arXiv subjects

Zoltán Vidnyánszky

Publications and source records attributed to Zoltán Vidnyánszky.

32 records · Page 2Linked to original sources

arXiv2018

The structure of random automorphisms of countable structures

In order to understand the structure of the `typical' element of an automorphism group, one has to study how large the conjugacy classes of the group are. When typical is meant in the sense of Baire category, a complete description of the size of the conjugacy classes has been given by Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behavior of the automorphisms is entirely different from the Baire category case. In this paper, we generalize the theorems of Dougherty and Mycielski about $S_\infty$ to arbitrary automorphism groups of countable structures isolating a new model theoretic property, the Cofinal Strong Amalgamation Property. As an application we show that a large class of automorphism groups can be decomposed into the union of a meager and a Haar null set.

math.LO↗

arXiv2018

On the existence of large antichains for definable quasi-orders

We generalize Harrington-Marker-Shelah's Dilworth-style characterization of the existence of non-empty perfect antichains to co-analytic quasi-orders, establish the analogous theorem at the next definable cardinal, and consider generalizations beyond the first level of the projective hierarchy.

math.LO↗

arXiv2018

The structure of random automorphisms of the random graph

We give a complete description of the size of the conjugacy classes of the automorphism group of the random graph with respect to Christensen's Haar null ideal. It is shown that every non-Haar null class contains a translated copy of a nonempty portion of every compact set and that there are continuum many non-Haar null conjugacy classes. Our methods also yield a new proof of an old result of Truss.

math.LO↗

arXiv2018

The structure of random homeomorphisms

In order to understand the structure of the "typical" element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood, see e.g. the works of Glasner and Weiss, and Kechris and Rosendal. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. When typical means random, that is, almost every with respect to this notion of Haar null sets, the behaviour of the homeomorphisms is entirely different from the generic case. For $\text{Homeo}^+([0,1])$ we describe the non-Haar null conjugacy classes and also show that their union is co-Haar null, for $\text{Homeo}^+(\mathbb{S}^1)$ we describe the non-Haar null conjugacy classes, and for $\mathcal{U}(\ell^2)$ we show that, apart from the classes of the multishifts, all conjugacy classes are Haar null. As an application we affirmatively answer the question whether these groups can be written as the union of a meagre and a Haar null set.

math.LO↗

arXiv2018

The structure of random automorphisms of the rational numbers

In order to understand the structure of the "typical" element of an automorphism group, one has to study how large the conjugacy classes of the group are. For the case when typical is meant in the sense of Baire category, Truss proved that there is a co-meagre conjugacy class in Aut(Q, <), the automorphism group of the rational numbers. Following Dougherty and Mycielski we investigate the measure theoretic dual of this problem, using Christensen's notion of Haar null sets. We give a complete description of the size of the conjugacy classes of Aut(Q, <) with respect to this notion. In particular, we show that there exist continuum many non-Haar null conjugacy classes, illustrating that the random behaviour is quite different from the typical one in the sense of Baire category.

math.LO↗

arXiv2018

Embeddability on functions: order and chaos

We study the quasi-order of topological embeddability on definable functions between Polish zero-dimensional spaces. We first study the descriptive complexity of this quasi-order restricted to the space of continuous functions. Our main result is the following dichotomy: the embeddability quasi-order restricted to continuous functions from a given compact space to another is either an analytic complete quasi-order or a well-quasi-order. We then turn to the existence of maximal elements with respect to embeddability in a given Baire class. It is proved that the class of continuous functions is the only Baire class to admit a maximal element. We prove that no Baire class admits a maximal element, except for the class of continuous functions which admits a maximum element.

math.LO↗

arXiv2017

Resting state fMRI functional connectivity-based classification using a convolutional neural network architecture

Machine learning techniques have become increasingly popular in the field of resting state fMRI (functional magnetic resonance imaging) network based classification. However, the application of convolutional networks has been proposed only very recently and has remained largely unexplored. In this paper we describe a convolutional neural network architecture for functional connectome classification called connectome-convolutional neural network (CCNN). Our results on simulated datasets and a publicly available dataset for amnestic mild cognitive impairment classification demonstrate that our CCNN model can efficiently distinguish between subject groups. We also show that the connectome-convolutional network is capable to combine information from diverse functional connectivity metrics and that models using a combination of different connectivity descriptors are able to outperform classifiers using only one metric. From this flexibility follows that our proposed CCNN model can be easily adapted to a wide range of connectome based classification or regression tasks, by varying which connectivity descriptor combinations are used to train the network.

stat.ML↗

arXiv2016

Characterization of order types of pointwise linearly ordered families of Baire class 1 functions

In the 1970s M. Laczkovich posed the following problem: Let $\mathcal{B}_1(X)$ denote the set of Baire class $1$ functions defined on an uncountable Polish space $X$ equipped with the pointwise ordering. \[\text{Characterize the order types of the linearly ordered subsets of $\mathcal{B}_1(X)$.} \]The main result of the present paper is a complete solution to this problem. We prove that a linear order is isomorphic to a linearly ordered family of Baire class $1$ functions iff it is isomorphic to a subset of the following linear order that we call $([0,1]^{<ω_1}_{\searrow 0},<_{altlex})$, where $[0,1]^{<ω_1}_{\searrow 0}$ is the set of strictly decreasing transfinite sequences of reals in $[0, 1]$ with last element $0$, and $<_{altlex}$, the so called \emph{alternating lexicographical ordering}, is defined as follows: if $(x_α)_{α\leq ξ}, (x'_α)_{α\leq ξ'} \in [0,1]^{<ω_1}_{\searrow 0}$, and $δ$ is the minimal ordinal where the two sequences differ then we say that \[ (x_α)_{α\leq ξ} <_{altlex} (x'_α)_{α\leq ξ'} \iff (δ\text{ is even and } x_δ x'_δ). \] Using this characterization we easily reprove all the known results and answer all the known open questions of the topic.

