1,721,125 research outputs found
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Reduktion und asymptotische Reduktion von Reaktionsgleichungen
No full textThis thesis deals with ordinary differential equations which model reacting systems obeying mass-action kinetics. We focus on the mathematical characterization and analysis of quasi-steady state. Frequently, singular perturbation methods are used to model this biochemical phenomenon. Essentially, the reduction of dimension goes back to Tikhonov and Fenichel and their fundamental results for singularly perturbed systems. In general, an explicit application of Tikhonov's theorem is impossible. But there is a way to directly compute a reduced system, which does not require a so-called Tikhonov standard form. Substantially this is a result of the geometrical interpretation of Fenichel. To determine a reduced system, Nöthen and Walcher give a straightforward procedure based on a projection, but this method is not feasible for practical purposes. We present a different approach in Chapter 2 for differential equations with rational right hand side, especially for reaction equations. Fundamentally, the method is based on a suitable local representation of the Slow Manifold. We construct a product representation for the slow system, which leads to a simple formula for the Tikhonov-Fenichel reduction. This formula only requires elementary algebraic operations. In particular, for slow and fast reactions the stoichiometry of the fast system leads to a canonical product representation. Moreover, we specify locally unique initial values for the reduced system on the slow manifold to complete the reduction. In general, these cannot be computed exactly. Therefore we develop an iterative procedure to generate approximate values. One area of application concerns enzymatic reactions with quasi-steady state assumptions for intermediates whose reaction equations are reduced to a real affine subspace. As described in Chapter 4, the special structure of the slow manifold yields a simplification of the reduction formula. Moreover, we are able to determine a domain of validity for an ad-hoc method that is commonly used in chemical literature. The reduction procedure in Chapter 2 is neither based on an explicit transformation of a system in Tikhonov standard form, nor does it make use of a parametrization of the slow manifold. In this respect, the new approach extends well-known reduction formulas in the literature. Frequently, a reduced equation is computable only via this extension. Given that a Tikhonov-Fenichel reduction exists, we are able to calculate a reduced equation explicitly for every autonomous differential equation with rational right hand side. Applications in Chapters 5 and 6 include generalizations and corrections of quasi-steady state reductions for several important examples. In particular, we are able to reduce reversible model extensions and higher dimensional systems. Moreover, Chapter 7 examines reaction-diffusion systems and the determination of a Tikhonov-Fenichel reduction for spatial discretization schemes. As a consequence, we obtain a heuristic to find candidates for reduced partial differential equations. Since there exists no conterpart to the Tikhonov-Fenichel reduction for systems of partial differential equations in the literature, even this step provides an approach to the solution of a nontrivial problem. A systematic method to classify all possible Tikhonov-Fenichel reductions for a parameter-dependent system is the central result of Part II. Thus, one is interested in points of phase-parameter space such that the conditions for Tikhonov-Fenichel hold for small perturbations along any fixed direction in the parameter space. We take this as a defining property of a so called Tikhonov parameter value. Moreover, for a strong Tikhonov parameter value there is an attractive slow manifold. The mathematical conditions for the existence of a reduction yield exact criteria for Tikhonov parameter values. Consequently, methods of algorithmic algebra are applicable. Among other results, all Tikhonov parameter values for the classical irreversible and reversible Michaelis-Menten dynamics are determined. In particular, in the irreversible Micha-elis--Menten-scheme a reduction with a small parameter is possible only for the well-known ones in the literature. Concerning the structure we show that the strong Tikhonov parameter values of a system form a semi-algebraic set. Furthermore, it is easy to apply necessary conditions for more complicated systems. Our computations for the applications in Chapter 9 show that only a small number of reductions is possible (apart from a solution preserving transformation)
Quasistationarität und fast-invariante Mengen gewöhnlicher Differentialgleichungen
No full textThe objective of the present paper is to analyse the concept of quasi-stationarity, which is motivated from bio-chemistry, and to investigate it from a matematical perspective. In Chapter 1 the concept of quasi-stationarity is presented in the context of chemical reactions. Examples, particularly the Michaelis-Menten-reaction, are given. We then give a first overview of relevant approaches to analyse given differential equation systems for chemical reactions. Chapter 2 presents approaches via singular perturbation theory, particularly Tikhonov's theorem. Given a differential equation system of particular form, which depends on a small parameter, then Tikhonov's theorem provides (asymptotically) a reduction to smaller dimension. Fenichel's theorem deepens this result in a broader context. The application of these theories to chemical systems is not without problems as these systems generally do not appear in the specific