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  - 
TY  - JOUR
A1  - de Wiljes, Jana
A1  - Reich, Sebastian
A1  - Stannat, Wilhelm
T1  - Long-Time stability and accuracy of the ensemble Kalman-Bucy Filter for fully observed processes and small measurement noise
JF  - SIAM Journal on Applied Dynamical Systems
N2  - The ensemble Kalman filter has become a popular data assimilation technique in the geosciences. However, little is known theoretically about its long term stability and accuracy. In this paper, we investigate the behavior of an ensemble Kalman-Bucy filter applied to continuous-time filtering problems. We derive mean field limiting equations as the ensemble size goes to infinity as well as uniform-in-time accuracy and stability results for finite ensemble sizes. The later results require that the process is fully observed and that the measurement noise is small. We also demonstrate that our ensemble Kalman-Bucy filter is consistent with the classic Kalman-Bucy filter for linear systems and Gaussian processes. We finally verify our theoretical findings for the Lorenz-63 system.
KW  - data assimilation
KW  - Kalman Bucy filter
KW  - ensemble Kalman filter
KW  - stability
KW  - accuracy
KW  - asymptotic behavior
Y1  - 2018
U6  - https://doi.org/10.1137/17M1119056
SN  - 1536-0040
VL  - 17
IS  - 2
SP  - 1152
EP  - 1181
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Hamm, Maximilian
A1  - Pelivan, Ivanka
A1  - Grott, Matthias
A1  - de Wiljes, Jana
T1  - Thermophysical modelling and parameter estimation of small solar system bodies via data assimilation
JF  - Monthly notices of the Royal Astronomical Society
N2  - Deriving thermophysical properties such as thermal inertia from thermal infrared observations provides useful insights into the structure of the surface material on planetary bodies. The estimation of these properties is usually done by fitting temperature variations calculated by thermophysical models to infrared observations. For multiple free model parameters, traditional methods such as least-squares fitting or Markov chain Monte Carlo methods become computationally too expensive. Consequently, the simultaneous estimation of several thermophysical parameters, together with their corresponding uncertainties and correlations, is often not computationally feasible and the analysis is usually reduced to fitting one or two parameters. Data assimilation (DA) methods have been shown to be robust while sufficiently accurate and computationally affordable even for a large number of parameters. This paper will introduce a standard sequential DA method, the ensemble square root filter, for thermophysical modelling of asteroid surfaces. This method is used to re-analyse infrared observations of the MARA instrument, which measured the diurnal temperature variation of a single boulder on the surface of near-Earth asteroid (162173) Ryugu. The thermal inertia is estimated to be 295 +/- 18 Jm(-2) K-1 s(-1/2), while all five free parameters of the initial analysis are varied and estimated simultaneously. Based on this thermal inertia estimate the thermal conductivity of the boulder is estimated to be between 0.07 and 0.12,Wm(-1) K-1 and the porosity to be between 0.30 and 0.52. For the first time in thermophysical parameter derivation, correlations and uncertainties of all free model parameters are incorporated in the estimation procedure that is more than 5000 times more efficient than a comparable parameter sweep.
KW  - radiation mechanisms: thermal
KW  - methods: data analysis
KW  - methods
KW  - statistical
KW  - minor planets, asteroids: individual: (162173) Ryugu
Y1  - 2020
U6  - https://doi.org/10.1093/mnras/staa1755
SN  - 0035-8711
SN  - 1365-2966
VL  - 496
IS  - 3
SP  - 2776
EP  - 2785
PB  - Oxford Univ. Press
CY  - Oxford
ER  - 
TY  - JOUR
A1  - Acevedo, Walter
A1  - De Wiljes, Jana
A1  - Reich, Sebastian
T1  - Second-order accurate ensemble transform particle filters
JF  - SIAM journal on scientific computing
N2  - Particle filters (also called sequential Monte Carlo methods) are widely used for state and parameter estimation problems in the context of nonlinear evolution equations. The recently proposed ensemble transform particle filter (ETPF) [S. Reich, SIAM T. Sci. Comput., 35, (2013), pp. A2013-A2014[ replaces the resampling step of a standard particle filter by a linear transformation which allows for a hybridization of particle filters with ensemble Kalman filters and renders the resulting hybrid filters applicable to spatially extended systems. However, the linear transformation step is computationally expensive and leads to an underestimation of the ensemble spread for small and moderate ensemble sizes. Here we address both of these shortcomings by developing second order accurate extensions of the ETPF. These extensions allow one in particular to replace the exact solution of a linear transport problem by its Sinkhorn approximation. It is also demonstrated that the nonlinear ensemble transform filter arises as a special case of our general framework. We illustrate the performance of the second-order accurate filters for the chaotic Lorenz-63 and Lorenz-96 models and a dynamic scene-viewing model. The numerical results for the Lorenz-63 and Lorenz-96 models demonstrate that significant accuracy improvements can be achieved in comparison to a standard ensemble Kalman filter and the ETPF for small to moderate ensemble sizes. The numerical results for the scene-viewing model reveal, on the other hand, that second-order corrections can lead to statistically inconsistent samples from the posterior parameter distribution.
KW  - Bayesian inference
KW  - data assimilation
KW  - particle filter
KW  - ensemble Kalman filter
KW  - Sinkhorn approximation
Y1  - 2017
U6  - https://doi.org/10.1137/16M1095184
SN  - 1064-8275
SN  - 1095-7197
SN  - 2168-3417
VL  - 39
IS  - 5
SP  - A1834
EP  - A1850
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - THES
A1  - Rafler, Mathias
T1  - Gaussian loop- and Pólya processes : a point process approach
T1  - Gaußsche Loop- and Pólya-Prozesse : ein Zugang via Punktprozessen
N2  - This thesis considers on the one hand the construction of point processes via conditional intensities, motivated by the partial Integration of the Campbell measure of a point process. Under certain assumptions on the intensity the existence of such a point process is shown. A fundamental example turns out to be the Pólya sum process, whose conditional intensity is a generalisation of the Pólya urn dynamics. A Cox process representation for that point process is shown. A further process considered is a Poisson process of Gaussian loops, which represents a noninteracting particle system derived from the discussion of indistinguishable particles. Both processes are used to define particle systems locally, for which thermodynamic limits are determined.
N2  - Betrachtet wird zum einen die Konstruktion von Punktprozessen mittels bedingter Intensitäten, motivert durch die partielle Integration des Campbell-Maßes eines Punktprozesses, die gerade bedingte Intensitäten liefert. Unter bestimmten Annahmen an die Intensitäten wird gezeigt, dass ein solcher Punktprozess existiert. Als ein fundamentaler Vertreter stellt sich der Pólyasche Summenprozess heraus, aus einer Verallgemeinerung der Dynamik der Pólyaschen Urne hervorgeht. Fuer ihn werden u.a. eine Darstellung als Cox-Prozess gezeigt. Mit einem Poissonprozess von Gaußschen Loops wird ein nicht wechselwirkendes Teilchensystem betrachtet, das aus der Diskussion von Systemen ununterscheidbarer Teilchen abgeleitet ist. Mit beiden Prozessen werden jeweils lokal Teilchensysteme konstuiert, fuer die die thermodynamischen Limiten identifiziert werden.
KW  - Punktprozesse
KW  - partielle Integration
KW  - Gaußsche Loopprozess
KW  - Papangelou-Prozess
KW  - Polyascher Prozess
KW  - Point Processes
KW  - Partial Integration
KW  - Gaussian Loop Processes
KW  - Papangelou Process
KW  - Polya Process
Y1  - 2009
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-38706
SN  - 978-3-86956-029-8
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Méléard, Sylvie
A1  - Roelly, Sylvie
T1  - Evolutive two-level population process and large population approximations
N2  - We are interested in modeling the Darwinian evolution of a population described by two levels of biological parameters: individuals characterized by an heritable phenotypic trait submitted to mutation and natural selection and cells in these individuals influencing their ability to consume resources and to reproduce. Our models are rooted in the microscopic description of a random (discrete) population of individuals characterized by one or several adaptive traits and cells characterized by their type. The population is modeled as a stochastic point process whose generator captures the probabilistic dynamics over continuous time of birth, mutation and death for individuals and birth and death for cells. The interaction between individuals (resp. between cells) is described by a competition between individual traits (resp. between cell types). We are looking for tractable large population approximations. By combining various scalings on population size, birth and death rates and mutation step, the single microscopic model is shown to lead to contrasting nonlinear macroscopic limits of different nature: deterministic approximations, in the form of ordinary, integro- or partial differential equations, or probabilistic ones, like stochastic partial differential equations or superprocesses.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 8 
KW  - Two-level interacting process
KW  - birth-death-mutation-competition point process
KW  - non-linear integro-differential equations
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-64604
SN  - 2193-6943
ER  - 
TY  - THES
A1  - Ludewig, Matthias
T1  - Path integrals on manifolds with boundary and their asymptotic expansions
T1  - Pfadintegrale auf Mannigfaltigkeiten mit Rand und ihre asymptotischen Entwicklungen
N2  - It is "scientific folklore" coming from physical heuristics that solutions to the heat equation on a Riemannian manifold can be represented by a path integral. However, the problem with such path integrals is that they are notoriously ill-defined. One way to make them rigorous (which is often applied in physics) is finite-dimensional approximation, or time-slicing approximation: Given a fine partition of the time interval into small subintervals, one restricts the integration domain to paths that are geodesic on each subinterval of the partition. These finite-dimensional integrals are well-defined, and the (infinite-dimensional) path integral then is defined as the limit of these (suitably normalized) integrals, as the mesh of the partition tends to zero.
In this thesis, we show that indeed, solutions to the heat equation on a general compact Riemannian manifold with boundary are given by such time-slicing path integrals. Here we consider the heat equation for general Laplace type operators, acting on sections of a vector bundle. We also obtain similar results for the heat kernel, although in this case, one has to restrict to metrics satisfying a certain smoothness condition at the boundary. One of the most important manipulations one would like to do with path integrals is taking their asymptotic expansions; in the case of the heat kernel, this is the short time asymptotic expansion. In order to use time-slicing approximation here, one needs the approximation to be uniform in the time parameter. We show that this is possible by giving strong error estimates.
