TY - JOUR A1 - Roos, Saskia T1 - The Dirac operator under collapse to a smooth limit space JF - Annals of global analysis and geometry N2 - Let (M-i, g(i))(i is an element of N) be a sequence of spin manifolds with uniform bounded curvature and diameter that converges to a lower-dimensional Riemannian manifold (B, h) in the Gromov-Hausdorff topology. Then, it happens that the spectrum of the Dirac operator converges to the spectrum of a certain first-order elliptic differential operator D-B on B. We give an explicit description of D-B and characterize the special case where D-B equals the Dirac operator on B. KW - Collapse KW - Dirac operator KW - Spin geometry Y1 - 2019 U6 - https://doi.org/10.1007/s10455-019-09691-8 SN - 0232-704X SN - 1572-9060 VL - 57 IS - 1 SP - 121 EP - 151 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Güneysu, Batu A1 - Keller, Matthias T1 - Scattering the Geometry of Weighted Graphs JF - Mathematical physics, analysis and geometry : an international journal devoted to the theory and applications of analysis and geometry to physics N2 - Given two weighted graphs (X, b(k), m(k)), k = 1, 2 with b(1) similar to b(2) and m(1) similar to m(2), we prove a weighted L-1-criterion for the existence and completeness of the wave operators W-+/- (H-2, H-1, I-1,I-2), where H-k denotes the natural Laplacian in l(2)(X, m(k)) w.r.t. (X, b(k), m(k)) and I-1,I-2 the trivial identification of l(2)(X, m(1)) with l(2) (X, m(2)). In particular, this entails a general criterion for the absolutely continuous spectra of H-1 and H-2 to be equal. KW - Graphs KW - Laplacian KW - Scattering theory Y1 - 2018 U6 - https://doi.org/10.1007/s11040-018-9285-1 SN - 1385-0172 SN - 1572-9656 VL - 21 IS - 3 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Fedchenko, Dmitry A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Rado theorem for the porous medium equation JF - Boletin de la Sociedad Matemática Mexicana N2 - We prove that if u is a locally Lipschitz continuous function on an open set chi subset of Rn + 1 satisfying the nonlinear heat equation partial derivative(t)u = Delta(vertical bar u vertical bar(p-1) u), p > 1, weakly away from the zero set u(-1) (0) in chi, then u is a weak solution to this equation in all of chi. KW - Quasilinear equations KW - Removable sets KW - Porous medium equation Y1 - 2017 U6 - https://doi.org/10.1007/s40590-017-0169-3 SN - 1405-213X SN - 2296-4495 VL - 24 IS - 2 SP - 427 EP - 437 PB - Springer CY - Cham ER - TY - JOUR A1 - Dimitrova, Ilinka A1 - Koppitz, Jörg T1 - On the semigroup of all partial fence-preserving injections on a finite set JF - Journal of Algebra and Its Applications N2 - For n∈N , let Xn={a1,a2,…,an} be an n-element set and let F=(Xn; infinity. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups. Y1 - 2017 U6 - https://doi.org/10.1017/etds.2017.9 SN - 0143-3857 SN - 1469-4417 VL - 38 SP - 3012 EP - 3041 PB - Cambridge Univ. Press CY - New York ER - TY - JOUR A1 - Perkins, Daniel M. A1 - Perna, Andrea A1 - Adrian, Rita A1 - Cermeno, Pedro A1 - Gaedke, Ursula A1 - Huete-Ortega, Maria A1 - White, Ethan P. A1 - Yvon-Durocher, Gabriel T1 - Energetic equivalence underpins the size structure of tree and phytoplankton communities JF - Nature Communications N2 - The size structure of autotroph communities - the relative abundance of small vs. large individuals - shapes the functioning of ecosystems. Whether common mechanisms underpin the size structure of unicellular and multicellular autotrophs is, however, unknown. Using a global data compilation, we show that individual body masses in tree and phytoplankton communities follow power-law distributions and that the average exponents of these individual size distributions (ISD) differ. Phytoplankton communities are characterized by an average ISD exponent consistent with three-quarter-power scaling of metabolism with body mass and equivalence in energy use among mass classes. Tree communities deviate from this pattern in a manner consistent with equivalence in energy use among diameter size classes. Our findings suggest that whilst universal metabolic constraints ultimately underlie the emergent size structure of autotroph communities, divergent aspects of body size (volumetric vs. linear dimensions) shape the ecological outcome of metabolic scaling in forest vs. pelagic ecosystems. Y1 - 2019 U6 - https://doi.org/10.1038/s41467-018-08039-3 SN - 2041-1723 VL - 10 PB - Nature Publ. Group CY - London ER - TY - JOUR A1 - Leung, Tsz Yan A1 - Leutbecher, Martin A1 - Reich, Sebastian A1 - Shepherd, Theodore G. T1 - Atmospheric Predictability: Revisiting the Inherent Finite-Time Barrier JF - Journal of the atmospheric sciences N2 - The accepted idea that there exists an inherent finite-time barrier in deterministically predicting atmospheric flows originates from Edward N. Lorenz’s 1969 work based on two-dimensional (2D) turbulence. Yet, known analytic results on the 2D Navier–Stokes (N-S) equations suggest that one can skillfully predict the 2D N-S system indefinitely far ahead should the initial-condition error become sufficiently small, thereby presenting a potential conflict with Lorenz’s theory. Aided by numerical simulations, the present work reexamines Lorenz’s model and reviews both sides of the argument, paying particular attention to the roles played by the slope of the kinetic energy spectrum. It is found that when this slope is shallower than −3, the Lipschitz continuity of analytic solutions (with respect to initial conditions) breaks down as the model resolution increases, unless the viscous range of the real system is resolved—which remains practically impossible. This breakdown leads to the inherent finite-time limit. If, on the other hand, the spectral slope is steeper than −3, then the breakdown does not occur. In this way, the apparent contradiction between the analytic results and Lorenz’s theory is reconciled. KW - Atmosphere KW - Turbulence KW - Error analysis KW - Spectral analysis KW - models KW - distribution KW - Numerical weather prediction KW - forecasting