TY - JOUR A1 - Stolle, Claudia A1 - Michaelis, Ingo A1 - Rauberg, Jan T1 - The role of high-resolution geomagnetic field models for investigating ionospheric currents at low Earth orbit satellites JF - Earth, planets and space N2 - Low Earth orbiting geomagnetic satellite missions, such as the Swarm satellite mission, are the only means to monitor and investigate ionospheric currents on a global scale and to make in situ measurements of F region currents. High-precision geomagnetic satellite missions are also able to detect ionospheric currents during quiet-time geomagnetic conditions that only have few nanotesla amplitudes in the magnetic field. An efficient method to isolate the ionospheric signals from satellite magnetic field measurements has been the use of residuals between the observations and predictions from empirical geomagnetic models for other geomagnetic sources, such as the core and lithospheric field or signals from the quiet-time magnetospheric currents. This study aims at highlighting the importance of high-resolution magnetic field models that are able to predict the lithospheric field and that consider the quiet-time magnetosphere for reliably isolating signatures from ionospheric currents during geomagnetically quiet times. The effects on the detection of ionospheric currents arising from neglecting the lithospheric and magnetospheric sources are discussed on the example of four Swarm orbits during very quiet times. The respective orbits show a broad range of typical scenarios, such as strong and weak ionospheric signal (during day- and nighttime, respectively) superimposed over strong and weak lithospheric signals. If predictions from the lithosphere or magnetosphere are not properly considered, the amplitude of the ionospheric currents, such as the midlatitude Sq currents or the equatorial electrojet (EEJ), is modulated by 10-15 % in the examples shown. An analysis from several orbits above the African sector, where the lithospheric field is significant, showed that the peak value of the signatures of the EEJ is in error by 5 % in average when lithospheric contributions are not considered, which is in the range of uncertainties of present empirical models of the EEJ. KW - Geomagnetic field KW - Ionospheric current KW - Geomagnetic models Y1 - 2016 U6 - https://doi.org/10.1186/s40623-016-0494-1 SN - 1880-5981 VL - 68 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - The partial clone of linear terms JF - Siberian Mathematical Journal N2 - Generalizing a linear expression over a vector space, we call a term of an arbitrary type tau linear if its every variable occurs only once. Instead of the usual superposition of terms and of the total many-sorted clone of all terms in the case of linear terms, we define the partial many-sorted superposition operation and the partial many-sorted clone that satisfies the superassociative law as weak identity. The extensions of linear hypersubstitutions are weak endomorphisms of this partial clone. For a variety V of one-sorted total algebras of type tau, we define the partial many-sorted linear clone of V as the partial quotient algebra of the partial many-sorted clone of all linear terms by the set of all linear identities of V. We prove then that weak identities of this clone correspond to linear hyperidentities of V. KW - linear term KW - clone KW - partial clone KW - linear hypersubstitution KW - linear identity KW - linear hyperidentity Y1 - 2016 U6 - https://doi.org/10.1134/S0037446616040030 SN - 0037-4466 SN - 1573-9260 VL - 57 SP - 589 EP - 598 PB - Pleiades Publ. CY - New York ER - TY - JOUR A1 - Kistner, Saskia A1 - Vollmeyer, Regina A1 - Burns, Bruce D. A1 - Kortenkamp, Ulrich T1 - Model development in scientific discovery learning with a computer-based physics task JF - Computers in human behavior N2 - Based on theories of scientific discovery learning (SDL) and conceptual change, this study explores students' preconceptions in the domain of torques in physics and the development of these conceptions while learning with a computer-based SDL task. As a framework we used a three-space theory of SDL and focused on model space, which is supposed to contain the current conceptualization/model of the learning domain, and on its change through hypothesis testing and experimenting. Three questions were addressed: (1) What are students' preconceptions of torques before learning about this domain? To do this a multiple-choice test for assessing students' models of torques was developed and given to secondary school students (N = 47) who learned about torques using computer simulations. (2) How do students' models of torques develop during SDL? Working with simulations led to replacement of some misconceptions with physically correct