@article{Denecke2000, author = {Denecke, Klaus-Dieter}, title = {Generalized hypersubstitutions and strongly solid varieties}, isbn = {3-8265- 7983-6}, year = {2000}, language = {en} } @book{OPUS4-19362, title = {General algebra and applications : proceedings of the 59th Workshop on General Algebra ; 15th Conference for Young Algebraists, Potsdam 2000}, series = {Berichte aus der Mathematik}, journal = {Berichte aus der Mathematik}, editor = {Denecke, Klaus-Dieter and Vogel, Hans-J{\"u}rgen}, publisher = {Shaker}, address = {Aachen}, isbn = {3-8265-7983-6}, pages = {VI, 202 S. : graph. Darst.}, year = {2000}, language = {en} } @article{Denecke2000, author = {Denecke, Klaus-Dieter}, title = {P-compatible hypersubstitutions and M_P -solid varieties}, year = {2000}, language = {en} } @article{DeneckeMahdavi2000, author = {Denecke, Klaus-Dieter and Mahdavi, Kazem}, title = {The order of normal form hypersubstitutions of type 2}, year = {2000}, language = {en} } @article{Denecke2000, author = {Denecke, Klaus-Dieter}, title = {Fluid, unsolid, and completely unsolid varieties}, year = {2000}, language = {en} } @article{DeneckePabhapote2001, author = {Denecke, Klaus-Dieter and Pabhapote, Nittiya}, title = {Tree-recognizers and tree-hyperrecognizers}, year = {2001}, language = {en} } @article{DeneckeArworn2000, author = {Denecke, Klaus-Dieter and Arworn, Srichan}, title = {Intervals defined by M-solid varieties}, isbn = {3-8265- 7983-6}, year = {2000}, language = {en} } @article{Denecke2001, author = {Denecke, Klaus-Dieter}, title = {Hyperidentities in semigroups}, isbn = {90-5199-490-7}, year = {2001}, language = {en} } @article{Denecke2000, author = {Denecke, Klaus-Dieter}, title = {The galois correspondence between subvariety lattices and monoids of hypersubstitutions}, year = {2000}, language = {en} } @article{ArwornDeneckeKoppitz2001, author = {Arworn, Srichan and Denecke, Klaus-Dieter and Koppitz, J{\"o}rg}, title = {Strongly luid and weakly unsolid varieties}, issn = {1346-0862}, year = {2001}, language = {en} } @article{ArwornDenecke2001, author = {Arworn, Srichan and Denecke, Klaus-Dieter}, title = {Tree Transformations defined by Hypersubstitutions}, issn = {1509 - 9415}, year = {2001}, language = {en} } @article{DeneckeKoppitzWismath2001, author = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Wismath, Shelly}, title = {The semantical hyperunification problem}, year = {2001}, language = {en} } @article{DeneckeLeeratanavalee2001, author = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak}, title = {M-solid polynomial varieties of semigroups}, year = {2001}, language = {en} } @article{DeneckeLueders2001, author = {Denecke, Klaus-Dieter and L{\"u}ders, Otfried}, title = {Categorical Equivalences and Invariant Relations}, year = {2001}, language = {en} } @article{DeneckeWismath2003, author = {Denecke, Klaus-Dieter and Wismath, Shelly}, title = {Valuations of Terms}, year = {2003}, abstract = {Let tau be a type of algebras. There are several commonly used measurements of the complexity of terms of type tau, including the depth or height of a term and the number of variable symbols appearing in a term. In this paper we formalize these various measurements, by defining a complexity or valuation mapping on terms. A valuation of terms is thus a mapping from the absolutely free term algebra of type tau into another algebra of the same type on which an order relation is defined. We develop the interconnections between such term valuations and the equational theory of Universal Algebra. The collection of all varieties of a given type forms a complete lattice which is very complex and difficult to study; valuations of terms offer a new method to study complete sublattices of this lattice}, language = {en} } @article{ArwornDenecke2002, author = {Arworn, Srichan and Denecke, Klaus-Dieter}, title = {Intervals and complete congruences defined by M-solid varieties}, year = {2002}, language = {en} } @article{DeneckeKoppitzNiwczyk2002, author = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Niwczyk, St.