Mathematik

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DFG Final Report: Validated computation of patterns in recurrent neural networks
(Hannover : Technische Informationsbibliothek, 2025) Queirolo, Elena
[no abstract available]
Final Report on the DFG project "On the analysis of a class of cross-diffusion Cahn-Hilliard systems"
(Hannover : Technische Informationsbibliothek, 2025) Marino, Greta
In this project we considered a class of cross-diffusion systems involving Cahn-Hilliard terms and gave exhaustive answers to some main questions related to it. This class arises when modeling mixtures composed of several species that interact with one another with cross-diffusion effects and also have the tendency to separate from each other. In the case under consideration, there is only one species (that accounts for the void) which does separate from all the others. The interest for such a model stems from the fact that in many real world applications there exist multiphase systems where miscible entities do coexist in one single phase of the system. Our project started from an existing model and, by making a combination of both numerical and theoretical approaches, provided a systematical analysis to some core topics related to it, in order to gain a better understanding of the dynamics of the model and of its stationary solutions.
Derivation of Moment Equations for the Theoretical Description of Electrons in Nonthermal Plasmas
(Irvine, Calif. : Scientific Research Publ., 2013) Becker, Markus M.; Loffhagen, Detlef
The derivation of moment equations for the theoretical description of electrons is of interest for modelling of gas discharge plasmas and semiconductor devices. Usually, certain artificial closure assumptions are applied in order to derive a closed system of moment equations from the electron Boltzmann equation. Here, a novel four-moment model for the description of electrons in nonthermal plasmas is derived by an expansion of the electron velocity distribution function in Legendre polynomials. The proposed system of partial differential equations is consistently closed by definition of transport coefficients that are determined by solving the electron Boltzmann equation and are then used in the fluid calculations as function of the mean electron energy. It is shown that the four-moment model can be simplified to a new drift-diffusion approximation for electrons without loss of accuracy, if the characteristic frequency of the electric field alteration in the discharge is small in comparison with the momentum dissipation frequency of the electrons. Results obtained by the proposed fluid models are compared to those of a conventional drift-diffusion approximation as well as to kinetic results using the example of low pressure argon plasmas. It is shown that the results provided by the new approaches are in good agreement with kinetic results and strongly improve the accuracy of fluid descriptions of gas discharges.
High-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with robin boundary conditions
(Berlin : De Gruyter, 2002) Hemker, Pieter W.; Shishkin, Grigorii I.; Shishkina, Lidia P.
The boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameterε. In contrast to the Dirichlet boundary-value problem, for the problem under consideration the errors of the well-known classical methods, generally speaking, grow without bound as ε≪N-1 where N defines the number of mesh points with respect to x. The order of convergence for the known ε-uniformly convergent schemes does not exceed 1. In this paper, using a defect correction technique, we construct ε-uniformly convergent schemes of highorder time-accuracy. The efficiency of the new defect-correction schemes is confirmed by numerical experiments. A new original technigue for experimental studying of convergence orders is developed for the cases where the orders of convergence in the x-direction and in the t-direction can be substantially different. © 2002, Institute of Mathematics, NAS of Belarus. All rights reserved.
DFG project report "A Unified Approach to Limit Theorems for Dual Objects in Probability and Number Theory"
(Hannover : Technische Informationsbibliothek, 2024-09-24) Indlekofer, Karl-Heinz; Klesov, Oleg I.; Steinebach, Josef G.
Several situations are known in mathematics, where similar statements hold for different mathematical objects. However, since the objects are different, the proofs of these statements as well as their assumptions are different. One of the basic ideas of the project is to suggest a general approach for fi nding certain similarities between two mathematical objects that allow one to provide the proof only for one of them and to transfer this to a similar result for the another one. Such objects are called dual in our language. An integral part of this approach is to develop tools for transforming necessary conditions for one of the objects into corresponding conditions for the another one.
DFG project report "Multidimensional moment problem and Schur algorithm."
(Hannover : Technische Informationsbibliothek, 2024) Kovalyov, Ivan
[no abstract available]
Categorical Linearly Ordered Structures
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Downey, Rod; Melnikov, Alexander; Ng, Keng Meng
We prove that for every computable limit ordinal α there exists a computable linear ordering A which is Δ^(0)_(α)-categorical and α is smallest such, but nonetheless for every isomorphic computable copy B of A there exists a β<α such that A≅Δ0βB. This answers a question left open in the earlier work of Downey, Igusa, and Melnikov. We also show that such examples can be found among ordered abelian groups and real-closed fields.
