TY  - JOUR
A1  - Boldrighini, Carlo
A1  - Frigio, Sandro
A1  - Maponi, Pierluigi
A1  - Pellegrinotti, Alessandro
A1  - Sinai, Yakov G.
T1  - 3-D incompressible Navier-Stokes equations: Complex blow-up and related real flows
JF  - Lectures in pure and applied mathematics
KW  - random point processes
KW  - statistical mechanics
KW  - stochastic analysis
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472201
SN  - 978-3-86956-485-2
SN  - 2199-4951
SN  - 2199-496X
IS  - 6
SP  - 185
EP  - 194
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - JOUR
A1  - Mariucci, Ester
A1  - Ray, Kolyan
A1  - Szabo, Botond
T1  - A Bayesian nonparametric approach to log-concave density estimation
JF  - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
N2  - The estimation of a log-concave density on R is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.
KW  - convergence rate
KW  - density estimation
KW  - Dirichlet mixture
KW  - log-concavity
KW  - nonparametric hypothesis testing
KW  - posterior distribution
Y1  - 2020
U6  - https://doi.org/10.3150/19-BEJ1139
SN  - 1350-7265
SN  - 1573-9759
VL  - 26
IS  - 2
SP  - 1070
EP  - 1097
PB  - International Statistical Institute
CY  - The Hague
ER  - 
TY  - INPR
A1  - Aizenberg, Lev A.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Bohr phenomenon for elliptic equations
N2  - In 1914 Bohr proved that there is an r ∈ (0, 1) such that if a power series converges in the unit disk and its sum has modulus less than 1 then, for |z| < r, the sum of absolute values of its terms is again less than 1. Recently analogous results were obtained for functions of several variables. The aim of this paper is to comprehend the theorem of Bohr in the context of solutions to second order elliptic equations meeting the maximum principle.
T3  - Preprint - (1999) 18 
Y1  - 1999
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25547
ER  - 
TY  - INPR
A1  - Witt, Ingo
T1  - A calculus for a class of finitely degenerate pseudodifferential operators
N2  - For a class of degenerate pseudodifferential operators, local parametrices are constructed. This is done in the framework of a pseudodifferential calculus upon adding conditions of trace and potential type, respectively, along the boundary on which the operators degenerate.
T3  - Preprint - (2002) 05 
Y1  - 2002
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26246
ER  - 
TY  - INPR
A1  - Rabinovich, Vladimir
A1  - Schulze, Bert-Wolfgang
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A calculus of boundary value problems in domains with Non-Lipschitz Singular Points
N2  - The paper is devoted to pseudodifferential boundary value problems in domains with singular points on the boundary. The tangent cone at a singular point is allowed to degenerate. In particular, the boundary may rotate and oscillate in a neighbourhood of such a point. We show a criterion for the Fredholm property of a boundary value problem and derive estimates of solutions close to singular points.
T3  - Preprint - (1997) 09 
KW  - pseudodifferential operators
KW  - boundary value problems
KW  - manifolds with cusps
Y1  - 1997
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-24957
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  - 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  - JOUR
A1  - Herzog, Moritz
A1  - Ehlert, Antje
A1  - Fritz, Annemarie
T1  - A Competency Model of Place Value Understanding in South African Primary School Pupils
JF  - African Journal of Research in Mathematics, Science and Technology Education
N2  - Although the general development of mathematical abilities in primary school has been the focus of many researchers, the development of place value understanding has rarely been investigated to date. This is possibly due to the lack of conceptual approaches and empirical studies related to this topic. To fill this gap, a theory-driven and empirically validated model was developed that describes five sequential conceptual levels of place value understanding. The level sequence model gives us the ability to estimate general abilities and difficulties in primary school pupils in the development of a conceptual place value understanding. The level sequence model was tried and tested in Germany, and given that number words are very differently constructed in German and in the languages used in South African classrooms, this study aims to investigate whether this level sequence model can be transferred to South Africa. The findings based on the responses of 198 Grade 2-4 learners show that the English translation of the test items results in the same item level allocation as the original German test items, especially for the three basic levels. Educational implications are provided, in particular concrete suggestions on how place value might be taught according to the model and how to collect specific empirical data related to place value understanding.
KW  - Place value
KW  - Rasch test modelling
KW  - mathematical concepts
Y1  - 2017
U6  - https://doi.org/10.1080/18117295.2017.1279453
SN  - 1811-7295
SN  - 2469-7656
VL  - 21
SP  - 37
EP  - 48
PB  - Routledge, Taylor & Francis Group
CY  - Abingdon
ER  - 
TY  - JOUR
A1  - Stauffer, Maxime
A1  - Mengesha, Isaak
A1  - Seifert, Konrad
A1  - Krawczuk, Igor
A1  - Fischer, Jens
A1  - Serugendo, Giovanna Di Marzo
T1  - A computational turn in policy process studies
BT  - coevolving network dynamics of policy change
JF  - Complexity
N2  - The past three decades of policy process studies have seen the emergence of a clear intellectual lineage with regard to complexity. Implicitly or explicitly, scholars have employed complexity theory to examine the intricate dynamics of collective action in political contexts. However, the methodological counterparts to complexity theory, such as computational methods, are rarely used and, even if they are, they are often detached from established policy process theory. Building on a critical review of the application of complexity theory to policy process studies, we present and implement a baseline model of policy processes using the logic of coevolving networks. Our model suggests that an actor's influence depends on their environment and on exogenous events facilitating dialogue and consensus-building. Our results validate previous opinion dynamics models and generate novel patterns. Our discussion provides ground for further research and outlines the path for the field to achieve a computational turn.
Y1  - 2022
U6  - https://doi.org/10.1155/2022/8210732
SN  - 1076-2787
SN  - 1099-0526
VL  - 2022
PB  - Wiley-Hindawi
CY  - London
ER  - 
TY  - INPR
A1  - Cattiaux, Patrick
A1  - Dai Pra, Paolo
A1  - Poelly, Sylvie
T1  - A constructive approach to a class of ergodic HJB equations with nonsmooth cost
N2  - We consider a class of ergodic Hamilton-Jacobi-Bellman (HJB) equations, related to large time asymptotics of non-smooth multiplicative functional of difusion processes. Under suitable ergodicity assumptions on the underlying difusion, we show existence of these asymptotics, and that they solve the related HJB equation in the viscosity sense.
T3  - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2007, 02 
Y1  - 2007
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49430
ER  - 
TY  - JOUR
A1  - Cvetković, Nada
A1  - Conrad, Tim
A1  - Lie, Han Cheng
T1  - A convergent discretization method for transition path theory for diffusion processes
JF  - Multiscale modeling & simulation : a SIAM interdisciplinary journal
N2  - Transition path theory (TPT) for diffusion processes is a framework for analyzing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretization of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretization. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.
KW  - ergodic diffusion processes
KW  - transition paths
KW  - rare events
KW  - Monte Carlo
KW  - methods
Y1  - 2021
U6  - https://doi.org/10.1137/20M1329354
SN  - 1540-3459
SN  - 1540-3467
VL  - 19
IS  - 1
SP  - 242
EP  - 266
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - INPR
A1  - Gauthier, Paul M.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A covering property of the Riemann zeta-function
N2  - For each compact subset K of the complex plane C which does not surround zero, the Riemann surface Sζ of the Riemann zeta function restricted to the critical half-strip 0 < Rs < 1/2 contains infinitely many schlicht copies of K lying ‘over’ K. If Sζ also contains at least one such copy, for some K which surrounds zero, then the Riemann hypothesis fails.
T3  - Preprint - (2004) 03 
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26683
ER  - 
TY  - INPR
A1  - Korey, Michael Brian
T1  - A decomposition of functions with vanishing mean oscillation
N2  - A function has vanishing mean oscillation (VMO) on R up(n) if its mean oscillation - the local average of its pointwise deviation from its mean value - both is uniformly bounded over all cubes within R up(n) and converges to zero with the volume of the cube. The more restrictive class of functions with vanishing lower oscillation (VLO) arises when the mean value is replaced by the minimum value in this definition. It is shown here that each VMO function is the difference of two functions in VLO.
T3  - Preprint - (2001) 02 
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25929
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  - Andjelkovic, Marko
A1  - Simevski, Aleksandar
A1  - Chen, Junchao
A1  - Schrape, Oliver
A1  - Stamenkovic, Zoran
A1  - Krstić, Miloš
A1  - Ilic, Stefan
A1  - Ristic, Goran
A1  - Jaksic, Aleksandar
A1  - Vasovic, Nikola
A1  - Duane, Russell
A1  - Palma, Alberto J.
A1  - Lallena, Antonio M.
A1  - Carvajal, Miguel A.
T1  - A design concept for radiation hardened RADFET readout system for space applications
JF  - Microprocessors and microsystems
N2  - Instruments for measuring the absorbed dose and dose rate under radiation exposure, known as radiation dosimeters, are indispensable in space missions. They are composed of radiation sensors that generate current or voltage response when exposed to ionizing radiation, and processing electronics for computing the absorbed dose and dose rate. Among a wide range of existing radiation sensors, the Radiation Sensitive Field Effect Transistors (RADFETs) have unique advantages for absorbed dose measurement, and a proven record of successful exploitation in space missions. It has been shown that the RADFETs may be also used for the dose rate monitoring. In that regard, we propose a unique design concept that supports the simultaneous operation of a single RADFET as absorbed dose and dose rate monitor. This enables to reduce the cost of implementation, since the need for other types of radiation sensors can be minimized or eliminated. For processing the RADFET's response we propose a readout system composed of analog signal conditioner (ASC) and a self-adaptive multiprocessing system-on-chip (MPSoC). The soft error rate of MPSoC is monitored in real time with embedded sensors, allowing the autonomous switching between three operating modes (high-performance, de-stress and fault-tolerant), according to the application requirements and radiation conditions.
KW  - RADFET
KW  - Radiation hardness
KW  - Absorbed dose
KW  - Dose rate
KW  - Self-adaptive MPSoC
Y1  - 2022
U6  - https://doi.org/10.1016/j.micpro.2022.104486
SN  - 0141-9331
SN  - 1872-9436
VL  - 90
PB  - Elsevier
CY  - Amsterdam
ER  - 
TY  - INPR
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A fixed point formula in one complex variable
N2  - We show a Lefschetz fixed point formula for holomorphic functions in a bounded domain D with smooth boundary in the complex plane. To introduce the Lefschetz number for a holomorphic map of D, we make use of the Bergman kernal of this domain. The Lefschetz number is proved to be the sum of usual contributions of fixed points of the map in D and contributions of boundary fixed points, these latter being different for attracting and repulsing fixed points.
T3  - Preprint - (2003) 01 
Y1  - 2003
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26495
ER  - 
TY  - INPR
A1  - Fedosov, Boris
A1  - Schulze, Bert-Wolfgang
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A general index formula on tropic manifolds with conical points
N2  - We solve the index problem for general elliptic pseudodifferential operators on toric manifolds with conical points.
T3  - Preprint - (1999) 15 
Y1  - 1999
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25501
ER  - 
TY  - INPR
A1  - Alsaedy, Ammar
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Hilbert boundary value problem for generalised Cauchy-Riemann equations
N2  - We elaborate a boundary Fourier method for studying an analogue of the Hilbert problem for analytic functions within the framework of generalised Cauchy-Riemann equations. The boundary value problem need not satisfy the Shapiro-Lopatinskij condition and so it fails to be Fredholm in Sobolev spaces. We show a solvability condition of the Hilbert problem, which looks like those for ill-posed
problems, and construct an explicit formula for approximate solutions.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 1 
KW  - Dirac operator
KW  - Clifford algebra
KW  - Riemann-Hilbert problem
KW  - Fredholm operator
Y1  - 2016
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-86109
SN  - 2193-6943
VL  - 5
IS  - 1
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - JOUR
A1  - Kempton, Mark
A1  - Münch, Florentin
A1  - Yau, Shing-Tung
T1  - A homology vanishing theorem for graphs with positive curvature
JF  - Communications in analysis and geometry
N2  - We prove a homology vanishing theorem for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Bochner on manifolds [3]. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau [11]. We moreover prove that the fundamental group is finite for graphs with positive Bakry-' Emery curvature, analogous to a classic result of Myers on manifolds [22]. The proofs draw on several separate areas of graph theory, including graph coverings, gain graphs, and cycle spaces, in addition to the Bakry-Emery curvature, path homology, and graph homotopy. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a graph with positive curvature cannot have a non-trivial infinite cover preserving 3-cycles and 4-cycles, and give a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. Furthermore, we relate gain graphs to graph homotopy and the fundamental group developed by Grigor'yan, Lin, Muranov, and Yau [12], and obtain an alternative proof of their result that the abelianization of the fundamental group of a graph is isomorphic to the first path homology over the integers.
Y1  - 2021
UR  - https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/2021/0029/0006/CAG-2021-0029-0006-a005.pdf
U6  - https://doi.org/10.4310/CAG.2021.v29.n6.a5
SN  - 1019-8385
SN  - 1944-9992
VL  - 29
IS  - 6
SP  - 1449
EP  - 1473
PB  - International Press of Boston
CY  - Somerville
ER  - 
TY  - INPR
A1  - Méléard, Sylvie
A1  - Roelly, Sylvie
T1  - A host-parasite multilevel interacting process and continuous approximations
N2  - We are interested in modeling some two-level population dynamics, resulting from the interplay of ecological interactions and phenotypic variation of individuals (or hosts) and the evolution of cells (or parasites) of two types living in these individuals. The ecological parameters of the individual dynamics depend on the number of cells of each type contained by the individual and the cell dynamics depends on the trait of the invaded individual. 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 look 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. The study of the long time behavior of these processes seems very hard and we only develop some simple cases enlightening the difficulties involved.
T3  - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2011, 01 
KW  - two-level interacting processes
KW  - birth-death-mutation-competition point process
KW  - host-parasite stochastic particle system
Y1  - 2011
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-51694
ER  - 
TY  - GEN
A1  - Wallenta, Daniel
T1  - A Lefschetz fixed point formula for elliptic quasicomplexes
T2  - Postprints der Universität Potsdam : Mathematisch Naturwissenschaftliche Reihe
N2  - In a recent paper, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 885 
KW  - elliptic complexes
KW  - Fredholm complexes
KW  - Lefschetz number
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-435471
SN  - 1866-8372
IS  - 885
SP  - 577
EP  - 587
ER  - 
TY  - INPR
A1  - Wallenta, Daniel
T1  - A Lefschetz fixed point formula for elliptic quasicomplexes
N2  - In a recent paper with N. Tarkhanov, the Lefschetz number for endomorphisms (modulo trace class operators) of sequences of trace class curvature was introduced. We show that this is a well defined, canonical extension of the classical Lefschetz number and establish the homotopy invariance of this number. Moreover, we apply the results to show that the Lefschetz fixed point formula holds for geometric quasiendomorphisms of elliptic quasicomplexes.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2(2013)12 
KW  - Perturbed complexes
KW  - curvature
KW  - Lefschetz number
KW  - fixed point formula
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-67016
ER  - 
TY  - INPR
A1  - Schulze, Bert-Wolfgang
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Lefschetz fixed point formula in the relative elliptic theory
N2  - A version of the classical Lefschetz fixed point formula is proved for the cohomology of the cone of a cochain mapping of elliptic complexes. As a particular case we show a Lefschetz formula for the relative de Rham cohomology.
T3  - Preprint - (1998) 01 
KW  - elliptic complexes
KW  - relative cohomology
KW  - Lefschetz number
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25159
ER  - 
TY  - INPR
A1  - Nazaikinskii, Vladimir
A1  - Schulze, Bert-Wolfgang
A1  - Sternin, Boris
A1  - Shatalov, Victor
T1  - A Lefschetz fixed point theorem for manifolds with conical singularities
N2  - We establish an Atiyah-Bott-Lefschetz formula for elliptic operators on manifolds with conical singular points.
T3  - Preprint - (1997) 20 
KW  - elliptic operator
KW  - Fredholm property
KW  - conical singularities
KW  - pseudo-diferential operators
KW  - Lefschetz fixed point formula
KW  - regularizer
Y1  - 1997
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25073
ER  - 
TY  - JOUR
A1  - Malem-Shinitski, Noa
A1  - Opper, Manfred
A1  - Reich, Sebastian
A1  - Schwetlick, Lisa
A1  - Seelig, Stefan A.
A1  - Engbert, Ralf
T1  - A mathematical model of local and global attention in natural scene viewing
JF  - PLoS Computational Biology : a new community journal
N2  - Author summary <br /> Switching between local and global attention is a general strategy in human information processing. We investigate whether this strategy is a viable approach to model sequences of fixations generated by a human observer in a free viewing task with natural scenes. Variants of the basic model are used to predict the experimental data based on Bayesian inference. Results indicate a high predictive power for both aggregated data and individual differences across observers. The combination of a novel model with state-of-the-art Bayesian methods lends support to our two-state model using local and global internal attention states for controlling eye movements. <br /> Understanding the decision process underlying gaze control is an important question in cognitive neuroscience with applications in diverse fields ranging from psychology to computer vision. The decision for choosing an upcoming saccade target can be framed as a selection process between two states: Should the observer further inspect the information near the current gaze position (local attention) or continue with exploration of other patches of the given scene (global attention)? Here we propose and investigate a mathematical model motivated by switching between these two attentional states during scene viewing. The model is derived from a minimal set of assumptions that generates realistic eye movement behavior. We implemented a Bayesian approach for model parameter inference based on the model's likelihood function. In order to simplify the inference, we applied data augmentation methods that allowed the use of conjugate priors and the construction of an efficient Gibbs sampler. This approach turned out to be numerically efficient and permitted fitting interindividual differences in saccade statistics. Thus, the main contribution of our modeling approach is two-fold; first, we propose a new model for saccade generation in scene viewing. Second, we demonstrate the use of novel methods from Bayesian inference in the field of scan path modeling.
Y1  - 2020
U6  - https://doi.org/10.1371/journal.pcbi.1007880
SN  - 1553-734X
SN  - 1553-7358
VL  - 16
IS  - 12
PB  - PLoS
CY  - San Fransisco
ER  - 
TY  - JOUR
A1  - Pornsawad, Pornsarp
A1  - Sapsakul, Nantawan
A1  - Böckmann, Christine
T1  - A modified asymptotical regularization of nonlinear ill-posed problems
JF  - Mathematics
N2  - In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.
KW  - nonlinear operator
KW  - regularization
KW  - discrepancy principle
KW  - asymptotic method
KW  - optimal rate
Y1  - 2019
U6  - https://doi.org/10.3390/math7050419
SN  - 2227-7390
VL  - 7
PB  - MDPI
CY  - Basel, Schweiz
ET  - 5
ER  - 
TY  - GEN
A1  - Pornsawad, Pornsarp
A1  - Sapsakul, Nantawan
A1  - Böckmann, Christine
T1  - A modified asymptotical regularization of nonlinear ill-posed problems
T2  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - In this paper, we investigate the continuous version of modified iterative Runge–Kutta-type methods for nonlinear inverse ill-posed problems proposed in a previous work. The convergence analysis is proved under the tangential cone condition, a modified discrepancy principle, i.e., the stopping time T is a solution of ∥𝐹(𝑥𝛿(𝑇))−𝑦𝛿∥=𝜏𝛿+ for some 𝛿+>𝛿, and an appropriate source condition. We yield the optimal rate of convergence.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1335 
KW  - nonlinear operator
KW  - regularization
KW  - discrepancy principle
KW  - asymptotic method
KW  - optimal rate
Y1  - 2019
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-473433
SN  - 1866-8372
IS  - 1335
ER  - 
TY  - THES
A1  - Zass, Alexander
T1  - A multifaceted study of marked Gibbs point processes
T1  - Facetten von markierten Gibbsschen Punktprozessen
N2  - This thesis focuses on the study of marked Gibbs point processes, in particular presenting some results on their existence and uniqueness, with ideas and techniques drawn from different areas of statistical mechanics: the entropy method from large deviations theory, cluster expansion and the Kirkwood--Salsburg equations, the Dobrushin contraction principle and disagreement percolation.