}, title = {Equational theories generated by generalized hypersubstitutions of type (n)}, year = {2002}, language = {en} } @book{DeneckeKoppitzShtraklov2001, author = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Shtraklov, Slavcho}, title = {The Depth of a Hypersubstitution}, year = {2001}, language = {en} } @book{DeneckeWismath2002, author = {Denecke, Klaus-Dieter and Wismath, Shelly}, title = {Universal algebra and applications in theoretical computer science}, publisher = {Chapman \& Hall/CRC}, address = {Boca Raton}, isbn = {1-584-88254-9}, pages = {383 S.}, year = {2002}, language = {en} } @article{DeneckeKoppitzWismath2002, author = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg and Wismath, Shelly}, title = {Solid Varietie of Arbitrary Type}, year = {2002}, language = {en} } @article{DeneckeKoppitz2001, author = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg}, title = {Fluid, unsolid and completely unsolid varieties}, year = {2001}, language = {en} } @article{DeneckeWismath2002, author = {Denecke, Klaus-Dieter and Wismath, Shelly}, title = {M-solidity testing systems}, year = {2002}, language = {en} } @article{DeneckeKoppitz2001, author = {Denecke, Klaus-Dieter and Koppitz, J{\"o}rg}, title = {Essential variables in hypersubstitution}, year = {2001}, language = {en} } @article{DeneckeLeeratanavalee2000, author = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak}, title = {Generalized hypersubstitutions and strongly solid varieties}, isbn = {3-8265- 7983-6}, year = {2000}, language = {en} } @article{DeneckeLeeratanavalee2000, author = {Denecke, Klaus-Dieter and Leeratanavalee, Sorasak}, title = {Solid polynomial varieties of semigroups which are definable by identities}, year = {2000}, language = {en} } @phdthesis{Trappmann2007, author = {Trappmann, Henryk}, title = {Arborescent numbers : higher arithmetic operations and division trees}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-15247}, school = {Universit{\"a}t Potsdam}, year = {2007}, abstract = {The overall program "arborescent numbers" is to similarly perform the constructions from the natural numbers (N) to the positive fractional numbers (Q+) to positive real numbers (R+) beginning with (specific) binary trees instead of natural numbers. N can be regarded as the associative binary trees. The binary trees B and the left-commutative binary trees P allow the hassle-free definition of arbitrary high arithmetic operations (hyper ... hyperpowers). To construct the division trees the algebraic structure "coppice" is introduced which is a group with an addition over which the multiplication is right-distributive. Q+ is the initial associative coppice. The present work accomplishes one step in the program "arborescent numbers". That is the construction of the arborescent equivalent(s) of the positive fractional numbers. These equivalents are the "division binary trees" and the "fractional trees". A representation with decidable word problem for each of them is given. The set of functions f:R1->R1 generated from identity by taking powers is isomorphic to P and can be embedded into a coppice by taking inverses.}, language = {en} } @article{HaferKiyLucke2014, author = {Hafer, J{\"o}rg and Kiy, Alexander and Lucke, Ulrike}, title = {Moodle \& Co. auf dem Weg zur Personal Learning Environment}, series = {eleed}, volume = {2014}, journal = {eleed}, number = {10}, issn = {1860-7470}, year = {2014}, abstract = {Ausgehend von der typischen IT-Infrastruktur f{\"u}r E-Learning an Hochschulen auf der einen Seite sowie vom bisherigen Stand der Forschung zu Personal Learning Environments (PLEs) auf der anderen Seite zeigt dieser Beitrag auf, wie bestehende Werkzeuge bzw. Dienste zusammengef{\"u}hrt und f{\"u}r die Anforderungen der modernen, rechnergest{\"u}tzten Pr{\"a}senzlehre aufbereitet werden k{\"o}nnen. F{\"u}r diesen interdisziplin{\"a}ren Entwicklungsprozess bieten sowohl klassische Softwareentwicklungsverfahren als auch bestehende PLE-Modelle wenig Hilfestellung an. Der Beitrag beschreibt die in einem campusweiten Projekt an der Universit{\"a}t Potsdam verfolgten Ans{\"a}tze und die damit erzielten Ergebnisse. Daf{\"u}r werden zun{\"a}chst typische Lehr-/Lern-bzw. Kommunikations-Szenarien identifiziert, aus denen Anforderungen an eine unterst{\"u}tzende Plattform abgeleitet werden. Dies f{\"u}hrt zu einer umfassenden Sammlung zu ber{\"u}cksichtigender Dienste und deren Funktionen, die gem{\"a}ß den Spezifika ihrer Nutzung in ein Gesamtsystem zu integrieren sind. Auf dieser Basis werden grunds{\"a}tzliche Integrationsans{\"a}tze und technische Details dieses Mash-Ups in einer Gesamtschau aller relevanten Dienste betrachtet und in eine integrierende Systemarchitektur {\"u}berf{\"u}hrt. Deren konkrete Realisierung mit Hilfe der Portal-Technologie Liferay wird dargestellt, wobei die eingangs definierten Szenarien aufgegriffen und exemplarisch vorgestellt werden. Erg{\"a}nzende Anpassungen im Sinne einer personalisierbaren bzw. adaptiven Lern-(und Arbeits-)Umgebung werden ebenfalls unterst{\"u}tzt und kurz aufgezeigt.