arXiv subjects

Balázs Maga

Publications and source records attributed to Balázs Maga.

At least 19 recordsLinked to original sources

arXiv2026

Microscopic sorting in a multispecies exclusion process

We study a one-dimensional multispecies TASEP model with each particle type having an associated speed parameter and faster particles overtaking slower ones with the rate being the positive part of their speed difference. After expressing the sorting time of a finite system given a reverse-speed initial condition as the maximum of sorting times for all different fast-slow cuts, we prove that forbidding fast-fast and slow-slow overtakes does not alter the law of the sorting time. This allows for an LPP formulation and an analytic expression of the sorting time.

math.PR↗

arXiv2026

Level sets of prevalent Weierstrass functions

The $α$-Weierstrass function is defined as $W_g^{α,b}(x) = \sum_{k=0}^{\infty} b^{-αk} g(b^k x)$, where $g$ is a Lipschitz function on the unit circle. For a prevalent $α$-Weierstrass function, we prove that the upper Minkowski dimension of every level set is at most $1-α$, and the Hausdorff dimension of almost every level set equals $1-α$ with respect to its occupation measure. We further demonstrate that the occupation measure of a prevalent $α$-Weierstrass function is absolutely continuous with respect to the Lebesgue measure. Consequently, the result on the Hausdorff dimension of level sets applies to a set of level sets with positive Lebesgue measure. A central tool in our analysis is the Weierstrass embedding. For a sufficiently large dimension $d$, we construct Lipschitz functions $g_0, g_1, \ldots, g_{d-1}$ such that the mapping $x \mapsto \big(W_{g_0}^{α,b}(x), W_{g_1}^{α,b}(x), \ldots, W_{g_{d-1}}^{α,b}(x)\big)$ is $α$-bi-Hölder. We also prove that such an embedding requires at least $1/α$ coordinate functions.

math.CA↗

arXiv2026

Preferential Attachment with Local Flexibility

From the formation of social ties to the budding quantum internet, growing networks often exhibit local flexibility upon new nodes attaching to an existing network. In our proposed model, a new node connects uniformly at random to a node within the proximity of the intended target, including, but not restricted to, the target itself. Through numerical simulations and rigorous stochastic analysis, we find this local flexibility to qualitatively change the global network behavior of nonlinear preferential attachment. Depending on whether the preferential attachment is superlinear or (sub)linear, two distinct classes of complex network architectures emerge. The superlinear phase leads to a layered hierarchy, with no stationary degree distribution. Although there is a stationary degree distribution in the linear and sublinear cases, it decays strictly faster than for the Barabási--Albert model. We interpret our results within a two-dimensional phase diagram of network growth models incorporating redirection, with broad implications.

quant-ph↗

arXiv2026

Letter frequency in shifts of finite type with one forbidden word

This work considers combinatorial and statistical aspects of {\em{shifts of finite type}}, which are families of words over a finite alphabet which avoid a fixed class of {\emph{forbidden}} sub-words. The overarching question we are interested in is: how do local statistics of a uniformly random element of the shift space depend on combinatorial features of the forbidden set? We focus on the binary alphabet $\{0,1\}$, the class of shift spaces where a single pattern is forbidden, and the average frequency of $1$s (equivalently, the probability of observing $1$ at a given position). In this case, the relevant combinatorial information is encoded by a two-variable auto-correlation polynomial associated to the forbidden word, which we call the {\em{border polynomial}}. We present several results and examples characterizing the ordering of all words by their letter frequencies: for example, we describe the set of patterns which, when forbidden, cause the frequency of $1$s to increase, decrease, or stay exactly $1/2$. Our methodologies include novel explicit local injections and bijections, generating function analysis, and a connection with a probabilistic notion of letter frequency.

math.CO↗

arXiv2026

Exact formula on upper box dimension of generic Hölder level sets

In the previous decades, the size of level sets of functions have been extensively studied in various setups involving different regularity properties and size notions. In the case of Hölder functions, the authors have provided various bounds, but to date no explicit formulae have been found for any studied dimension and the results were valid only about very specific fractals. In this paper, for the first time, we have a result valid for a large class of self-similar sets, namely we prove that for these fractals Lebesgue almost every level set of the generic 1-Hölder-$α$ function defined on $F\subseteq \mathbb{R}^p$ has upper box dimension $\dim_H F - α$.

math.CA↗

arXiv2026

Chebyshev centers and radius of the set of permutons

We study the metric geometry of the set of permutons under the rectangular distance $d_{\square}$. We determine the Chebyshev radius to be 1/4 and characterize all Chebyshev centers: a permuton is a center if and only if it is 1/2- periodic in each coordinate. We also describe permutons that attain the extremal distance 1/4 from a given center.