math.LO↗

arXiv2015

Almost disjoint refinements and mixing reals

We investigate families of subsets of $ω$ with almost disjoint refinements in the classical case as well as with respect to given ideals on $ω$. More precisely, we study the following topics and questions: 1) Examples of projective ideals. 2) We prove the following generalization of a result due to J. Brendle: If $V\subseteq W$ are transitive models, $ω_1^W\subseteq V$, $\mathcal{P}(ω)\cap V\not = \mathcal{P}(ω)\cap W$, and $\mathcal{I}$ is an analytic or coanalytic ideal coded in $V$, then there is an $\mathcal{I}$-almost disjoint refinement ($\mathcal{I}$-ADR) of $\mathcal{I}^+\cap V$ in $W$, that is, a family $\{A_X:X\in\mathcal{I}^+\cap V\}\in W$ such that (i) $A_X\subseteq X$, $A_X\in \mathcal{I}^+$ for every $X$ and (ii) $A_X\cap A_Y\in\mathcal{I}$ for every distinct $X$ and $Y$. 3) The existence of perfect $\mathcal{I}$-almost disjoint ($\mathcal{I}$-AD) families, and the existence of a "nice" ideal $\mathcal{I}$ on $ω$ with the property: Every $\mathcal{I}$-AD family is countable but $\mathcal{I}$ is nowhere maximal. 4) The existence of $(\mathcal{I},\text{Fin})$-almost disjoint refinements of families of $\mathcal{I}$-positive sets in the case of everywhere meager (e.g. analytic or coanalytic) ideals. We prove a positive result under Martin's Axiom. 5) Connections between classical properties of forcing notions and adding mixing reals (and mixing injections), that is, a (one-to-one) function $f:ω\toω$ such that $|f[X]\cap Y|=ω$ for every $X,Y\in [ω]^ω\cap V$.

math.LO↗

arXiv2015

Naively Haar null sets in Polish groups

Let $(G,\cdot)$ be a Polish group. We say that a set $X \subset G$ is Haar null if there exists a universally measurable set $U \supset X$ and a Borel probability measure $μ$ such that for every $g, h \in G$ we have $μ(gUh)=0$. We call a set $X$ naively Haar null if there exists a Borel probability measure $μ$ such that for every $g, h \in G$ we have $μ(gXh)=0$. Generalizing a result of Elekes and Steprāns, which answers the first part of Problem FC from Fremlin's list, we prove that in every abelian Polish group there exists a naively Haar null set that is not Haar null.

math.LO↗

arXiv2015

Ranks on the Baire class $ξ$ functions

In 1990 Kechris and Louveau developed the theory of three very natural ranks on the Baire class $1$ functions. A rank is a function assigning countable ordinals to certain objects, typically measuring their complexity. We extend this theory to the case of Baire class $ξ$ functions, and generalize most of the results from the Baire class 1 case. We also show that their assumption of the compactness of the underlying space can be eliminated. As an application, we solve a problem concerning the so called solvability cardinals of systems of difference equations, arising from the theory of geometric decompositions. We also show that certain other very natural generalizations of the ranks of Kechris and Louveau surprisingly turn out to be bounded in $ω_1$. Finally, we prove a general result showing that all ranks satisfying some natural properties coincide for bounded functions.

math.LO↗

arXiv2014

Unions of regular polygons with large perimeter-to-area ratio

T. Keleti asked, whether the ratio of the perimeter and the area of a finite union of unit squares is always at most 4. In this paper we present an example where the ratio is greater than 4. We also answer the analogous question for regular triangles negatively and list a number of open problems.

math.MG↗

arXiv2014

Haar null sets without $G_δ$ hulls

Let $G$ be an abelian Polish group, e.g. a separable Banach space. A subset $X \subset G$ is called Haar null (in the sense of Christensen) if there exists a Borel set $B \supset X$ and a Borel probability measure $μ$ on $G$ such that $μ(B+g)=0$ for every $g \in G$. The term shy is also commonly used for Haar null, and co-Haar null sets are often called prevalent. Answering an old question of Mycielski we show that if $G$ is not locally compact then there exists a Borel Haar null set that is not contained in any $G_δ$ Haar null set. We also show that $G_δ$ can be replaced by any other class of the Borel hierarchy, which implies that the additivity of the $σ$-ideal of Haar null sets is $ω_1$. The definition of a generalised Haar null set is obtained by replacing the Borelness of $B$ in the above definition by universal measurability. We give an example of a generalised Haar null set that is not Haar null, more precisely we construct a coanalytic generalised Haar null set without a Borel Haar null hull. This solves Problem GP from Fremlin's problem list. Actually, all our results readily generalise to all Polish groups that admit a two-sided invariant metric.

math.LO↗

arXiv2014

Transfinite inductions producing coanalytic sets

A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that in $V=L$ there exists an uncountable coanalytic subset of the plane that intersects every $C^1$ curve in a countable set.

math.LO↗