Y1 - 2019 U6 - https://doi.org/10.1175/JAS-D-19-0057.1 SN - 0022-4928 SN - 1520-0469 VL - 76 IS - 12 SP - 3883 EP - 3892 PB - American Meteorological Soc. CY - Boston ER - TY - GEN A1 - Beckus, Siegfried A1 - Bellissard, Jean A1 - De Nittis, Giuseppe T1 - Corrigendum to: Spectral continuity for aperiodic quantum systems I. General theory. - [Journal of functional analysis. - 275 (2018), 11, S. 2917 – 2977] T2 - Journal of functional analysis N2 - A correct statement of Theorem 4 in [1] is provided. The change does not affect the main results. KW - Haar system Y1 - 2019 U6 - https://doi.org/10.1016/j.jfa.2019.06.001 SN - 0022-1236 SN - 1096-0783 VL - 277 IS - 9 SP - 3351 EP - 3353 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Fernandes, Vitor H. A1 - Koppitz, Jörg A1 - Musunthia, Tiwadee T1 - The Rank of the Semigroup of All Order-Preserving Transformations on a Finite Fence JF - Bulletin of the Malaysian Mathematical Sciences Society volume N2 - A zig-zag (or fence) order is a special partial order on a (finite) set. In this paper, we consider the semigroup TFn of all order-preserving transformations on an n-element zig-zag-ordered set. We determine the rank of TFn and provide a minimal generating set for TFn. Moreover, a formula for the number of idempotents in TFn is given. KW - Transformation semigroups KW - Rank of semigroup KW - Idempotents KW - Order-preserving KW - Fence KW - Zig-zag order Y1 - 2019 U6 - https://doi.org/10.1007/s40840-017-0598-1 SN - 0126-6705 SN - 2180-4206 VL - 42 IS - 5 SP - 2191 EP - 2211 PB - Malaysian mathematical sciences sciences soc CY - Pulau Punang ER - TY - JOUR A1 - Shcherbakov, Robert A1 - Zhuang, Jiancang A1 - Zöller, Gert A1 - Ogata, Yosihiko T1 - Forecasting the magnitude of the largest expected earthquake JF - Nature Communications N2 - The majority of earthquakes occur unexpectedly and can trigger subsequent sequences of events that can culminate in more powerful earthquakes. This self-exciting nature of seismicity generates complex clustering of earthquakes in space and time. Therefore, the problem of constraining the magnitude of the largest expected earthquake during a future time interval is of critical importance in mitigating earthquake hazard. We address this problem by developing a methodology to compute the probabilities for such extreme earthquakes to be above certain magnitudes. We combine the Bayesian methods with the extreme value theory and assume that the occurrence of earthquakes can be described by the Epidemic Type Aftershock Sequence process. We analyze in detail the application of this methodology to the 2016 Kumamoto, Japan, earthquake sequence. We are able to estimate retrospectively the probabilities of having large subsequent earthquakes during several stages of the evolution of this sequence. Y1 - 2019 U6 - https://doi.org/10.1038/s41467-019-11958-4 SN - 2041-1723 VL - 10 PB - Nature Publishing Group CY - London ER - TY - JOUR A1 - Conforti, Giovanni A1 - Kosenkova, Tetiana A1 - Roelly, Sylvie T1 - Conditioned Point Processes with Application to Levy Bridges JF - Journal of theoretical probability N2 - Our first result concerns a characterization by means of a functional equation of Poisson point processes conditioned by the value of their first moment. It leads to a generalized version of Mecke’s formula. En passant, it also allows us to gain quantitative results about stochastic domination for Poisson point processes under linear constraints. Since bridges of a pure jump Lévy process in Rd with a height a can be interpreted as a Poisson point process on space–time conditioned by pinning its first moment to a, our approach allows us to characterize bridges of Lévy processes by means of a functional equation. The latter result has two direct applications: First, we obtain a constructive and simple way to sample Lévy bridge dynamics; second, it allows us to estimate the number of jumps for such bridges. We finally show that our method remains valid for linearly perturbed Lévy processes like periodic Ornstein–Uhlenbeck processes driven by Lévy noise. KW - Ornstein-Uhlenbeck Y1 - 2019 U6 - https://doi.org/10.1007/s10959-018-0863-8 SN - 0894-9840 SN - 1572-9230 VL - 32 IS - 4 SP - 2111 EP - 2134 PB - Springer CY - New York ER - TY - JOUR A1 - Staniforth, Andrew A1 - Wood, Nigel A1 - Reich, Sebastian T1 - A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations JF - Quarterly journal of the Royal Meteorological Society N2 - A time-staggered semi-Lagrangian discretization of the rotating shallow-water equations is proposed and analysed. Application of regularization to the geopotential field used in the momentum equations leads to an unconditionally stable scheme. The analysis, together with a fully nonlinear example application, suggests that this approach is a promising, efficient, and accurate alternative to traditional schemes. KW - regularization KW - temporal discretization Y1 - 2006 U6 - https://doi.org/10.1256/qj.06.30 SN - 0035-9009 VL - 132 IS - 621C SP - 3107 EP - 3116 PB - Wiley CY - Weinheim ER - TY - JOUR A1 - Kirsche, Andreas A1 - Böckmann, Christine T1 - Pade iteration method for regularization JF - Applied mathematics and computation N2 - In this study we present iterative regularization methods using rational approximations, in particular, Pade approximants, which work well for ill-posed problems. We prove that the (k,j)-Pade method is a convergent and order optimal iterative regularization method in using the discrepancy principle of Morozov. Furthermore, we present a hybrid Pade method, compare it with other well-known methods and found that it is faster than the Landweber method. It is worth mentioning that this study is a completion of the paper [A. Kirsche, C. Bockmann, Rational approximations for ill-conditioned equation systems, Appl. Math. Comput. 