conceptions. (3) Are there differential patterns of model development and if so, how do they relate to students’ use of the simulations? By analyzing individual differences in model development, we found that an intensive use of the simulations was associated with the acquisition of correct conceptions. Thus, the three-space theory provided a useful framework for understanding conceptual change in SDL. KW - Scientific discovery learning KW - Multiple problem spaces KW - Computer simulations KW - Physics concepts KW - Misconceptions KW - Conceptual change Y1 - 2016 U6 - https://doi.org/10.1016/j.chb.2016.02.041 SN - 0747-5632 SN - 1873-7692 VL - 59 SP - 446 EP - 455 PB - Elsevier CY - Oxford ER - TY - JOUR A1 - Sinclair, Nathalie A1 - Bussi, Maria G. Bartolini A1 - de Villiers, Michael A1 - Jones, Keith A1 - Kortenkamp, Ulrich A1 - Leung, Allen A1 - Owens, Kay T1 - Recent research on geometry education: an ICME-13 survey team report JF - ZDM : The International Journal on Mathematics Education N2 - This survey on the theme of Geometry Education (including new technologies) focuses chiefly on the time span since 2008. Based on our review of the research literature published during this time span (in refereed journal articles, conference proceedings and edited books), we have jointly identified seven major threads of contributions that span from the early years of learning (pre-school and primary school) through to post-compulsory education and to the issue of mathematics teacher education for geometry. These threads are as follows: developments and trends in the use of theories; advances in the understanding of visuo spatial reasoning; the use and role of diagrams and gestures; advances in the understanding of the role of digital technologies; advances in the understanding of the teaching and learning of definitions; advances in the understanding of the teaching and learning of the proving process; and, moving beyond traditional Euclidean approaches. Within each theme, we identify relevant research and also offer commentary on future directions. KW - Geometry KW - Technology KW - Diagrams KW - Definitions KW - Gestures KW - Proving KW - Digital technology KW - Visuospatial reasoning Y1 - 2016 U6 - https://doi.org/10.1007/s11858-016-0796-6 SN - 1863-9690 SN - 1863-9704 VL - 48 SP - 691 EP - 719 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Kistner, Saskia A1 - Burns, Bruce D. A1 - Vollmeyer, Regina A1 - Kortenkamp, Ulrich T1 - The importance of understanding: Model space moderates goal specificity effects JF - The quarterly journal of experimental psychology N2 - The three-space theory of problem solving predicts that the quality of a learner's model and the goal specificity of a task interact on knowledge acquisition. In Experiment 1 participants used a computer simulation of a lever system to learn about torques. They either had to test hypotheses (nonspecific goal), or to produce given values for variables (specific goal). In the good- but not in the poor-model condition they saw torque depicted as an area. Results revealed the predicted interaction. A nonspecific goal only resulted in better learning when a good model of torques was provided. In Experiment 2 participants learned to manipulate the inputs of a system to control its outputs. A nonspecific goal to explore the system helped performance when compared to a specific goal to reach certain values when participants were given a good model, but not when given a poor model that suggested the wrong hypothesis space. Our findings support the three-space theory. They emphasize the importance of understanding for problem solving and stress the need to study underlying processes. KW - Goal specificity KW - Problem solving KW - Three-space theory KW - Scientific discovery learning Y1 - 2016 U6 - https://doi.org/10.1080/17470218.2015.1076865 SN - 1747-0218 SN - 1747-0226 VL - 69 SP - 1179 EP - 1196 PB - Optical Society of America CY - Abingdon ER - TY - JOUR A1 - Kortenkamp, Ulrich A1 - Monaghan, John A1 - Trouche, Luc T1 - Jonathan M Borwein (1951-2016): exploring, experiencing and experimenting in mathematics - an inspiring journey in mathematics JF - Educational studies in mathematics : an international journal Y1 - 2016 U6 - https://doi.org/10.1007/s10649-016-9729-0 SN - 0013-1954 SN - 1573-0816 VL - 93 SP - 131 EP - 136 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Rado theorem for p-harmonic functions JF - Boletin de la Sociedad Matemática Mexicana N2 - Let A be a nonlinear differential operator on an open set X subset of R-n and S a closed subset of X. Given a class F of functions in X, the set S is said to be removable for F relative to A if any weak solution of A(u) = 0 in XS of class F satisfies this equation weakly in all of X. For the