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;<f) be a fence, also called a zigzag poset. As usual, we denote by In the symmetric inverse semigroup on Xn. We say that a transformation α∈In is fence-preserving if x<fy implies that xα<fyα, for all x,y in the domain of α. In this paper, we study the semigroup PFIn of all partial fence-preserving injections of Xn and its subsemigroup IFn={α∈PFIn:α−1∈PFIn}. Clearly, IFn is an inverse semigroup and contains all regular elements of PFIn. We characterize the Green’s relations for the semigroup IFn. Further, we prove that the semigroup IFn is generated by its elements with rank ≥n−2. Moreover, for n∈2N, we find the least generating set and calculate the rank of IFn.
KW  - Finite transformation semigroup
KW  - fence-preserving transformations
KW  - inverse semigroup
KW  - Green´s Relations
KW  - generators
KW  - rank
Y1  - 2017
U6  - https://doi.org/10.1142/S0219498817502231
SN  - 0219-4988
SN  - 1793-6829
VL  - 16
IS  - 12
PB  - World Scientific
CY  - Singapore
ER  - 
TY  - JOUR
A1  - Salamat, Mona
A1  - Zöller, Gert
A1  - Zare, Mehdi
A1  - Amini, Mortaza
T1  - The maximum expected earthquake magnitudes in different future time intervals of six seismotectonic zones of Iran and its surroundings
JF  - Journal of seismology
N2  - One of the crucial components in seismic hazard analysis is the estimation of the maximum earthquake magnitude and associated uncertainty. In the present study, the uncertainty related to the maximum expected magnitude mu is determined in terms of confidence intervals for an imposed level of confidence. Previous work by Salamat et al. (Pure Appl Geophys 174:763-777, 2017) shows the divergence of the confidence interval of the maximum possible magnitude m(max) for high levels of confidence in six seismotectonic zones of Iran. In this work, the maximum expected earthquake magnitude mu is calculated in a predefined finite time interval and imposed level of confidence. For this, we use a conceptual model based on a doubly truncated Gutenberg-Richter law for magnitudes with constant b-value and calculate the posterior distribution of mu for the time interval T-f in future. We assume a stationary Poisson process in time and a Gutenberg-Richter relation for magnitudes. The upper bound of the magnitude confidence interval is calculated for different time intervals of 30, 50, and 100 years and imposed levels of confidence alpha = 0.5, 0.1, 0.05, and 0.01. The posterior distribution of waiting times T-f to the next earthquake with a given magnitude equal to 6.5, 7.0, and7.5 are calculated in each zone. In order to find the influence of declustering, we use the original and declustered version of the catalog. The earthquake catalog of the territory of Iran and surroundings are subdivided into six seismotectonic zones Alborz, Azerbaijan, Central Iran, Zagros, Kopet Dagh, and Makran. We assume the maximum possible magnitude m(max) = 8.5 and calculate the upper bound of the confidence interval of mu in each zone. The results indicate that for short time intervals equal to 30 and 50 years and imposed levels of confidence 1 - alpha = 0.95 and 0.90, the probability distribution of mu is around mu = 7.16-8.23 in all seismic zones.
KW  - Maximum expected earthquake magnitude
KW  - Future time interval
KW  - Level of confidence
KW  - Iran
Y1  - 2018
U6  - https://doi.org/10.1007/s10950-018-9780-7
SN  - 1383-4649
SN  - 1573-157X
VL  - 22
IS  - 6
SP  - 1485
EP  - 1498
PB  - Springer
CY  - Dordrecht
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  - 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  - 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  - 
TY  - JOUR
A1  - Clavier, Pierre J.
T1  - Double shuffle relations for arborified zeta values
JF  - Journal of algebra
N2  - Arborified zeta values are defined as iterated series and integrals using the universal properties of rooted trees. This approach allows to study their convergence domain and to relate them to multiple zeta values. Generalisations to rooted trees of the stuffle and shuffle products are defined and studied. It is further shown that arborified zeta values are algebra morphisms for these new products on trees.
KW  - Rooted trees
KW  - Multiple zeta values
KW  - Shuffle products
KW  - Rota-Baxter
KW  - algebras
Y1  - 2020
U6  - https://doi.org/10.1016/j.jalgebra.2019.10.015
SN  - 0021-8693
SN  - 1090-266X
VL  - 543
SP  - 111
EP  - 155
PB  - Elsevier
CY  - San Diego
ER  - 
TY  - GEN
A1  - Serth, Sebastian
A1  - Podlesny, Nikolai
A1  - Bornstein, Marvin
A1  - Lindemann, Jan
A1  - Latt, Johanna
A1  - Selke, Jan
A1  - Schlosser, Rainer
A1  - Boissier, Martin
A1  - Uflacker, Matthias
T1  - An interactive platform to simulate dynamic pricing competition on online marketplaces
T2  - 2017 IEEE 21st International Enterprise Distributed Object Computing Conference (EDOC)
N2  - E-commerce marketplaces are highly dynamic with constant competition. While this competition is challenging for many merchants, it also provides plenty of opportunities, e.g., by allowing them to automatically adjust prices in order to react to changing market situations. For practitioners however, testing automated pricing strategies is time-consuming and potentially hazardously when done in production. Researchers, on the other side, struggle to study how pricing strategies interact under heavy competition. As a consequence, we built an open continuous time framework to simulate dynamic pricing competition called Price Wars. The microservice-based architecture provides a scalable platform for large competitions with dozens of merchants and a large random stream of consumers. Our platform stores each event in a distributed log. This allows to provide different performance measures enabling users to compare profit and revenue of various repricing strategies in real-time. For researchers, price trajectories are shown which ease evaluating mutual price reactions of competing strategies. Furthermore, merchants can access historical marketplace data and apply machine learning. By providing a set of customizable, artificial merchants, users can easily simulate both simple rule-based strategies as well as sophisticated data-driven strategies using demand learning to optimize their pricing strategies.
Y1  - 2017
SN  - 978-1-5090-3045-3
U6  - https://doi.org/10.1109/EDOC.2017.17
SN  - 2325-6354
SP  - 61
EP  - 66
PB  - Institute of Electrical and Electronics Engineers
CY  - New York
ER  - 
TY  - JOUR
A1  - Taghvaei, Amirhossein
A1  - de Wiljes, Jana
A1  - Mehta, Prashant G.
A1  - Reich, Sebastian
T1  - Kalman filter and its modern extensions for the continuous-time nonlinear filtering problem
JF  - Journal of dynamic systems measurement and control
N2  - This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman-Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman-Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman-Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.
Y1  - 2017
U6  - https://doi.org/10.1115/1.4037780
SN  - 0022-0434
SN  - 1528-9028
VL  - 140
IS  - 3
PB  - ASME
CY  - New York
ER  -