}, language = {en} } @article{FischerKeller2021, author = {Fischer, Florian and Keller, Matthias}, title = {Riesz decompositions for Schr{\"o}dinger operators on graphs}, series = {Journal of mathematical analysis and applications}, volume = {495}, journal = {Journal of mathematical analysis and applications}, number = {1}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0022-247X}, doi = {10.1016/j.jmaa.2020.124674}, pages = {22}, year = {2021}, abstract = {We study superharmonic functions for Schrodinger operators on general weighted graphs. Specifically, we prove two decompositions which both go under the name Riesz decomposition in the literature. The first one decomposes a superharmonic function into a harmonic and a potential part. The second one decomposes a superharmonic function into a sum of superharmonic functions with certain upper bounds given by prescribed superharmonic functions. As application we show a Brelot type theorem.}, language = {en} } @article{Zoeller2022, author = {Z{\"o}ller, Gert}, title = {A note on the estimation of the maximum possible earthquake magnitude based on extreme value theory for the Groningen Gas Field}, series = {The bulletin of the Seismological Society of America : BSSA}, volume = {112}, journal = {The bulletin of the Seismological Society of America : BSSA}, number = {4}, publisher = {Seismological Society of America}, address = {El Cerito, Calif.}, issn = {0037-1106}, doi = {10.1785/0120210307}, pages = {1825 -- 1831}, year = {2022}, abstract = {Extreme value statistics is a popular and frequently used tool to model the occurrence of large earthquakes. The problem of poor statistics arising from rare events is addressed by taking advantage of the validity of general statistical properties in asymptotic regimes. In this note, I argue that the use of extreme value statistics for the purpose of practically modeling the tail of the frequency-magnitude distribution of earthquakes can produce biased and thus misleading results because it is unknown to what degree the tail of the true distribution is sampled by data. Using synthetic data allows to quantify this bias in detail. The implicit assumption that the true M-max is close to the maximum observed magnitude M-max,M-observed restricts the class of the potential models a priori to those with M-max = M-max,M-observed + Delta M with an increment Delta M approximate to 0.5... 1.2. This corresponds to the simple heuristic method suggested by Wheeler (2009) and labeled :M-max equals M-obs plus an increment." The incomplete consideration of the entire model family for the frequency-magnitude distribution neglects, however, the scenario of a large so far unobserved earthquake.}, language = {en} } @article{Denecke2020, author = {Denecke, Klaus-Dieter}, title = {Partial clones}, series = {Asian-European journal of mathematics : AEJM}, volume = {13}, journal = {Asian-European journal of mathematics : AEJM}, number = {8}, publisher = {World Scientific}, address = {Singapore}, issn = {1793-5571}, doi = {10.1142/S1793557120501612}, pages = {19}, year = {2020}, abstract = {A set C of operations defined on a nonempty set A is said to be a clone if C is closed under composition of operations and contains all projection mappings. The concept of a clone belongs to the algebraic main concepts and has important applications in Computer Science. A clone can also be regarded as a many-sorted algebra where the sorts are the n-ary operations defined on set A for all natural numbers n >= 1 and the operations are the so-called superposition operations S-m(n) for natural numbers m, n >= 1 and the projection operations as nullary operations. Clones generalize monoids of transformations defined on set A and satisfy three clone axioms. The most important axiom is the superassociative law, a generalization of the associative law. If the superposition operations are partial, i.e. not everywhere defined, instead of the many-sorted clone algebra, one obtains partial many-sorted algebras, the partial clones. Linear terms, linear tree languages or linear formulas form partial clones. In this paper, we give a survey on partial clones and their properties.