We first present an existence result for infinite-volume marked Gibbs point processes. More precisely, we use the so-called entropy method (and large-deviation tools) to construct marked Gibbs point processes in R^d under quite general assumptions. In particular, the random marks belong to a general normed space S and are not bounded. Moreover, we allow for interaction functionals that may be unbounded and whose range is finite but random. The entropy method relies on showing that a family of finite-volume Gibbs point processes belongs to sequentially compact entropy level sets, and is therefore tight.

We then present infinite-dimensional Langevin diffusions, that we put in interaction via a Gibbsian description. In this setting, we are able to adapt the general result above to show the existence of the associated infinite-volume measure. We also study its correlation functions via cluster expansion techniques, and obtain the uniqueness of the Gibbs process for all inverse temperatures β and activities z below a certain threshold. This method relies in first showing that the correlation functions of the process satisfy a so-called Ruelle bound, and then using it to solve a fixed point problem in an appropriate Banach space. The uniqueness domain we obtain consists then of the model parameters z and β for which such a problem has exactly one solution.

Finally, we explore further the question of uniqueness of infinite-volume Gibbs point processes on R^d, in the unmarked setting. We present, in the context of repulsive interactions with a hard-core component, a novel approach to uniqueness by applying the discrete Dobrushin criterion to the continuum framework. We first fix a discretisation parameter a>0 and then study the behaviour of the uniqueness domain as a goes to 0. With this technique we are able to obtain explicit thresholds for the parameters z and β, which we then compare to existing results coming from the different methods of cluster expansion and disagreement percolation.

Throughout this thesis, we illustrate our theoretical results with various examples both from classical statistical mechanics and stochastic geometry.
N2  - Diese Arbeit konzentriert sich auf die Untersuchung von markierten Gibbs-Punkt-Prozessen und stellt insbesondere einige Ergebnisse zu deren Existenz und Eindeutigkeit vor. Dabei werden Ideen und Techniken aus verschiedenen Bereichen der statistischen Mechanik verwendet: die Entropie-Methode aus der Theorie der großen Abweichungen, die Cluster-Expansion und die Kirkwood-Salsburg-Gleichungen, das Dobrushin-Kontraktionsprinzip und die Disagreement-Perkolation.