A Well-Posedness Result for Viscous Compressible Fluids with Only Bounded Density
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Danchin, Raphaël; Fanelli, Francesco; Paicu, Marius
We are concerned with the existence and uniqueness of solutions with only bounded density for the barotropic compressible Navier-Stokes equations. Assuming that the initial velocity has slightly sub-critical regularity and that the initial density is a small perturbation (in th L^∞ norm) of a positive constant, we prove the existence of local-in-time solutions. In the case where the density takes two constant values across a smooth interface (or, more generally, has striated regularity with respect to some nondegenerate family of vector-fields), we get uniqueness. This latter result supplements the work by D. Hoff in [26] with a uniqueness statement, and is valid in any dimension d≥2 and for general pressure laws.
Max-Linear Models on Infinite Graphs Generated by Bernoulli Bond Percolation
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Klüppelberg, Claudia; Sönmez, Ercan
We extend previous work of max-linear models on finite directed acyclic graphs to infinite graphs, and investigate their relations to classical percolation theory. We formulate results for the oriented square lattice graph Z² and nearest neighbor bond percolation. Focus is on the dependence introduced by this graph into the max-linear model. As a natural application we consider communication networks, in particular, the distribution of extreme opinions in social networks.
Homogenization of a nonlinear monotone problem with nonlinear Signorini boundary conditions in a domain with highly rough boundary
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Gaudiello, Antonio; Mel'nek, Taras A.
We consider a domain Ωε⊂Rⁿ, N≥2, with a very rough boundary depending on ~ε. For instance, if N=3 the domain Ωε has the form of a brush with an ε-periodic distribution of thin cylinders with fixed height and a small diameter of order ε. In Ωε a nonlinear monotone problem with nonlinear Signorini boundary conditions, depending on ε, on the lateral boundary of the cylinders is considered. We study the asymptotic behavior of this problem, as ε vanishes, i.e. when the number of thin attached cylinders increases unboundedly, while their cross sections tend to zero. We identify the limit problem which is a nonstandard homogenized problem. Namely, in the region filled up by the thin cylinders the limit problem is given by a variational inequality coupled to an algebraic system.
On the Gauss Algebra of Toric Algebras
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Herzog, Jürgen; Jafari, Raheleh; Nasrollah Nejad, Abbas
Let A be a K-subalgebra of the polynomial ring S=K[x₁,…,xd] of dimension d, generated by finitely many monomials of degree r. Then the Gauss algebra G(A) of A is generated by monomials of degree (r−1)d in S. We describe the generators and the structure of G(A), when A is a Borel fixed algebra, a squarefree Veronese algebra, generated in degree 2, or the edge ring of a bipartite graph with at least one loop. For a bipartite graph G with one loop, the embedding dimension of G(A) is bounded by the complexity of the graph G.
The Tutte Polynomial of Ideal Arrangements
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Randriamaro, Hery
The Tutte polynomial is originally a bivariate polynomial enumerating the colorings of a graph and of its dual graph. But it reveals more of the internal structure of the graph like its number of forests, of spanning subgraphs, and of acyclic orientations. In 2007, Ardila extended the notion of Tutte polynomial to hyperplane arrangements, and computed the Tutte polynomials of the classical root systems for a certain prime power of the first variable. In this article, we compute the Tutte polynomials of ideal arrangements. Those arrangements were introduced in 2006 by Sommers and Tymoczko, and are defined for ideals of root systems. For the ideals of the classical root systems, we bring a slight improvement of the finite field method showing that it can applied on any finite field whose cardinality is not a minor of the matrix associated to a hyperplane arrangement. Computing the minor set associated to an ideal of a classical root system permits us particularly to deduce the Tutte polynomials of the classical root systems. For the ideals of the exceptional root systems of type G2, F4, and E6, we use the formula of Crapo.
The Sylow Structure of Scalar Automorphism Groups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Herfort, Wolfgang; Hofmann, Karl Heinrich; Kramer, Linus; Russo, Francesco G.