}, language = {en} } @article{Baer2021, author = {B{\"a}r, Christian}, title = {The Faddeev-LeVerrier algorithm and the Pfaffian}, series = {Linear algebra and its applications}, volume = {630}, journal = {Linear algebra and its applications}, publisher = {Elsevier}, address = {New York}, issn = {0024-3795}, doi = {10.1016/j.laa.2021.07.023}, pages = {39 -- 55}, year = {2021}, abstract = {We adapt the Faddeev-LeVerrier algorithm for the computation of characteristic polynomials to the computation of the Pfaffian of a skew-symmetric matrix. This yields a very simple, easy to implement and parallelize algorithm of computational cost O(n(beta+1)) where nis the size of the matrix and O(n(beta)) is the cost of multiplying n x n-matrices, beta is an element of [2, 2.37286). We compare its performance to that of other algorithms and show how it can be used to compute the Euler form of a Riemannian manifold using computer algebra.}, language = {en} } @article{KistnerVollmeyerBurnsetal.2016, author = {Kistner, Saskia and Vollmeyer, Regina and Burns, Bruce D. and Kortenkamp, Ulrich}, title = {Model development in scientific discovery learning with a computer-based physics task}, series = {Computers in human behavior}, volume = {59}, journal = {Computers in human behavior}, publisher = {Elsevier}, address = {Oxford}, issn = {0747-5632}, doi = {10.1016/j.chb.2016.02.041}, pages = {446 -- 455}, year = {2016}, abstract = {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.}, language = {en} } @article{SinclairBussideVilliersetal.2016, author = {Sinclair, Nathalie and Bussi, Maria G. Bartolini and de Villiers, Michael and Jones, Keith and Kortenkamp, Ulrich and Leung, Allen and Owens, Kay}, title = {Recent research on geometry education: an ICME-13 survey team report}, series = {ZDM : The International Journal on Mathematics Education}, volume = {48}, journal = {ZDM : The International Journal on Mathematics Education}, publisher = {Springer}, address = {Heidelberg}, issn = {1863-9690}, doi = {10.1007/s11858-016-0796-6}, pages = {691 -- 719}, year = {2016}, abstract = {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.}, language = {en} } @article{KistnerBurnsVollmeyeretal.2016, author = {Kistner, Saskia and Burns, Bruce D. and Vollmeyer, Regina and Kortenkamp, Ulrich}, title = {The importance of understanding: Model space moderates goal specificity effects}, series = {The quarterly journal of experimental psychology}, volume = {69}, journal = {The quarterly journal of experimental psychology}, publisher = {Optical Society of America}, address = {Abingdon}, issn = {1747-0218}, doi = {10.1080/17470218.2015.1076865}, pages = {1179 -- 1196}, year = {2016}, abstract = {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.}, language = {en} } @article{KortenkampMonaghanTrouche2016, author = {Kortenkamp, Ulrich and Monaghan, John and Trouche, Luc}, title = {Jonathan M Borwein (1951-2016): exploring, experiencing and experimenting in mathematics - an inspiring journey in mathematics}, series = {Educational studies in mathematics : an international journal}, volume = {93}, journal = {Educational studies in mathematics : an international journal}, publisher = {Springer}, address = {Dordrecht}, issn = {0013-1954}, doi = {10.1007/s10649-016-9729-0}, pages = {131 -- 136}, year = {2016}, language = {en} } @misc{KleinRosenberger2018, author = {Klein, Markus and Rosenberger, Elke}, title = {The tunneling effect for a class of difference operators}, series = {Reviews in Mathematical Physics}, volume = {30}, journal = {Reviews in Mathematical Physics}, number = {4}, publisher = {World Scientific}, address = {Singapore}, issn = {0129-055X}, doi = {10.1142/S0129055X18300029}, pages = {42}, year = {2018}, abstract = {We analyze a general class of self-adjoint difference operators H-epsilon = T-epsilon + V-epsilon on l(2)((epsilon Z)(d)), where V-epsilon is a multi-well potential and v(epsilon) is a small parameter. We give a coherent review of our results on