form required. Moreover the small parameter should be motivated from chemical consideration. Next, we present an iterative method due to Fraser and Roussel who approximate an invariant manifold for a reduction of the differential equation. Their original system is in standard form for application of singular perturbation theory. Chapter 2 concludes with the methods of Heinrich and Schauer (resp. Stiefenhofer) for reacting systems split up into slow and fast reactions. They perform a transformation of the differential equation to obtain the reduction via Tikhonov. None of the methods presented above is particularly appropriate for a systematic analysis of quasi-stationarity. In Chapter 3 the chemical notion of quasi-stationarity is combined with the concept of near-invariance. We obtain a method which, starting from a biochemically motivated assumption, provides locally necessary conditions on the parameters for quasi-steady state behaviour. These conditions can be determined by using the near-invariance of a certain set. In addition "small parameters" can be detected. Moreover various approaches to reduction are presented which are motivated by quasi-stationarity. The iterative method of Chapter 2 is discussed again, now from the perspective of near-invariance. In Chapter 4 we analyse the long-term behaviour of solutions. For chemical reactions we frequently find a dynamical equilibrium in the limit. Mathematically this equilibrium is reflected by a stationary point in the differential equation system. It will be shown that the long-term behaviour in an asymptotically stable stationary point is determined by the linear part alone. In examples we analyse whether the long-term behaviour is consistent with the nearly-invariant set discussed in Chapter 3. Finally in Chapter 5 we discuss how the theory of Tikhonov and Fenichel can be applied to chemical reaction systems with a quasi-steady state assumption. Identifying a small parameter is essential, and this can be done via nearly-invariant sets as discussed in Chapter 3. Then we show how a transformation should be designed to obtain from a given system a system in standard form of singular perturbation theory. Conditions for applicability of Tikhonov and Fenichel can be tested and, if applicable, a reduced system can be identified. The motivation for this approach goes back to Heinrich and Schauer as outlined in Chapter 2. However, the formulation here is more general and the range of applications is larger, particularly by including quasi-steady state behaviour. With this approach the local statements from Chapter 3 are becoming global statements. We illustrate this systematic approach by considering several examples
Elementare Wege zur mathematischen Modellbildung : Fallbeispiele aus Biowissenschaften und Chemie
No full textThe role of mathematical models in high school education is often restricted to the application of model equations, or to fitting parameters. The modelling process is usually not a focus topic, and rarely the derivation of a model is discussed extensively. In the present thesis, using examples from biology and chemistry present approaches to modelling for various age groups on the high school level. In biology we develop conceptual models for the competition within one species, which are extracted from observations of competition in nature. These simple conceptual models can be transferred to stochastic models and simulation tools that allow experimentation and observation in silico. Evaluating the simulation results, requiring only tools from high school mathematics, yields, among others, the well-known Ricker model. A conceptual model for simple bimolecular chemical reactions allows the simulation and visualization of the systems behaviour. Again, high school mathematics is sufficient to analyse the simulation results and derive the mass action law for the equilibrium state. For both classes of examples we also develop a purely mathematical approach, using combinatorics. The mathematics used have exceed high school level, but are well within reach of first-year university level.The simulation-tools can be found under the following URL:www.matha.rwth-aachen.de/lehre/lehramtsausbildung/Tools.htm
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Wechselwirkung von Populationen in einem begrenzten Lebensraum : Modellierung, Simulation und mathematische Analyse im Unterricht
No full textThe thesis introduces an authentic modelling process for two interacting species which is well accessible to high school students. Based on an analysis of ecological systems, a simple conceptual model leads to simulation software tools and the derivation of a mathematical model. A wide range of systems, e.g. predator-prey, competition or parasitism can be investigated. The approach also allows independent modelling activities and in silico experimentation by students. As the presented modelling process builds on authentic research by Johannson and Sumpter (2003) it allows to give students an insight into current research of Theoretical Biology
Reduktion und asymptotische Reduktion von Reaktionsgleichungen
