Oberwolfach TEST Repository
http://publications.mfo.de:8080
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Wed, 15 Nov 2017 05:46:22 GMT2017-11-15T05:46:22ZOberwolfach TEST Repositoryhttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:8080
Looking Back on Inverse Scattering Theory
http://publications.mfo.de/handle/mfo/1323
Looking Back on Inverse Scattering Theory
Colton, David; Kress, Rainer
We present an essay on the mathematical development of inverse scattering theory for time-harmonic waves during the past fifty years together with some personal memories of our participation in these
events.
Thu, 05 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13232017-10-05T00:00:00ZColton, DavidKress, RainerWe present an essay on the mathematical development of inverse scattering theory for time-harmonic waves during the past fifty years together with some personal memories of our participation in these
events.Geometry of Free Loci and Factorization of Noncommutative Polynomials
http://publications.mfo.de/handle/mfo/1322
Geometry of Free Loci and Factorization of Noncommutative Polynomials
Helton, J. William; Klep, Igor; Volčič, Jurij
The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.
MSC 2010: 13J30; 15A22; 47A56 (Primary) | 14P10; 16U30; 16R30 (Secondary)
Mon, 02 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13222017-10-02T00:00:00ZHelton, J. WilliamKlep, IgorVolčič, JurijThe free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.GAP Functionality for Zariski Dense Groups
http://publications.mfo.de/handle/mfo/1321
GAP Functionality for Zariski Dense Groups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
In this document we describe the functionality of GAP [4] routines for Zariski dense or arithmetic groups that are developed in [1, 2, 3]. The research underlying the software was supported through the programme "Research in Pairs", at the Mathematisches Forschungsinstitut Oberwolfach, and part of the software was written during a stay in June 2017. The hospitality we received has been greatly appreciated. Our research was also supported by a Marie Sklodowska-Curie Individual Fellowship grant under Horizon 2020 (EU Framework Programme for Research and Innovation), and a Simons Foundation Collaboration Grant Nr. 244502. All support is acknowledged with gratitude.
Thu, 14 Sep 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13212017-09-14T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderIn this document we describe the functionality of GAP [4] routines for Zariski dense or arithmetic groups that are developed in [1, 2, 3]. The research underlying the software was supported through the programme "Research in Pairs", at the Mathematisches Forschungsinstitut Oberwolfach, and part of the software was written during a stay in June 2017. The hospitality we received has been greatly appreciated. Our research was also supported by a Marie Sklodowska-Curie Individual Fellowship grant under Horizon 2020 (EU Framework Programme for Research and Innovation), and a Simons Foundation Collaboration Grant Nr. 244502. All support is acknowledged with gratitude.Composition of Irreducible Morphisms in Coils
http://publications.mfo.de/handle/mfo/1320
Composition of Irreducible Morphisms in Coils
Chaio, Claudia; Malicki, Piotr
We study the non-zero composition of n irreducible morphisms between modules lying in coils in relation with the powers of the radical of their module category.
Mon, 30 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13202017-10-30T00:00:00ZChaio, ClaudiaMalicki, PiotrWe study the non-zero composition of n irreducible morphisms between modules lying in coils in relation with the powers of the radical of their module category.Experimenting with Zariski Dense Subgroups
http://publications.mfo.de/handle/mfo/1319
Experimenting with Zariski Dense Subgroups
Detinko, Alla; Flannery, Dane; Hulpke, Alexander
Tue, 24 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13192017-10-24T00:00:00ZDetinko, AllaFlannery, DaneHulpke, AlexanderOn an Effective Variation of Kronecker’s Approximation Theorem Avoiding Algebraic Sets
http://publications.mfo.de/handle/mfo/1316
On an Effective Variation of Kronecker’s Approximation Theorem Avoiding Algebraic Sets
Fukshansky, Lenny; German, Oleg; Moshchevitin, Nikolay
Let $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such that $\Lambda \nsubseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $\Lambda$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $\Lambda$, and let $L_1,\dots,L_t$ be linear forms in $n$ variables with algebraic coefficients satisfying an appropriate linear independence condition over $K_1$. For each $\varepsilon > 0$ and $\boldsymbol a \in \mathbb R^n$, we prove the existence of a vector $\boldsymbol x \in \Lambda \setminus \mathcal Z$ of explicitly bounded sup-norm such that
$$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$
for each $1 \leq i \leq t$, where $\|\ \|$ stands for the distance to the nearest integer. The bound on sup-norm of $\boldsymbol x$ depends on $\varepsilon$, as well as on $\Lambda$, $K$, $\mathcal Z$ and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of $\Lambda \setminus \mathcal Z$ under the linear forms $L_1,\dots,L_t$ in the $t$-torus~$\mathbb R^t/\mathbb Z^t$. In the appendix, we also discuss a construction of badly approximable matrices, a subject closely related to our proof of effective Kronecker's theorem, via Liouville-type inequalities and algebraic transference principles.