Finally, we apply these results to obtain short time asymptotic expansions of the heat kernel also in degenerate cases (i.e. at the cut locus). Furthermore, our results allow to relate the asymptotic expansion of the heat kernel to a formal asymptotic expansion of the infinite-dimensional path integral, which gives relations between geometric quantities on the manifold and on the loop space. In particular, we show that the lowest order term in the asymptotic expansion of the heat kernel is essentially given by the Fredholm determinant of the Hessian of the energy functional. We also investigate how this relates to the zeta-regularized determinant of the Jacobi operator along minimizing geodesics.
N2  - Es ist "wissenschaftliche Folklore", abgeleitet von der physikalischen Anschauung, dass Lösungen der Wärmeleitungsgleichung auf einer riemannschen Mannigfaltigkeit als Pfadintegrale dargestellt werden können. Das Problem mit Pfadintegralen ist allerdings, dass schon deren Definition Mathematiker vor gewisse Probleme stellt. Eine Möglichkeit, Pfadintegrale rigoros zu definieren ist endlich-dimensionale Approximation, oder time-slicing-Approximation: Für eine gegebene Unterteilung des Zeitintervals in kleine Teilintervalle schränkt man den Integrationsbereich auf diejenigen Pfade ein, die auf jedem Teilintervall geodätisch sind. Diese endlichdimensionalen Integrale sind wohldefiniert, und man definiert das (unendlichdimensionale) Pfadintegral als den Limes dieser (passend normierten) Integrale, wenn die Feinheit der Unterteilung gegen Null geht.
In dieser Arbeit wird gezeigt, dass Lösungen der Wärmeleitungsgleichung auf einer allgemeinen riemannschen Mannigfaltigkeit tatsächlich durch eine solche endlichdimensionale Approximation gegeben sind. Hierbei betrachten wir die Wärmeleitungsgleichung für allgemeine Operatoren von Laplace-Typ, die auf Schnitten in Vektorbündeln wirken. Wir zeigen auch ähnliche Resultate für den Wärmekern, wobei wir uns allerdings auf Metriken einschränken müssen, die eine gewisse Glattheitsbedingung am Rand erfüllen.
Eine der wichtigsten Manipulationen, die man an Pfadintegralen vornehmen möchte, ist das Bilden ihrer asymptotischen Entwicklungen; in Falle des Wärmekerns ist dies die Kurzzeitasymptotik. Um die endlich-dimensionale Approximation hier nutzen zu können, ist es nötig, dass die Approximation uniform im Zeitparameter ist. Dies kann in der Tat erreicht werden; zu diesem Zweck geben wir starke Fehlerabschätzungen an.
Schließlich wenden wir diese Resultate an, um die Kurzzeitasymptotik des Wärmekerns (auch im degenerierten Fall, d.h. am Schittort) herzuleiten. Unsere Resultate machen es außerdem möglich, die asymptotische Entwicklung des Wärmekerns mit einer formalen asymptotischen Entwicklung der unendlichdimensionalen Pfadintegrale in Verbindung zu bringen. Auf diese Weise erhält man Beziehungen zwischen geometrischen Größen der zugrundeliegenden Mannigfaltigkeit und solchen des Pfadraumes. Insbesondere zeigen wir, dass der Term niedrigster Ordnung in der asymptotischen Entwicklung des Wärmekerns im Wesentlichen durch die Fredholm-Determinante der Hesseschen des Energie-Funktionals gegeben ist. Weiterhin untersuchen wir die Verbindung zur zeta-regularisierten Determinante des Jakobi-Operators entlang von minimierenden Geodätischen.
KW  - heat equation
KW  - heat kernel
KW  - path integral
KW  - determinant
KW  - asymptotic expansion
KW  - Laplace expansion
KW  - heat asymptotics
KW  - Wiener measure
KW  - Wärmeleitungsgleichung
KW  - Wärmekern
KW  - Pfadintegrale
KW  - asymptotische Entwicklung
KW  - Determinante
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-94387
ER  - 
TY  - THES
A1  - Scharrer, Christian
T1  - Relating diameter and mean curvature for varifolds
T1  - Relativer Diameter und mittlere Krümmung für Varifaltigkeiten
N2  - The main results of this thesis are formulated in a class of surfaces (varifolds) generalizing closed and connected smooth submanifolds of Euclidean space which allows singularities. Given an indecomposable varifold with dimension at least two in some Euclidean space such that the first variation is locally bounded, the total variation is absolutely continuous with respect to the weight measure, the density of the weight measure is at least one outside a set of weight measure zero and the generalized mean curvature is locally summable to a natural power (dimension of the varifold minus one) with respect to the weight measure. The thesis presents an improved estimate of the set where the lower density is small in terms of the one dimensional Hausdorff measure. Moreover, if the support of the weight measure is compact, then the intrinsic diameter with respect to the support of the weight measure is estimated in terms of the generalized mean curvature. This estimate is in analogy to the diameter control for closed connected manifolds smoothly immersed in some Euclidean space of Peter Topping. Previously, it was not known whether the hypothesis in this thesis implies that two points in the support of the weight measure have finite geodesic distance.
N2  - Die wichtigsten Ergebnisse dieser Arbeit sind formuliert für eine Klasse von Oberflächen (Varifaltigkeiten), welche geschlossene glatte Untermannigfaltigkeiten des euklidischen Raums verallgemeinern und Singularitäten erlauben. Gegeben sei eine mindestens zwei-dimensionale unzerlegbare Varifaltigkeit im euklidischen Raum, sodass die erste Variation lokal beschränkt ist, die totale Variation absolut stetig bezüglich dem Gewichtsmaß ist, die Dichte des Gewichtsmaßes außerhalb einer Nullmenge mindesten eins ist, und die verallgemeinerte mittlere Krümmung bezüglich dem Gewichtsmaß lokal summierbar zu einer natürlichen Potenz (Dimension der Varifaltigkeit minus eins) ist. Es wird die Menge, wo die untere Dichte klein ist, durch das ein-dimensionale Hausdorff-Maß abgeschätzt. Das Ergebnis ist eine neue, stark verbesserte untere Dichte-Schranke. Ist der Träger des Gewichtsmaßes kompakt, so wird der intrinsische Diameter des Trägers des Gewichtsmaßes abgeschätzt durch ein Integral der verallgemeinerten mittleren Krümmung. Diese Ungleichung ist analog zu der Ungleichung von Peter Topping für geschlossene zusammenhängende Mannigfaltigkeit, welche durch eine glatte Immersion in den euklidischen Raum eingebettet sind. Bisher war nicht bekannt, dass die oben genannten Annahmen an die Varifaltigkeit implizieren, dass der geodätische Abstand zweier Punkte aus dem Träger des Gewichtsmaßes endlich ist.
KW  - varifold
KW  - rectifiable varifold
KW  - indecomposable varifold
KW  - first variation
KW  - mean curvature
KW  - isoperimetric inequality
KW  - density of a measure
KW  - geodesic distance
KW  - intrinsic diameter
KW  - Varifaltigkeit
KW  - rektifizierbare Varifaltigkeit
KW  - unzerlegbare Varifaltigkeit
KW  - erste Variation
KW  - mittlere Krümmung
KW  - isoperimetrische Ungleichung
KW  - Dichte eines Maßes
KW  - geodätischer Abstand
KW  - intrinsischer Diameter
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-97013
ER  - 
TY  - INPR
A1  - Shlapunov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - An open mapping theorem for the Navier-Stokes equations
N2  - We consider the Navier-Stokes equations in the layer R^n x [0,T] over R^n with finite T > 0. Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form (I+K) u = f, where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all t in [0,T]. On using the particular properties of the de Rham complex we conclude that the Fréchet derivative (I+K)' is continuously invertible at each point of the Banach space under consideration and the map I+K is open and injective in the space. In this way the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016)10 
KW  - Navier-Stokes equations
KW  - weighted Hölder spaces
KW  - integral representation method
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-98687
SN  - 2193-6943
VL  - 5
IS  - 10
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Blanchard, Gilles
A1  - Krämer, Nicole
T1  - Convergence rates of kernel conjugate gradient for random design regression
N2  - We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true
regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 8 
KW  - nonparametric regression
KW  - reproducing kernel Hilbert space
KW  - conjugate gradient
KW  - partial least squares
KW  - minimax convergence rates
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-94195
SN  - 2193-6943
VL  - 5
IS  - 8
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Beinrucker, Andre
T1  - Variable selection in high dimensional data analysis with applications
Y1  - 2015
ER  - 
TY  - INPR
A1  - Fedosov, Boris
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Deformation quantisation and boundary value problems
N2  - We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 5 
KW  - symplectic manifold
KW  - star product
KW  - trace
KW  - index
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-77150
SN  - 2193-6943
VL  - 4
IS  - 5
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Elin, Mark
A1  - Shoikhet, David
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Analytic semigroups of holomorphic mappings and composition operators
N2  - In this paper we study the problem of analytic extension in parameter for a semigroup of holomorphic self-mappings of the unit ball in a complex Banach space and its relation to the linear continuous semigroup of composition operators. We also provide a brief review around this topic.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 6 
KW  - nonlinear semigroup
KW  - composition operator
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-77914
SN  - 2193-6943
VL  - 4
IS  - 6
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - CHAP
A1  - Petsche, Hans-Joachim
T1  - Einführung
T2  - Raum und Zahl im Fokus der Wissenschaften : eine multidisziplinäre Vorlesungsreihe
Y1  - 2015
SN  - 978-3-86464-082-7
SP  - 9
EP  - 14
PB  - Trafo
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Petsche, Hans-Joachim
T1  - Raum und Zahl - philosophische Kontexte
JF  - Raum und Zahl im Fokus der Wissenschaften : eine multidisziplinäre Vorlesungsreihe
Y1  - 2015
SN  - 978-3-86464-082-7
SP  - 15
EP  - 33
PB  - Trafo
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Sixtus, Elena
A1  - Fischer, Martin H.