No full textThis thesis deals with ordinary differential equations which model reacting systems obeying mass-action kinetics. We focus on the mathematical characterization and analysis of quasi-steady state. Frequently, singular perturbation methods are used to model this biochemical phenomenon. Essentially, the reduction of dimension goes back to Tikhonov and Fenichel and their fundamental results for singularly perturbed systems. In general, an explicit application of Tikhonov's theorem is impossible. But there is a way to directly compute a reduced system, which does not require a so-called Tikhonov standard form. Substantially this is a result of the geometrical interpretation of Fenichel. To determine a reduced system, Nöthen and Walcher give a straightforward procedure based on a projection, but this method is not feasible for practical purposes. We present a different approach in Chapter 2 for differential equations with rational right hand side, especially for reaction equations. Fundamentally, the method is based on a suitable local representation of the Slow Manifold. We construct a product representation for the slow system, which leads to a simple formula for the Tikhonov-Fenichel reduction. This formula only requires elementary algebraic operations. In particular, for slow and fast reactions the stoichiometry of the fast system leads to a canonical product representation. Moreover, we specify locally unique initial values for the reduced system on the slow manifold to complete the reduction. In general, these cannot be computed exactly. Therefore we develop an iterative procedure to generate approximate values. One area of application concerns enzymatic reactions with quasi-steady state assumptions for intermediates whose reaction equations are reduced to a real affine subspace. As described in Chapter 4, the special structure of the slow manifold yields a simplification of the reduction formula. Moreover, we are able to determine a domain of validity for an ad-hoc method that is commonly used in chemical literature. The reduction procedure in Chapter 2 is neither based on an explicit transformation of a system in Tikhonov standard form, nor does it make use of a parametrization of the slow manifold. In this respect, the new approach extends well-known reduction formulas in the literature. Frequently, a reduced equation is computable only via this extension. Given that a Tikhonov-Fenichel reduction exists, we are able to calculate a reduced equation explicitly for every autonomous differential equation with rational right hand side. Applications in Chapters 5 and 6 include generalizations and corrections of quasi-steady state reductions for several important examples. In particular, we are able to reduce reversible model extensions and higher dimensional systems. Moreover, Chapter 7 examines reaction-diffusion systems and the determination of a Tikhonov-Fenichel reduction for spatial discretization schemes. As a consequence, we obtain a heuristic to find candidates for reduced partial differential equations. Since there exists no conterpart to the Tikhonov-Fenichel reduction for systems of partial differential equations in the literature, even this step provides an approach to the solution of a nontrivial problem. A systematic method to classify all possible Tikhonov-Fenichel reductions for a parameter-dependent system is the central result of Part II. Thus, one is interested in points of phase-parameter space such that the conditions for Tikhonov-Fenichel hold for small perturbations along any fixed direction in the parameter space. We take this as a defining property of a so called Tikhonov parameter value. Moreover, for a strong Tikhonov parameter value there is an attractive slow manifold. The mathematical conditions for the existence of a reduction yield exact criteria for Tikhonov parameter values. Consequently, methods of algorithmic algebra are applicable. Among other results, all Tikhonov parameter values for the classical irreversible and reversible Michaelis-Menten dynamics are determined. In particular, in the irreversible Micha-elis--Menten-scheme a reduction with a small parameter is possible only for the well-known ones in the literature. Concerning the structure we show that the strong Tikhonov parameter values of a system form a semi-algebraic set. Furthermore, it is easy to apply necessary conditions for more complicated systems. Our computations for the applications in Chapter 9 show that only a small number of reductions is possible (apart from a solution preserving transformation)
Homöostase in Neuronen
No full textNeurons maintain their electrical activity patterns over long periods of time despite of ongoing channel turnover, cell growth and varying extracellular conditions. In order to maintain these fixed patterns Abbott et al. suggested in 1993 that neurons may regulate the maximal conductances for various currents, which leads to a variant of the classical Hodgkin-Huxley model (1952), where the maximal conductance of each ionic current was assumed to be a fixed parameter rather than a dynamical. The regulation process requires feedback systems capable of reacting to changes of electrical activity on different time scales. In the model under consideration, the intracellular calcium concentration serves as such a regulatory feedback element for the regulation of maximal conductances, because this concentration links neuronal conductances to electrical activity. If the activity pattern leaves the equilibrium state the calcium sensors ensure that the values of the conductances are adapted to the modified activity level. Abbott et al. investigated this model and several variants, mostly with the help of numerical simulations. The purpose of this work is a mathematical analysis of these types of models. Based on the models of Abbott et al. a five-dimensional system of differential equations is developed. Intermediate steps of independent interest are(1) Reduction of dimension, that is common approaches to reduce the four-dimensional system of Hodgkin and Huxley to a system of dimension two, as already done by FitzHugh and Nagumo, are described and confronted by a new approach based on the criterion of nearly-invariance. A related approach from singular perturbation theory is also discussed.(2) Qualitative analysis of the reduced system, that is the functions in the model equations are generally characterized by properties like positivity or monotonicity, rather than by concrete functional expressions and the analysis proceeds accordingly. Results as well for the existence and the number of stationary points and their stability as global properties can be received. (3) Maximal conductances are treated as parameters with regard to a possible Hopf bifurcation.(4) Critical analysis of the Abbott model and the alternative model for calcium target concentrations, that is the Abbott model is analysed and due to some inconsistencies an alternative model for the control of homeostasis in neurons is developed