171 (2005) 385-397] where this method was treated to solve ill-conditioned equation systems. (c) 2006 Elsevier Inc. All rights reserved. KW - Pade approximants KW - iterative regularization KW - ill-posed problem Y1 - 2006 U6 - https://doi.org/10.1016/j.amc.2006.01.011 SN - 0096-3003 VL - 180 IS - 2 SP - 648 EP - 663 PB - Elsevier CY - New York ER - TY - JOUR A1 - Reich, Sebastian T1 - Linearly implicit time stepping methods for numerical weather prediction JF - BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians N2 - The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer-Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations. KW - numerical weather prediction KW - linearly implicit time stepping methods KW - semi-Lagrangian method KW - Stormer-Verlet method KW - shallow-water equations Y1 - 2006 U6 - https://doi.org/10.1007/s10543-006-0065-0 SN - 0006-3835 VL - 46 SP - 607 EP - 616 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Bär, Christian A1 - Strohmaier, Alexander T1 - An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary JF - American Journal of Mathematics N2 - We show that the Dirac operator on a compact globally hyperbolic Lorentzian spacetime with spacelike Cauchy boundary is a Fredholm operator if appropriate boundary conditions are imposed. We prove that the index of this operator is given by the same expression as in the index formula of Atiyah-Patodi-Singer for Riemannian manifolds with boundary. The index is also shown to equal that of a certain operator constructed from the evolution operator and a spectral projection on the boundary. In case the metric is of product type near the boundary a Feynman parametrix is constructed. Y1 - 2019 U6 - https://doi.org/10.1353/ajm.2019.0037 SN - 0002-9327 SN - 1080-6377 VL - 141 IS - 5 SP - 1421 EP - 1455 PB - Johns Hopkins Univ. Press CY - Baltimore ER - TY - JOUR A1 - Lewandowski, Max T1 - Hadamard states for bosonic quantum field theory on globally hyperbolic spacetimes JF - Journal of mathematical physics N2 - According to Radzikowski’s celebrated results, bisolutions of a wave operator on a globally hyperbolic spacetime are of the Hadamard form iff they are given by a linear combination of distinguished parametrices i2(G˜aF−G˜F+G˜A−G˜R) in the sense of Duistermaat and Hörmander [Acta Math. 128, 183–269 (1972)] and Radzikowski [Commun. Math. Phys. 179, 529 (1996)]. Inspired by the construction of the corresponding advanced and retarded Green operator GA, GR as done by Bär, Ginoux, and Pfäffle {Wave Equations on Lorentzian Manifolds and Quantization [European Mathematical Society (EMS), Zürich, 2007]}, we construct the remaining two Green operators GF, GaF locally in terms of Hadamard series. Afterward, we provide the global construction of i2(G˜aF−G˜F), which relies on new techniques such as a well-posed Cauchy problem for bisolutions and a patching argument using Čech cohomology. This leads to global bisolutions of the Hadamard form, each of which can be chosen to be a Hadamard two-point-function, i.e., the smooth part can be adapted such that, additionally, the symmetry and the positivity condition are exactly satisfied. Y1 - 2022 U6 - https://doi.org/10.1063/5.0055753 SN - 0022-2488 SN - 1089-7658 VL - 63 IS - 1 PB - American Institute of Physics CY - Melville ER - TY - JOUR A1 - Denecke, Klaus-Dieter A1 - Koppitz, Jörg A1 - Štrakov, Slavčo T1 - Multi-hypersubstitutions and colored solid varieties JF - International journal of algebra and computation N2 - Hypersubstitutions are mappings which map operation symbols to terms. Terms can be visualized by trees. Hypersubstitutions can be extended to mappings defined on sets of trees. The nodes of the trees, describing terms, are labelled by operation symbols and by colors, i.e. certain positive integers. We are interested in mappings which map differently-colored operation symbols to different terms. In this paper we extend the theory of hypersubstitutions and solid varieties to multi-hypersubstitutions and colored solid varieties. We develop the interconnections between such colored terms and multihypersubstitutions 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; multi-hypersubstitutions and colored solid varieties offer a new method to study complete sublattices of this lattice. KW - coloration of terms KW - multi-hypersubstitutions KW - colored solid varieties Y1 - 2006 U6 - https://doi.org/10.1142/S0218196706003189 SN - 0218-1967 VL - 16 IS - 4 SP - 797 EP - 815 PB - World Scient. Publ. CY - Singapore ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - The partial clone of linear formulas JF - Siberian mathematical journal N2 - A term t is linear if no variable occurs more than once in t. An identity s ≈ t is said to be linear if s and t are linear terms. Identities are particular formulas. As for terms superposition operations can be defined for formulas too. We define the arbitrary linear formulas and seek for a condition for the set of all linear formulas to be closed under superposition. This will be used to define the partial superposition operations on the set of linear formulas and a partial many-sorted algebra Formclonelin(τ, τ′). This algebra has similar properties with the partial many-sorted clone of all linear terms. We extend the concept of a hypersubstitution of type τ to the linear hypersubstitutions of type (τ, τ′) for algebraic systems. The extensions of linear hypersubstitutions of type (τ, τ′) send linear formulas to linear formulas, presenting weak endomorphisms of Formclonelin(τ, τ′). KW - term KW - formula KW - superposition KW - linear term KW - linear formula KW - clone KW - partial clone KW - linear hypersubstitution Y1 - 2019 U6 - https://doi.org/10.1134/S0037446619040037 SN - 0037-4466 SN - 1573-9260 VL - 60 IS - 4 SP - 572 EP - 584 PB - Pleiades Publ. CY - New York ER - TY - JOUR A1 - Lekkoksung, Nareupanat A1 - Denecke, Klaus-Dieter T1 - The partial clone of linear tree languages JF - Siberian mathematical journal N2 - A term, also called a tree, is said to be linear, if each variable occurs in the term only once. The linear terms and sets of linear terms, the so-called linear tree languages, play some role in automata theory and in the theory of formal languages in connection with recognizability. We define a partial superposition