T1  - Eine kognitionswissenschaftliche Betrachtung der Konzepte "Raum" und "Zahl"
JF  - Raum und Zahl im Fokus der Wissenschaften : eine multidisziplinäre Vorlesungsreihe
Y1  - 2015
SN  - 978-3-86464-082-7
SP  - 35
EP  - 62
PB  - Trafo
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Pröve, Ralf
T1  - Numerische Zeichen und die Repräsentation von Sinn
BT  - Zahlen und Zeit-Räume im Spiegel der Wissenschaft von vergangenen Zeiten
JF  - Raum und Zahl im Fokus der Wissenschaften : eine multidisziplinäre Vorlesungsreihe
Y1  - 2015
SN  - 978-3-86464-082-7
SP  - 63
EP  - 76
PB  - Trafo
CY  - Berlin
ER  - 
TY  - JOUR
A1  - Heimann, Heinz-Dieter
T1  - Kreise - Punkte - Linien
BT  - Zum Wandel der Raumkonzeption und der Raumerfassung im Bild spätmittelalterlicher Welt- und Territorialkarten
JF  - Raum und Zahl im Fokus der Wissenschaften : eine multidisziplinäre Vorlesungsreihe
Y1  - 2015
SN  - 978-3-86464-082-7
SP  - 111
EP  - 127
PB  - Trafo
CY  - Berlin
ER  - 
TY  - THES
A1  - Chutsagulprom, Nawinda
T1  - Ensemble-based filters dealing with non-Gaussianity and nonlinearity
Y1  - 2016
ER  - 
TY  - INPR
A1  - Roelly, Sylvie
A1  - Vallois, Pierre
T1  - Convoluted Brownian motion
BT  - a semimartingale approach
N2  - In this paper we analyse semimartingale properties of a class of Gaussian periodic processes, called convoluted Brownian motions, obtained by convolution between a deterministic function and a Brownian motion. A classical
example in this class is the periodic Ornstein-Uhlenbeck process. We compute their characteristics and show that in general, they are neither
Markovian nor satisfy a time-Markov field property. Nevertheless, by enlargement
of filtration and/or addition of a one-dimensional component, one can in some case recover the Markovianity. We treat exhaustively the case of the bidimensional trigonometric convoluted Brownian motion and the higher-dimensional monomial convoluted Brownian motion.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 9 
KW  - periodic Gaussian process
KW  - periodic Ornstein-Uhlenbeck process
KW  - Markov-field property
KW  - enlargement of filtration
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-96339
SN  - 2193-6943
VL  - 5
IS  - 9
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Fedchenko, Dmitry
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Radó Theorem for the Porous Medium Equation
T2  - Preprints des Instituts für Mathematik der Universität Potsdam
N2  - We prove that each locally Lipschitz continuous function satisfying the porous medium equation away from the set of its zeroes is actually a weak solution of this equation in the whole domain.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 1 
KW  - quasilinear equation
KW  - removable set
KW  - porous medium equation
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102735
VL  - 6
IS  - 1
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Shlapunov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Golusin-Krylov Formulas in Complex Analysis
T2  - Preprints des Instituts für Mathematik der Universität Potsdam
N2  - This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy-Smirnov spaces.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 2 
KW  - analytic continuation
KW  - integral formulas
KW  - Cauchy problem
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-102774
VL  - 6
IS  - 2
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Vasiliev, Serguei
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Construction of series of perfect lattices by layer superposition
N2  - We construct a new series of perfect lattices in n dimensions by the layer superposition method of Delaunay-Barnes.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016)11 
KW  - lattice packing and covering
KW  - polyhedra and polytopes
KW  - regular figures
KW  - division of spaces
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-100591
SN  - 2193-6943
VL  - 5
IS  - 11
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Bartels, Knut
T1  - Tests zur Modellspezifikation in der nichtlinearen Regression
N2  - Als Grundlage vieler statistischer Verfahren wird der Prozess der Entstehung von Daten modelliert, um dann weitere Schätz- und Testverfahren anzuwenden. Diese Arbeit befasst sich mit der Frage, wie diese Spezifikation für parametrische Modelle selbst getestet werden kann. In Erweiterung bestehender Verfahren werden Tests mit festem Kern eingeführt und ihre asymptotischen Eigenschaften werden analysiert. Es wird gezeigt, dass die Bestimmung der kritischen Werte mit mehreren Stichprobenwiederholungsverfahren möglich ist. Von diesen ist eine neue Monte-Carlo-Approximation besonders wichtig, da sie die Komplexität der Berechnung deutlich verringern kann. Ein bedingter Kleinste-Quadrate-Schätzer für nichtlineare parametrische Modelle wird definiert und seine wesentlichen asymptotischen Eigenschaften werden hergeleitet. Sämtliche Versionen der Tests und alle neuen Konzepte wurden in Simulationsstudien untersucht, deren wichtigste Resultate präsentiert werden. Die praktische Anwendbarkeit der Testverfahren wird an einem Datensatz zur Produktwahl dargelegt, der mit multinomialen Logit-Modellen analysiert werden soll.
N2  - The data generating process often is modeled as a basis for many subsequent statistical estimation and testing procedures. In this work the question is studied, how this specification of parametric models itself can be tested. In generalization of existing methods, tests with fixed kernel are introduced and their asymptotics are analyzed. It is shown that the determination of critical values is possible using several resampling procedures. Of these a new Monte-Carlo-approximation is of special importance, since it can reduce the complexity of calculation substantially. A conditional least squares estimator for nonlinear models is defined and its essential asymptotic properties are derived. All versions of the tests and all new concepts were studied in simulation studies and the most important results are presented. The applicability of the tests is demonstrated with a dataset on product choice that is to be analyzed with multinomial logit models.
KW  - nichtlineare Modelle
KW  - Spezifikationstests
KW  - Resampling
KW  - Simulationsstudien
Y1  - 1999
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-0000171
ER  - 
TY  - JOUR
A1  - Meyerhöfer, Wolfram
T1  - Was misst TIMSS?
BT  - Einige Überlegungen zum Problem der Interpretierbarkeit der erhobenen Daten
N2  - Bei der Erstellung und Interpretation mathematischer Leistungstests steht die Frage, was eine Aufgabe mißt. Der Artikel stellt mit der strukturalen oder objektiven Hermeneutik eine Methode vor, mit der die verschiedenen Dimensionen der von einer Aufgabe erfassten Fähigkeiten herausgearbeitet werden können. Dabei werden fachliche Anforderungen, Irritationsmomente und das durch die Aufgabe transportierte Bild vom jeweiligen Fach ebenso erfasst wie Momente, die man eher als Testfähigkeit bezeichnen würde.Am Beispiel einer TIMSS-Aufgabe wird diskutiert, dass das von den Testerstellern benutzte theoretische Konstrukt kaum geeignet ist, nachhaltig zu beschreiben, was eine Aufgabe misst.
N2  - The design and interpretation of aptitude tests in mathematics provoke questions as to what each of the set tasks actually measures. With structural or objective hermeneutics, this article introduces a methodology capable of discerning the various dimensions of skills required for a particular task. Not only does this approach allow for the recognition of the technical requirements of the task, its off-putting factors and the image of the subject conveyed. The methodology is also able to locate the elements addressing the kind of skill that can more accurately be clas-sified as 'test ability'. Focusing on an example selected from a TIMSS aptitude test, the discussion seeks to de-monstrate that the theoretical construction employed in setting the test is hardly suited to define with any sense of permanence what is measured by each task.
KW  - TIMSS
KW  - Leistungstests
KW  - Testfähigkeit
KW  - Hermeneutik
KW  - TIMSS
KW  - aptitude tests
KW  - test ability
KW  - hermeneutics
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-224
SP  - 1
EP  - 11
ER  - 
TY  - INPR
A1  - Roelly, Sylvie
A1  - Fradon, Myriam
T1  - Infinite system of Brownian balls : equilibrium measures are canonical Gibbs
N2  - We consider a system of infinitely many hard balls in R<sup>d undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional stochastic differential equation with a local time term. We prove that the set of all equilibrium measures, solution of a detailed balance equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.
KW  - Stochastic Differential Equation
KW  - hard core potential
KW  - Canonical Gibbs measure
KW  - detailed balance equation
KW  - reversible measure
Y1  - 2006
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6720
ER  - 
TY  - THES
A1  - Rosenberger, Elke
T1  - Asymptotic spectral analysis and tunnelling for a class of difference operators
T1  - Asymptotische Spektralanalyse und Tunneleffekt für eine Klasse von Differenzen-Operatoren
N2  - We analyze the asymptotic behavior in the limit epsilon to zero for a wide class of difference operators H_epsilon = T_epsilon + V_epsilon with underlying multi-well potential. They act on the square summable functions on the lattice (epsilon Z)^d. We start showing the validity of an harmonic approximation and construct WKB-solutions at the wells. Then we construct a Finslerian distance d induced by H and show that short integral curves are geodesics and d gives the rate for the exponential decay of Dirichlet eigenfunctions. In terms of this distance, we give sharp estimates for the interaction between the wells and construct the interaction matrix.
N2  - Wir analysieren das asymptotische Verhalten im Grenzwert epsilon gegen null von einer weiten Klasse von Differenzen operatoren H_epsilon = T_epsilon + V_epsilon mit unterliegendem Potential. Sie wirken auf die quadrat-summierbaren Funktionen auf dem Gitter (epsilon Z)^d. Zunächst zeigen wir die Gültigkeit einer harmonischen Approximation und konstruieren WKB-Lösungen an den Töpfen. Dann konstruieren wir eine Finslersche Abstandsfunktion d, die durch H induziert wird und zeigen, daß kurze Integralkurven Geodäten sind und daß d die Rate des exponentiellen Abfallverhaltens von Dirichlet-Eigenfunktionen beschreibt. Bezügliche dieses Abstands geben wir scharfe Abschätzungen für die Wechselwirkung zwischen den Töpfen und konstruieren die Wechselwirkungs-Matrix.
KW  - Mathematische Physik
KW  - Operatortheorie
KW  - Generalized translation operator
KW  - Tunneleffekt
KW  - Spektraltheorie
KW  - Asymptotische Entwicklung
KW  - Semi-klasische Abschätzung
KW  - Finsler-Abstand
KW  - Pseudodifferentialoperatoren auf dem Torus
KW  - Kontinuumsgrenzwert
KW  - Differenzenoperator
KW  - tunneling
KW  - semi-classical spectral estimates
KW  - Finsler-distance
KW  - difference operator
KW  - scaled lattice
Y1  - 2006
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-7393
ER  - 
TY  - INPR
A1  - Mekler, A. A.
A1  - Böckmann, Christine
A1  - Sokolovskaia, N.