Wir präsentieren zunächst ein Existenzergebnis für unendlich-volumige markierte Gibbs-Punkt-Prozesse. Genauer gesagt verwenden wir die sogenannte Entropie-Methode (und Werkzeuge der großen Abweichung), um markierte Gibbs-Punkt-Prozesse in R^d unter möglichst allgemeinen Annahmen zu konstruieren. Insbesondere gehören die zufälligen Markierungen zu einem allgemeinen normierten Raum und sind nicht beschränkt. Außerdem lassen wir Interaktionsfunktionale zu, die unbeschränkt sein können und deren Reichweite endlich, aber zufällig ist. Die Entropie-Methode beruht darauf, zu zeigen, dass eine Familie von endlich-volumigen Gibbs-Punkt-Prozessen zu sequentiell kompakten Entropie-Niveau-Mengen gehört, und daher dicht ist.

Wir stellen dann unendlich-dimensionale Langevin-Diffusionen vor, die wir über eine Gibbssche Beschreibung in Wechselwirkung setzen. In dieser Umgebung sind wir in der Lage, das vorangehend vorgestellte allgemeine Ergebnis anzupassen, um die Existenz des zugehörigen unendlich-dimensionalen Maßes zu zeigen. Wir untersuchen auch seine Korrelationsfunktionen über Cluster-Expansions Techniken und erhalten die Eindeutigkeit des Gibbs-Prozesses für alle inversen Temperaturen β und Aktivitäten z unterhalb einer bestimmten Schwelle. Diese Methode beruht darauf, zunächst zu zeigen, dass die Korrelationsfunktionen des Prozesses eine so genannte Ruelle-Schranke erfüllen, um diese dann zur Lösung eines Fixpunktproblems in einem geeigneten Banach-Raum zu verwenden. Der Eindeutigkeitsbereich, den wir erhalten, wird dann aus den Modellparametern z und β definiert, für die ein solches Problem genau eine Lösung hat.

Schließlich untersuchen wir die Frage nach der Eindeutigkeit von unendlich-volumigen Gibbs-Punkt-Prozessen auf R^d im unmarkierten Fall weiter. Im Zusammenhang mit repulsiven Wechselwirkungen basierend auf einer Hartkernkomponente stellen wir einen neuen Ansatz zur Eindeutigkeit vor, indem wir das diskrete Dobrushin-Kriterium im kontinuierlichen Rahmen anwenden. Wir legen zunächst einen Diskretisierungsparameter a>0 fest und untersuchen dann das Verhalten des Bereichs der Eindeutigkeit, wenn a gegen 0 geht. Mit dieser Technik sind wir in der Lage, explizite Schwellenwerte für die Parameter z und β zu erhalten, die wir dann mit bestehenden Ergebnissen aus den verschiedenen Methoden der Cluster-Expansion und der Disagreement-Perkolation vergleichen.