operation on sets of linear trees of a given type and study the properties of some many-sorted partial clones that have sets of linear trees as elements and partial superposition operations as fundamental operations. The endomorphisms of those algebras correspond to nondeterministic linear hypersubstitutions. KW - linear term KW - linear tree language KW - clone KW - partial clone KW - linear hypersubstitution KW - nondeterministic linear hypersubstitution Y1 - 2019 U6 - https://doi.org/10.1134/S0037446619030121 SN - 0037-4466 SN - 1573-9260 VL - 60 IS - 3 SP - 497 EP - 507 PB - Pleiades Publ. CY - New York ER - TY - THES A1 - Supaporn, Worakrit T1 - Categorical equivalence of clones Y1 - 2014 ER - TY - THES A1 - Trappmann, Henryk T1 - Arborescent numbers : higher arithmetic operations and division trees T1 - Baumartige Zahlen : höhere arithmetische Operationen und Divisionsbäume N2 - 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. N2 - Baumartige Zahlen und höhere arithmetische Operationen Von Schülern und Laienmathematikern wird oft die Frage gestellt, warum nach den Operationen Addition (1. Stufe), Multiplikation (2. Stufe), Potenzieren (3. Stufe) keine Operationen der 4. oder höheren Stufen betrachtet werden. Jede Operation der nächsthöheren Stufe ist die Wiederholung der vorhergehenden Operation, z.B. n * x = x + x + ... + x x^n = x * x * ... * x Das offensichtliche Problem mit der Wiederholung des Potenzierens besteht darin, dass das Potenzieren nicht assoziativ ist und es somit mehrere Klammerungsmöglichkeiten für die Wiederholung dieser Operation gibt. Wählt man eine spezifische Klammerungsmöglichkeit aus, z.B. x^^n = (x^(x^(x^(......)))), gibt es jedoch wieder verschiedene Möglichkeiten, diese Operation auf rationale oder reelle n fortzusetzen. In der Tat kann man im Internet verschiedene solcher Fortsetzungen beschrieben finden und keine scheint besonders ausgezeichnet zu sein. Das ganze Dilemma der verschiedenen Klammerungen kann man jedoch überwinden, in dem man den Zahlenbereich abstrakter macht. So dass statt nur der Anzahl auch eine Klammerungsstruktur in einer Zahl kodiert wird. Die ganz natürliche Verallgemeinerung der natürlichen Zahlen in dieser Hinsicht sind die Binärbäume. Und in der Tat lassen sich die 4. und höhere Operationen in einer eindeutigen Weise auf den Binärbäumen erklären. Vielmehr stellt sich sogar heraus, dass die Binärbäume zu viel Information mit sich tragen, wenn es nur darum geht, die höheren Operationen zu definieren. Es gibt eine Spezialisierung der Binärbäume, die aber allgemeiner als die natürlichen Zahlen (die die assoziative Spezialisierung der Binärbäume sind) ist, und die die passende Informationsmenge zur Definition der höheren Operationen kodiert. Dies sind die so genannten linkskommutativen Binärbäume. Es stellt sich heraus, dass die (linkskommutativen) Binärbäume viele Eigenschaften der natürlichen Zahlen teilen, so z.B. die Assoziativität der Multiplikation (die Operation der 2. Stufe) und eine eindeutige Primzahlzerlegung. Dies motiviert die Frage, ob man die Erweiterungskonstruktionen der Zahlen: „natürliche Zahlen zu gebrochenen Zahlen“ (macht die Multiplikation umkehrbar) „gebrochene Zahlen zu positiven reellen Zahlen“ (macht das Potenzieren umkehrbar und erlaubt Grenzwertbildung) auch ausgehend von (linkskommutativen) Binärbäumen vornehmen kann. In der vorliegenden Arbeit wird (neben unzähligen anderen Resultaten) gezeigt, dass die Zahlenbereichserweiterung „natürliche Zahlen zu gebrochenen Zahlen“ auch analog für (linkskommutative) Binärbäume möglich ist. Das Ergebnis dieser Konstruktion sind die Divisionsbinärbäume (bzw. die gebrochenen Bäume). Letztere lassen sich unerwartet in der Form von Brüchen darstellen, sind jedoch als Verallgemeinerung der gebrochenen Zahlen sehr viel komplexer als dieser. (Das kann man live nachprüfen mit dem dafür erstellten Online-Rechner für gebrochene Bäume (auf englisch): http://math.eretrandre.org/cgi-bin/ftc/ftc.pl ) Damit wird ein Programm „baumartige Zahlen“ gestartet, indem es darum geht, auch die Erweiterung „gebrochene Zahlen zu positiven reellen Zahlen“ für die Divisionsbinärbäume (bzw. die gebrochenen Bäume) durchzuführen, wobei die höheren Operationen auf dieser Erweiterung definiert werden könnten und umkehrbar sein müssten. Ob dies wirklich möglich ist, ist derzeit unklar (neben diversen anderen direkt aus der Dissertation sich ergebenden Fragen) und eröffnet damit ein enorm umfangreiches Feld für weitere Forschungen. KW - Tetration KW - höhere Operationen KW - strukturierte Zahlen KW - Divisionsbäume KW - tetration KW - higher operations KW - structured numbers KW - division trees Y1 - 2007 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15247 ER - TY - JOUR A1 - Fischer, Florian A1 - Keller, Matthias T1 - Riesz decompositions for Schrödinger operators on graphs JF - Journal of mathematical analysis and applications N2 - 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. KW - Potential theory KW - Green's function KW - Schrödinger operator KW - Weighted KW - graph KW - Subcritical KW - Greatest harmonic minorant Y1 - 2021 U6 - https://doi.org/10.1016/j.jmaa.2020.124674 SN - 0022-247X SN - 1096-0813 VL - 495 IS - 1 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Denecke, Klaus-Dieter T1 - Partial clones JF - Asian-European journal of mathematics : AEJM N2 - 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. KW - Operation KW - term KW - formula KW - superposition of operations KW - terms and KW - formulas KW - linear term KW - linear formula KW - linear tree language KW - clone KW - partial clone KW - linear hypersubstitution KW - dht-symmetric category KW - partial KW - theory Y1 - 2020 U6 - https://doi.org/10.1142/S1793557120501612 SN - 1793-5571 SN - 1793-7183 VL - 13 IS - 8 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Bär, Christian T1 - The Faddeev-LeVerrier algorithm and the Pfaffian JF - Linear algebra and its applications N2 - 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. KW - Characteristic polynomial KW - Determinant KW - Pfaffian KW - Gauss-Bonnet-Chern KW - theorem Y1 - 2021 U6 - https://doi.org/10.1016/j.laa.2021.07.023 SN - 0024-3795 SN - 1873-1856 VL - 630 SP - 39 EP - 55 PB - Elsevier CY - New York ER - TY - JOUR A1 - Klein, Markus A1 - Rosenberger, Elke T1 - The tunneling effect for a class of difference operators JF - Reviews in Mathematical Physics N2 - 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]. KW - Semiclassical difference operator KW - tunneling KW - interaction matrix KW - asymptotic expansion KW - multi-well potential KW - Finsler distance KW - Agmon estimates Y1 - 2018 U6 - https://doi.org/10.1142/S0129055X18300029 SN - 0129-055X SN - 1793-6659 VL - 30 IS - 4 PB - World Scientific CY - Singapore ER - TY - GEN A1 - Zöller, Gert A1 - Holschneider, Matthias T1 - 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 Zöller and Matthias Holschneider” by Mathias Raschke T2 - Bulletin of the Seismological Society of America Y1 - 2018 U6 - https://doi.org/10.1785/0120170131 SN - 0037-1106 SN - 1943-3573 VL - 108 IS - 2 SP - 1029 EP - 1030 PB - Seismological Society of America CY - Albany ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt T1 - Convergence of dynamics and the Perron-Frobenius operator JF - Israel Journal of Mathematics N2 - We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system. Y1 - 2018 U6 - https://doi.org/10.1007/s11856-018-1671-7 SN - 0021-2172 SN - 1565-8511 VL - 225 IS - 1 SP - 451 EP - 463 PB - Hebrew univ magnes press CY - Jerusalem ER - TY - JOUR A1 - Liu, Shiping A1 - Münch, Florentin A1 - Peyerimhoff, Norbert T1 - Bakry-Emery curvature and diameter bounds on graphs JF - Calculus of variations and partial differential equations N2 - We prove finiteness and diameter bounds for graphs having a positive Ricci-curvature bound in the Bakry–Émery 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 für 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–Émery curvature functions of graphs, 2016). Y1 - 2018 U6 - https://doi.org/10.1007/s00526-018-1334-x SN - 0944-2669 SN - 1432-0835 VL - 57 IS - 2 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Keller, Matthias A1 - Schwarz, Michael T1 - The Kazdan-Warner equation on canonically compactifiable graphs JF - Calculus of variations and partial differential equations N2 - 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. Y1 - 2018 U6 - https://doi.org/10.1007/s00526-018-1329-7 SN - 0944-2669 SN - 1432-0835 VL - 57 IS - 2 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Lesur, Vincent A1 - Wardinski, Ingo A1 - Bärenzung, Julien A1 - Holschneider, Matthias T1 - On the frequency spectra of the core magnetic field Gauss coefficients JF - Physics of the earth and planetary interiors N2 - 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. KW - Geomagnetism KW - Core field KW - Secular variation rate of change KW - Geomagnetic jerks KW - Correlation based modelling Y1 - 2017 U6 - https://doi.org/10.1016/j.pepi.2017.05.017 SN - 0031-9201 SN - 1872-7395 VL - 276 SP - 145 EP - 158 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Gerlach, Moritz Reinhardt A1 - Glück, Jochen T1 - On a convergence theorem for semigroups of positive integral operators JF - Comptes Rendus Mathematique N2 - We give a new and very short proof of a theorem of Greiner asserting that a positive and contractive -semigroup on an -space is strongly convergent in case it has a strictly positive fixed point and contains an integral operator. Our proof is a streamlined version of a much more general approach to the asymptotic theory of positive semigroups developed recently by the authors. Under the assumptions of Greiner's theorem, this approach becomes particularly elegant and simple. We also give an outlook on several generalisations of this result. Y1 - 2017 U6 - https://doi.org/10.1016/j.crma.2017.07.017 SN - 1631-073X SN - 1778-3569 VL - 355 SP - 973 EP - 976 PB - Elsevier CY - Paris ER - TY - JOUR A1 - Brungs, Hans H. A1 - Gräter, Joachim T1 - On central extensions of SL(2, F) admitting left-orderings JF - Journal of Algebra N2 - 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. KW - Universal covering group KW - Central extensions of groups KW - Perfect groups KW - Ordered fields KW - Left-ordered groups KW - Order-preserving bijections KW - Euclidean fields Y1 - 2017 U6 - https://doi.org/10.1016/j.jalgebra.2017.05.025 SN - 0021-8693 SN - 1090-266X VL - 486 SP - 288 EP - 327 PB - Elsevier CY - San Diego ER - TY - JOUR A1 - Somogyvári, Márk A1 - Reich, Sebastian T1 - Convergence tests for transdimensional Markov chains in geoscience imaging JF - Mathematical geosciences : the official journal of the International Association for Mathematical Geosciences N2 - 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. KW - transdimensional inversion KW - MCMC modelling KW - convergence assessment Y1 - 2019 U6 - https://doi.org/10.1007/s11004-019-09811-x SN - 1874-8961 SN - 1874-8953 VL - 52 IS - 5 SP - 651 EP - 668 PB - Springer CY - Heidelberg ER - TY - JOUR A1 - Ly, Ibrahim T1 - A Cauchy problem for the Cauchy-Riemann operator JF - Afrika Matematika N2 - 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. KW - nonlinear PDI KW - Cauchy problem KW - Zaremba problem Y1 - 2020 U6 - https://doi.org/10.1007/s13370-020-00810-4 SN - 1012-9405 SN - 2190-7668 VL - 32 IS - 1-2 SP - 69 EP - 76 PB - Springer CY - Heidelberg ER - TY - INPR A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolai Nikolaevich T1 - A Radó theorem for p-harmonic functions N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 3 KW - Quasilinear equations KW - removable sets KW - p-Laplace Operator Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-71492 SN - 2193-6943 VL - 4 IS - 3 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - INPR A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolai Nikolaevich T1 - Asymptotic expansions at nonsymmetric cuspidal points N2 - 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. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 7 KW - the Dirichlet problem KW - singular point KW - asymptotic expansion Y1 - 2015 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-78199 SN - 2193-6943 VL - 4 IS - 7 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Malass, Ihsane A1 - Tarkhanov, Nikolaj Nikolaevič T1 - A perturbation of the de Rham complex T1 - Возмущение комплекса де Рама JF - Journal of Siberian Federal University : Mathematics & Physics JF - Žurnal Sibirskogo Federalʹnogo Universiteta : Matematika i fizika N2 - 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. N2 - Рассмотрим возмущение комплекса де Рама на компактном многообразии с краем. Это возмущение выходит за рамки комплексов, и поэтому когомологии к нему не относятся. С другой стороны, его кривизна "мала", поэтому существует естественный способ ввести характеристику Эйлера и разработать теорию Лефшеца для возмущения. Данная работа предназначена для попытки разработать теорию когомологий для произвольных последовательностей линейных отображений. KW - de Rham complex KW - cohomology KW - Hodge theory KW - Neumann problem KW - комплекс де Рама KW - когомологии KW - теория Ходжа KW - проблема Неймана Y1 - 2020 U6 - https://doi.org/10.17516/1997-1397-2020-13-5-519-532 SN - 1997-1397 SN - 2313-6022 VL - 13 IS - 5 SP - 519 EP - 532 PB - Siberian Federal University CY - Krasnojarsk ER - TY - THES A1 - Jakobs, Friedrich T1 - Dubrovin–rings and their connection to Hughes–free skew fields of fractions T1 - Dubrovinringe und ihre Verbindung zu Hughes-freien Quotientenschiefkörpern N2 - 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. Gräter 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]). N2 - In dieser Arbeit beschäftigen wir uns mit Quotientenschiefkörpern von verschränkten Produkten F [G; η, σ], wobei G eine Gruppe und F ein Schiefkörper ist. Eine Methode Gruppen in Schiefkörper einzubetten stammt von A. I. Mal’tsev und B. H. Neumann. Ist G eine beidseitig geordnete Gruppe, so lässt sich die Menge der formalen Potenzreihen F ((G)) über F in G mit wohlgeordnetem Träger als Schiefkörper interpretieren. In diesen lässt sich jedes verschränkte Produkt F [G; η, σ] einbetten. Möchte man die Klasse der einzubettenden Gruppen erweitern, so bieten sich links–geordnete Gruppen an. In diesem Fall hat F ((G)) keine natürliche Ringstruktur, aber man kann nutzen, dass F ((G)) ein rechter F–Vektorraum ist und seinen Endomorphismenring untersuchen. Jedes Verschränkte Produkt F [G; η, σ] lässt sich derart in den Endomorphismenring einbetten, dass die zugehörigen von Null verschiedenen Endomorphismen Automorphismen sind. Der rationale Abschluss von F [G; η, σ] in End(F ((G))), den wir Dubrovinring von F [G; η, σ] nennen, ist somit ein potentieller Quotientenschiefkörper von F [G; η, σ]. Neben der Existenz von Quotientenschiefkörpern ist deren Eindeutigkeit (bis auf Isomorphie) von Interesse. Im Gegensatz zum kommutativen Fall sind Quotientenschiefkörper im Allgemeinen nicht eindeutig. So lässt sich beispielsweise die freie Gruppe mit zwei Erzeugenden in nicht–isomorphe Quotientenschiefkörper einbetten. Eine große Klasse an Ringen, die eindeutige Quotientenschiefkörper besitzen, sind Ore–Bereiche. Vermutlich lässt sich diese Klasse nicht erweitern, ohne zusätzliche Eigenschaften der Quotientenschiefkörper zu verlangen. Eine solche Eigenschaft, im Folgenden Hughes–frei genannt, wurde von I. Hughes vorgeschlagen. Er konnte beweisen, dass Hughes–freie Quotientenschiefkörper eindeutig sind, wenn sie existieren. In dieser Arbeit verbinden wir die Ideen von I. Hughes und N. I. Dubrovin. Wir zeigen, dass die Elemente von Hughes–freien Quotientenschiefkörpern als formale schiefe Laurent–Reihen dargestellt werden können und dass diese Darstellungen in gewisser Weise eindeutig sind. Dieses Ergebnis nutzen wir um zu beweisen, dass die Aussagen “F [G; η, σ] besitzt einen Hughes–freien Quotientenschiefkörper” und “Der Dubrovinring von F [G; η, σ] ist ein Schiefkörper” äquivalent sind, wenn G eine links–geordnete Gruppe von Conrad–Typ mit maximalem Rang ist. Wir stellen den nötigen Begriffsapparat zur Verfügung. Dieser basiert vorwiegend auf den Arbeiten von N. I. Dubrovin und umfasst spezielle Eigenschaften der Endomorphismen von F ((G)) sowie die Komplexität von Elementen in rationalen Abschlüssen. Des Weiteren gehen wir auf links–geordnete Gruppen von Conrad–Typ ein und untersuchen ihren Zusammenhang mit lokal indizierbaren Gruppen, die eine grundlegende Rolle für Hughes–freie Quotientenschiefkörper spielen. Wir werden zeigen können, dass Dubrovinringe, die Schiefkörper sind, stark Hughes–freie Quotientenschiefkörper sind, was die offene Frage beantwortet, ob Hughes–freie Quotientenschiefkörper stark Hughes–frei sind. Außerdem werden wir einen alternativen Beweis der Eindeutigkeit von Hughes–freien Quotientenschiefkörpern präsentieren und die Fortsetzbarkeit von Automorphismen eines verschränkten Produkts auf Hughes–freie Quotientenschiefkörper untersuchen. KW - Hughes-free KW - Dubrovinring KW - left ordered groups KW - Conradian ordered groups KW - skew field of fraction KW - locally indicable KW - series representation KW - strongly Hughes-free KW - Hughes-frei KW - Dubrovinring KW - linksgeordnete Gruppen KW - geordnete Gruppen von Conrad-Typ KW - Quotientenschiefkörper KW - lokal indizierbar KW - Reihendarstellungen KW - stark Hughes-frei Y1 - 2019 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-435561 ER - TY - GEN A1 - Benini, Marco A1 - Schenkel, Alexander T1 - Quantum field theories on categories fibered in groupoids T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - We introduce an abstract concept of quantum field theory on categories fibered in groupoids over the category of spacetimes. This provides us with a general and flexible framework to study quantum field theories defined on spacetimes with extra geometric structures such as bundles, connections and spin structures. Using right Kan extensions, we can assign to any such theory an ordinary quantum field theory defined on the category of spacetimes and we shall clarify under which conditions it satisfies the axioms of locally covariant quantum field theory. The same constructions can be performed in a homotopy theoretic framework by using homotopy right Kan extensions, which allows us to obtain first toy-models of homotopical quantum field theories resembling some aspects of gauge theories. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 895 KW - C-asterisk-algebra KW - observables KW - covariance KW - locality Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-431541 SN - 1866-8372 IS - 895 ER - TY - GEN A1 - Karpuz, Eylem Guzel A1 - Çevik, Ahmet Sinan A1 - Koppitz, Jörg A1 - Cangul, Ismail Naci T1 - Some fixed-point results on (generalized) Bruck-Reilly ∗-extensions of monoids T2 - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe N2 - In this paper, we determine necessary and sufficient conditions for Bruck-Reilly and generalized Bruck-Reilly ∗-extensions of arbitrary monoids to be regular, coregular and strongly π-inverse. These semigroup classes have applications in various field of mathematics, such as matrix theory, discrete mathematics and p-adic analysis (especially in operator theory). In addition, while regularity and coregularity have so many applications in the meaning of boundaries (again in operator theory), inverse monoids and Bruck-Reilly extensions contain a mixture fixed-point results of algebra, topology and geometry within the purposes of this journal. T3 - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 942 KW - Bruck-Reilly extension KW - generalized Bruck-Reilly ∗-extension KW - π -inverse monoid KW - regular monoid Y1 - 2020 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-432701 SN - 1866-8372 IS - 942 ER - TY - JOUR A1 - Al-Saedy, Ammar Jaffar Muhesin A1 - Tarchanov, Nikolaj Nikolaevič T1 - A degree theory for Lagrangian boundary value problems JF - Žurnal Sibirskogo Federalʹnogo Universiteta = Journal of Siberian Federal University; mathematics & physics N2 - We study those nonlinear partial differential equations which appear as Euler-Lagrange equations of variational problems. On defining weak boundary values of solutions to such equations we initiate the theory of Lagrangian boundary value problems in spaces of appropriate smoothness. We also analyse if the concept of mapping degree of current importance applies to Lagrangian problems. N2 - Мы изучаем те нелинейные уравнения с частными производными, которые возникают как уравнения Эйлера-Лагранжа вариационных задач. Определяя слабые граничные значения решений таких уравнений, мы инициируем теорию лагранжевых краевых задач в функциональных пространствах подходящей гладкости. Мы также анализируем, применяется ли современная концепция степени отображения к лагранжевым проблемам. KW - nonlinear equations KW - Lagrangian system KW - weak boundary values KW - quasilinear Fredholm operators KW - mapping degree Y1 - 2020 U6 - https://doi.org/10.17516/1997-1397-2020-13-1-5-25 SN - 1997-1397 SN - 2313-6022 VL - 13 IS - 1 SP - 5 EP - 25 PB - Sibirskij Federalʹnyj Universitet CY - Krasnojarsk ER - TY - JOUR A1 - Clavier, Pierre J. A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - An algebraic formulation of the locality principle in renormalisation JF - European Journal of Mathematics N2 - We study the mathematical structure underlying the concept of locality which lies at the heart of classical and quantum field theory, and develop a machinery used to preserve locality during the renormalisation procedure. Viewing renormalisation in the framework of Connes and Kreimer as the algebraic Birkhoff factorisation of characters on a Hopf algebra with values in a Rota-Baxter algebra, we build locality variants of these algebraic structures, leading to a locality variant of the algebraic Birkhoff factorisation. This provides an algebraic formulation of the conservation of locality while renormalising. As an application in the context of the Euler-Maclaurin formula on lattice cones, we renormalise the exponential generating function which sums over the lattice points in a lattice cone. As a consequence, for a suitable multivariate regularisation, renormalisation from the algebraic Birkhoff factorisation amounts to composition by a projection onto holomorphic multivariate germs. KW - Locality KW - Renormalisation KW - Algebraic Birkhoff factorisation KW - Partial algebra KW - Hopf algebra KW - Rota-Baxter algebra KW - Multivariate meromorphic functions KW - Lattice cones Y1 - 2019 U6 - https://doi.org/10.1007/s40879-018-0255-8 SN - 2199-675X SN - 2199-6768 VL - 5 IS - 2 SP - 356 EP - 394 PB - Springer CY - Cham ER - TY - CHAP A1 - Clavier, Pierre J. A1 - Guo, Li A1 - Paycha, Sylvie A1 - Zhang, Bin T1 - Renormalisation and locality BT - branched zeta values T2 - Algebraic Combinatorics, Resurgence, Moulds and Applications (CARMA) Volume 2 Y1 - 2020 SN - 978-3-03719-205-4 print SN - 978-3-03719-705-9 online U6 - https://doi.org/10.4171/205 SP - 85 EP - 132 PB - European Mathematical Society Publishing House CY - Zürich ER - TY - CHAP A1 - Audin, Michèle A1 - Ducourtioux, Catherine A1 - Ouédraogo, Françoise A1 - Schulz, René A1 - Delgado, Julio A1 - Ruzhansky, Michael A1 - Lebeau, Gilles ED - Paycha, Sylvie T1 - Integral Fourier operators T1 - Fourier Integraloperatoren BT - proceedings of a summer school, Ouagadougou 14–25 September 2015 BT - Akten einer Sommerschule, Ouagadougou, Burkina Faso, 14-26. September 2015 N2 - This volume of contributions based on lectures delivered at a school on Fourier Integral Operators held in Ouagadougou, Burkina Faso, 14–26 September 2015, provides an introduction to Fourier Integral Operators (FIO) for a readership of Master and PhD students as well as any interested layperson. Considering the wide spectrum of their applications and the richness of the mathematical tools they involve, FIOs lie the cross-road of many a field. This volume offers the necessary background, whether analytic or geometric, to get acquainted with FIOs, complemented by more advanced material presenting various aspects of active research in that area. N2 - Dieser Band basiert auf Vorlesungen, die in einer Schule