For any locally compact abelian periodic group A its automorphism group contains as a subgroup those automorphisms that leave invariant every closed subgroup of A, to be denoted by SAut(A). This subgroup is again a locally compact abelian periodic group in its natural topology and hence allows a decomposition into its p-primary subgroups for p the primes for which topological p-elements in this automorphism subgroup exist. The interplay between the p-primary decomposition of SAut(A) and A can be encoded in a bipartite graph, the mastergraph of A. Properties and applications of this concept are discussed.
Exceptional Legendrian Torus Knots
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Geiges, Hansjörg; Onaran, Sinem
We present classification results for exceptional Legendrian realisations of torus knots. These are the first results of that kind for non-trivial topological knot types. Enumeration results of Ding-Li-Zhang concerning tight contact structures on certain Seifert fibred manifolds with boundary allow us to place upper bounds on the number of tight contact structures on the complements of torus knots; the classification of exceptional realisations of these torus knots is then achieved by exhibiting suffciently many realisations in terms of contact surgery diagrams. We also discuss a couple of general theorems about the existence of exceptional Legendrian knots.
Spectral Continuity for Aperiodic Quantum Systems II. Periodic Approximations in 1D
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Beckus, Siegfried; Bellissard, Jean; De Nittis, Giuseppe
The existence and construction of periodic approximations with convergent spectra is crucial in solid state physics for the spectral study of corresponding Schrödinger operators. In a forthcoming work [9] this task was boiled down to the existence and construction of periodic approximations of the underlying dynamical systems in the Hausdorff topology. As a result the one-dimensional systems admitting such approximations are completely classified in the present work. In addition explicit constructions are provided for dynamical systems defined by primitive substitutions covering all studied examples such as the Fibonacci sequence or the Golay-Rudin-Shapiro sequence. One main tool is the description of the Hausdorff topology by the local pattern topology on the dictionaries as well as the GAP-graphs describing the local structure. The connection of branching vertices in the GAP-graphs and defects is discussed.
Demailly's Notion of Algebraic Hyperbolicity: Geometricity, Boundedness, Moduli of Maps (Revised Edition)
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Javanpeykar, Ariyan; Kamenova, Ljudmila
Demailly's conjecture, which is a consequence of the Green-Griffths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence for Demailly's conjecture by verifying several predictions it makes. We first define what an algebraically hyperbolic projective variety is, extending Demailly's definition to (not necessarily smooth) projective varieties over an arbitrary algebraically closed field of characteristic zero, and we prove that this property is stable under extensions of algebraically closed fields. Furthermore, we show that the set of (not necessarily surjective) morphisms from a projective variety Y to a projective algebraically hyperbolic variety X that map a fixed closed subvariety of Y onto a fixed closed subvariety of X is finite. As an application, we obtain that Aut(X) is finite and that every surjective endomorphism of X is an automorphism. Finally, we explore "weaker" notions of hyperbolicity related to boundedness of moduli spaces of maps, and verify similar predictions made by the Green-Griffths-Lang conjecture on hyperbolic projective varieties.
Sur le Minimum de la Fonction de Brjuno
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Balazard, Michel; Martin, Bruno
The Brjuno function attains a strict global minimum at the golden section.
Global Variants of Hartogs' Theorem
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Bochnak, Jacek; Kucharz, Wojciech
Hartogs' theorem asserts that a separately holomorphic function, defined on an open subset of Cⁿ, is holomorphic in all the variables. We prove a global variant of this theorem for functions defined on an open subset of the product of complex algebraic manifolds. We also obtain global Hartogs-type theorems for complex Nash functions and complex regular functions.
Affine Space Fibrations
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Gurjar, Rajendra V.; Masuda, Kayo; Miyanishi, Masayoshi
We discuss various aspects of affine space fibrations. Our interest will be focused in the singular fibers, the generic fiber and the propagation of properties of a given smooth special fiber to nearby fibers.
The Martin Boundary of Relatively Hyperbolic Groups with Virtually Abelian Parabolic Subgroups
(Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2018) Dussaule, Mattieu; Gekhtman, Ilya; Gerasimov, Victor; Potyagailo, Leonid
Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space Hⁿ, we show that the Martin boundary coincides with the CAT(0) boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.

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