In dieser Arbeit illustrieren wir unsere theoretischen Ergebnisse mit verschiedenen Beispielen sowohl aus der klassischen statistischen Mechanik als auch aus der stochastischen Geometrie.
KW  - marked Gibbs point processes
KW  - Langevin diffusions
KW  - Dobrushin criterion
KW  - Entropy method
KW  - Cluster expansion
KW  - Kirkwood--Salsburg equations
KW  - DLR equations
KW  - Markierte Gibbs-Punkt-Prozesse
KW  - Entropiemethode
KW  - Cluster-Expansion
KW  - DLR-Gleichungen
KW  - Dobrushin-Kriterium
KW  - Kirkwood-Salsburg-Gleichungen
KW  - Langevin-Diffusions
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-512775
ER  - 
TY  - JOUR
A1  - Hedayat Mahmoudi, Mahdi
A1  - Schulze, Bert-Wolfgang
T1  - A new approach to the second order edge calculus
JF  - Journal of pseudo-differential operators and applications
N2  - We establish essential steps of an iterative approach to operator algebras, ellipticity and Fredholm property on stratified spaces with singularities of second order. We cover, in particular, corner-degenerate differential operators. Our constructions are focused on the case where no additional conditions of trace and potential type are posed, but this case works well and will be considered in a forthcoming paper as a conclusion of the present calculus.
KW  - Operators on singular manifolds
KW  - Mellin transform
KW  - Stratified spaces
KW  - Ellipticity and parametrices
Y1  - 2018
U6  - https://doi.org/10.1007/s11868-017-0191-2
SN  - 1662-9981
SN  - 1662-999X
VL  - 9
IS  - 2
SP  - 265
EP  - 300
PB  - Springer
CY  - Basel
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Münch, Florentin
T1  - A new discrete Hopf-Rinow theorem
JF  - Discrete Mathematics
N2  - We prove a version of the Hopf-Rinow theorem with respect to path metrics on discrete spaces. The novel aspect is that we do not a priori assume local finiteness but isolate a local finiteness type condition, called essentially locally finite, that is indeed necessary. As a side product we identify the maximal weight, called the geodesic weight, generating the path metric in the situation when the space is complete with respect to any of the equivalent notions of completeness proven in the Hopf-Rinow theorem. As an application we characterize the graphs for which the resistance metric is a path metric induced by the graph structure.
Y1  - 2019
U6  - https://doi.org/10.1016/j.disc.2019.03.014
SN  - 0012-365X
SN  - 1872-681X
VL  - 342
IS  - 9
SP  - 2751
EP  - 2757
PB  - Elsevier
CY  - Amsterdam
ER  - 
TY  - JOUR
A1  - Makhmudov, K. O.
A1  - Makhmudov, O. I.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A nonstandard Cauchy problem for the heat equation
JF  - Mathematical Notes
N2  - We consider the Cauchy problem for the heat equation in a cylinder C (T) = X x (0, T) over a domain X in R (n) , with data on a strip lying on the lateral surface. The strip is of the form S x (0, T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S, we derive an explicit formula for solutions of this problem.
Y1  - 2017
U6  - https://doi.org/10.1134/S0001434617070264
SN  - 0001-4346
SN  - 1573-8876
VL  - 102
SP  - 250
EP  - 260
PB  - Pleiades Publ.
CY  - New York
ER  - 
TY  - INPR
A1  - Makhmudov, K. O.
A1  - Makhmudov, O. I.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A nonstandard Cauchy problem for the heat equation
N2  - We consider a Cauchy problem for the heat equation in a cylinder X x (0,T) over a domain X in the n-dimensional space with data on a strip lying on the lateral surface. The strip is of the form 
S x (0,T), where S is an open subset of the boundary of X. The problem is ill-posed. Under natural restrictions on the configuration of S we derive an explicit formula for solutions of this problem.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015)11 
KW  - heat equation
KW  - Cauchy problem
KW  - Carleman formulas
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-83830
SN  - 2193-6943
VL  - 4
IS  - 11
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Liero, Hannelore
T1  - A Note on : testing the Copula Based on Densities
N2  - We consider the problem of testing whether the density of a mul- tivariate random variable can be expressed by a prespecified copula function and the marginal densities. The proposed test procedure is based on the asymptotic normality of the properly standardized integrated squared distance between a multivariate kernel density estimator and an estimator of its expectation under the hypothesis. The test of independence is a special case of this approach.
T3  - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2006, 02 
Y1  - 2006
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49393
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Liu, Shiping
A1  - Peyerimhoff, Norbert
T1  - A note on eigenvalue bounds for non-compact manifolds
JF  - Mathematische Nachrichten
N2  - In this article we prove upper bounds for the Laplace eigenvalues lambda(k) below the essential spectrum for strictly negatively curved Cartan-Hadamard manifolds. Our bound is given in terms of k(2) and specific geometric data of the manifold. This applies also to the particular case of non-compact manifolds whose sectional curvature tends to -infinity, where no essential spectrum is present due to a theorem of Donnelly/Li. The result stands in clear contrast to Laplacians on graphs where such a bound fails to be true in general.
KW  - Cheeger inequality
KW  - eigenvalues
KW  - Laplacian
KW  - negative curvature
KW  - Riemannian manifold
Y1  - 2021
U6  - https://doi.org/10.1002/mana.201900209
SN  - 0025-584X
SN  - 1522-2616
VL  - 294
IS  - 6
SP  - 1134
EP  - 1139
PB  - Wiley-VCH
CY  - Weinheim
ER  - 
TY  - JOUR
A1  - Dimitrova, Ilinka
A1  - Fernandes, Vitor H.
A1  - Koppitz, Jörg
T1  - A note on generators of the endomorphism semigroup of an infinite countable chain
JF  - Journal of Algebra and its Applications
N2  - In this note, we consider the semigroup O(X) of all order endomorphisms of an infinite chain X and the subset J of O(X) of all transformations alpha such that vertical bar Im(alpha)vertical bar = vertical bar X vertical bar. For an infinite countable chain X, we give a necessary and sufficient condition on X for O(X) = < J > to hold. We also present a sufficient condition on X for O(X) = < J > to hold, for an arbitrary infinite chain X.
KW  - Infinite chain
KW  - endomorphism semigroup
KW  - generators
KW  - relative rank
Y1  - 2016
U6  - https://doi.org/10.1142/S0219498817500311
SN  - 0219-4988
SN  - 1793-6829
VL  - 16
IS  - 2
PB  - World Scientific
CY  - Singapore
ER  - 
TY  - JOUR
A1  - Hinz, Michael
A1  - Schwarz, Michael
T1  - A note on Neumann problems on graphs
JF  - Positivity
N2  - We discuss Neumann problems for self-adjoint Laplacians on (possibly infinite) graphs. Under the assumption that the heat semigroup is ultracontractive we discuss the unique solvability for non-empty subgraphs with respect to the vertex boundary and provide analytic and probabilistic representations for Neumann solutions. A second result deals with Neumann problems on canonically compactifiable graphs with respect to the Royden boundary and provides conditions for unique solvability and analytic and probabilistic representations.
KW  - Graphs
KW  - Discrete Dirichlet forms
KW  - Neumann problem
KW  - Royden boundary
Y1  - 2022
U6  - https://doi.org/10.1007/s11117-022-00930-0
SN  - 1385-1292
SN  - 1572-9281
VL  - 26
IS  - 4
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - INPR
A1  - Camalès, Renaud
T1  - A note on the ramified Cauchy problem
N2  - In this paper, the ramified Cauchy problem in C² for operator with multiple characteristics of constant multiplicity and second member ramified around some analytic set is studied.
T3  - Preprint - (2003) 21 
KW  - Ramified Cauchy problem
KW  - analytic continuation
Y1  - 2003
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26646
ER  - 
TY  - INPR
A1  - Nehring, Benjamin
A1  - Zessin, Hans
T1  - A path integral representation of the moment measures of the general ideal Bose gas
N2  - We reconsider the fundamental work of Fichtner ([2]) and exhibit the permanental structure of the ideal Bose gas again, using another approach which combines a characterization of infinitely divisible random measures (due to Kerstan,Kummer and Matthes [5, 6] and Mecke [8, 9]) with a decomposition of the moment measures into its factorial measures due to Krickeberg [4]. To be more precise, we exhibit the moment measures of all orders of the general ideal Bose gas in terms of certain path integrals. This representation can be considered as a point process analogue of the old idea of Symanzik [11] that local times and self-crossings of the Brownian motion can be used as a tool in quantum field theory. Behind the notion of a general ideal Bose gas there is a class of infinitely divisible point processes of all orders with a Levy-measure belonging to some large class of measures containing the one of the classical ideal Bose gas considered by Fichtner. It is well known that the calculation of moments of higher order of point processes are notoriously complicated. See for instance Krickeberg's calculations for the Poisson or the Cox process in [4].
T3  - Mathematische Statistik und Wahrscheinlichkeitstheorie : Preprint - 2010, 10 
Y1  - 2010
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-49635
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  - INPR
A1  - Keller, Peter
A1  - Roelly, Sylvie
A1  - Valleriani, Angelo
T1  - A quasi-random-walk to model a biological transport process
N2  - Transport Molecules play a crucial role for cell viability. Amongst others, linear motors transport cargos along rope-like structures from one location of the cell to another in a stochastic fashion. Thereby each step of the motor, either forwards or backwards, bridges a fixed distance. While moving along the rope the motor can also detach and is lost. We give here a mathematical formalization of such dynamics as a random process which is an extension of Random Walks, to which we add an absorbing state to model the detachment of the motor from the rope. We derive particular properties of such processes that have not been available before. Our results include description of the maximal distance reached from the starting point and the position from which detachment takes place. Finally, we apply our theoretical results to a concrete established model of the transport molecule Kinesin V.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 3 
KW  - Markov chain
KW  - random walk
KW  - molecular motor
KW  - step process
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63582
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  - 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  - 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  - Fedosov, Boris
A1  - Schulze, Bert-Wolfgang
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A remark on the index of symmetric operators
N2  - We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.
T3  - Preprint - (1998) 04 
KW  - manifolds with singularities
KW  - differential operators
KW  - index
KW  - 'eta' invariant
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25169
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  - INPR
A1  - Polkovnikov, Alexander
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A Riemann-Hilbert problem for the Moisil-Teodorescu system
N2  - In a bounded domain with smooth boundary in R^3 we consider the stationary Maxwell equations
for a function u with values in R^3 subject to a nonhomogeneous condition 
(u,v)_x = u_0 on
the boundary, where v is a given vector field and u_0 a function on the boundary. We specify this problem within the framework of the Riemann-Hilbert boundary value problems for the Moisil-Teodorescu system. This latter is proved to satisfy the Shapiro-Lopaniskij condition if an only if the vector v is at no point tangent to the boundary. The Riemann-Hilbert problem for the Moisil-Teodorescu system fails to possess an adjoint boundary value problem with respect to the Green formula, which satisfies the Shapiro-Lopatinskij condition. We develop the construction of Green formula to get a proper concept of adjoint boundary value problem.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 6 (2017) 3 
KW  - Dirac operator
KW  - Riemann-Hilbert problem
KW  - Fredholm operators
Y1  - 2017
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-397036
VL  - 6
IS  - 3
ER  - 
TY  - JOUR
A1  - Hanisch, Florian
A1  - Ludewig, Matthias
T1  - A rigorous construction of the supersymmetric path integral associated to a compact spin manifold
JF  - Communications in mathematical physics
N2  - We give a rigorous construction of the path integral in N = 1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler-Jones-Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Guneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah-Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaume, Atiyah, Bismut and Witten.
Y1  - 2022
U6  - https://doi.org/10.1007/s00220-022-04336-7
SN  - 0010-3616
SN  - 1432-0916
VL  - 391
IS  - 3
SP  - 1209
EP  - 1239
PB  - Springer
CY  - Berlin ; Heidelberg
ER  - 
TY  - JOUR
A1  - Kucharski, Maciej
A1  - Ergintav, Arzu
A1  - Ahmad, Wael Abdullah
A1  - Krstić, Miloš
A1  - Ng, Herman Jalli
A1  - Kissinger, Dietmar
T1  - A Scalable 79-GHz Radar Platform Based on Single-Channel Transceivers
JF  - IEEE Transactions on Microwave Theory and Techniques
N2  - This paper presents a scalable E-band radar platform based on single-channel fully integrated transceivers (TRX) manufactured using 130-nm silicon-germanium (SiGe) BiCMOS technology. The TRX is suitable for flexible radar systems exploiting massive multiple-input-multipleoutput (MIMO) techniques for multidimensional sensing. A fully integrated fractional-N phase-locked loop (PLL) comprising a 39.5-GHz voltage-controlled oscillator is used to generate wideband frequency-modulated continuous-wave (FMCW) chirp for E-band radar front ends. The TRX is equipped with a vector modulator (VM) for high-speed carrier modulation and beam-forming techniques. A single TRX achieves 19.2-dBm maximum output power and 27.5-dB total conversion gain with input-referred 1-dB compression point of -10 dBm. It consumes 220 mA from 3.3-V supply and occupies 3.96 mm(2) silicon area. A two-channel radar platform based on full-custom TRXs and PLL was fabricated to demonstrate high-precision and high-resolution FMCW sensing. The radar enables up to 10-GHz frequency ramp generation in 74-84-GHz range, which results in 1.5-cm spatial resolution. Due to high output power, thus high signal-to-noise ratio (SNR), a ranging precision of 7.5 mu m for a target at 2 m was achieved. The proposed architecture supports scalable multichannel applications for automotive FMCW using a single local oscillator (LO).
KW  - Automotive
KW  - E-band
KW  - frequency-modulated continuous-wave (FMCW)
KW  - patch antenna
KW  - phase-locked loop (PLL)
KW  - power amplifier (PA)
KW  - radar
KW  - scalable
KW  - transceiver (TRX)
Y1  - 2019
U6  - https://doi.org/10.1109/TMTT.2019.2914104
SN  - 0018-9480
SN  - 1557-9670
VL  - 67
IS  - 9
SP  - 3882
EP  - 3896
PB  - Inst. of Electr. and Electronics Engineers
CY  - Piscataway
ER  - 
TY  - INPR
A1  - Schulze, Bert-Wolfgang
A1  - Nazaikinskii, Vladimir
A1  - Sternin, Boris
T1  - A semiclassical quantization on manifolds with singularities and the Lefschetz Formula for Elliptic Operators
N2  - For general endomorphisms of elliptic complexes on manifolds with conical singularities, the semiclassical asymptotics of the Atiyah-Bott-Lefschetz number is calculated in terms of fixed points of the corresponding canonical transformation of the symplectic space.
T3  - Preprint - (1998) 19 
KW  - elliptic operator
KW  - Fredholm property
KW  - conical singularities
KW  - pseudodiferential operators
KW  - Lefschetz fixed point formula
KW  - regularizer
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25296
ER  - 
TY  - INPR
A1  - Schrohe, Elmar
T1  - A short introduction to Boutet de Monvel's calculus
N2  - This paper provides an introduction to Boutet de Monvel's calculus on the half-space IRn (positiv) in the framework of a pseudodifferential calculus with operator-valued symbols.
T3  - Preprint - (2000) 03 
Y1  - 2000
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25696
ER  - 
TY  - INPR
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - A simple numerical approach to the Riemann hypothesis
N2  - The Riemann hypothesis is equivalent to the fact the the reciprocal function 1/zeta (s) extends from the interval (1/2,1) to an analytic function in the quarter-strip 1/2 < Re s < 1 and Im s > 0. Function theory allows one to rewrite the condition of analytic continuability in an elegant form amenable to numerical experiments.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 9 
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-57645
SN  - 2193-6943
ER  - 
TY  - INPR
A1  - Flandoli, Franco
A1  - Högele, Michael
T1  - A solution selection problem with small stable perturbations
N2  - The zero-noise limit of differential equations with singular coefficients is investigated for the first time in the case when the noise is a general alpha-stable process. It is proved that extremal solutions are selected and the probability of selection is computed. Detailed analysis of the characteristic function of an exit time form on the half-line is performed, with a suitable decomposition in small and large jumps adapted to the singular drift.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 8 
KW  - stochastic differential equations
KW  - singular drifts
KW  - zero-noise limit
KW  - Peano phenomena
KW  - non-uniqueness
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-71205
SN  - 2193-6943
VL  - 3
IS  - 8
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  - JOUR
A1  - Zöller, Gert
T1  - A statistical model for earthquake recurrence based on the assimilation of paleoseismicity, historic seismicity, and instrumental seismicity
JF  - Journal of geophysical research : Solid earth
N2  - Paleoearthquakes and historic earthquakes are the most important source of information for the estimation of long-term earthquake recurrence intervals in fault zones, because corresponding sequences cover more than one seismic cycle. However, these events are often rare, dating uncertainties are enormous, and missing or misinterpreted events lead to additional problems. In the present study, I assume that the time to the next major earthquake depends on the rate of small and intermediate events between the large ones in terms of a clock change model. Mathematically, this leads to a Brownian passage time distribution for recurrence intervals. I take advantage of an earlier finding that under certain assumptions the aperiodicity of this distribution can be related to the Gutenberg-Richter b value, which can be estimated easily from instrumental seismicity in the region under consideration. In this way, both parameters of the Brownian passage time distribution can be attributed with accessible seismological quantities. This allows to reduce the uncertainties in the estimation of the mean recurrence interval, especially for short paleoearthquake sequences and high dating errors. Using a Bayesian framework for parameter estimation results in a statistical model for earthquake recurrence intervals that assimilates in a simple way paleoearthquake sequences and instrumental data. I present illustrative case studies from Southern California and compare the method with the commonly used approach of exponentially distributed recurrence times based on a stationary Poisson process.
KW  - statistical seismology
KW  - paleoearthquakes
KW  - stochastic models
KW  - seismic hazard
Y1  - 2018
U6  - https://doi.org/10.1029/2017JB015099
SN  - 2169-9313
SN  - 2169-9356
VL  - 123
IS  - 6
SP  - 4906
EP  - 4921
PB  - American Geophysical Union
CY  - Washington
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  - Edlund, Helena
A1  - Grisic, Ana-Marija
A1  - Steenholdt, Casper
A1  - Ainsworth, Mark Andrew
A1  - Brynskov, Torn
A1  - Huisinga, Wilhelm
A1  - Kloft, Charlotte
T1  - Absence of Relationship Between Crohn's Disease Activity Index or C-Reactive Protein and Infliximab Exposure Calls for Objective Crohn's Disease Activity Measures for the Evaluation of Treatment Effects at Treatment Failure
JF  - Therapeutic drug monitoring : official journal of the International Association of Therapeutic Drug Monitoring and Clinical Toxicology
N2  - Background: Circulating infliximab (IFX) concentrations correlate with clinical outcomes, forming the basis of the IFX concentration monitoring in patients with Crohn's disease. This study aims to investigate and refine the exposure-response relationship by linking the disease activity markers "Crohn's disease activity index" (CDAI) and C-reactive protein (CRP) to IFX exposure. In addition, we aim to explore the correlations between different disease markers and exposure metrics.

Methods: Data from 47 Crohn's disease patients of a randomized controlled trial were analyzed post hoc. All patients had secondary treatment failure at inclusion and had received intensified IFX of 5 mg/kg every 4 weeks for up to 20 weeks. Graphical analyses were performed to explore exposure-response relationships. Metrics of exposure included area under the concentration-time curve (AUC) and trough concentrations (Cmin). Disease activity was measured by CDAI and CRP values, their change from baseline/last visit, and response/remission outcomes at week 12.

