Oberwolfach Publications
http://publications.mfo.de:8080
The DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Fri, 20 Sep 2019 18:01:11 GMT2019-09-20T18:01:11ZOberwolfach Publicationshttp://publications.mfo.de/themes/Mirage2/images/apple-touch-icon.png
http://publications.mfo.de:8080
Limits of graph sequences
http://publications.mfo.de/handle/mfo/2516
Limits of graph sequences
Klimošová, Tereza
Graphs are simple mathematical structures used to
model a wide variety of real-life objects. With the
rise of computers, the size of the graphs used for
these models has grown enormously. The need to efficiently
represent and study properties of extremely
large graphs led to the development of the theory of
graph limits.
Wed, 04 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25162019-09-04T00:00:00ZKlimošová, TerezaGraphs are simple mathematical structures used to
model a wide variety of real-life objects. With the
rise of computers, the size of the graphs used for
these models has grown enormously. The need to efficiently
represent and study properties of extremely
large graphs led to the development of the theory of
graph limits.On Logic, Choices and Games
http://publications.mfo.de/handle/mfo/2515
On Logic, Choices and Games
Oliva, Paulo
Can we always mathematically formalise our taste
and preferences? We discuss how this has been done
historically in the field of game theory, and how recent
ideas from logic and computer science have brought
an interesting twist to this beautiful theory.
Wed, 04 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25152019-09-04T00:00:00ZOliva, PauloCan we always mathematically formalise our taste
and preferences? We discuss how this has been done
historically in the field of game theory, and how recent
ideas from logic and computer science have brought
an interesting twist to this beautiful theory.Nonlinear Acoustics
http://publications.mfo.de/handle/mfo/2514
Nonlinear Acoustics
Kaltenbacher, Barbara; Brunnhuber, Rainer
Nonlinear acoustics has been a topic of research for
more than 250 years. Driven by a wide range and a
large number of highly relevant industrial and medical
applications, this area has expanded enormously
in the last few decades. Here, we would like to give
a glimpse of the mathematical modeling techniques
that are commonly employed to tackle problems in
this area of research, with a selection of references
for the interested reader to further their knowledge
into this mathematically interesting field.
Wed, 04 Sep 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25142019-09-04T00:00:00ZKaltenbacher, BarbaraBrunnhuber, RainerNonlinear acoustics has been a topic of research for
more than 250 years. Driven by a wide range and a
large number of highly relevant industrial and medical
applications, this area has expanded enormously
in the last few decades. Here, we would like to give
a glimpse of the mathematical modeling techniques
that are commonly employed to tackle problems in
this area of research, with a selection of references
for the interested reader to further their knowledge
into this mathematically interesting field.Group-Graded Rings Satisfying the Strong Rank Condition
http://publications.mfo.de/handle/mfo/2513
Group-Graded Rings Satisfying the Strong Rank Condition
Kropholler, Peter H.; Lorensen, Karl
A ring $R$ satisfies the $\textit{strong rank condition}$ (SRC) if, for every natural number $n$, the free $R$-submodules of $R^n$ all have rank $\leq n$. Let $G$ be a group and $R$ a ring strongly graded by $G$ such that the base ring $R_1$ is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that $R$ satisfies SRC if and only if
$R_1$ satisfies SRC and $G$ is amenable. The special case of this result for group rings allows us to prove a characterization of amenability involving the group von Neumann algebra that was conjectured by Wolfgang Lück. In addition, we include two applications to the study of group rings and their modules.
Fri, 16 Aug 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25132019-08-16T00:00:00ZKropholler, Peter H.Lorensen, KarlA ring $R$ satisfies the $\textit{strong rank condition}$ (SRC) if, for every natural number $n$, the free $R$-submodules of $R^n$ all have rank $\leq n$. Let $G$ be a group and $R$ a ring strongly graded by $G$ such that the base ring $R_1$ is a domain. Using an argument originated by Laurent Bartholdi for studying cellular automata, we prove that $R$ satisfies SRC if and only if
$R_1$ satisfies SRC and $G$ is amenable. The special case of this result for group rings allows us to prove a characterization of amenability involving the group von Neumann algebra that was conjectured by Wolfgang Lück. In addition, we include two applications to the study of group rings and their modules.A Cheeger Type Inequality in Finite Cayley Sum Graphs
http://publications.mfo.de/handle/mfo/2512
A Cheeger Type Inequality in Finite Cayley Sum Graphs
Biswas, Arindam; Saha, Jyoti Prakash
Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}{\eta}, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_{\Sigma}(G,S)$ and $\eta = 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.
