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
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.
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 - JOUR
A1 - Hanisch, Florian
A1 - Strohmaier, Alexander
A1 - Waters, Alden
T1 - A relative trace formula for obstacle scattering
JF - Duke mathematical journal
N2 - We consider the case of scattering by several obstacles in Rd for d ≥ 2.
In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator Δ0 are unitarily equivalent.
For suitable functions that decay sufficiently fast, we have that the difference g(Δ) - g(Δ0) is a trace-class operator and its trace is described by the Krein spectral shift function.
In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators Δ1 and Δ2 obtained by imposing Dirichlet boundary conditions only on one of the objects.
Our main result in this case states that then g(Δ) - g(Δ1) - g(Δ2) C g(Δ0) is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman–Krein formula. In case g(x) D x 2 , 1 the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators.
Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization.
KW - Birman–Krein formula
KW - Casimir energy
KW - obstacle scattering
KW - trace formula
Y1 - 2022
U6 - https://doi.org/10.1215/00127094-2022-0053
SN - 0012-7094
SN - 1547-7398
VL - 171
IS - 11
SP - 2233
EP - 2274
PB - Duke Univ. Press
CY - Durham, NC
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 -