T1  - Particle distribution from spectral Mie-scattering: kernel representation and singular-value spectrum
N2  - This paper deals with the Mie scattering kernels for multi-spectral data. The kernels may be represented in form of power series. Furthermore, the singular-value spectrum and the degree of ill-posedness in dependence on the refractive index of the particles are numerically approximated. A special hybrid regularization technique allows us to determine via inversion the particle distributions of different types.
T3  - NLD Preprints - 58 
Y1  - 2000
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14948
ER  - 
TY  - THES
A1  - Ong, James Kwan Yau
T1  - The predictability problem
T1  - Das Vorhersagbarkeitsproblem
N2  - Wir versuchen herauszufinden, ob das subjektive Maß der Cloze-Vorhersagbarkeit mit der Kombination objektiver Maße (semantische und n-gram-Maße) geschätzt werden kann, die auf den statistischen Eigenschaften von Textkorpora beruhen. Die semantischen Maße werden entweder durch Abfragen von Internet-Suchmaschinen oder durch die Anwendung der Latent Semantic Analysis gebildet, während die n-gram-Wortmaße allein auf den Ergebnissen von Internet-Suchmaschinen basieren. Weiterhin untersuchen wir die Rolle der Cloze-Vorhersagbarkeit in SWIFT, einem Modell der Blickkontrolle, und wägen ab, ob andere Parameter den der Vorhersagbarkeit ersetzen können. Unsere Ergebnisse legen nahe, dass ein computationales Modell, welches Vorhersagbarkeitswerte berechnet, nicht nur Maße beachten muss, die die Relatiertheit eines Wortes zum Kontext darstellen; das Vorhandensein eines Maßes bezüglich der Nicht-Relatiertheit ist von ebenso großer Bedeutung. Obwohl hier jedoch nur Relatiertheits-Maße zur Verfügung stehen, sollte SWIFT ebensogute Ergebnisse liefern, wenn wir Cloze-Vorhersagbarkeit mit unseren Maßen ersetzen.
N2  - We try to determine whether it is possible to approximate the subjective Cloze predictability measure with two types of objective measures, semantic and word n-gram measures, based on the statistical properties of text corpora. The semantic measures are constructed either by querying Internet search engines or by applying Latent Semantic Analysis, while the word n-gram measures solely depend on the results of Internet search engines. We also analyse the role of Cloze predictability in the SWIFT eye movement model, and evaluate whether other parameters might be able to take the place of predictability. Our results suggest that a computational model that generates predictability values not only needs to use measures that can determine the relatedness of a word to its context; the presence of measures that assert unrelatedness is just as important. In spite of the fact, however, that we only have similarity measures, we predict that SWIFT should perform just as well when we replace Cloze predictability with our measures.
KW  - Cloze-Vorhersagbarkeit
KW  - Blickbewegungen
KW  - Latente-Semantische-Analyse
KW  - Wort-n-Gramme-Wahrscheinlichkeit
KW  - Ähnlichkeit-Masse
KW  - Cloze predictability
KW  - eye movements
KW  - Latent Semantic Analysis
KW  - word n-gram probability
KW  - similarity measures
Y1  - 2007
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-15025
ER  - 
TY  - THES
A1  - Koh, Dennis
T1  - The evolution equation for closed magnetic geodesics
T1  - Die Evolutionsgleichung für geschlossene magnetische Geodäten
N2  - Orbits of charged particles under the effect of a magnetic field are mathematically described by magnetic geodesics. They appear as solutions to a system of (nonlinear) ordinary differential equations of second order. But we are only interested in periodic solutions. To this end, we study the corresponding system of (nonlinear) parabolic equations for closed magnetic geodesics and, as a main result, eventually prove the existence of long time solutions. As generalization one can consider a system of elliptic nonlinear partial differential equations whose solutions describe the orbits of closed p-branes under the effect of a "generalized physical force". For the corresponding evolution equation, which is a system of parabolic nonlinear partial differential equations associated to the elliptic PDE, we can establish existence of short time solutions.
N2  - Bahnen von geladenen Teilchen, die sich unter dem Einfluss eines Magnetfeldes bewegen, werden in der Mathematik durch magnetische Geodäten beschrieben. Sie ergeben sich als Lösungen eines Systems (nichtlinearer) gewöhnlicher Differentialgleichungen zweiter Ordnung. Wir interessieren uns ausschließich für periodische Lösungen. Dazu studieren wir das zugehörige System (nichtlinearer) parabolischer Differentialgleichungen für geschlossene magnetische Geodäten. Als Hauptresultat beweisen wir die Existenz von Langzeitlösungen. Verallgemeinernd betrachten wir noch ein System von elliptischen nichtlinearen partiellen Differentialgleichungen, dessen Lösungen die Orbiten von geschlossenen p-Branen unter dem Einfluss einer verallgemeinerten physikalischen Kraft beschreiben. Für die entsprechende Evolutionsgleichung, welche ein System von parabolischen nichtlinearen partiellen Differentialgleichungen ist, das dem elliptischen Problem zugeordnet ist, können wir die Existenz von Kurzzeitlösungen beweisen.
KW  - magnetisch
KW  - Geodäten
KW  - Evolutionsgleichung
KW  - Strings
KW  - p-Branen
KW  - magnetic
KW  - geodesics
KW  - evolution equation
KW  - p-branes
Y1  - 2008
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-16647
SN  - 978-3-940793-24-9
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Murr, Rüdiger
T1  - Dualitätsformeln für Brownsche Bewegung und für eine Irrfahrt mit Anwendung am Konvergenzergebnis von Donsker
N2  - Aus dem Inhalt: 0.1 Danksagung 0.2 Einleitung 1 Allgemeines und Grundlagen 1.1 Die Brownsche Bewegung 2 Die Dualitätsformel des Wienermaßes 2.1 Wienermaß erfüllt Dualitätsformel 2.2 Dualitätsformel charakterisiert Wienermaß 3 Die diskrete Dualitätsformel der Irrfahrt 3.1 Verallgemeinerte symmetrische Irrfahrt erfüllt diskrete Dualitätsformel 3.2 Diskrete Dualitätsformel charakterisiert verallgemeinerte symmetrische Irrfahrt 4 Donskers Theorem und die Dualitätsformeln 4.1 Straffheit der renormierten stetigen Irrfahrt 4.2 Konvergenz der Irrfahrt 5 Anhang
T3  - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2008, 05 
Y1  - 2008
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49476
ER  - 
TY  - INPR
A1  - Fedchenko, Dmitry
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - An index formula for Toeplitz operators
N2  - We prove a Fedosov index formula for the index of Toeplitz operators connected with the Hardy space of solutions to an elliptic system of first order partial differential equations in a bounded domain of Euclidean space with infinitely differentiable boundary.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3(2014)12 
KW  - Toeplitz operators
KW  - Fredholm property
KW  - index
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72499
SN  - 2193-6943
VL  - 3
IS  - 12
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Dyachenko, Evgeniya
T1  - Elliptic problems with small parameter
T1  - Elliptische Problemen mit kleinem Parameter
N2  - In this thesis we consider diverse aspects of existence and correctness of asymptotic solutions to elliptic differential and pseudodifferential equations. We begin our studies with the case of a general elliptic boundary value problem in partial derivatives. A small parameter enters the coefficients of the main equation as well as into the boundary conditions. Such equations have already been investigated satisfactory, but there still exist certain theoretical deficiencies. Our aim is to present the general theory of elliptic problems with a small parameter. For this purpose we examine in detail the case of a bounded domain with a smooth boundary. First of all, we construct formal solutions as power series in the small parameter. Then we examine their asymptotic properties. It suffices to carry out sharp two-sided \emph{a priori} estimates for the operators of boundary value problems which are uniform in the small parameter. Such estimates failed to hold in functional spaces used in classical elliptic theory. To circumvent this limitation we exploit norms depending on the small parameter for the functions defined on a bounded domain. Similar norms are widely used in literature, but their properties have not been investigated extensively. Our theoretical investigation shows that the usual elliptic technique can be correctly carried out in these norms. The obtained results also allow one to extend the norms to compact manifolds with boundaries. We complete our investigation by formulating algebraic conditions on the operators and showing their equivalence to the existence of a priori estimates. In the second step, we extend the concept of ellipticity with a small parameter to more general classes of operators. Firstly, we want to compare the difference in asymptotic patterns between the obtained series and expansions for similar differential problems. Therefore we investigate the heat equation in a bounded domain with a small parameter near the time derivative. In this case the characteristics touch the boundary at a finite number of points. It is known that the solutions are not regular in a neighbourhood of such points in advance. We suppose moreover that the boundary at such points can be non-smooth but have cuspidal singularities. We find a formal asymptotic expansion and show that when a set of parameters comes through a threshold value, the expansions fail to be asymptotic. The last part of the work is devoted to general concept of ellipticity with a small parameter. Several theoretical extensions to pseudodifferential operators have already been suggested in previous studies. As a new contribution we involve the analysis on manifolds with edge singularities which allows us to consider wider classes of perturbed elliptic operators. We examine that introduced classes possess a priori estimates of elliptic type. As a further application we demonstrate how developed tools can be used to reduce singularly perturbed problems to regular ones.