most extensively studied classes F, we show conditions on S which guarantee that S is removable for F relative to A. KW - Quasilinear equations KW - Removable sets KW - p-Laplace equation Y1 - 2016 U6 - https://doi.org/10.1007/s40590-016-0109-7 SN - 1405-213X SN - 2296-4495 VL - 22 SP - 461 EP - 472 PB - Springer CY - Basel ER - TY - THES A1 - Ludewig, Matthias T1 - Path integrals on manifolds with boundary and their asymptotic expansions T1 - Pfadintegrale auf Mannigfaltigkeiten mit Rand und ihre asymptotischen Entwicklungen N2 - It is "scientific folklore" coming from physical heuristics that solutions to the heat equation on a Riemannian manifold can be represented by a path integral. However, the problem with such path integrals is that they are notoriously ill-defined. One way to make them rigorous (which is often applied in physics) is finite-dimensional approximation, or time-slicing approximation: Given a fine partition of the time interval into small subintervals, one restricts the integration domain to paths that are geodesic on each subinterval of the partition. These finite-dimensional integrals are well-defined, and the (infinite-dimensional) path integral then is defined as the limit of these (suitably normalized) integrals, as the mesh of the partition tends to zero. In this thesis, we show that indeed, solutions to the heat equation on a general compact Riemannian manifold with boundary are given by such time-slicing path integrals. Here we consider the heat equation for general Laplace type operators, acting on sections of a vector bundle. We also obtain similar results for the heat kernel, although in this case, one has to restrict to metrics satisfying a certain smoothness condition at the boundary. One of the most important manipulations one would like to do with path integrals is taking their asymptotic expansions; in the case of the heat kernel, this is the short time asymptotic expansion. In order to use time-slicing approximation here, one needs the approximation to be uniform in the time parameter. We show that this is possible by giving strong error estimates. Finally, we apply these results to obtain short time asymptotic expansions of the heat kernel also in degenerate cases (i.e. at the cut locus). Furthermore, our results allow to relate the asymptotic expansion of the heat kernel to a formal asymptotic expansion of the infinite-dimensional path integral, which gives relations between geometric quantities on the manifold and on the loop space. In particular, we show that the lowest order term in the asymptotic expansion of the heat kernel is essentially given by the Fredholm determinant of the Hessian of the energy functional. We also investigate how this relates to the zeta-regularized determinant of the Jacobi operator along minimizing geodesics. N2 - Es ist "wissenschaftliche Folklore", abgeleitet von der physikalischen Anschauung, dass Lösungen der Wärmeleitungsgleichung auf einer riemannschen Mannigfaltigkeit als Pfadintegrale dargestellt werden können. Das Problem mit Pfadintegralen ist allerdings, dass schon deren Definition Mathematiker vor gewisse Probleme stellt. Eine Möglichkeit, Pfadintegrale rigoros zu definieren ist endlich-dimensionale Approximation, oder time-slicing-Approximation: Für eine gegebene Unterteilung des Zeitintervals in kleine Teilintervalle schränkt man den Integrationsbereich auf diejenigen Pfade ein, die auf jedem Teilintervall geodätisch sind. Diese endlichdimensionalen Integrale sind wohldefiniert, und man definiert das (unendlichdimensionale) Pfadintegral als den Limes dieser (passend normierten) Integrale, wenn die Feinheit der Unterteilung gegen Null geht. In dieser Arbeit wird gezeigt, dass Lösungen der Wärmeleitungsgleichung auf einer allgemeinen riemannschen Mannigfaltigkeit tatsächlich durch eine solche endlichdimensionale Approximation gegeben sind. Hierbei betrachten wir die Wärmeleitungsgleichung für allgemeine Operatoren von Laplace-Typ, die auf Schnitten in Vektorbündeln wirken. Wir zeigen auch ähnliche Resultate für den Wärmekern, wobei wir uns allerdings auf Metriken einschränken müssen, die eine gewisse Glattheitsbedingung am Rand erfüllen. Eine der wichtigsten Manipulationen, die man an Pfadintegralen vornehmen möchte, ist das Bilden ihrer asymptotischen Entwicklungen; in Falle des Wärmekerns ist dies die Kurzzeitasymptotik. Um die endlich-dimensionale Approximation hier nutzen zu können, ist es nötig, dass die Approximation uniform im Zeitparameter ist. Dies kann in der Tat erreicht werden; zu diesem Zweck geben wir starke Fehlerabschätzungen an. Schließlich wenden wir diese Resultate an, um die Kurzzeitasymptotik des Wärmekerns (auch im degenerierten Fall, d.h. am Schittort) herzuleiten. Unsere Resultate machen es