tunneling up to new sharp results on the level of complete asymptotic expansions (see [30-35]). Our emphasis is on general ideas and strategy, possibly of interest for a broader range of readers, and less on detailed mathematical proofs. The wells are decoupled by introducing certain Dirichlet operators on regions containing only one potential well. Then the eigenvalue problem for the Hamiltonian H-epsilon is treated as a small perturbation of these comparison problems. After constructing a Finslerian distance d induced by H-epsilon, we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by this distance to the well. It follows with microlocal techniques that the first n eigenvalues of H-epsilon converge to the first n eigenvalues of the direct sum of harmonic oscillators on R-d located at several wells. In a neighborhood of one well, we construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low-lying eigenvalues of H-epsilon. These are obtained from eigenfunctions or quasimodes for the operator H-epsilon acting on L-2(R-d), via restriction to the lattice (epsilon Z)(d). Tunneling is then described by a certain interaction matrix, similar to the analysis for the Schrodinger operator (see [22]), the remainder is exponentially small and roughly quadratic compared with the interaction matrix. We give weighted l(2)-estimates for the difference of eigenfunctions of Dirichlet-operators in neighborhoods of the different wells and the associated WKB-expansions at the wells. In the last step, we derive full asymptotic expansions for interactions between two "wells" (minima) of the potential energy, in particular for the discrete tunneling effect. Here we essentially use analysis on phase space, complexified in the momentum variable. These results are as sharp as the classical results for the Schrodinger operator in [22].}, language = {en} } @misc{ZoellerHolschneider2018, author = {Z{\"o}ller, Gert and Holschneider, Matthias}, title = {Reply to "Comment on 'The Maximum Possible and the Maximum Expected Earthquake Magnitude for Production-Induced Earthquakes at the Gas Field in Groningen, The Netherlands' by Gert Z{\"o}ller and Matthias Holschneider" by Mathias Raschke}, series = {Bulletin of the Seismological Society of America}, volume = {108}, journal = {Bulletin of the Seismological Society of America}, number = {2}, publisher = {Seismological Society of America}, address = {Albany}, issn = {0037-1106}, doi = {10.1785/0120170131}, pages = {1029 -- 1030}, year = {2018}, language = {en} } @article{LiuMuenchPeyerimhoff2018, author = {Liu, Shiping and M{\"u}nch, Florentin and Peyerimhoff, Norbert}, title = {Bakry-Emery curvature and diameter bounds on graphs}, series = {Calculus of variations and partial differential equations}, volume = {57}, journal = {Calculus of variations and partial differential equations}, number = {2}, publisher = {Springer}, address = {Heidelberg}, issn = {0944-2669}, doi = {10.1007/s00526-018-1334-x}, pages = {9}, year = {2018}, abstract = {We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry-{\´E}mery sense. Our first result using only curvature and maximal vertex degree is sharp in the case of hypercubes. The second result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from Fathi and Shu (Bernoulli 24(1):672-698, 2018) and Horn et al. (J f{\"u}r die reine und angewandte Mathematik (Crelle's J), 2017, https://doi.org/10.1515/crelle-2017-0038) and solve a conjecture from Cushing et al. (Bakry-{\´E}mery curvature functions of graphs, 2016).}, language = {en} } @article{KellerSchwarz2018, author = {Keller, Matthias and Schwarz, Michael}, title = {The Kazdan-Warner equation on canonically compactifiable graphs}, series = {Calculus of variations and partial differential equations}, volume = {57}, journal = {Calculus of variations and partial differential equations}, number = {2}, publisher = {Springer}, address = {Heidelberg}, issn = {0944-2669}, doi = {10.1007/s00526-018-1329-7}, pages = {18}, year = {2018}, abstract = {We study the Kazdan-Warner equation on canonically compactifiable graphs. These graphs are distinguished as analytic properties of Laplacians on these graphs carry a strong resemblance to Laplacians on open pre-compact manifolds.