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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.
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.
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.
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.
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.
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.
We construct equivariant KK-theory with coefficients in and R/Z as suitable inductive limits over II1-factors. We show that the Kasparov product, together with its usual functorial properties, extends to KK-theory with real coefficients. Let Gamma be a group. We define a Gamma-algebra A to be K-theoretically free and proper (KFP) if the group trace tr of Gamma acts as the unit element in KKR Gamma (A, A). We show that free and proper Gamma-algebras (in the sense of Kasparov) have the (KFP) property. Moreover, if Gamma is torsion free and satisfies the KK Gamma-form of the Baum-Connes conjecture, then every Gamma-algebra satisfies (KFP). If alpha : Gamma -> U-n is a unitary representation and A satisfies property (KFP), we construct in a canonical way a rho class rho(A)(alpha) is an element of KKR/Z1,Gamma (A A) This construction generalizes the Atiyah-Patodi-Singer K-theory class with R/Z-coefficients associated to alpha. (C) 2015 Elsevier Inc. All rights reserved.
We study graphs whose vertex degree tends to infinity and which are, therefore, called rapidly branching. We prove spectral estimates, discreteness of spectrum, first order eigenvalue and Weyl asymptotics solely in terms of the vertex degree growth. The underlying techniques are estimates on the isoperimetric constant. Furthermore, we give lower volume growth bounds and we provide a new criterion for stochastic incompleteness. (C) 2016 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim