und findet insbesondere in der Quantenphysik eine breite Anwendung, ist aber auch in der Theorie der q-orthogonalen Polynome und speziellen Funktionen von großer Bedeutung. Die betrachteten mathematischen Objekte aus der q-Welt weisen meist eine recht komplizierte Struktur auf und es liegt daher nahe, sie mit Computeralgebrasystemen zu behandeln. In der vorliegenden Dissertation werden Algorithmen für q-holonome Funktionen und q-hypergeometrische Reihen vorgestellt. Alle Algorithmen sind in dem Maple-Package qFPS, welches integraler Bestandteil der Arbeit ist, implementiert. Nachdem in den ersten beiden Kapiteln Grundlagen geschaffen werden, werden im dritten Kapitel Algorithmen präsentiert, mit denen man zu einer q-holonomen Funktion q-holonome Rekursionsgleichungen durch Kenntnis derer q-Shifts aufstellen kann. Operationen mit q-holonomen Rekursionen werden ebenfalls behandelt. Im vierten Kapitel werden effiziente Methoden zur Bestimmung polynomialer, rationaler und q-hypergeometrischer Lösungen von q-holonomen Rekursionen beschrieben. Das fünfte Kapitel beschäftigt sich mit q-hypergeometrischen Potenzreihen bzgl. spezieller Polynombasen. Wir formulieren einen neuen Algorithmus, der zu einer q-holonomen Rekursionsgleichung einer q-hypergeometrischen Reihe mit nichttrivialem Entwicklungspunkt die entsprechende q-holonome Rekursionsgleichung für die Koeffizienten ermittelt. Ferner können wir einen neuen Algorithmus angeben, der umgekehrt zu einer q-holonomen Rekursionsgleichung für die Koeffizienten eine q-holonome Rekursionsgleichung der Reihe bestimmt und der nützlich ist, um q-holonome Rekursionen für bestimmte verallgemeinerte q-hypergeometrische Funktionen aufzustellen. Mit Formulierung des q-Taylorsatzes haben wir schließlich alle Zutaten zusammen, um das Hauptergebnis dieser Arbeit, das q-Analogon des FPS-Algorithmus zu erhalten. Wolfram Koepfs FPS-Algorithmus aus dem Jahre 1992 bestimmt zu einer gegebenen holonomen Funktion die entsprechende hypergeometrische Reihe. Wir erweitern den Algorithmus dahingehend, dass sogar Linearkombinationen q-hypergeometrischer Potenzreihen bestimmt werden können. ________________________________________________________________________________________________________________

Use of the Maple System in Math Tuition at Universities

The following article explores the application of educational technologies at a University level and their contribution in enhancing the educational effectiveness. It discusses the capabilities of computer algebra systems, such as Maple. It is integrated in the math tuition of the Technical University (TU) in Varna and is used by its students during laboratory exercises.

Teaching University-Level Mathematics Using Mathematica

This paper considers the use of the computer algebra system Mathematica for teaching university-level mathematics subjects. Outlined are basic Mathematica concepts, connected with different mathematics areas: algebra, linear algebra, geometry, calculus and analysis, complex functions, numerical analysis and scientific computing, probability and statistics. The course “Information technologies in mathematics”, which involves the use of Mathematica, is also presented - discussed are the syllabus, aims, approaches and outcomes.

Truth in complex adaptive systems models should be based on truth by constructive verification

It is argued that the truth status of emergent properties of complex adaptive systems models should be based on an epistemology of proof by constructive verification and therefore on the ontological axioms of a non-realist logical system such as constructivism or intuitionism. ‘Emergent’ properties of complex adaptive systems (CAS) models create particular epistemological and ontological challenges. These challenges bear directly on current debates in the philosophy of mathematics and in theoretical computer science. CAS research, with its emphasis on computer simulation, is heavily reliant on models which explore the entailments of Formal Axiomatic Systems (FAS). The incompleteness results of Gödel, the incomputability results of Turing, and the Algorithmic Information Theory results of Chaitin, undermine a realist (platonic) truth model of emergent properties. These same findings support the hegemony of epistemology over ontology and point to alternative truth models such as intuitionism, constructivism and quasi-empiricism.

Computing and Visualizing Solution Sets of Interval Linear Systems

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006

One-Parameter Bifurcation Analysis of Dynamical Systems using Maple

This paper presents two algorithms for one-parameter local bifurcations of equilibrium points of dynamical systems. The algorithms are implemented in the computer algebra system Maple 13 © and designed as a package. Some examples are reported to demonstrate the package’s facilities.