}, language = {en} } @article{LesurWardinskiBaerenzungetal.2017, author = {Lesur, Vincent and Wardinski, Ingo and B{\"a}renzung, Julien and Holschneider, Matthias}, title = {On the frequency spectra of the core magnetic field Gauss coefficients}, series = {Physics of the earth and planetary interiors}, volume = {276}, journal = {Physics of the earth and planetary interiors}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0031-9201}, doi = {10.1016/j.pepi.2017.05.017}, pages = {145 -- 158}, year = {2017}, abstract = {From monthly mean observatory data spanning 1957-2014, geomagnetic field secular variation values were calculated by annual differences. Estimates of the spherical harmonic Gauss coefficients of the core field secular variation were then derived by applying a correlation based modelling. Finally, a Fourier transform was applied to the time series of the Gauss coefficients. This process led to reliable temporal spectra of the Gauss coefficients up to spherical harmonic degree 5 or 6, and down to periods as short as 1 or 2 years depending on the coefficient. We observed that a k(-2) slope, where k is the frequency, is an acceptable approximation for these spectra, with possibly an exception for the dipole field. The monthly estimates of the core field secular variation at the observatory sites also show that large and rapid variations of the latter happen. This is an indication that geomagnetic jerks are frequent phenomena and that significant secular variation signals at short time scales - i.e. less than 2 years, could still be extracted from data to reveal an unexplored part of the core dynamics.}, language = {en} } @article{BrungsGraeter2017, author = {Brungs, Hans H. and Gr{\"a}ter, Joachim}, title = {On central extensions of SL(2, F) admitting left-orderings}, series = {Journal of Algebra}, volume = {486}, journal = {Journal of Algebra}, publisher = {Elsevier}, address = {San Diego}, issn = {0021-8693}, doi = {10.1016/j.jalgebra.2017.05.025}, pages = {288 -- 327}, year = {2017}, abstract = {For an arbitrary euclidean field F we introduce a central extension (G(F), Phi) of SL(2, F) admitting a left-ordering and study its algebraic properties. The elements of G(F) are order preserving bijections of the convex hull of Q in F. If F = R then G(F) is isomorphic to the classical universal covering group of the Lie group SL(2, R). Among other results we show that G(F) is a perfect group which possesses a rank 1 cone of exceptional type. We also prove that its centre is an infinite cyclic group and investigate its normal subgroups.}, language = {en} } @article{SomogyvariReich2020, author = {Somogyv{\´a}ri, M{\´a}rk and Reich, Sebastian}, title = {Convergence tests for transdimensional Markov chains in geoscience imaging}, series = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences}, volume = {52}, journal = {Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences}, number = {5}, publisher = {Springer}, address = {Heidelberg}, issn = {1874-8961}, doi = {10.1007/s11004-019-09811-x}, pages = {651 -- 668}, year = {2020}, abstract = {Classic inversion methods adjust a model with a predefined number of parameters to the observed data. With transdimensional inversion algorithms such as the reversible-jump Markov chain Monte Carlo (rjMCMC), it is possible to vary this number during the inversion and to interpret the observations in a more flexible way. Geoscience imaging applications use this behaviour to automatically adjust model resolution to the inhomogeneities of the investigated system, while keeping the model parameters on an optimal level. The rjMCMC algorithm produces an ensemble as result, a set of model realizations, which together represent the posterior probability distribution of the investigated problem. The realizations are evolved via sequential updates from a randomly chosen initial solution and converge toward the target posterior distribution of the inverse problem. Up to a point in the chain, the realizations may be strongly biased by the initial model, and must be discarded from the final ensemble. With convergence assessment techniques, this point in the chain can be identified. Transdimensional MCMC methods produce ensembles that are not suitable for classic convergence assessment techniques because of the changes in parameter numbers. To overcome this hurdle, three solutions are introduced to convert model realizations to a common dimensionality while maintaining the statistical characteristics of the ensemble. A scalar, a vector and a matrix representation for models is presented, inferred from tomographic subsurface investigations, and three classic convergence assessment techniques are applied on them. It is shown that appropriately chosen scalar conversions of the models could retain similar statistical ensemble properties as geologic projections created by rasterization.