N2  - In dieser Dissertation betrachten wir verschiedene Aspekte der Existenz und Korrektheit asymptotischer Lösungen für elliptische Differentialgleichungen und Pseudodifferentialgleichungen. Am Anfang betrachtet die Arbeit den Fall eines allgemeinen elliptischen Grenzwertproblems in partiellen Ableitungen. Hierbei hängen die Koeffizienten von einem kleinen Parameter ab. Solche Gleichungen wurden schon reichlich untersucht, aber es gibt immer noch theoretische Lücken. Unser Ziel ist eine allgemeine Theorie elliptischer Operatorklassen mit kleinen Parametern. Zu diesem Zweck untersuchen wir im Detail den Fall eines beschränkten Gebietes mit glattem Rand. Zuerst konstruieren wir formale Lösungen als Potenzreihe einer kleinen Variablen. Weiter untersuchen wir ihre asymptotischen Eigenschaften. Dazu reicht es aus, beidseitige A-Priori Abschätzungen für diejenigen Randwertproblemoperatoren zu bestimmen, die gleichmäßig stetig von den kleinen Parametern abhängen. Solche Abschätzungen gelten nicht in Funktionenräumen, die in der klassischen elliptischen Theorie benutzt werden. Um diese Beschränkungen zu überwinden, nutzen wir Normen abhängig vom kleinen Parameter. Änliche Normen finden sich oft in der Literatur, aber ihre Eigenschaften wurden unzureichend untersucht. Unsere theoretische Forschung zeigt, dass die gewöhnliche elliptische Methode korrekt durchgeführt werden kann. Die erhaltenen Abschätzungen erlauben das Fortsetzen der Normen auf kompakte Mannigfältigkeiten mit Rand. Unsere Forschung wird mit algebraischen Bedingungen für die Operatoren abgeschlossen. Wir zeigen, dass diese Bedingungen äquivalent zu der Existenz der A-Priori-Abschätzungen sind. Im zweiten Schritt erweitern wir das Konzept der Elliptizität mit kleinen Parametern zu allgemeineren Operatorklassen. Zuerst wollen wir den Unterschied in asymptotischen Mustern zwischen der erhaltenen Reihe und Lösungen ähnlicher Probleme untersuchen. Deshalb untersuchen wir die Wärmeleitungsgleichung in einem beschränkten Gebiet mit einem kleinen Parameter in der Zeitableitung. In diesem Fall tangiert der Rand die Charakteristik endlich oft. Es ist bekannt, dass die Lösungen unregulär im Allgemeinen in Umgebungen solcher Stellen sind. Wir nehmen an, dass der Rand an solchen Stellen nicht glätt sein kann und kaspydalische Singularitäten hat. Wir haben eine formale asymptotische Zerlegung gefunden und einen Schwellenwert gezeigt, sodass die asymptotische Eigenschaft der Reihe nicht mehr gilt, wenn der Randparameter diesen Schwellenwert übersteigt. Der letze Teil der Arbeit führt ein allgemeines Konzept der Elliptizit\"at mit einem kleinen Parameter ein. Mehrere theoretische Erweiterungen auf Pseudodifferentialoperatoren wurden schon in früheren Studien vorgeschlagen. Als neuen Beitrag wenden wir die Analysis auf Manigfältigkeiten mit Kantensingularitäten an. Dies lässt es zu, allgemeinere gestörte Operatorklassen zu betrachten. Wir beobachten, dass die eingef\"uhrten Klassen A-Priori-Abschätzungen elliptischer Gestalt haben. Als weitere Anwendung demonstrieren wir, wie die entwickelten Mittel zum Reduzieren singular gestörter Probleme zu regulären Fällen benutzt werden können.
KW  - elliptische Gleichungen
KW  - kleine Parameter
KW  - elliptic problems
KW  - small parameter
KW  - boundary layer
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72056
ER  - 
TY  - INPR
A1  - Alsaedy, Ammar
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - The method of Fischer-Riesz equations for elliptic boundary value problems
N2  - We develop the method of Fischer-Riesz equations for general boundary value problems elliptic in the sense of Douglis-Nirenberg. To this end we reduce them to a boundary problem for a (possibly overdetermined) first order system whose classical symbol has a left inverse. For such a problem there is a uniquely determined boundary value problem which is adjoint to the given one with respect to the Green formula. On using a well elaborated theory of approximation by solutions of the adjoint problem, we find the Cauchy data of solutions of our problem.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1(2012)24 
KW  - Boundary value problems for first order systems
KW  - Green formula
KW  - Fischer-Riesz equations
KW  - regularisation
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-61792
ER  - 
TY  - THES
A1  - Kröncke, Klaus
T1  - Stability of Einstein Manifolds
T1  - Stabilität von Einstein-Mannigfaltigkeiten
N2  - This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum.
N2  - Die vorliegende Arbeit beschäftigt sich mit Einsteinmetriken und Ricci-Fluss auf kompakten Mannigfaltigkeiten. Wir studieren die zweite Variation des Einstein- Hilbert Funktionals auf Einsteinmetriken. Im ersten Teil der Arbeit finden wir Krümmungsbedingungen, die die Stabilität von Einsteinmannigfaltigkeiten bezüglich des Einstein-Hilbert Funktionals sicherstellen, d.h. die zweite Varia- tion des Einstein-Hilbert Funktionals ist nichtpositiv in Richtung transversaler spurfreier Tensoren. Der zweite Teil der Arbeit widmet sich dem Studium des Ricci-Flusses und wie dessen Verhalten in der Nähe von Einsteinmetriken durch das Variationsver- halten des Einstein-Hilbert Funktionals beeinflusst wird. Wir finden Bedinun- gen, die dynamische Stabilität oder Instabilität von Einsteinmetriken bezüglich des Ricci-Flusses implizieren und wir drücken diese Bedingungen in Termen der Stabilität der Metrik bezüglich des Einstein-Hilbert Funktionals und Eigen- schaften des Spektrums des Laplaceoperators aus.
KW  - Einstein-Mannigfaltigkeiten
KW  - Ricci-Fluss
KW  - Variationsstabilität
KW  - Einstein-Hilbert-Wirkung
KW  - Einstein manifolds
KW  - Ricci flow
KW  - variational stability
KW  - Einstein-Hilbert action
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-69639
ER  - 
TY  - INPR
A1  - Gairing, Jan
A1  - Högele, Michael
A1  - Kosenkova, Tetiana
A1  - Kulik, Alexei Michajlovič
T1  - Coupling distances between Lévy measures and applications to noise sensitivity of SDE
N2  - We introduce the notion of coupling distances on the space of Lévy measures in order to quantify rates of convergence towards a limiting Lévy jump diffusion in terms of its characteristic triplet, in particular in terms of the tail of the Lévy measure. The main result yields an estimate of the Wasserstein-Kantorovich-Rubinstein distance on path space between two Lévy diffusions in terms of the couping distances. We want to apply this to obtain precise rates of convergence for Markov chain approximations and a statistical goodness-of-fit test for low-dimensional conceptual climate models with paleoclimatic data.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)16 
KW  - Lévy diffusion approximation
KW  - coupling methods
KW  - Skorokhod' s invariance principle
KW  - statistical model selection
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68886
ER  - 
TY  - INPR
A1  - Fedchenko, Dmitry
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Class of Toeplitz Operators in Several Variables
N2  - We introduce the concept of Toeplitz operator associated with the Laplace-Beltrami operator on a compact Riemannian manifold with boundary. We characterise those Toeplitz operators which are Fredholm, thus initiating the index theory.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)17 
KW  - Cauchy data spaces
KW  - Laplace-Beltrami operator
KW  - Toeplitz operators
KW  - Fredholm property
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-68932
SN  - 2193-6943
ER  - 
TY  - THES
A1  - Di Gesù, Giacomo
T1  - Semiclassical spectral analysis of discrete Witten Laplacians
T1  - Semiklassische Spektraltheorie von diskreten Witten-Laplace-Operatoren
N2  - A discrete analogue of the Witten Laplacian on the n-dimensional integer lattice is considered. After rescaling of the operator and the lattice size we analyze the tunnel effect between different wells, providing sharp asymptotics of the low-lying spectrum. Our proof, inspired by work of B. Helffer, M. Klein and F. Nier in continuous setting, is based on the construction of a discrete Witten complex and a semiclassical analysis of the corresponding discrete Witten Laplacian on 1-forms. The result can be reformulated in terms of metastable Markov processes on the lattice.
N2  - In dieser Arbeit wird auf dem n-dimensionalen Gitter der ganzen Zahlen ein Analogon des Witten-Laplace-Operatoren eingeführt. Nach geeigneter Skalierung des Gitters und des Operatoren analysieren wir den Tunneleffekt zwischen verschiedenen Potentialtöpfen und erhalten vollständige Aymptotiken für das tiefliegende Spektrum. Der Beweis (nach Methoden, die von B. Helffer, M. Klein und F. Nier im Falle des kontinuierlichen Witten-Laplace-Operatoren entwickelt wurden) basiert auf der Konstruktion eines diskreten Witten-Komplexes und der Analyse des zugehörigen Witten-Laplace-Operatoren auf 1-Formen. Das Resultat kann im Kontext von metastabilen Markov Prozessen auf dem Gitter reformuliert werden und ermöglicht scharfe Aussagen über metastabile Austrittszeiten.
KW  - Semiklassische Spektralasymptotik
KW  - Metastabilität
KW  - diskreter Witten-Laplace-Operator
KW  - Eyring-Kramers Formel
KW  - Tunneleffekt
KW  - semiclassical spectral asymptotics
KW  - metastability
KW  - low-lying eignvalues
KW  - discrete Witten complex
KW  - rescaled lattice
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-65286
ER  - 
TY  - INPR
A1  - Conforti, Giovanni
A1  - Roelly, Sylvie
T1  - Reciprocal class of random walks on an Abelian group
N2  - Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, we compute explicitly its reciprocal characteristics and we present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke's formula from the point process theory, which allows us to study, as transformation on the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 1 
KW  - reciprocal class
KW  - stochastic bridge
KW  - random walk on Abelian group
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72604
SN  - 2193-6943
VL  - 4
IS  - 1
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Bettenbühl, Mario
T1  - Microsaccades
T1  - Mikrosakkaden
BT  - Symbols in fixational eye movements
BT  - Symbole in den Fixationsbewegungen der Augen
N2  - The first thing we do upon waking is open our eyes. Rotating them in our eye sockets, we scan our surroundings and collect the information into a picture in our head. Eye movements can be split into saccades and fixational eye movements, which occur when we attempt to fixate our gaze. The latter consists of microsaccades, drift and tremor. Before we even lift our eye lids, eye movements – such as saccades and microsaccades that let the eyes jump from one to another position – have partially been prepared in the brain stem. Saccades and microsaccades are often assumed to be generated by the same mechanisms. But how saccades and microsaccades can be classified according to shape has not yet been reported in a statistical manner. Research has put more effort into the investigations of microsaccades’ properties and generation only since the last decade. Consequently, we are only beginning to understand the dynamic processes governing microsaccadic eye movements. Within this thesis, the dynamics governing the generation of microsaccades is assessed and the development of a model for the underlying processes. Eye movement trajectories from different experiments are used, recorded with a video-based eye tracking technique, and a novel method is proposed for the scale-invariant detection of saccades (events of large amplitude) and microsaccades (events of small amplitude). Using a time-frequency approach, the method is examined with different experiments and validated against simulated data. A shape model is suggested that allows for a simple estimation of saccade- and microsaccade related properties. For sequences of microsaccades, in this thesis a time-dynamic Markov model is proposed, with a memory horizon that changes over time and which can best describe sequences of microsaccades.