Wed, 31 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25122019-07-31T00:00:00ZBiswas, ArindamSaha, Jyoti PrakashLet $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator is bounded away from $-1$. We also establish an explicit lower bound for the spectrum of these graphs, namely, the non-trivial eigenvalues of the normalised adjacency operator lies in the interval $\left(-1+\frac{h(G)^{4}}{\eta}, 1-\frac{h(G)^{2}}{2d^{2}}\right]$, where $h(G)$ denotes the (vertex) Cheeger constant of the $d$-regular graph $C_{\Sigma}(G,S)$ and $\eta = 2^{9}d^{8}$. Further, we improve upon a recently obtained bound on the non-trivial spectrum of the normalised adjacency operator of the non-bipartite Cayley graph $C(G,S)$.On a Cheeger Type Inequality in Cayley Graphs of Finite Groups
http://publications.mfo.de/handle/mfo/2511
On a Cheeger Type Inequality in Cayley Graphs of Finite Groups
Biswas, Arindam
Let $G$ be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h(\mathbb{G})^{4}}{\gamma}, 1-\frac{h(\mathbb{G})^{2}}{2d^{2}}\right]$, where $h(\mathbb{G})$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma = 2^{9}d^{6}(d+1)^{2}$.
Mon, 22 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25112019-07-22T00:00:00ZBiswas, ArindamLet $G$ be a finite group. It was remarked by Breuillard-Green-Guralnick-Tao that if the Cayley graph $C(G,S)$ is an expander graph and is non-bipartite then the spectrum of the adjacency operator $T$ is bounded away from $-1$. In this article we are interested in explicit bounds for the spectrum of these graphs. Specifically, we show that the non-trivial spectrum of the adjacency operator lies in the interval $\left[-1+\frac{h(\mathbb{G})^{4}}{\gamma}, 1-\frac{h(\mathbb{G})^{2}}{2d^{2}}\right]$, where $h(\mathbb{G})$ denotes the (vertex) Cheeger constant of the $d$ regular graph $C(G,S)$ with respect to a symmetric set $S$ of generators and $\gamma = 2^{9}d^{6}(d+1)^{2}$.Random permutations
http://publications.mfo.de/handle/mfo/2510
Random permutations
Betz, Volker
100 people leave their hats at the door at a party and
pick up a completely random hat when they leave.
How likely is it that at least one of them will get
back their own hat? If the hats carry name tags,
how difficult is it to arrange for all hats to be returned
to their owner? These classical questions of
probability theory can be answered relatively easily.
But if a geometric component is added, answering
the same questions immediately becomes very hard,
and little is known about them. We present some
of the open questions and give an overview of what
current research can say about them.
Fri, 12 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25102019-07-12T00:00:00ZBetz, Volker100 people leave their hats at the door at a party and
pick up a completely random hat when they leave.
How likely is it that at least one of them will get
back their own hat? If the hats carry name tags,
how difficult is it to arrange for all hats to be returned
to their owner? These classical questions of
probability theory can be answered relatively easily.
But if a geometric component is added, answering
the same questions immediately becomes very hard,
and little is known about them. We present some
of the open questions and give an overview of what
current research can say about them.On Co-Minimal Pairs in Abelian Groups
http://publications.mfo.de/handle/mfo/2509
On Co-Minimal Pairs in Abelian Groups
Biswas, Arindam; Saha, Jyoti Prakash
A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is a complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also construct infinite sets forming co-minimal pairs. Finally, we remark that a result of Kwon on the existence of minimal self-complements in $\mathbb{Z}$, also holds in any abelian group.