}, language = {en} } @phdthesis{Ly2009, author = {Ly, Ibrahim}, title = {Asymptotic solutions of the Cauchy problem for nonlinear elliptic differential equations}, address = {Potsdam}, pages = {VI, 100 S.}, year = {2009}, language = {en} } @article{Ly2020, author = {Ly, Ibrahim}, title = {A Cauchy problem for the Cauchy-Riemann operator}, series = {Afrika Matematika}, volume = {32}, journal = {Afrika Matematika}, number = {1-2}, publisher = {Springer}, address = {Heidelberg}, issn = {1012-9405}, doi = {10.1007/s13370-020-00810-4}, pages = {69 -- 76}, year = {2020}, abstract = {We study the Cauchy problem for a nonlinear elliptic equation with data on a piece S of the boundary surface partial derivative X. By the Cauchy problem is meant any boundary value problem for an unknown function u in a domain X with the property that the data on S, if combined with the differential equations in X, allows one to determine all derivatives of u on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We discuss a variational setting of the Cauchy problem which always possesses a generalized solution.}, language = {en} } @unpublished{LyTarkhanov2013, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {Generalised Beltrami equations}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus-67416}, year = {2013}, abstract = {We enlarge the class of Beltrami equations by developping a stability theory for the sheaf of solutions of an overdetermined elliptic system of first order homogeneous partial differential equations with constant coefficients in the Euclidean space.}, language = {en} } @unpublished{LyTarkhanov2015, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {A Rad{\´o} theorem for p-harmonic functions}, volume = {4}, number = {3}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-71492}, pages = {10}, year = {2015}, abstract = {Let A be a nonlinear differential operator on an open set X in 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 the complement of S 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.}, language = {en} } @unpublished{LyTarkhanov2015, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {Asymptotic expansions at nonsymmetric cuspidal points}, volume = {4}, number = {7}, publisher = {Universit{\"a}tsverlag Potsdam}, address = {Potsdam}, issn = {2193-6943}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-78199}, pages = {11}, year = {2015}, abstract = {We study asymptotics of solutions to the Dirichlet problem in a domain whose boundary contains a nonsymmetric conical point. We establish a complete asymptotic expansion of solutions near the singular point.}, language = {en} } @article{MalassTarkhanov2020, author = {Malass, Ihsane and Tarkhanov, Nikolaj Nikolaevič}, title = {A perturbation of the de Rham complex}, series = {Journal of Siberian Federal University : Mathematics \& Physics}, volume = {13}, journal = {Journal of Siberian Federal University : Mathematics \& Physics}, number = {5}, publisher = {Siberian Federal University}, address = {Krasnojarsk}, issn = {1997-1397}, doi = {10.17516/1997-1397-2020-13-5-519-532}, pages = {519 -- 532}, year = {2020}, abstract = {We consider a perturbation of the de Rham complex on a compact manifold with boundary. This perturbation goes beyond the framework of complexes, and so cohomology does not apply to it. On the other hand, its curvature is "small", hence there is a natural way to introduce an Euler characteristic and develop a Lefschetz theory for the perturbation. This work is intended as an attempt to develop a cohomology theory for arbitrary sequences of linear mappings.}, language = {en} } @article{LyTarkhanov2016, author = {Ly, Ibrahim and Tarkhanov, Nikolai Nikolaevich}, title = {A Rado theorem for p-harmonic functions}, series = {Boletin de la Sociedad Matem{\~A}!'tica Mexicana}, volume = {22}, journal = {Boletin de la Sociedad Matem{\~A}!'tica Mexicana}, publisher = {Springer}, address = {Basel}, issn = {1405-213X}, doi = {10.1007/s40590-016-0109-7}, pages = {461 -- 472}, year = {2016}, abstract = {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.