Results: Although trends toward lower Cmin and lower AUC in nonresponders were observed, neither CDAI nor CRP showed consistent trends of lower disease activity with higher IFX exposure across the 30 evaluated relationships. As can be expected, Cmin and AUC were strongly correlated with each other. Contrarily, the disease activity markers were only weakly correlated with each other.

Conclusions: No significant relationship between disease activity, as evaluated by CDAI or CRP, and IFX exposure was identified. AUC did not add benefit compared with Cmin. These findings support the continued use of Cmin and call for stringent objective disease activity (bio-)markers (eg, endoscopy) to form the basis of personalized IFX therapy for Crohn's disease patients with IFX treatment failure.
Y1  - 2019
U6  - https://doi.org/10.1097/FTD.0000000000000590
SN  - 0163-4356
SN  - 1536-3694
VL  - 41
IS  - 2
SP  - 235
EP  - 242
PB  - Lippincott Williams & Wilkins
CY  - Philadelphia
ER  - 
TY  - JOUR
A1  - Guggenberger, Tobias
A1  - Chechkin, Aleksei
A1  - Metzler, Ralf
T1  - Absence of stationary states and non-Boltzmann distributions of fractional Brownian motion in shallow external potentials
JF  - New journal of physics : the open-access journal for physics
N2  - We study the diffusive motion of a particle in a subharmonic potential of the form U(x) = |x|( c ) (0 < c < 2) driven by long-range correlated, stationary fractional Gaussian noise xi ( alpha )(t) with 0 < alpha <= 2. In the absence of the potential the particle exhibits free fractional Brownian motion with anomalous diffusion exponent alpha. While for an harmonic external potential the dynamics converges to a Gaussian stationary state, from extensive numerical analysis we here demonstrate that stationary states for shallower than harmonic potentials exist only as long as the relation c > 2(1 - 1/alpha) holds. We analyse the motion in terms of the mean squared displacement and (when it exists) the stationary probability density function. Moreover we discuss analogies of non-stationarity of Levy flights in shallow external potentials.
KW  - diffusion
KW  - Boltzmann distribution
KW  - fractional Brownian motion
Y1  - 2022
U6  - https://doi.org/10.1088/1367-2630/ac7b3c
SN  - 1367-2630
VL  - 24
IS  - 7
PB  - Dt. Physikalische Ges.
CY  - [Bad Honnef]
ER  - 
TY  - JOUR
A1  - Jansen, Sabine
A1  - Kolesnikov, Leonid
T1  - Activity expansions for Gibbs correlation functions
JF  - Lectures in pure and applied mathematics
KW  - random point processes
KW  - statistical mechanics
KW  - stochastic analysis
Y1  - 2020
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-472121
SN  - 978-3-86956-485-2
SN  - 2199-4951
SN  - 2199-496X
IS  - 6
SP  - 145
EP  - 154
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - JOUR
A1  - Carpentier, Alexandra
A1  - Klopp, Olga
A1  - Löffler, Matthias
A1  - Nickl, Richard
T1  - Adaptive confidence sets for matrix completion
JF  - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
N2  - In the present paper, we study the problem of existence of honest and adaptive confidence sets for matrix completion. We consider two statistical models: the trace regression model and the Bernoulli model. In the trace regression model, we show that honest confidence sets that adapt to the unknown rank of the matrix exist even when the error variance is unknown. Contrary to this, we prove that in the Bernoulli model, honest and adaptive confidence sets exist only when the error variance is known a priori. In the course of our proofs, we obtain bounds for the minimax rates of certain composite hypothesis testing problems arising in low rank inference.
KW  - adaptivity
KW  - confidence sets
KW  - low rank recovery
KW  - matrix completion
KW  - minimax hypothesis testing
KW  - unknown variance
Y1  - 2018
U6  - https://doi.org/10.3150/17-BEJ933
SN  - 1350-7265
SN  - 1573-9759
VL  - 24
IS  - 4A
SP  - 2429
EP  - 2460
PB  - International Statistical Institute
CY  - Voorburg
ER  - 
TY  - INPR
A1  - Junker, Wolfgang
A1  - Schrohe, Elmar
T1  - Adiabatic vacuum states on general spacetime manifolds : definition, construction, and physical properties
N2  - Adiabatic vacuum states are a well-known class of physical states for linear quantum fields n Robertson-Walker spacetimes. We extend the definition of adiabatic vacua to general spacetime manifolds by using the notion of the Sobolev wavefront set. This definition is also applicable to interacting field theories. Hadamard states form a special subclass of the adiabatic vacua. We analyze physical properties of adiabatic vacuum representations of the Klein-Gordon field on globally hyperbolic spacetme manifolds (factoriality, quasiequivalence, local definteness, Haag duality) and construct them explicitly, if the manifold has a compact Cauchy surface.
T3  - Preprint - (2001) 27 
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26100
ER  - 
TY  - THES
A1  - Hecher, Markus
T1  - Advanced tools and methods for treewidth-based problem solving
N2  - In the last decades, there was a notable progress in solving the well-known Boolean satisfiability (Sat) problem, which can be witnessed by powerful Sat solvers. One of the reasons why these solvers are so fast are structural properties of instances that are utilized by the solver’s interna. This thesis deals with the well-studied structural property treewidth, which measures the closeness of an instance to being a tree. In fact, there are many problems parameterized by treewidth that are solvable in polynomial time in the instance size when parameterized by treewidth.
In this work, we study advanced treewidth-based methods and tools for problems in knowledge representation and reasoning (KR). Thereby, we provide means to establish precise runtime results (upper bounds) for canonical problems relevant to KR. Then, we present a new type of problem reduction, which we call decomposition-guided (DG) that
allows us to precisely monitor the treewidth when reducing from one problem to another problem. This new reduction type will be the basis for a long-open lower bound result for quantified Boolean formulas and allows us to design a new methodology for establishing runtime lower bounds for problems parameterized by treewidth.
Finally, despite these lower bounds, we provide an efficient implementation of algorithms that adhere to treewidth. Our approach finds suitable abstractions of instances, which are subsequently refined in a recursive fashion, and it uses Sat solvers for solving subproblems. It turns out that our resulting solver is quite competitive for two canonical counting problems related to Sat.
N2  - In den letzten Jahrzehnten konnte ein beachtlicher Fortschritt im Bereich der Aussagenlogik verzeichnet werden. Dieser äußerte sich dadurch, dass für das wichtigste Problem in diesem Bereich, genannt „Sat“, welches sich mit der Fragestellung befasst, ob eine gegebene aussagenlogische Formel erfüllbar ist oder nicht, überwältigend schnelle Computerprogramme („Solver“) entwickelt werden konnten. Interessanterweise liefern diese Solver eine beeindruckende Leistung, weil sie oft selbst Probleminstanzen mit mehreren Millionen von Variablen spielend leicht lösen können. Auf der anderen Seite jedoch glaubt man in der Wissenschaft weitgehend an die Exponentialzeithypothese (ETH), welche besagt, dass man im schlimmsten Fall für das Lösen einer Instanz in diesem Bereich exponentielle Laufzeit in der Anzahl der Variablen benötigt. Dieser vermeintliche Widerspruch ist noch immer nicht vollständig geklärt, denn wahrscheinlich gibt es viele ineinandergreifende Gründe für die Schnelligkeit aktueller Sat Solver. Einer dieser Gründe befasst sich weitgehend mit strukturellen Eigenschaften von Probleminstanzen, die wohl indirekt und intern von diesen Solvern ausgenützt werden.

Diese Dissertation beschäftigt sich mit solchen strukturellen Eigenschaften, nämlich mit der sogenannten Baumweite. Die Baumweite ist sehr gut erforscht und versucht zu messen, wie groß der Abstand von Probleminstanzen zu Bäumen ist (Baumnähe). Allerdings ist dieser Parameter sehr generisch und bei Weitem nicht auf Problemstellungen der Aussagenlogik beschränkt. Tatsächlich gibt es viele weitere Probleme, die parametrisiert mit Baumweite in polynomieller Zeit gelöst werden können. Interessanterweise gibt es auch viele Probleme in der Wissensrepräsentation (KR), von denen man davon ausgeht, dass sie härter sind als das Problem Sat, die bei beschränkter Baumweite in polynomieller Zeit gelöst werden können. Ein prominentes Beispiel solcher Probleme ist das Problem QSat, welches sich für die Gültigkeit einer gegebenen quantifizierten, aussagenlogischen Formel (QBF), das sind aussagenlogische Formeln, wo gewisse Variablen existenziell bzw. universell quantifiziert werden können, befasst. Bemerkenswerterweise wird allerdings auch im Zusammenhang mit Baumweite, ähnlich zu Methoden der klassischen Komplexitätstheorie, die tatsächliche Komplexität (Härte) solcher Problemen quantifiziert, wo man die exakte Laufzeitabhängigkeit beim Problemlösen in der Baumweite (Stufe der Exponentialität) beschreibt.

Diese Arbeit befasst sich mit fortgeschrittenen, Baumweite-basierenden Methoden und Werkzeugen für Probleme der Wissensrepräsentation und künstlichen Intelligenz (AI). Dabei präsentieren wir Methoden, um präzise Laufzeitresultate (obere Schranken) für prominente Fragmente der Antwortmengenprogrammierung (ASP), welche ein kanonisches Paradigma zum Lösen von Problemen der Wissensrepräsentation darstellt, zu erhalten. Unsere Resultate basieren auf dem Konzept der dynamischen Programmierung, die angeleitet durch eine sogenannte Baumzerlegung und ähnlich dem Prinzip „Teile-und-herrsche“ funktioniert. Solch eine Baumzerlegung ist eine konkrete, strukturelle Zerlegung einer Probleminstanz, die sich stark an der Baumweite orientiert.

Des Weiteren präsentieren wir einen neuen Typ von Problemreduktion, den wir als „decomposition-guided (DG)“, also „zerlegungsangeleitet“, bezeichnen. Dieser Reduktionstyp erlaubt es, Baumweiteerhöhungen und -verringerungen während einer Problemreduktion von einem bestimmten Problem zu einem anderen Problem präzise zu untersuchen und zu kontrollieren. Zusätzlich ist dieser neue Reduktionstyp die Basis, um ein lange offen gebliebenes Resultat betreffend quantifizierter, aussagenlogischer Formeln zu zeigen. Tatsächlich sind wir damit in der Lage, präzise untere Schranken, unter der Annahme der Exponentialzeithypothese, für das Problem QSat bei beschränkter Baumweite zu zeigen. Genauer gesagt können wir mit diesem Konzept der DG Reduktionen zeigen, dass das Problem QSat, beschränkt auf Quantifizierungsrang ` und parametrisiert mit Baumweite k, im Allgemeinen nicht besser als in einer Laufzeit, die `-fach exponentiell in der Baumweite und polynomiell in der Instanzgröße ist1, lösen. Dieses Resultat hebt auf nicht-inkrementelle Weise ein bekanntes Ergebnis für Quantifizierungsrang 2 auf beliebige Quantifizierungsränge, allerdings impliziert es auch sehr viele weitere Konsequenzen.