N2  - Beim Aufwachen jeden Morgen, ist es das erste, was wir tun: wir öffnen unsere Augen. Wir lassen die Augen rotieren und suchen unsere Umgebung ab. Gleichzeitig wird die gesammelte Information in unserem Gehirn zu einem Bild vereint. Augenbewegungen können getrennt werden in Sakkaden, welche sprunghafte Augenbewegungen darstellen, und Fixationsbewegungen der Augen, letztere bestehend aus Mikrosakkaden, Tremor und Drift. Bevor wir unsere Augen aufschlagen, wurden die Bewegungen bereits teilweise im Hirnstamm vorprogrammiert. So ist dieser Teil unseres Gehirns verantwortlich für die Auslösung einer Sakkade oder Mikrosakkade, worin man versuchen kann auch gleichzeitig einen Zusammenhang für die Generierung dieser Bewegung herzustellen. Es wird vermutet, dass Mikrosakkaden auch als kleinskaligere Sakkade verstanden werden können, welche auftreten, wenn wir versuchen unsere Augen still auf einen Punkt zu fixieren. Bisher gibt es keine statistische Untersuchung bezüglich einer Klassifizierung von Sakkaden und Mikrosakkaden aufgrund ihrer Form, d.h. ihrer räumlichen Entwicklung über die Zeit. Seit Beginn des neuen Milleniums verstärkte sich die Forschung wieder auf die Eigenschaften und Entstehung von Mikrosakkaden. Demnach stehen wir immer noch am Anfang diese Phänomene mit dynamischen Prozessen beschreiben zu können. Der Fokus dieser Arbeit konzentriert sich auf das Verstehen der generierenden Dynamik von Mikrosakkaden. Es wird ein Model für den unterliegenden Prozess entwickelt und getestet. Es wurden Aufzeichnungen von Augenbewegungen aus verschiedenen Experimenten genutzt, jeweils aufgenommen mit einem videobasiertem System. Es wird eine neue Methode zur amplitudenunabhängigen Detektion von Sakkaden eingeführt, um die Zeitpunkte des Auftretens von Mikrosakkaden und Sakkaden zu bestimmen. Dabei werden für Daten verschiedener Experimente Methoden der Zeit-Frequenz-Analyse genutzt und anschließend die Methode validiert mit simulierten Daten. Außerdem wird ein Modell vorgestellt für die formabhängigen Ausprägungen von Sakkaden und Mikrosakkaden, um die Schätzung ihrer beiden physikalisch relevanten Eigenschaften zu erleichtern. Zum Ende der Arbeit wird ein zeitdynamisches Modell für Sequenzen von Mikrosakkadensymbolen aufgezeigt. Mithilfe der Beschreibung der in Symbolsequenzen übersetzten Mikrosakkadensequenzen als Markovketten, wird diese Form der Augenbewegung durch einen stochastischen Prozess beschrieben. Hierbei bestehen zeitliche und räumliche Abhängigkeiten zwischen den aufeinanderfolgenden zeitdiskreten Symbolen und erlauben somit, ein Referenzmodell für einen Teil der Fixationsbewegungen der Augen zu haben.
T3  - Potsdam Cognitive Science Series - 5 
KW  - Mikrosakkaden
KW  - Fixationsbewegungen der Augen
KW  - Sakkadendetektion
KW  - Mikrosakkadensequenzen
KW  - Waveletanalyse
KW  - microsaccades
KW  - fixational eye movements
KW  - saccade detection
KW  - sequences of microsaccades
KW  - wavelet analysis
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72622
SN  - 978-3-86956-122-6
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Alsaedy, Ammar
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Weak boundary values of solutions of Lagrangian problems
N2  - We define weak boundary values of solutions to those nonlinear differential equations which appear as Euler-Lagrange equations of variational problems. As a result 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 the study of Lagrangian problems.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 2 
KW  - nonlinear equations
KW  - Lagrangian system
KW  - weak boundary values
KW  - quasilinear Fredholm operator
KW  - mapping degree
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-72617
SN  - 2193-6943
VL  - 4
IS  - 2
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A spectral theorem for deformation quantisation
N2  - We present a construction of the eigenstate at a noncritical level of the Hamiltonian function. Moreover, we evaluate the contributions of Morse critical points to the spectral decomposition.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 4 
KW  - star product
KW  - WKB method
KW  - spectral theorem
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-72425
SN  - 2193-6943
VL  - 4
IS  - 4
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Schorsch, Andrea
T1  - Statistische Eigenschaften von Clusterverfahren
T1  - Statistical properties of cluster procedures
N2  - Die vorliegende Diplomarbeit beschäftigt sich mit zwei Aspekten der statistischen Eigenschaften von Clusterverfahren. Zum einen geht die Arbeit auf die Frage der Existenz von unterschiedlichen Clusteranalysemethoden zur Strukturfindung und deren unterschiedlichen Vorgehensweisen ein. Die Methode des Abstandes zwischen Mannigfaltigkeiten und die K-means Methode liefern ausgehend von gleichen Daten unterschiedliche Endclusterungen. Der zweite Teil dieser Arbeit beschäftigt sich näher mit den asymptotischen Eigenschaften des K-means Verfahrens. Hierbei ist die Menge der optimalen Clusterzentren konsistent. Bei Vergrößerung des Stichprobenumfangs gegen Unendlich konvergiert diese in Wahrscheinlichkeit gegen die Menge der Clusterzentren, die das Varianzkriterium minimiert. Ebenfalls konvergiert die Menge der optimalen Clusterzentren für n gegen Unendlich gegen eine Normalverteilung. Es hat sich dabei ergeben, dass die einzelnen Clusterzentren voneinander abhängen.
N2  - The following thesis describes two different views onto the statistical characterics of clustering procedures. At first it adresses the questions whether different clustering methods exist to ascertain the structure of clusters and in what ays the strategies of these methods differ from each other. The method of distance between the manifolds as well as the k-means method provide different final clusters based on equal initial data. The second part of the thesis concentrates on asymptotic properties of the k-means procedure. Here the amount of optimal clustering centres is consistent. If the size of the sample range is enlarged towards infinity, it also converges in probability towards the amount of clustering centres which minimized the whithin cluster sum of squares. Likewise the amount of optimal clustering centres converges for infinity towards the normal distribution. The main result shows that the individual clustering centres are dependent on each other.
KW  - Clusteranalyse
KW  - K-Means Verfahren
KW  - asymptotische Normalverteilung
KW  - cluster analysis
KW  - k-means clustering
KW  - asymptotical normal distribution
Y1  - 2008
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29026
ER  - 
TY  - THES
A1  - Pénisson, Sophie
T1  - Conditional limit theorems for multitype branching processes and illustration in epidemiological risk analysis
T1  - Bedingte Grenzwertsätze für Mehrtyp-Verzweigungsprozesse und Illustration in epidemiologischen Risikoanalyse
N2  - This thesis is concerned with the issue of extinction of populations composed of different types of individuals, and their behavior before extinction and in case of a very late extinction. We approach this question firstly from a strictly probabilistic viewpoint, and secondly from the standpoint of risk analysis related to the extinction of a particular model of population dynamics. In this context we propose several statistical tools. The population size is modeled by a branching process, which is either a continuous-time multitype Bienaymé-Galton-Watson process (BGWc), or its continuous-state counterpart, the multitype Feller diffusion process. We are interested in different kinds of conditioning on non-extinction, and in the associated equilibrium states. These ways of conditioning have been widely studied in the monotype case. However the literature on multitype processes is much less extensive, and there is no systematic work establishing connections between the results for BGWc processes and those for Feller diffusion processes. In the first part of this thesis, we investigate the behavior of the population before its extinction by conditioning the associated branching process X_t on non-extinction (X_t≠0), or more generally on non-extinction in a near future 0≤θ<∞ (X_{t+θ}≠0), and by letting t tend to infinity. We prove the result, new in the multitype framework and for θ>0, that this limit exists and is non-degenerate. This reflects a stationary behavior for the dynamics of the population conditioned on non-extinction, and provides a generalization of the so-called Yaglom limit, corresponding to the case θ=0. In a second step we study the behavior of the population in case of a very late extinction, obtained as the limit when θ tends to infinity of the process conditioned by X_{t+θ}≠0. The resulting conditioned process is a known object in the monotype case (sometimes referred to as Q-process), and has also been studied when X_t is a multitype Feller diffusion process. We investigate the not yet considered case where X_t is a multitype BGWc process and prove the existence of the associated Q-process. In addition, we examine its properties, including the asymptotic ones, and propose several interpretations of the process. Finally, we are interested in interchanging the limits in t and θ, as well as in the not yet studied commutativity of these limits with respect to the high-density-type relationship between BGWc processes and Feller processes. We prove an original and exhaustive list of all possible exchanges of limit (long-time limit in t, increasing delay of extinction θ, diffusion limit). The second part of this work is devoted to the risk analysis related both to the extinction of a population and to its very late extinction. We consider a branching population model (arising notably in the epidemiological context) for which a parameter related to the first moments of the offspring distribution is unknown. We build several estimators adapted to different stages of evolution of the population (phase growth, decay phase, and decay phase when extinction is expected very late), and prove moreover their asymptotic properties (consistency, normality). In particular, we build a least squares estimator adapted to the Q-process, allowing a prediction of the population development in the case of a very late extinction. This would correspond to the best or to the worst-case scenario, depending on whether the population is threatened or invasive. These tools enable us to study the extinction phase of the Bovine Spongiform Encephalopathy epidemic in Great Britain, for which we estimate the infection parameter corresponding to a possible source of horizontal infection persisting after the removal in 1988 of the major route of infection (meat and bone meal). This allows us to predict the evolution of the spread of the disease, including the year of extinction, the number of future cases and the number of infected animals. In particular, we produce a very fine analysis of the evolution of the epidemic in the unlikely event of a very late extinction.