}, language = {en} } @phdthesis{Jakobs2019, author = {Jakobs, Friedrich}, title = {Dubrovin-rings and their connection to Hughes-free skew fields of fractions}, doi = {10.25932/publishup-43556}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:517-opus4-435561}, school = {Universit{\"a}t Potsdam}, pages = {ix, 62}, year = {2019}, abstract = {One method of embedding groups into skew fields was introduced by A. I. Mal'tsev and B. H. Neumann (cf. [18, 19]). If G is an ordered group and F is a skew field, the set F((G)) of formal power series over F in G with well-ordered support forms a skew field into which the group ring F[G] can be embedded. Unfortunately it is not suficient that G is left-ordered since F((G)) is only an F-vector space in this case as there is no natural way to define a multiplication on F((G)). One way to extend the original idea onto left-ordered groups is to examine the endomorphism ring of F((G)) as explored by N. I. Dubrovin (cf. [5, 6]). It is possible to embed any crossed product ring F[G; η, σ] into the endomorphism ring of F((G)) such that each non-zero element of F[G; η, σ] defines an automorphism of F((G)) (cf. [5, 10]). Thus, the rational closure of F[G; η, σ] in the endomorphism ring of F((G)), which we will call the Dubrovin-ring of F[G; η, σ], is a potential candidate for a skew field of fractions of F[G; η, σ]. The methods of N. I. Dubrovin allowed to show that specific classes of groups can be embedded into a skew field. For example, N. I. Dubrovin contrived some special criteria, which are applicable on the universal covering group of SL(2, R). These methods have also been explored by J. Gr{\"a}ter and R. P. Sperner (cf. [10]) as well as N.H. Halimi and T. Ito (cf. [11]). Furthermore, it is of interest to know if skew fields of fractions are unique. For example, left and right Ore domains have unique skew fields of fractions (cf. [2]). This is not the general case as for example the free group with 2 generators can be embedded into non-isomorphic skew fields of fractions (cf. [12]). It seems likely that Ore domains are the most general case for which unique skew fields of fractions exist. One approach to gain uniqueness is to restrict the search to skew fields of fractions with additional properties. I. Hughes has defined skew fields of fractions of crossed product rings F[G; η, σ] with locally indicable G which fulfill a special condition. These are called Hughes-free skew fields of fractions and I. Hughes has proven that they are unique if they exist [13, 14]. This thesis will connect the ideas of N. I. Dubrovin and I. Hughes. The first chapter contains the basic terminology and concepts used in this thesis. We present methods provided by N. I. Dubrovin such as the complexity of elements in rational closures and special properties of endomorphisms of the vector space of formal power series F((G)). To combine the ideas of N.I. Dubrovin and I. Hughes we introduce Conradian left-ordered groups of maximal rank and examine their connection to locally indicable groups. Furthermore we provide notations for crossed product rings, skew fields of fractions as well as Dubrovin-rings and prove some technical statements which are used in later parts. The second chapter focuses on Hughes-free skew fields of fractions and their connection to Dubrovin-rings. For that purpose we introduce series representations to interpret elements of Hughes-free skew fields of fractions as skew formal Laurent series. This 1 Introduction allows us to prove that for Conradian left-ordered groups G of maximal rank the statement "F[G; η, σ] has a Hughes-free skew field of fractions" implies "The Dubrovin ring of F [G; η, σ] is a skew field". We will also prove the reverse and apply the results to give a new prove of Theorem 1 in [13]. Furthermore we will show how to extend injective ring homomorphisms of some crossed product rings onto their Hughes-free skew fields of fractions. At last we will be able to answer the open question whether Hughes--free skew fields are strongly Hughes-free (cf. [17, page 53]).}, language = {en} }