Das Resultat über die untere Schranke des Problems QSat erlaubt es, eine neue Methodologie zum Zeigen unterer Schranken einer Vielzahl von Problemen der Wissensrepräsentation und künstlichen Intelligenz, zu etablieren. In weiterer Konsequenz können wir damit auch zeigen, dass die oberen Schranken sowie die DG Reduktionen dieser Arbeit unter der Hypothese ETH „eng“ sind, d.h., sie können wahrscheinlich nicht mehr signifikant verbessert werden. Die Ergebnisse betreffend der unteren Schranken für QSat und die dazugehörige Methodologie konstituieren in gewisser Weise eine Hierarchie von über Baumweite parametrisierte Laufzeitklassen. Diese Laufzeitklassen können verwendet werden, um die Härte von Problemen für das Ausnützen von Baumweite zu quantifizieren und diese entsprechend ihrer Laufzeitabhängigkeit bezüglich Baumweite zu kategorisieren.

Schlussendlich und trotz der genannten Resultate betreffend unterer Schranken sind wir im Stande, eine effiziente Implementierung von Algorithmen basierend auf dynamischer Programmierung, die entlang einer Baumzerlegung angeleitet wird, zur Verfügung zu stellen. Dabei funktioniert unser Ansatz dahingehend, indem er probiert, passende Abstraktionen von Instanzen zu finden, die dann im Endeffekt sukzessive und auf rekursive Art und Weise verfeinert und verbessert werden. Inspiriert durch die enorme Effizienz und Effektivität der Sat Solver, ist unsere Implementierung ein hybrider Ansatz, weil sie den starken Gebrauch von Sat Solvern zum Lösen diverser Subprobleme, die während der dynamischen Programmierung auftreten, pflegt. Dabei stellt sich heraus, dass der resultierende Solver unserer Implementierung im Bezug auf Effizienz beim Lösen von zwei kanonischen, Sat-verwandten Zählproblemen mit bestehenden Solvern locker mithalten kann. Tatsächlich sind wir im Stande, Instanzen, wo die oberen Schranken von Baumweite 260 übersteigen, zu lösen. Diese überraschende Beobachtung zeigt daher, dass Baumweite ein wichtiger Parameter sein könnte, der wohl in modernen Designs von Solvern berücksichtigt werden sollte.
KW  - Treewidth
KW  - Dynamic Programming
KW  - Knowledge Representation and Reasoning
KW  - Artificial Intelligence
KW  - Computational Complexity
KW  - Parameterized Complexity
KW  - Answer Set Programming
KW  - Exponential Time Hypothesis
KW  - Lower Bounds
KW  - Algorithms
KW  - Algorithmen
KW  - Antwortmengenprogrammierung
KW  - Künstliche Intelligenz
KW  - Komplexitätstheorie
KW  - Dynamische Programmierung
KW  - Exponentialzeit Hypothese
KW  - Wissensrepräsentation und Schlussfolgerung
KW  - Untere Schranken
KW  - Parametrisierte Komplexität
KW  - Baumweite
Y1  - 2021
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-512519
ER  - 
TY  - JOUR
A1  - Garbuno-Inigo, Alfredo
A1  - Nüsken, Nikolas
A1  - Reich, Sebastian
T1  - Affine invariant interacting Langevin dynamics for Bayesian inference
JF  - SIAM journal on applied dynamical systems
N2  - We propose a computational method (with acronym ALDI) for sampling from a given target distribution based on first-order (overdamped) Langevin dynamics which satisfies the property of affine invariance. The central idea of ALDI is to run an ensemble of particles with their empirical covariance serving as a preconditioner for their underlying Langevin dynamics. ALDI does not require taking the inverse or square root of the empirical covariance matrix, which enables application to high-dimensional sampling problems. The theoretical properties of ALDI are studied in terms of nondegeneracy and ergodicity. Furthermore, we study its connections to diffusion on Riemannian manifolds and Wasserstein gradient flows. Bayesian inference serves as a main application area for ALDI. In case of a forward problem with additive Gaussian measurement errors, ALDI allows for a gradient-free approximation in the spirit of the ensemble Kalman filter. A computational comparison between gradient-free and gradient-based ALDI is provided for a PDE constrained Bayesian inverse problem.
KW  - Langevin dynamics
KW  - interacting particle systems
KW  - Bayesian inference
KW  - gradient flow
KW  - multiplicative noise
KW  - affine invariance
KW  - gradient-free
Y1  - 2020
U6  - https://doi.org/10.1137/19M1304891
SN  - 1536-0040
VL  - 19
IS  - 3
SP  - 1633
EP  - 1658
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  - 
TY  - INPR
A1  - Klein, Markus
A1  - Léonard, Christian
A1  - Rosenberger, Elke
T1  - Agmon-type estimates for a class of jump processes
N2  - In the limit we analyze the generators of families of reversible jump processes in the n-dimensional space associated with a class of symmetric non-local Dirichlet forms and show exponential decay of the eigenfunctions. The exponential rate function is a Finsler distance, given as solution of certain eikonal equation. Fine results are sensitive to the rate functions being twice differentiable or just Lipschitz. Our estimates are similar to the semiclassical Agmon estimates for differential operators of second order. They generalize and strengthen previous results on the lattice.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 1 (2012) 6 
KW  - finsler distance
KW  - decay of eigenfunctions
KW  - jump process
KW  - Dirichlet form
Y1  - 2012
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-56995
ER  - 
TY  - JOUR
A1  - Guo, Li
A1  - Paycha, Sylvie
A1  - Zhang, Bin
T1  - Algebraic Birkhoff factorization and the Euler–Maclaurin formula on cones
JF  - Duke mathematical journal
N2  - We equip the space of lattice cones with a coproduct which makes it a cograded, coaugmented, connnected coalgebra. The exponential generating sum and exponential generating integral on lattice cones can be viewed as linear maps on this space with values in the space of meromorphic germs with linear poles at zero. We investigate the subdivision properties-reminiscent of the inclusion-exclusion principle for the cardinal on finite sets-of such linear maps and show that these properties are compatible with the convolution quotient of maps on the coalgebra. Implementing the algebraic Birkhoff factorization procedure on the linear maps under consideration, we factorize the exponential generating sum as a convolution quotient of two maps, with each of the maps in the factorization satisfying a subdivision property. A direct computation shows that the polar decomposition of the exponential generating sum on a smooth lattice cone yields an Euler-Maclaurin formula. The compatibility with subdivisions of the convolution quotient arising in the algebraic Birkhoff factorization then yields the Euler-Maclaurin formula for any lattice cone. This provides a simple formula for the interpolating factor by means of a projection formula.
Y1  - 2017
U6  - https://doi.org/10.1215/00127094-3715303
SN  - 0012-7094
SN  - 1547-7398
VL  - 166
IS  - 3
SP  - 537
EP  - 571
PB  - Duke Univ. Press
CY  - Durham
ER  - 
TY  - GEN
A1  - Reich, Sebastian
T1  - Algebrodifferentialgleichungen und Vektorfelder auf Mannigfaltigkeiten
N2  - In diesem Beitrag wird der Zusammenhang zwischen Algebrodifferentialgleichungen (ADGL) und Vektorfeldern auf Mannigfaltigkeiten untersucht. Dazu wird zunächst der Begriff der regulären ADGL eingeführt, wobei unter eirter regulären ADGL eine ADGL verstanden wird, deren Lösungsmenge identisch mit der Lösungsmenge eines Vektorfeldes ist. Ausgehend von bekannten Aussagen über die Lösungsmenge eines Vektorfeldes werden analoge Aussagen für die Lösungsmenge einer regulären ADGL abgeleitet. Es wird eine Reduktionsmethode angegeben, die auf ein Kriterium für die Begularität einer ADGL und auf die Definition des Index einer nichtlinearen ADGL führt. Außerdem wird gezeigt, daß beliebige Vektorfelder durch reguläre ADGL so realisiert werden können, daß die Lösungsmenge des Vektorfeldes mit der der realisierenden ADGL identisch ist. Abschließend werden die für autonome ADGL gewonnenen Aussagen auf den Fall der nichtautonomen ADGL übertragen.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - paper 160 
Y1  - 1980
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-47290
ER  - 
TY  - INPR
A1  - Schulze, Bert-Wolfgang
T1  - An algebra of boundary value problems not requiring Shapiro-Lopatinskil conditions
N2  - We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro-Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces.
T3  - Preprint - (1999) 24 
Y1  - 1999
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25596
ER  - 
TY  - INPR
A1  - Maniccia, L.
A1  - Schulze, Bert-Wolfgang
T1  - An algebra of meromorphic corner symbols
N2  - Operators on manifolds with corners that have base configurations with geometric singularities can be analysed in the frame of a conormal symbolic structure which is in spirit similar to the one for conical singularities of Kondrat'ev's work. Solvability of elliptic equations and asymptotics of solutions are determined by meromorphic conormal symbols. We study the case when the base has edge singularities which is a natural assumption in a number of applications. There are new phenomena, caused by a specific kind of higher degeneracy of the underlying symbols. We introduce an algebra of meromorphic edge operators that depend on complex parameters and investigate meromorphic inverses in the parameter-dependent elliptic case. Among the examples are resolvents of elliptic differential operators on manifolds with edges.
T3  - Preprint - (2002) 18 
Y1  - 2002
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26360
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  - JOUR
A1  - Mera, Azal Jaafar Musa
A1  - Tarkhanov, Nikolai
T1  - An elliptic equation of finite index in a domain
JF  - Boletin de la Sociedad Matemática Mexicana
N2  - We give an example of first order elliptic equation for a complex-valued function in a plane domain which has a finite number of linearly independent solutions for any right-hand side. No boundary value conditions are thus required.
KW  - elliptic equation
KW  - Fredholm operator
KW  - index
Y1  - 2022
U6  - https://doi.org/10.1007/s40590-022-00442-7
SN  - 1405-213X
SN  - 2296-4495
VL  - 28
IS  - 2
PB  - Springer International
CY  - New York [u.a.]
ER  - 
TY  - GEN
A1  - Roelly, Sylvie
A1  - Dai Pra, Paolo
T1  - An existence result for infinite-dimensional Brownian diffusions with non- regular and non Markovian drift
N2  - We prove in this paper an existence result for infinite-dimensional stationary interactive Brownian diffusions. The interaction is supposed to be small in the norm ||.||∞ but otherwise is very general, being possibly non-regular and non-Markovian. Our method consists in using the characterization of such diffusions as space-time Gibbs fields so that we construct them by space-time cluster expansions in the small coupling parameter.
KW  - infinite-dimensional Brownian diffusion
KW  - space-time Gibbs field
KW  - cluster expansion
Y1  - 2004
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-6684
ER  - 
TY  - JOUR
A1  - Houdebert, Pierre
A1  - Zass, Alexander
T1  - An explicit Dobrushin uniqueness region for Gibbs point processes with repulsive interactions
JF  - Journal of applied probability / Applied Probability Trust
N2  - We present a uniqueness result for Gibbs point processes with interactions that come from a non-negative pair potential; in particular, we provide an explicit uniqueness region in terms of activity z and inverse temperature beta. The technique used relies on applying to the continuous setting the classical Dobrushin criterion. We also present a comparison to the two other uniqueness methods of cluster expansion and disagreement percolation, which can also be applied for this type of interaction.
KW  - Gibbs point process
KW  - DLR equations
KW  - uniqueness
KW  - Dobrushin criterion;
KW  - cluster expansion
KW  - disagreement percolation