außerdem möglich, die asymptotische Entwicklung des Wärmekerns mit einer formalen asymptotischen Entwicklung der unendlichdimensionalen Pfadintegrale in Verbindung zu bringen. Auf diese Weise erhält man Beziehungen zwischen geometrischen Größen der zugrundeliegenden Mannigfaltigkeit und solchen des Pfadraumes. Insbesondere zeigen wir, dass der Term niedrigster Ordnung in der asymptotischen Entwicklung des Wärmekerns im Wesentlichen durch die Fredholm-Determinante der Hesseschen des Energie-Funktionals gegeben ist. Weiterhin untersuchen wir die Verbindung zur zeta-regularisierten Determinante des Jakobi-Operators entlang von minimierenden Geodätischen. KW - heat equation KW - heat kernel KW - path integral KW - determinant KW - asymptotic expansion KW - Laplace expansion KW - heat asymptotics KW - Wiener measure KW - Wärmeleitungsgleichung KW - Wärmekern KW - Pfadintegrale KW - asymptotische Entwicklung KW - Determinante Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-94387 ER - TY - THES A1 - Scharrer, Christian T1 - Relating diameter and mean curvature for varifolds T1 - Relativer Diameter und mittlere Krümmung für Varifaltigkeiten N2 - The main results of this thesis are formulated in a class of surfaces (varifolds) generalizing closed and connected smooth submanifolds of Euclidean space which allows singularities. Given an indecomposable varifold with dimension at least two in some Euclidean space such that the first variation is locally bounded, the total variation is absolutely continuous with respect to the weight measure, the density of the weight measure is at least one outside a set of weight measure zero and the generalized mean curvature is locally summable to a natural power (dimension of the varifold minus one) with respect to the weight measure. The thesis presents an improved estimate of the set where the lower density is small in terms of the one dimensional Hausdorff measure. Moreover, if the support of the weight measure is compact, then the intrinsic diameter with respect to the support of the weight measure is estimated in terms of the generalized mean curvature. This estimate is in analogy to the diameter control for closed connected manifolds smoothly immersed in some Euclidean space of Peter Topping. Previously, it was not known whether the hypothesis in this thesis implies that two points in the support of the weight measure have finite geodesic distance. N2 - Die wichtigsten Ergebnisse dieser Arbeit sind formuliert für eine Klasse von Oberflächen (Varifaltigkeiten), welche geschlossene glatte Untermannigfaltigkeiten des euklidischen Raums verallgemeinern und Singularitäten erlauben. Gegeben sei eine mindestens zwei-dimensionale unzerlegbare Varifaltigkeit im euklidischen Raum, sodass die erste Variation lokal beschränkt ist, die totale Variation absolut stetig bezüglich dem Gewichtsmaß ist, die Dichte des Gewichtsmaßes außerhalb einer Nullmenge mindesten eins ist, und die verallgemeinerte mittlere Krümmung bezüglich dem Gewichtsmaß lokal summierbar zu einer natürlichen Potenz (Dimension der Varifaltigkeit minus eins) ist. Es wird die Menge, wo die untere Dichte klein ist, durch das ein-dimensionale Hausdorff-Maß abgeschätzt. Das Ergebnis ist eine neue, stark verbesserte untere Dichte-Schranke. Ist der Träger des Gewichtsmaßes kompakt, so wird der intrinsische Diameter des Trägers des Gewichtsmaßes abgeschätzt durch ein Integral der verallgemeinerten mittleren Krümmung. Diese Ungleichung ist analog zu der Ungleichung von Peter Topping für geschlossene zusammenhängende Mannigfaltigkeit, welche durch eine glatte Immersion in den euklidischen Raum eingebettet sind. Bisher war nicht bekannt, dass die oben genannten Annahmen an die Varifaltigkeit implizieren, dass der geodätische Abstand zweier Punkte aus dem Träger des Gewichtsmaßes endlich ist. KW - varifold KW - rectifiable varifold KW - indecomposable varifold KW - first variation KW - mean curvature KW - isoperimetric inequality KW - density of a measure KW - geodesic distance KW - intrinsic diameter KW - Varifaltigkeit KW - rektifizierbare Varifaltigkeit KW - unzerlegbare Varifaltigkeit KW - erste Variation KW - mittlere Krümmung KW - isoperimetrische Ungleichung KW - Dichte eines Maßes KW - geodätischer Abstand KW - intrinsischer Diameter Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-97013 ER - TY - INPR A1 - Shlapunov, Alexander A1 - Tarkhanov, Nikolai Nikolaevich T1 - An open mapping theorem for the Navier-Stokes equations N2 - We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016)10 KW - Navier-Stokes equations KW - weighted Hölder spaces KW - integral representation method Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-98687 SN - 2193-6943 VL - 5 IS - 10 PB - Universitätsverlag Potsdam CY - Potsdam ER -