N2  - Diese Arbeit befasst sich mit der Frage des Aussterbens von Populationen verschiedener Typen von Individuen. Uns interessiert das Verhalten vor dem Aussterben sowie insbesondere im Falle eines sehr späten Aussterbens. Wir untersuchen diese Fragestellung zum einen von einer rein wahrscheinlichkeitstheoretischen Sicht und zum anderen vom Standpunkt der Risikoanalyse aus, welche im Zusammenhang mit dem Aussterben eines bestimmten Modells der Populationsdynamik steht. In diesem Kontext schlagen wir mehrere statistische Werkzeuge vor. Die Populationsgröße wird entweder durch einen zeitkontinuierlichen mehrtyp-Bienaymé-Galton-Watson Verzweigungsprozess (BGWc) oder durch sein Analogon mit kontinuierlichem Zustandsraum, den Feller Diffusionsprozess, modelliert. Wir interessieren uns für die unterschiedlichen Arten auf Überleben zu bedingen sowie für die hierbei auftretenden Gleichgewichtszustände. Diese Bedingungen wurden bereits weitreichend im Falle eines einzelnen Typen studiert. Im Kontext von mehrtyp-Verzweigungsprozessen hingegen ist die Literatur weniger umfangreich und es gibt keine systematischen Arbeiten, welche die Ergebnisse von BGWc Prozessen mit denen der Feller Diffusionsprozesse verbinden. Wir versuchen hiermit diese Lücke zu schliessen. Im ersten Teil dieser Arbeit untersuchen wir das Verhalten von Populationen vor ihrem Aussterben, indem wir das zeitasymptotysche Verhalten des auf Überleben bedingten zugehörigen Verzweigungsprozesses (X_t|X_t≠0)_t betrachten (oder allgemeiner auf Überleben in naher Zukunft 0≤θ<∞, (X_t|X_{t+θ}≠0)_t). Wir beweisen das Ergebnis, neuartig im mehrtypen Rahmen und für θ>0, dass dieser Grenzwert existiert und nicht-degeneriert ist. Dies spiegelt ein stationäres Verhalten für auf Überleben bedingte Bevölkerungsdynamiken wider und liefert eine Verallgemeinerung des sogenannten Yaglom Grenzwertes (welcher dem Fall θ=0 entspricht). In einem zweiten Schritt studieren wir das Verhalten der Populationen im Falle eines sehr späten Aussterbens, welches wir durch den Grenzübergang auf θ→∞ erhalten. Der resultierende Grenzwertprozess ist ein bekanntes Objekt im eintypen Fall (oftmals als Q-Prozess bezeichnet) und wurde ebenfalls im Fall von mehrtyp-Feller-Diffusionsprozessen studiert. Wir untersuchen den bisher nicht betrachteten Fall, in dem X_t ein mehrtyp-BGWc Prozess ist und beweisen die Existenz des zugehörigen Q-Prozesses. Darüber hinaus untersuchen wir seine Eigenschaften einschließlich der asymptotischen und weisen auf mehrere Auslegungen hin. Schließlich interessieren wir uns für die Austauschbarkeit der Grenzwerte in t und θ, und die Vertauschbarkeit dieser Grenzwerte in Bezug auf die Beziehung zwischen BGWc und Feller Prozessen. Wir beweisen die Durchführbarkeit aller möglichen Grenzwertvertauschungen (Langzeitverhalten, wachsende Aussterbeverzögerung, Diffusionslimit). Der zweite Teil dieser Arbeit ist der Risikoanalyse in Bezug auf das Aussterben und das sehr späte Aussterben von Populationen gewidmet. Wir untersuchen ein Modell einer verzweigten Bevölkerung (welches vor allem im epidemiologischen Rahmen erscheint), für welche ein Parameter der Reproduktionsverteilung unbekannt ist. Wir konstruieren Schätzer, die an die jeweiligen Stufen der Evolution adaptiert sind (Wachstumsphase, Verfallphase sowie die Verfallphase, wenn das Aussterben sehr spät erwartet wird), und beweisen zudem deren asymptotische Eigenschaften (Konsistenz, Normalverteiltheit). Im Besonderen bauen wir einen für Q-Prozesse adaptierten kleinste-Quadrate-Schätzer, der eine Vorhersage der Bevölkerungsentwicklung im Fall eines sehr späten Aussterbens erlaubt. Dies entspricht dem Best- oder Worst-Case-Szenario, abhängig davon, ob die Bevölkerung bedroht oder invasiv ist. Diese Instrumente ermöglichen uns die Betrachtung der Aussterbensphase der Bovinen spongiformen Enzephalopathie Epidemie in Großbritannien. Wir schätzen den Infektionsparameter in Bezug auf mögliche bestehende Quellen der horizontalen Infektion nach der Beseitigung des primären Infektionsweges (Tiermehl) im Jahr 1988. Dies ermöglicht uns eine Vorhersage des Verlaufes der Krankheit inklusive des Jahres des Aussterbens, der Anzahl von zukünftigen Fällen sowie der Anzahl infizierter Tiere. Insbesondere ermöglicht es uns die Erstellung einer sehr detaillierten Analyse des Epidemieverlaufs im unwahrscheinlichen Fall eines sehr späten Aussterbens.
KW  - Mehrtyp-Verzweigungsprozesse
KW  - Feller Diffusionsprozesse
KW  - Schätzung von Verzweigungsprozessen
KW  - Epidemiologie
KW  - Risikoanalyse
KW  - Multitype branching processes
KW  - Feller diffusion processes
KW  - Estimation for branching processes
KW  - Epidemiology
KW  - Risk analysis
Y1  - 2010
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-45307
ER  - 
TY  - THES
A1  - Jankowski, £ukasz
T1  - Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres
N2  - Kunststofflichtwellenleiter (POFs) stellen ein verhältnismäßig neues Medium zur optische Datenkommunikation über kurzen Strecken dar. Während ihrer Einsatzdauer unterliegen POFs unterschiedlichen Arten von Umweltbeanspruchungen, hauptsächlich durch hohe Temperatur, hohe Feuchtigkeit und mechanischen Belastungen. Zahlreiche experimentelle Forschungen beschäftigten sich mit der standardisierten Prüfung der Zuverlässigkeit von im Handel erhältlichen Fasern. Jedoch gab es bisher wenig Erfolg bei der Bemühung, zwei grundlegende optische Erscheinungen, Absorption und Streuung, die die Lichtausbreitung in Fasern stark beeinflussen, zu verstehen und praktisch zu modellieren: Diese beiden Effekte beschreiben nicht nur die Qualität neuer Fasern, sondern sie werden auch stark durch die Alterungsprozess beeinflusst.  Der Hauptzweck dieser Doktorarbeit war es, ein praktisch verwendbares und theoretisch gut fundiertes Modell der Lichtausbreitung in nicht gealterten und gealterten POFs zu entwickeln und es durch optische Experimente zu verifizieren. Dabei wurden anwendungsorientierte Aspekte mit theoretischer POF-Modellierung kombiniert. Die Arbeit enthält die erste bekannte Anwendung der Wellenanalyse zur Untersuchung der winkelabhängigen Eigenschaften der Streuung in Lichtwellenleitern.  Für die praktischen Experimente wurden mehrere POF-Proben unterschiedlicher Hersteller künstlich gealtert, indem sie bis 4500 Stunden bei 100 °C gelagert wurden (ohne Feuchtekontrolle). Die Parameter der jeweiligen Simulationen wurden mittels einer systematischen Optimierung an die gemessen optischen Eigenschaften der gealterten Proben angeglichen. Die Resultate deuten an, dass der Übertragungsverlust der gealterten Fasern in den ersten Tagen und Wochen der Alterung am stärksten durch eine wesentliche physikalische Verschlechterung der Kern-Mantel-Grenzfläche verursacht wird. Chemische Effekte des Alterungsprozesses scheinen im Faserkernmaterial zuerst nach einigen Monaten aufzutreten.
N2  - This thesis discusses theoretical and practical aspects of modelling of light propagation in non-aged and aged step-index polymer optical fibres (POFs). Special attention has been paid in describing optical characteristics of non-ideal fibres, scattering and attenuation, and in combining application-oriented and theoretical approaches. The precedence has been given to practical issues, but much effort has been also spent on the theoretical analysis of basic mechanisms governing light propagation in cylindrical waveguides.  As a result a practically usable general POF model based on the raytracing approach has been developed and implemented. A systematic numerical optimisation of its parameters has been performed to obtain the best fit between simulated and measured optical characteristics of numerous non-aged and aged fibre samples. The model was verified by providing good agreement, especially for the non-aged fibres. The relations found between aging time and optimal values of model parameters contribute to a better understanding of the aging mechanisms of POFs.
T2  - Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres
KW  - Modellierung
KW  - Lichtwellenleiter
KW  - Kunststofflichtwellenleiter
KW  - optische Fasern
KW  - POF
KW  - Streuung
KW  - Temperatur
KW  - Alterung
KW  - modelling optical fibres waveguides pof scattering temperature aging ageing
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-0001649
ER  - 
TY  - BOOK
A1  - Jankowski, £ukasz
T1  - Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres. - [überarb. Diss.]
N2  - Kunststofflichtwellenleiter (POFs) stellen ein verhältnismäßig neues Medium zur optische Datenkommunikation über kurzen Strecken dar. Während ihrer Einsatzdauer unterliegen POFs unterschiedlichen Arten von Umweltbeanspruchungen, hauptsächlich durch hohe Temperatur, hohe Feuchtigkeit und mechanischen Belastungen. Zahlreiche experimentelle Forschungen beschäftigten sich mit der standardisierten Prüfung der Zuverlässigkeit von im Handel erhältlichen Fasern. Jedoch gab es bisher wenig Erfolg bei der Bemühung, zwei grundlegende optische Erscheinungen, Absorption und Streuung, die die Lichtausbreitung in Fasern stark beeinflussen, zu verstehen und praktisch zu modellieren: Diese beiden Effekte beschreiben nicht nur die Qualität neuer Fasern, sondern sie werden auch stark durch die Alterungsprozess beeinflusst. Der Hauptzweck dieser Doktorarbeit war es, ein praktisch verwendbares und theoretisch gut fundiertes Modell der Lichtausbreitung in nicht gealterten und gealterten POFs zu entwickeln und es durch optische Experimente zu verifizieren. Dabei wurden anwendungsorientierte Aspekte mit theoretischer POF-Modellierung kombiniert. Die Arbeit enthält die erste bekannte Anwendung der Wellenanalyse zur Untersuchung der winkelabhängigen Eigenschaften der Streuung in Lichtwellenleitern. Für die praktischen Experimente wurden mehrere POF-Proben unterschiedlicher Hersteller künstlich gealtert, indem sie bis 4500 Stunden bei 100 °C gelagert wurden (ohne Feuchtekontrolle). Die Parameter der jeweiligen Simulationen wurden mittels einer systematischen Optimierung an die gemessen optischen Eigenschaften der gealterten Proben angeglichen. Die Resultate deuten an, dass der Übertragungsverlust der gealterten Fasern in den ersten Tagen und Wochen der Alterung am stärksten durch eine wesentliche physikalische Verschlechterung der Kern-Mantel-Grenzfläche verursacht wird. Chemische Effekte des Alterungsprozesses scheinen im Faserkernmaterial zuerst nach einigen Monaten aufzutreten.