Thu, 19 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13162017-10-19T00:00:00ZFukshansky, LennyGerman, OlegMoshchevitin, NikolayLet $\Lambda \subset \mathbb R^n$ be an algebraic lattice, coming from a projective module over the ring of integers of a number field $K$. Let $\mathcal Z \subset \mathbb R^n$ be the zero locus of a finite collection of polynomials such that $\Lambda \nsubseteq \mathcal Z$ or a finite union of proper full-rank sublattices of $\Lambda$. Let $K_1$ be the number field generated over $K$ by coordinates of vectors in $\Lambda$, and let $L_1,\dots,L_t$ be linear forms in $n$ variables with algebraic coefficients satisfying an appropriate linear independence condition over $K_1$. For each $\varepsilon > 0$ and $\boldsymbol a \in \mathbb R^n$, we prove the existence of a vector $\boldsymbol x \in \Lambda \setminus \mathcal Z$ of explicitly bounded sup-norm such that
$$\| L_i(\boldsymbol x) - a_i \| < \varepsilon$$
for each $1 \leq i \leq t$, where $\|\ \|$ stands for the distance to the nearest integer. The bound on sup-norm of $\boldsymbol x$ depends on $\varepsilon$, as well as on $\Lambda$, $K$, $\mathcal Z$ and heights of linear forms. This presents a generalization of Kronecker's approximation theorem, establishing an effective result on density of the image of $\Lambda \setminus \mathcal Z$ under the linear forms $L_1,\dots,L_t$ in the $t$-torus~$\mathbb R^t/\mathbb Z^t$. In the appendix, we also discuss a construction of badly approximable matrices, a subject closely related to our proof of effective Kronecker's theorem, via Liouville-type inequalities and algebraic transference principles.Review of the Methods of Reflections
http://publications.mfo.de/handle/mfo/1315
Review of the Methods of Reflections
Ciaramella, Gabriele; Gander, Martin J.; Halpern, Laurence; Salomon, Julien
The methods of reflections were invented to obtain approximate solutions of the motion of more than one particle in a given environment, provided that one can represent the solution for one particle rather easily. This motivation is quite similar to the motivation of the Schwarz domain decomposition method, which was invented to prove existence and uniqueness of solutions of the Laplace equation on complicated domains, which are composed of simpler ones, for which existence and uniqueness of solutions was known. Like for Schwarz methods, there is also an alternating and a parallel method of reflections, but interestingly, the parallel method is not always convergent. We carefully trace in this paper the historical development of these methods of reflections, give several precise mathematical formulations, an equivalence result with the alternating Schwarz method for two particles, and also an analysis for a one dimensional model problem with three particles of the alternating, parallel, and a recent averaged parallel method of reflections.
Wed, 18 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13152017-10-18T00:00:00ZCiaramella, GabrieleGander, Martin J.Halpern, LaurenceSalomon, JulienThe methods of reflections were invented to obtain approximate solutions of the motion of more than one particle in a given environment, provided that one can represent the solution for one particle rather easily. This motivation is quite similar to the motivation of the Schwarz domain decomposition method, which was invented to prove existence and uniqueness of solutions of the Laplace equation on complicated domains, which are composed of simpler ones, for which existence and uniqueness of solutions was known. Like for Schwarz methods, there is also an alternating and a parallel method of reflections, but interestingly, the parallel method is not always convergent. We carefully trace in this paper the historical development of these methods of reflections, give several precise mathematical formulations, an equivalence result with the alternating Schwarz method for two particles, and also an analysis for a one dimensional model problem with three particles of the alternating, parallel, and a recent averaged parallel method of reflections.Detecting Ineffective Features for Pattern Recognition
http://publications.mfo.de/handle/mfo/1314
Detecting Ineffective Features for Pattern Recognition
Györfi, László; Walk, Harro
For a binary classification problem, the hypothesis testing is studied, that a component of the observation vector is not effective, i.e., that component carries no information for the classification. We introduce nearest neighbor and partitioning estimates of the Bayes error probability, which result in a strongly consistent test.
Tue, 17 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13142017-10-17T00:00:00ZGyörfi, LászlóWalk, HarroFor a binary classification problem, the hypothesis testing is studied, that a component of the observation vector is not effective, i.e., that component carries no information for the classification. We introduce nearest neighbor and partitioning estimates of the Bayes error probability, which result in a strongly consistent test.Molecular Quantum Dynamics
http://publications.mfo.de/handle/mfo/1313
Molecular Quantum Dynamics
Hagedorn, George A.; Lasser, Caroline
We provide a brief introduction to some basic ideas
of Molecular Quantum Dynamics. We discuss the
scope, strengths and main applications of this field
of science. Finally, we also mention open problems
of current interest in this exciting subject.
Tue, 24 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13132017-10-24T00:00:00ZHagedorn, George A.Lasser, CarolineWe provide a brief introduction to some basic ideas
of Molecular Quantum Dynamics. We discuss the
scope, strengths and main applications of this field
of science. Finally, we also mention open problems
of current interest in this exciting subject.Exact Rate of Convergence of k-Nearest-Neighbor Classification Rule
http://publications.mfo.de/handle/mfo/1312
Exact Rate of Convergence of k-Nearest-Neighbor Classification Rule
Györfi, László; Döring, Maik; Walk, Harro
A binary classification problem is considered. The excess error probability of the k-nearest neighbor classification rule according to the error probability of the Bayes decision is revisited by a decomposition of the excess error probability into approximation and estimation error. Under a weak margin condition and under a modified Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded.
Mon, 16 Oct 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/13122017-10-16T00:00:00ZGyörfi, LászlóDöring, MaikWalk, HarroA binary classification problem is considered. The excess error probability of the k-nearest neighbor classification rule according to the error probability of the Bayes decision is revisited by a decomposition of the excess error probability into approximation and estimation error. Under a weak margin condition and under a modified Lipschitz condition, tight upper bounds are presented such that one avoids the condition that the feature vector is bounded.