Y1  - 2022
U6  - https://doi.org/10.1017/jpr.2021.70
SN  - 0021-9002
SN  - 1475-6072
VL  - 59
IS  - 2
SP  - 541
EP  - 555
PB  - Cambridge Univ. Press
CY  - Cambridge
ER  - 
TY  - INPR
A1  - Dereudre, David
A1  - Mazzonetto, Sara
A1  - Roelly, Sylvie
T1  - An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers
N2  - In this paper we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover we propose a rejection method to simulate this density in an exact way.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 4 (2015) 9 
KW  - skew Brownian motion
KW  - semipermeable barriers
KW  - distorted Brownian motion
KW  - local time
KW  - rejection sampling
KW  - exact simulation
Y1  - 2015
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-80613
SN  - 2193-6943
VL  - 4
IS  - 9
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - INPR
A1  - Makhmudov, Olimdjan
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - An extremal problem related to analytic continuation
N2  - We show that the usual variational formulation of the problem of analytic continuation from an arc on the boundary of a plane domain does not lead to a relaxation of this overdetermined problem. To attain such a relaxation, we bound the domain of the functional, thus changing the Euler equations.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 2 (2013) 4 
KW  - Extremal problem
KW  - Euler equations
KW  - p-Laplace operator
KW  - mixed problems
Y1  - 2013
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-63634
ER  - 
TY  - JOUR
A1  - Keller, Matthias
A1  - Pinchover, Yehuda
A1  - Pogorzelski, Felix
T1  - An improved discrete hardy inequality
JF  - The American mathematical monthly : an official publication of the Mathematical Association of America
N2  - In this note, we prove an improvement of the classical discrete Hardy inequality. Our improved Hardy-type inequality holds with a weight w which is strictly greater than the classical Hardy weight w(H)(n) : 1/(2n)(2), where N.
KW  - Primary 26D15
Y1  - 2018
U6  - https://doi.org/10.1080/00029890.2018.1420995
SN  - 0002-9890
SN  - 1930-0972
VL  - 125
IS  - 4
SP  - 347
EP  - 350
PB  - Taylor & Francis Group
CY  - Philadelphia
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  - 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  - INPR
A1  - Aizenberg, Lev A.
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - An integral formula for the number of lattice points in a domain
N2  - Using the multidimensional logarithmic residue we show a simple formula for the difference between the number of integer points in a bounded domain of R^n and the volume of this domain. The difference proves to be the integral of an explicit differential form over the boundary of the domain.
T3  - Preprints des Instituts für Mathematik der Universität Potsdam - 3 (2014) 3 
KW  - logarithmic residue
KW  - lattice point
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70453
SN  - 2193-6943
VL  - 3
IS  - 3
PB  - Universitätsverlag Potsdam
CY  - Potsdam
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  - BOOK
A1  - Pilipenko, Andrey
T1  - An introduction to stochastic differential equations with reflection
T3  - Lectures in pure and applied mathematics
N2  - These lecture notes are intended as a short introduction to diffusion processes on a domain with a reflecting boundary for graduate students, researchers in stochastic analysis and interested readers. Specific results on stochastic differential equations with reflecting boundaries such as existence and uniqueness, continuity and Markov properties, relation to partial differential equations and submartingale problems are given. An extensive list of references to current literature is included. This book has its origins in a mini-course the author gave at the University of Potsdam and at the Technical University of Berlin in Winter 2013.
T3  - Lectures in pure and applied mathematics - 1 
KW  - Diffusionsprozess
KW  - Reflektierende Randbedingungen
KW  - stochastische Differentialgleichungen
KW  - diffusion process
KW  - reflecting boundary
KW  - stochastic differential equations
Y1  - 2014
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-70782
SN  - 978-3-86956-297-1
SN  - 2199-4951
SN  - 2199-496X
IS  - 1
PB  - Universitätsverlag Potsdam
CY  - Potsdam
ER  - 
TY  - THES
A1  - Abed, Jamil
T1  - An iterative approach to operators on manifolds with singularities
T1  - Ein iterativer Zugang zu Operatoren auf Mannigfaltigkeiten mit Singularitäten
N2  - We establish elements of a new approach to ellipticity and parametrices within operator algebras on manifolds with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaes. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The "full" calculus involves two separate theories, one near the tip of the corner and another one at the conical exit to infinity. However, concerning the conical exit to infinity, we establish here a new concrete calculus of edge-degenerate operators which can be iterated to higher singularities.
N2  - Wir führen einen neuen Zugang ein zu Elliptizität und Parametrices in Operatorenalgebren auf Mannigfaltigkeiten mit höheren Singularitäten, nur basierend auf allgemeinen axiomatischen Voraussetzungen über parameter-abhängige Operatoren in geeigneten Skalen von Räumen. Die Idee besteht darin, ein iteratives Verfahren zu modellieren mit neuen Generationen von parameter-abhängigen Operatortheorien, zusammen mit neuen Skalen von Räumen, die analoge Voraussetzungen erfüllen wie die ursprünglichen Objekte, jetzt auf dem entsprechenden höheren Niveau. Der „volle“ Kalkül besteht aus zwei separaten Theorien, eine nahe der Spitze der Ecke und eine andere am konischen Ausgang nach Unendlich. Allerdings, bezüglich des konischen Ausgangs nach Unendlich, bauen wir hier einen neuen konkreten Kalkül von kanten-entarteten Operatoren auf, der für höhere Singularitäten iteriert werden kann.
KW  - Pseudo-Differentialoperatoren
KW  - kanten- und ecken-entartete Symbole
KW  - Elliptizität
KW  - Parametrices
KW  - höhere Singularitäten
KW  - Pseudo-differential operators
KW  - edge- and corner-degenerate symbols
KW  - ellipticity
KW  - parametrices
KW  - higher singularities
Y1  - 2010
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-44757
ER  - 
TY  - JOUR
A1  - Carpentier, Alexandra
A1  - Kim, Arlene K. H.
T1  - An iterative hard thresholding estimator for low rank matrix recovery with explicit limiting distribution
JF  - Statistica Sinica
N2  - We consider the problem of low rank matrix recovery in a stochastically noisy high-dimensional setting. We propose a new estimator for the low rank matrix, based on the iterative hard thresholding method, that is computationally efficient and simple. We prove that our estimator is optimal in terms of the Frobenius risk and in terms of the entry-wise risk uniformly over any change of orthonormal basis, allowing us to provide the limiting distribution of the estimator. When the design is Gaussian, we prove that the entry-wise bias of the limiting distribution of the estimator is small, which is of interest for constructing tests and confidence sets for low-dimensional subsets of entries of the low rank matrix.
KW  - High dimensional statistical inference
KW  - inverse problem
KW  - limiting distribution
KW  - low rank matrix recovery
KW  - numerical methods
KW  - uncertainty quantification
Y1  - 2018
U6  - https://doi.org/10.5705/ss.202016.0103
SN  - 1017-0405
SN  - 1996-8507
VL  - 28
IS  - 3
SP  - 1371
EP  - 1393
PB  - Statistica Sinica, Institute of Statistical Science, Academia Sinica
CY  - Taipei
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  - JOUR
A1  - Shlapunov, Alexander
A1  - Tarchanov, Nikolaj Nikolaevič
T1  - An open mapping theorem for the Navier-Stokes type equations associated with the de Rham complex over R-n
JF  - Siberian electronic mathematical reports = Sibirskie ėlektronnye matematičeskie izvestija
N2  - We consider an initial problem for the Navier-Stokes type equations associated with the de Rham complex over R-n x[0, T], n >= 3, with a positive time T. We prove that the problem induces an open injective mappings on the scales of specially constructed function spaces of Bochner-Sobolev type. In particular, the corresponding statement on the intersection of these classes gives an open mapping theorem for smooth solutions to the Navier-Stokes equations.
KW  - Navier-Stokes equations
KW  - de Rham complex
KW  - open mapping theorem
Y1  - 2021
U6  - https://doi.org/10.33048/semi.2021.18.108
SN  - 1813-3304
VL  - 18
IS  - 2
SP  - 1433
EP  - 1466
PB  - Institut Matematiki Imeni S. L. Soboleva
CY  - Novosibirsk
ER  - 
TY  - GEN
A1  - Wiljes, Jana de
A1  - Tong, Xin T.
T1  - Analysis of a localised nonlinear ensemble Kalman Bucy filter with complete and accurate observations
T2  - Postprints der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe
N2  - Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.
T3  - Zweitveröffentlichungen der Universität Potsdam : Mathematisch-Naturwissenschaftliche Reihe - 1221 
KW  - data assimilation
KW  - stability and accuracy
KW  - dimension independent bound
KW  - localisation
KW  - high dimensional
KW  - filter
KW  - nonlinear
Y1  - 2022
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-540417
SN  - 1866-8372
VL  - 33
IS  - 9
SP  - 4752
EP  - 4782
PB  - IOP Publ.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Wiljes, Jana de
A1  - Tong, Xin T.
T1  - Analysis of a localised nonlinear ensemble Kalman Bucy filter with complete and accurate observations
JF  - Nonlinearity
N2  - Concurrent observation technologies have made high-precision real-time data available in large quantities. Data assimilation (DA) is concerned with how to combine this data with physical models to produce accurate predictions. For spatial-temporal models, the ensemble Kalman filter with proper localisation techniques is considered to be a state-of-the-art DA methodology. This article proposes and investigates a localised ensemble Kalman Bucy filter for nonlinear models with short-range interactions. We derive dimension-independent and component-wise error bounds and show the long time path-wise error only has logarithmic dependence on the time range. The theoretical results are verified through some simple numerical tests.
KW  - data assimilation
KW  - stability and accuracy
KW  - dimension independent bound
KW  - localisation
KW  - high dimensional
KW  - filter
KW  - nonlinear
Y1  - 2020
U6  - https://doi.org/10.1088/1361-6544/ab8d14
SN  - 0951-7715
SN  - 1361-6544
VL  - 33
IS  - 9
SP  - 4752
EP  - 4782
PB  - IOP Publ.
CY  - Bristol
ER  - 
TY  - JOUR
A1  - Chang, Der-Chen
A1  - Khalil, Sara
A1  - Schulze, Bert-Wolfgang
T1  - Analysis on regular corner spaces
JF  - The journal of geometric analysis
N2  - We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel's calculus of operators with the transmission property, see Boutet de Monvel (Acta Math 126:11-51, 1971) for the case of smooth boundary. On corner configuration operators act in spaces with multiple weights. We mainly study the case of upper left entries in the respective 2 x 2 operator block-matrices of such a calculus. Green operators in the sense of Boutet de Monvel (Acta Math 126:11-51, 1971) analogously appear in singular cases, and they are complemented by contributions of Mellin type. We formulate a result on ellipticity and the Fredholm property in weighted corner spaces, with parametrices of analogous kind.
KW  - Boutet de Monvel's calculus
KW  - Pseudo-differential operators
KW  - Singular cones
KW  - Mellin symbols with values in the edge calculus
KW  - Parametrices of elliptic operators
KW  - Kegel space
Y1  - 2021
U6  - https://doi.org/10.1007/s12220-021-00614-3
SN  - 1050-6926
SN  - 1559-002X
VL  - 31
IS  - 9
SP  - 9199
EP  - 9240