N2  - This thesis discusses theoretical and practical aspects of modelling of light propagation in non-aged and aged step-index polymer optical fibres (POFs). Special attention has been paid in describing optical characteristics of non-ideal fibres, scattering and attenuation, and in combining application-oriented and theoretical approaches. The precedence has been given to practical issues, but much effort has been also spent on the theoretical analysis of basic mechanisms governing light propagation in cylindrical waveguides.As a result a practically usable general POF model based on the raytracing approach has been developed and implemented. A systematic numerical optimisation of its parameters has been performed to obtain the best fit between simulated and measured optical characteristics of numerous non-aged and aged fibre samples. The model was verified by providing good agreement, especially for the non-aged fibres. The relations found between aging time and optimal values of model parameters contribute to a better understanding of the aging mechanisms of POFs.
T2  - Modelling and simulation of light propagation in non-aged and aged step-index polymer optical fibres
KW  - Modellierung
KW  - Lichtwellenleiter
KW  - Kunststofflichtwellenleiter
KW  - optische Fasern
KW  - POF
KW  - Streuung
KW  - Temperatur
KW  - Alterung
KW  - modelling optical fibres waveguides pof scattering temperature aging ageing
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-0001659
ER  - 
TY  - THES
A1  - Becker, Christian
T1  - On the Riemannian geometry of Seiberg-Witten moduli spaces
T1  - Über die Riemannsche Geometrie von Seiberg-Witten-Modulräumen
N2  - In this thesis, we give two constructions for Riemannian metrics on Seiberg-Witten moduli spaces. Both these constructions are naturally induced from the L2-metric on the configuration space. The construction of the so called quotient L2-metric is very similar to the one construction of an L2-metric on Yang-Mills moduli spaces as given by Groisser and Parker. To construct a Riemannian metric on the total space of the Seiberg-Witten bundle in a similar way, we define the reduced gauge group as a subgroup of the gauge group. We show, that the quotient of the premoduli space by the reduced gauge group is isomorphic as a U(1)-bundle to the quotient of the premoduli space by the based gauge group. The total space of this new representation of the Seiberg-Witten bundle carries a natural quotient L2-metric, and the bundle projection is a Riemannian submersion with respect to these metrics. We compute explicit formulae for the sectional curvature of the moduli space in terms of Green operators of the elliptic complex associated with a monopole. Further, we construct a Riemannian metric on the cobordism between moduli spaces for different perturbations. The second construction of a Riemannian metric on the moduli space uses a canonical global gauge fixing, which represents the total space of the Seiberg-Witten bundle as a finite dimensional submanifold of the configuration space. We consider the Seiberg-Witten moduli space on a simply connected K&auml;uhler surface. We show that the moduli space (when nonempty) is a complex projective space, if the perturbation does not admit reducible monpoles, and that the moduli space consists of a single point otherwise. The Seiberg-Witten bundle can then be identified with the Hopf fibration. On the complex projective plane with a special Spin-C structure, our Riemannian metrics on the moduli space are Fubini-Study metrics. Correspondingly, the metrics on the total space of the Seiberg-Witten bundle are Berger metrics. We show that the diameter of the moduli space shrinks to 0 when the perturbation approaches the wall of reducible perturbations. Finally we show, that the quotient L2-metric on the Seiberg-Witten moduli space on a K&auml;hler surface is a K&auml;hler metric.
N2  - In dieser Dissertationsschrift geben wir zwei Konstruktionen Riemannscher Metriken auf Seiberg-Witten-Modulr&auml;umen an. Beide Metriken werden in nat&uuml;rlicher Weise durch die L2-Metrik des Konfiguartionsraumes induziert. Die Konstruktion der sogenannten Quotienten-L2-Metrik entspricht der durch Groisser und Parker angegebenen Konstruktion einer L2-Metrik auf Yang-Mills-Modulr&auml;umen. Zur Konstruktion einer Quotienten-Metrik auf dem Totalraum des Seiberg-Witten-B&uuml;ndels f&uuml;hren wir die sogenannte reduzierte Eichgruppe ein. Wir zeigen, dass der Quotient des Pr&auml;modulraumes nach der reduzierten Eichgruppe als U(1)-B&uuml;ndel isomorph ist zu dem Quotienten nach der basierten Eichgruppe. Dadurch tr&auml;gt der Totalraum des Seiberg-Witten B&uuml;ndels eine nat&uuml;rliche Quotienten-L2-Metrik, bzgl. derer die B&uuml;ndelprojektion eine Riemannsche Submersion ist. Wir berechnen explizite Formeln f&uuml;r die Schnittr&uuml;mmung des Modulraumes in Ausdr&uuml;cken der Green-Operatoren des zu einem Monopol geh&ouml;rigen elliptischen Komplexes. Ferner konstruieren wir eine Riemannsche Metrik auf dem Kobordismus zwischen Modulr&auml;umen zu verschiedenen St&ouml;rungen. Die zweite Konstruktion einer Riemannschen Metrik auf Seiberg-Witten-Modulr&auml;umen benutzt eine kanonische globale Eichfixierung, verm&ouml;ge derer der Totalraum des Seiberg-Witten-B&uuml;ndels als endlich-dimensionale Untermannigfaltigkeit des Konfigurationsraumes dargestellt werden kann. Wir betrachten speziell die Seiberg-Witten-Modulr&auml;ume auf einfach zusammenh&auml;ngenden K&auml;hler-Mannigfaltigkeiten. Wir zeigen, dass der Seiberg-Witten-Modulraum (falls nicht-leer) im irreduziblen Fall ein komplex projektiver Raum its und im reduziblen Fall aus einem einzelnen Punkt besteht. Das Seiberg-Witten-B&uuml;ndel l&auml;&szlig;t sich mit der Hopf-Faserung identifizieren. Die L2-Metrik des Modulraumes auf der komplex projektiven Fl&auml;che CP2 (mit einer speziellen Spin-C-Struktur) ist die Fubini-Study-Metrik; entsprechend sind die Metriken auf dem Totalraum Berger-Metriken. Wir zeigen, dass der Durchmesser des Modulraumes gegen 0 konvergiert, wenn die St&ouml;rung sich dem reduziblen Fall n&auml;hert. Schlie&szlig;lich zeigen wir, dass die Quotienten-L2-Metrik auf dem Seiberg-Witten-Modulraum einer K&auml;hlerfl&auml;che eine K&auml;hler-Metrik ist.
KW  - Eichtheorie
KW  - Seiberg-Witten-Invariante
KW  - Modulraum
KW  - Riemannsche Geometrie
KW  - Kähler-Mannigfaltigkeit
KW  - Unendlichdimensionale Mannigfaltigkeit
KW  - L2-Metrik
KW  - 4-Mannigfaltigkeiten
KW  - Gauge theory
KW  - Seiberg-Witten theory
KW  - Moduli spaces
KW  - Infinite dimensional manifolds
KW  - L2 metrics
Y1  - 2005
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-5425
ER  - 
TY  - GEN
A1  - Ginoux, Nicolas
T1  - Une nouvelle estimation extrinsèque du spectre de l'opérateur de Dirac
T1  - A new extrinsic estimate for the spectrum of the Dirac operator
N2  - Nous établissons une nouvelle majoration optimale pour les plus petites valeurs propres de l'opérateur de Dirac sur une hypersurface compacte de l'espace hyperbolique.
N2  - We prove a new upper bound for the smallest eigenvalues of the Dirac operator on a compact hypersurface of the hyperbolic space.
KW  - bounds
KW  - Eigenvalues
Y1  - 2003
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-5644
ER  - 
TY  - GEN
A1  - Roelly, Sylvie
A1  - Sortais, Michel
T1  - Space-time asymptotics of an infinite-dimensional diffusion having a long- range memory
N2  - We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory; these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.
KW  - Random Field Ising Model
KW  - Langevin Dynamics
KW  - Interacting Diffusion Processes
KW  - Space-Time Cluster Expansions
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6700
ER  - 
TY  - GEN
A1  - Roelly, Sylvie
A1  - Thieullen, Michèle
T1  - Duality formula for the bridges of a Brownian diffusion : application to gradient drifts
N2  - In this paper, we consider families of time Markov fields (or reciprocal classes) which have the same bridges as a Brownian diffusion. We characterize each class as the set of solutions of an integration by parts formula on the space of continuous paths C[0; 1]; R-d) Our techniques provide a characterization of gradient diffusions by a duality formula and, in case of reversibility, a generalization of a result of Kolmogorov.
KW  - reciprocal processes
KW  - stochastic bridge
KW  - mixture of bridges
KW  - integration by parts formula
KW  - Malliavin calculus
KW  - entropy
KW  - time reversal
Y1  - 2005
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6710
ER  - 
TY  - GEN
A1  - Louis, Pierre-Yves
T1  - Increasing coupling for probabilistic cellular automata
N2  - We give a necessary and sufficient condition for the existence of an increasing coupling of N (N >= 2) synchronous dynamics on S-Zd (PCA). Increasing means the coupling preserves stochastic ordering. We first present our main construction theorem in the case where S is totally ordered; applications to attractive PCAs are given. When S is only partially ordered, we show on two examples that a coupling of more than two synchronous dynamics may not exist. We also prove an extension of our main result for a particular class of partially ordered spaces.
KW  - Wahrscheinlichkeitstheorie
KW  - stochastische Anordnung
KW  - stochastische Zellulare Automaten
KW  - Kopplung
KW  - stochastic ordering
KW  - Probabilistic Cellular Automata
KW  - monotone coupling
Y1  - 2005
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6593
ER  -