über Fourier Integraloperatoren in Ouagadougou, Burkina Faso, 14. - 26. September 2015 gehalten wurden. Es bietet eine Einführung in die Fourier Integraloperatoren (FIO) und richtet sich sowohl an Masterstudierende und Promovenden als auch an interessierte Laien. Aufgrund der Breite des Spektrums ihrer Anwendungen und der Vielfalt der mathematischen Werkzeuge, die sie ins Spiel bringen, liegen FIO an der Grenze zwischen mehreren Gebieten. Dieses Band bietet sowohl die analytisch und geometrisch nötigen Kenntnisse, um sich mit dem Begriff der FIO vertraut zu machen als auch fortgeschrittenes Material für einen Einblick in verschiedene Aspekte der gegenwärtigen Forschung dieses Gebietes an. T3 - Lectures in pure and applied mathematics - 3 KW - pseudodifferentiale Operatoren KW - Fourier Integraloperatoren KW - Lagrange Distributionen KW - microlokale Analysis KW - pseudodifferential operators KW - integral Fourier operators KW - Lagrangian submanifolds KW - microlocal analysis Y1 - 2017 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-402657 SN - 978-3-86956-413-5 SN - 2199-4951 SN - 2199-496X PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Chelkh, W. A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolai T1 - A remark on the Laplace transform JF - Siberian Mathematical Journal N2 - The study of the Cauchy problem for solutions of the heat equation in a cylindrical domain with data on the lateral surface by the Fourier method raises the problem of calculating the inverse Laplace transform of the entire function cos root z. This problem has no solution in the standard theory of the Laplace transform. We give an explicit formula for the inverse Laplace transform of cos root z using the theory of analytic functionals. This solution suits well to efficiently develop the regularization of solutions to Cauchy problems for parabolic equations with data on noncharacteristic surfaces. KW - Fourier-Laplace transform KW - distributions with one-sided support KW - holomorphic function KW - analytic functional Y1 - 2020 U6 - https://doi.org/10.1134/S0037446620040151 SN - 0037-4466 SN - 1573-9260 VL - 61 IS - 4 SP - 755 EP - 762 PB - Consultants Bureau, Springer CY - New York ER - TY - JOUR A1 - Kaya, Adem A1 - Freitag, Melina A. T1 - Conditioning analysis for discrete Helmholtz problems JF - Computers and mathematics with applications : an international journal N2 - In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions. KW - Helmholtz problem KW - Condition number KW - Ill-conditioning KW - Indefinite KW - matrices Y1 - 2022 U6 - https://doi.org/10.1016/j.camwa.2022.05.016 SN - 0898-1221 SN - 1873-7668 VL - 118 SP - 171 EP - 182 PB - Elsevier Science CY - Amsterdam ER - TY - JOUR A1 - Keller, Matthias A1 - Schwarz, Michael T1 - Courant’s nodal domain theorem for positivity preserving forms JF - Journal of spectral theory N2 - We introduce a notion of nodal domains for positivity preserving forms. This notion generalizes the classical ones for Laplacians on domains and on graphs. We prove the Courant nodal domain theorem in this generalized setting using purely analytical methods. KW - Nodal domain KW - eigenfunction KW - Dirichlet form KW - compact resolvent Y1 - 2020 U6 - https://doi.org/10.4171/JST/292 SN - 1664-039X SN - 1664-0403 VL - 10 IS - 1 SP - 271 EP - 309 PB - EMS Publishing House CY - Zürich ER - TY - JOUR A1 - Kolasinski, Slawomir A1 - Menne, Ulrich T1 - Decay rates for the quadratic and super-quadratic tilt-excess of integral varifolds JF - Nonlinear Differential Equations and Applications NoDEA N2 - This paper concerns integral varifolds of arbitrary dimension in an open subset of Euclidean space satisfying integrability conditions on their first variation. Firstly, the study of pointwise power decay rates almost everywhere of the quadratic tilt-excess is completed by establishing the precise decay rate for two-dimensional integral varifolds of locally bounded first variation. In order to obtain the exact decay rate, a coercive estimate involving a height-excess quantity measured in Orlicz spaces is established. Moreover, counter-examples to pointwise power decay rates almost everywhere of the super-quadratic tilt-excess are obtained. These examples are optimal in terms of the dimension of the varifold and the exponent of the integrability condition in most cases, for example if the varifold is not two-dimensional. These examples also demonstrate that within the scale of Lebesgue spaces no local higher integrability of the second fundamental form, of an at least two-dimensional curvature varifold, may be deduced from boundedness of its generalised mean curvature vector. Amongst the tools are Cartesian products of curvature varifolds. KW - Integral varifold KW - First variation KW - Generalised mean curvature vector KW - Quadratic tilt-excess KW - Super-quadratic tilt-excess KW - Orlicz space height-excess KW - Curvature varifold KW - Second fundamental form KW - Cartesian product of varifolds Y1 - 2017 U6 - https://doi.org/10.1007/s00030-017-0436-z SN - 1021-9722 SN - 1420-9004 VL - 24 PB - Springer CY - Basel ER - TY - JOUR A1 - Ly, Ibrahim A1 - Tarkhanov, Nikolaj Nikolaevič T1 - Asymptotic expansions at nonsymmetric cuspidal points JF - Mathematical notes N2 - We study the asymptotics of solutions to the Dirichlet problem in a domain X subset of R3 whose boundary contains a singular point O. In a small neighborhood of this point, the domain has the form {z > root x(2) + y(4)}, i.e., the origin is a nonsymmetric conical point at the boundary. So far, the behavior of solutions to elliptic boundary-value problems has not been studied sufficiently in the case of nonsymmetric singular points. This problem was posed by V.A. Kondrat'ev in 2000. We establish a complete asymptotic expansion of solutions near the singular point. KW - Dirichlet problem KW - singular points KW - asymptotic expansions Y1 - 2020 U6 - https://doi.org/10.1134/S0001434620070238 SN - 0001-4346 SN - 1573-8876 VL - 108 IS - 1-2 SP - 219 EP - 228 PB - Springer Science CY - New York ER -