Orthogonalität und beste Approximation
No full textThe topic of the present thesis is the determination of good approximations through orthonormal basis expansion. This method is based on fundamental mathematical ideas which transcend traditional boundaries between disciplines. In recent decades the method turned out to be exceptionally viable and (even commercially) effective in applications. To make this interesting mathematical topic more easily accessible for school teaching is the purpose of this work. At the same time I would like to present successful mathematics of the past decades and demonstrate that considering mathematical structure can be of great benefit for insight as well as applications. The subject builds on students' experience with and knowledge of elementary geometry. A fundamental perception is the fact that by dropping the perpendicular one finds the point on a given straight line or plane which is closest to a given point in the space. In combination with the ability to compute orthogonal projections by means of the inner product known from analytic geometry, this insight can be generalized and put to good use far beyond the boundaries of concrete geometry. While perpendicular lines are relevant only in two or three dimensional space, good approximations are in high demand in many areas of science and technology, even more so to handle mounds of data in the present era. High school students will be able to experience and verify in an exemplary manner how approximation methods in the Fourier analysis of audio signals or image processing in JPEG format proceed in the same way as distance calculations in geometric space. The common mathematical foundation is given by the structure of Euclidean vector spaces. The notion of 'vector space' in its most general form has taken shape during the second half of the 19th century and has proven to be extremely viable. Particularly important steps were the transition to n-dimensional space on the one hand and to function spaces on the other hand. These transitions in particular play a key role in the present work. Real vector spaces in which an inner product is defined are called Euclidean. As is the case in Euclidean geometry, terms such as 'length', 'distance' and 'orthogonality' are linked to the inner product, as well as facts related to the triangle inequality or the Pythagorean theorem. Therefore, one can also transfer methods such as the determination of good approximations via orthogonal projection. In Euclidian vector spaces the orthogonal projection of a point onto a subspace provides its best approximation in this subspace, in the sense of the Euclidean norm. This projection can be decomposed into projections on mutually orthogonal, one-dimensional subspaces and thus determined via orthogonal basis expansion. These mathematical ideas stand in the focus of the present work. They can be extended to a universal method of systematic approximation or analysis of complicated mathematical objects, and can be used wherever the structure of an Euclidean vector space is given and the Euclidean norm is sensible to measure the similarity of the objects. Generally speaking, the thesis starts from a solid anchoring of concepts and relationships in elementary geometry to first lead to some applications involving column vectors in R^n, n>3, with various interpretations. For such problems the level of abstraction is moderate and alternative solution methods are available. Thus they allow to familiarize with the general method and to give an impression of its potential. Next a different view of column vectors as lists of values for piecewise constant functions is presented and emphasized. Keeping the standard inner product one has to reconsider the notion of 'orthogonality' and 'distance', as well as the question which orthogonal bases could prove useful. In the limiting case this interpretation leads to an inner product of piecewise continuous functions defined by an integral, and to the associated L^2-norm. The transition to function spaces makes applications in signal processing accessible; in this area the determination of good approximations is of particular relevance. As practically important examples the compression of digital signals via Haar wavelets and the analysis of continuous signals via Fourier expansion are treated in detail. These applications also highlight the importance of choosing subspaces in a judicious manner, and the special role of certain orthogonal systems. Within the framework of the present thesis, motivating examples, exercises and interactive Maple worksheets for all the covered topics were developed. In addition, a collection of experiments concerning the processing of optic and acoustic signals enables students to link the theoretical findings to auditory and visual perception. Theory presentation, computer materials and experiments were tested in workshops with high school students. A discussion of practicability in class and possible curriculum connections concludes the thesis
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