PB  - Springer
CY  - New York
ER  - 
TY  - INPR
A1  - Kytmanov, Alexander
A1  - Myslivets, Simona
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Analytic representation of CR Functions on hypersurfaces with singularities
N2  - We prove a theorem on analytic representation of integrable CR functions on hypersurfaces with singular points. Moreover, the behaviour of representing analytic functions near singular points is investigated. We are aimed at explaining the new effect caused by the presence of a singularity rather than at treating the problem in full generality.
T3  - Preprint - (1999) 29 
Y1  - 1999
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25631
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  - JOUR
A1  - Elin, Mark
A1  - Shoikhet, David
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Analytic Semigroups of Holomorphic Mappings and Composition Operators
JF  - Computational Methods and Function Theory
N2  - In this manuscript we provide a review on the classical and resent results related to 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.
KW  - Non-linear semigroups
KW  - Composition operators
KW  - Analytic extension
KW  - Holomorphic mappings
Y1  - 2017
U6  - https://doi.org/10.1007/s40315-017-0227-x
SN  - 1617-9447
SN  - 2195-3724
VL  - 18
IS  - 2
SP  - 269
EP  - 294
PB  - Springer
CY  - Heidelberg
ER  - 
TY  - INPR
A1  - Tarkhanov, Nikolai Nikolaevich
T1  - Anisotropic edge problems
N2  - We investigate elliptic pseudodifferential operators which degenerate in an anisotropic way on a submanifold of arbitrary codimension. To find Fredholm problems for such operators we adjoint to them boundary and coboundary conditions on the submanifold.The algebra obtained this way is a far reaching generalisation of Boutet de Monvel's algebra of boundary value problems with transmission property. We construct left and right regularisers and prove theorems on hypoellipticity and local solvability.
T3  - Preprint - (2002) 09 
Y1  - 2002
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26280
ER  - 
TY  - INPR
A1  - Paneah, Boris
T1  - Another approach to the stability of linear functional operators
N2  - Contents: 1 Introduction. 2 The main notations and notions. 3 Statement of results 4 Proofs of the main results.
T3  - Preprint - (2006) 13 
Y1  - 2006
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-30139
ER  - 
TY  - THES
A1  - Le, Tuan Anh
T1  - Applying realistic mathematics education in Vietnam : teaching middle school geometry
T1  - Der Einsatz von ‘Realistic Mathematics Education’ in Vietnam : Geometrieunterricht an Mittelschulen
N2  - Since 1971, the Freudenthal Institute has developed an approach to mathematics education named Realistic Mathematics Education (RME). The philosophy of RME is based on Hans Freudenthal’s concept of ‘mathematics as a human activity’. Prof. Hans Freudenthal (1905-1990), a mathematician and educator, believes that ‘ready-made mathematics’ should not be taught in school. By contrast, he urges that students should be offered ‘realistic situations’ so that they can rediscover from informal to formal mathematics. Although mathematics education in Vietnam has some achievements, it still encounters several challenges. Recently, the reform of teaching methods has become an urgent task in Vietnam. It appears that Vietnamese mathematics education lacks necessary theoretical frameworks. At first sight, the philosophy of RME is suitable for the orientation of the teaching method reform in Vietnam. However, the potential of RME for mathematics education as well as the ability of applying RME to teaching mathematics is still questionable in Vietnam. The primary aim of this dissertation is to research into abilities of applying RME to teaching and learning mathematics in Vietnam and to answer the question “how could RME enrich Vietnamese mathematics education?”. This research will emphasize teaching geometry in Vietnamese middle school. More specifically, the dissertation will implement the following research tasks: • Analyzing the characteristics of Vietnamese mathematics education in the ‘reformed’ period (from the early 1980s to the early 2000s) and at present; • Implementing a survey of 152 middle school teachers’ ideas from several Vietnamese provinces and cities about Vietnamese mathematics education; • Analyzing RME, including Freudenthal’s viewpoints for RME and the characteristics of RME; • Discussing how to design RME-based lessons and how to apply these lessons to teaching and learning in Vietnam; • Experimenting RME-based lessons in a Vietnamese middle school; • Analyzing the feedback from the students’ worksheets and the teachers’ reports, including the potentials of RME-based lessons for Vietnamese middle school and the difficulties the teachers and their students encountered with RME-based lessons; • Discussing proposals for applying RME-based lessons to teaching and learning mathematics in Vietnam, including making suggestions for teachers who will apply these lessons to their teaching and designing courses for in-service teachers and teachers-in training. This research reveals that although teachers and students may encounter some obstacles while teaching and learning with RME-based lesson, RME could become a potential approach for mathematics education and could be effectively applied to teaching and learning mathematics in Vietnamese school.
N2  - Seit 1971 wurde an dem renommierten Freudenthal Institut in Utrecht ein als Realistic Mathematics Education (RME) bezeichneter mathematikdidaktischer Ansatz entwickelt. Die Philosophie von RME beruht auf Hans Freudenthals Auffassung von Mathematik als menschlicher Aktivität. Der Mathematiker und Didaktiker Prof. Hans Freudenthal (1905 – 1990) plädierte dafür, dass Mathematik an den Schulen nicht als Fertigprodukt unterrichtet werden sollte. Im Gegensatz dazu forderte er, den Schülern an ‚realistischen’ Situationen nicht-formale und formale Mathematik wieder entdecken zu lassen. Obwohl die mathematische Schulbildung in Vietnam in den letzten Jahrzehnten schon einige Fortschritte gemacht hat, steht sie noch vor großen Herausforderungen. Derzeit ist die Reform der Unterrichtsmethoden eine dringliche Aufgabe in Vietnam. Augenscheinlich ermangelt es der Mathematikdidaktik in Vietnam an dem dazu notwendigen theoretischen Rahmen. Die Philosophie von RME eignet sich grundsätzlich als Orientierung für die Reform der Unterrichtsmethoden in Vietnam. Allerdings ist die Potenz von RME für die mathematische Schulbildung in Vietnam und die Möglichkeiten, RME im Mathematikunterricht anzuwenden, noch zu klären. Das Hauptziel dieser Arbeit war zu erforschen, wie RME beim Mathematik-Lernen und -Lehren in Vietnam eingesetzt werden kann und die Frage zu beantworten: Wie kann RME den Mathematikunterricht in Vietnam bereichern? Dazu wurde insbesondere der Geometrieunterricht in der Sekundarstufe I betrachtet. Im Einzelnen beinhaltet die Untersuchung: • eine Analyse der vietnamesischen Mathematikdidaktik in der ‘Reformperiode’ (etwa von 1980 bis 2000) • die Konzeption, Durchführung und Auswertung einer Befragung von 152 Mittelschullehrern aus verschiedenen vietnamesischen Provinzen und Städten zum Mathematikunterricht in Vietnam • eine Analyse von RME einschließlich der Freudenthalschen Sicht von RME und der Charakteristika von RME • die Diskussion, wie man RME-basierten Unterrichtseinheiten gestalten und diese in den Mathematikunterricht in Vietnam integrieren kann • Test solcher Einheiten in vietnamesischen Mittelschulen • Analyse der Rückmeldungen anhand der Schülerarbeitsblätter und der Lehrerberichte • Diskussion der Chancen und Probleme von RME-basierten Unterrichtseinheiten im Geometrieunterricht vietnamesischer Mittelschulen • Diskussion von Vorschläge zur Entwicklung und zum Einsatz RME- basierter Unterrichtseinheiten in Vietnam, einschließlich von Hinweisen für Lehrende und der Konzeption von Ausbildungs- und Fortbildungskursen zu RME Die Untersuchung zeigt, dass – obwohl Lehrer wie Schüler zunächst einige Hindernisse beim Lehren und Lernen mit RME- basierten Unterrichtseinheiten zu bewältigen haben werden – RME ein mächtiger mathematikdidaktischer Ansatz ist, der wirkungsvoll im Lehren und Lernen von Mathematik in vietnamesischen Schulen angewandt werden kann.
KW  - Didaktik der Mathematik
KW  - Vietnam
KW  - Geometrieunterricht
KW  - Sekundarstufe I
KW  - Realistic Mathematics Education
KW  - Vietnam
KW  - middle school
KW  - geometry
Y1  - 2006
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-13480
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  - Kuzle, Ana
T1  - Assessing metacognition of grade 2 and grade 4 students using an adaptation of multi-method interview approach during mathematics problem-solving
JF  - Mathematics Education Research Journal
N2  - The important role that metacognition plays as a predictor for student mathematical learning and for mathematical problem-solving, has been extensively documented. But only recently has attention turned to primary grades, and more research is needed at this level. The goals of this paper are threefold: (1) to present metacognitive framework during mathematics problem-solving, (2) to describe their multi-method interview approach developed to study student mathematical metacognition, and (3) to empirically evaluate the utility of their model and the adaptation of their approach in the context of grade 2 and grade 4 mathematics problem-solving. The results are discussed not only with regard to further development of the adapted multi-method interview approach, but also with regard to their theoretical and practical implications.
KW  - Metacognition
KW  - Multi-method interview approach
KW  - Mathematics problem-solving
KW  - Elementary school students
Y1  - 2018
U6  - https://doi.org/10.1007/s13394-017-0227-1
SN  - 1033-2170
SN  - 2211-050X
VL  - 30
IS  - 2
SP  - 185
EP  - 207
PB  - Springer
CY  - Dordrecht
ER  - 
TY  - INPR
A1  - Witt, Ingo
T1  - Asymptotic algebras
N2  - The concept of asymptotic type that primarily appears in singular and asymptotic analysis is developed. Especially, asymptotic algebras are introduced.
T3  - Preprint - (2001) 23 
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26069
ER  - 
TY  - INPR
A1  - Fang, Daoyuan
A1  - Xu, Jiang
T1  - Asymptotic behavior of solutions to multidimensional nonisentropic hydrodynamic model for semiconductors
N2  - In this paper, a global existence result of smooth solutions to the multidimen- sional nonisentropic hydrodynamic model for semiconductors is proved, under the assumption that the initial data is a perturbation of the stationary solutions for the thermal equilibrium state. The resulting evolutionary solutions converge to the stationary solutions in time asymptotically exponentially fast.
T3  - Preprint - (2005) 02 
KW  - Multidimensional nonisentropic hydrodynamic model
KW  - semiconductors
KW  - asymptotic behavior
KW  - global solutions
Y1  - 2005
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-29767
ER  - 
TY  - INPR
A1  - Chen, Hua
A1  - Yu, Chun
T1  - Asymptotic behaviour of the trace for Schrödinger operator on fractal drums
T3  - Preprint - (2001) 32 
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-26157
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  - 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  - INPR
A1  - Xiaochun, Liu
A1  - Witt, Ingo
T1  - Asymptotic expansions for bounded solutions to semilinear Fuchsian equations
N2  - It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.
T3  - Preprint - (2001) 01 
KW  - Calculus of conormal symbols
KW  - conormal asymptotic expansions
KW  - discrete saymptotic types
KW  - weighted Sobolev spaces with discrete saymptotics
Y1  - 2001
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-25912
ER  -