Tue, 09 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25092019-07-09T00:00:00ZBiswas, ArindamSaha, Jyoti PrakashA pair of non-empty subsets $(W,W')$ in an abelian group $G$ is a complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a group, the existence of minimal complement(s) depends on its structure. The dual problem asks that given such a set, if it is a minimal complement to some subset. We study tightness property of complement pairs $(W,W')$ such that both $W$ and $W'$ are minimal to each other. These are termed co-minimal pairs and we show that any non-empty finite set in an arbitrary free abelian group belongs to some co-minimal pair. We also construct infinite sets forming co-minimal pairs. Finally, we remark that a result of Kwon on the existence of minimal self-complements in $\mathbb{Z}$, also holds in any abelian group.A Quantitative Analysis of the “Lion-Man” Game
http://publications.mfo.de/handle/mfo/2508
A Quantitative Analysis of the “Lion-Man” Game
Kohlenbach, Ulrich; López-Acedo, Genaro; Nicolae, Adriana
In this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an $\varepsilon$-capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of
convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.
Mon, 08 Jul 2019 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25082019-07-08T00:00:00ZKohlenbach, UlrichLópez-Acedo, GenaroNicolae, AdrianaIn this paper we analyze, based on an interplay between ideas and techniques from logic and geometric analysis, a pursuit-evasion game. More precisely, we focus on a discrete lion and man game with an $\varepsilon$-capture criterion. We prove that in uniformly convex bounded domains the lion always wins and, using ideas stemming from proof mining, we extract a uniform rate of
convergence for the successive distances between the lion and the man. As a byproduct of our analysis, we study the relation among different convexity properties in the setting of geodesic spaces.Mathematical Instruments between Material Artifacts and Ideal Machines: Their Scientific and Social Role before 1950
http://publications.mfo.de/handle/mfo/2507
Mathematical Instruments between Material Artifacts and Ideal Machines: Their Scientific and Social Role before 1950
Since 1950, mathematicians have become increasingly familiar with the digital computer in their professional practice. Previously, however, many other instruments, now mostly forgotten, were commonly used to compute numerical solutions, generate geometrical objects, investigate mathematical problems, derive new results, and apply mathematics in a variety of scientific contexts. The problem of characterizing the mathematical objects that can be constructed with a given set of instruments frequently prompted deep theoretical investigations, from the Euclidean geometry of constructions with straightedge and compass, to Shannon’s theorem which, in 1941, stated that the functions constructible with a differential analyzer are exactly the solutions of algebraic differential equations. Beyond these mathematical considerations, instruments should also be viewed as social objects of a given time period and cultural tradition that can amalgamate the perspectives of the inventor, the maker, the user, and the collector; in this sense, mathematical instruments are an important part of the mathematical cultural heritage and are thus widely used in many science museums to demonstrate the cultural value of mathematics to the public. This workshop brought together mathematicians, historians, philosophers, collection curators, and scholars of education to address the various approaches to the history of mathematical instruments and compare the definition and role of these instruments over time, with the following fundamental questions in mind – What is mathematical in a mathematical instrument? What kind of mathematics is involved? What does it mean to embody mathematics in a material artefact, and how do non-mathematicians engage with this kind of embodied mathematics?
Sun, 01 Jan 2017 00:00:00 GMThttp://publications.mfo.de/handle/mfo/25072017-01-01T00:00:00ZSince 1950, mathematicians have become increasingly familiar with the digital computer in their professional practice. Previously, however, many other instruments, now mostly forgotten, were commonly used to compute numerical solutions, generate geometrical objects, investigate mathematical problems, derive new results, and apply mathematics in a variety of scientific contexts. The problem of characterizing the mathematical objects that can be constructed with a given set of instruments frequently prompted deep theoretical investigations, from the Euclidean geometry of constructions with straightedge and compass, to Shannon’s theorem which, in 1941, stated that the functions constructible with a differential analyzer are exactly the solutions of algebraic differential equations. Beyond these mathematical considerations, instruments should also be viewed as social objects of a given time period and cultural tradition that can amalgamate the perspectives of the inventor, the maker, the user, and the collector; in this sense, mathematical instruments are an important part of the mathematical cultural heritage and are thus widely used in many science museums to demonstrate the cultural value of mathematics to the public. This workshop brought together mathematicians, historians, philosophers, collection curators, and scholars of education to address the various approaches to the history of mathematical instruments and compare the definition and role of these instruments over time, with the following fundamental questions in mind – What is mathematical in a mathematical instrument? What kind of mathematics is involved? What does it mean to embody mathematics in a material artefact, and how do non-mathematicians engage with this kind of embodied mathematics?