math.CO↗

arXiv2026

On the approximation of permutons

We study the optimal rectangular-discrepancy approximation of permutons by finite permutations. We transfer bounds from discrepancy theory to this more restricted setup. Moreover, we show that superlinear approximation can occur only for permutons supported by graphs of measure-preserving functions, and demonstrate how the local regularity of this function obstructs approximability. We also consider the biased Brownian separable permuton and prove a lower bound on its approximation error by showing that its supporting measure-preserving function has Lipschitz points almost surely.

math.CO↗

arXiv2025

On the accuracy of population level approximation of network processes

The individual-based model of simple contagion processes is considered on regular graphs. This model explicitly incorporates the adjacency matrix of the network enabling us to study the effect of network structure on the dynamic of the propagation process. While the asymptotic behaviour of the model is well known, the transient behaviour has been less studied. Our goal in this paper is to give a theoretical estimate on the accuracy of the one-dimensional population-level approximation. This is carried out for arbitrary simple contagion processes and regular Turán graphs. Numerical evidence is shown that the theoretical estimate is rather sharp for dense graphs.

physics.soc-ph↗

arXiv2025

Matroid products via submodular coupling

The study of matroid products traces back to the 1970s, when Lovász and Mason studied the existence of various types of matroid products with different strengths. Among these, the tensor product is arguably the most important, which can be considered as an extension of the tensor product from linear algebra. However, Las Vergnas showed that the tensor product of two matroids does not always exist. Over the following four decades, matroid products remained surprisingly underexplored, regaining attention only in recent years due to applications in tropical geometry and the limit theory of matroids. In this paper, inspired by the concept of coupling in probability theory, we introduce the notion of coupling for matroids -- or, more generally, for submodular set functions. This operation can be viewed as a relaxation of the tensor product. Unlike the tensor product, however, we prove that a coupling always exists for any two submodular functions and can be chosen to be increasing if the original functions are increasing. As a corollary, we show that two matroids always admit a matroid coupling, leading to a novel operation on matroids. Our construction is algorithmic, providing an oracle for the coupling matroid through a polynomial number of oracle calls to the original matroids. We apply this construction to derive new necessary conditions for matroid representability and establish connection between tensor products and Ingleton's inequality. Additionally, we verify the existence of set functions that are universal with respect to a given property, meaning any set function over a finite domain with that property can be obtained as a quotient.

math.CO↗

arXiv2025

On the sampling entropy of permutons

For a permuton $μ$ let $H_n(μ)$ denote the Shannon entropy of the sampling distribution of $μ$ on $n$ points. We investigate the asymptotic growth of $H_n(μ)$ for a wide class of permutons. We prove that if $μ$ has a non-vanishing absolutely continuous part, then $H_n(μ)$ has a growth rate $Θ(n \log n)$. We show that if $μ$ is the graph of a piecewise continuously differentiable, measure-preserving function $f$, then $H_n(μ)/n$ tends to the Kolmogorov--Sinai entropy of $f$. Using genericity arguments, we also prove the existence of function permutons for which $H_n(μ)$ does not converge either after normalizing by $n$ or by $n\log n$. We study the sampling entropy of a natural family of random fractal-like permutons determined by a sequence of i.i.d. choices. It turns out that for every $n$, $H_n(μ)/n$ is heavily concentrated. We prove that the sequence $H_n(μ)/n$ either converges or has deterministic log-periodic oscillations almost surely, and argue towards the conjecture that in nondegenerate case, oscillation holds. On the other hand, for a straightforward random perturbation of the model $\tildeμ$ of $μ$, we prove the almost sure convergence of $H_n(\tildeμ)/n$.

math.PR↗

arXiv2024

Better estimates of Hölder thickness of fractals

Dimensions of level sets of generic continuous functions and generic Hölder functions defined on a fractal $F$ encode information about the geometry, ``the thickness" of $F$. While in the continuous case this quantity is related to a reasonably tame dimension notion which is called the topological Hausdorff dimension of $F$, the Hölder case seems to be highly nontrivial. A number of earlier papers attempted to deal with this problem, carrying out investigation in the case of Hausdorff dimension and box dimension. In this paper we continue our study of the Hausdorff dimension of almost every level set of generic $1$-Hölder-$ α$ functions, denoted by $D_{*}( α, F)$. We substantially improve previous lower and upper bounds on $D_{*}( α, Δ)$, where $Δ$ is the Sierpiński triangle, achieving asymptotically equal bounds as $α\to 0+$. Using a similar argument, we also give an even stronger lower bound on the generic lower box dimension of level sets. Finally, we construct a connected fractal $F$ on which there is a phase transition of $D_{*}( α, F)$, thus providing the first example exhibiting this behavior.

math.CA↗

arXiv2023

Box dimension of generic Hölder level sets

Hausdorff dimension of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections" "network" corresponding to a fractal set, $F$. This lead to the definition of the topological Hausdorff dimension of fractals. Finer information might be obtained by considering the Hausdorff dimension of level sets of generic $1$-Hölder-$α$ functions, which has a stronger dependence on the geometry of the fractal, as displayed in our previous papers. In this paper, we extend our investigations to the lower and upper box-counting dimension as well: while the former yields results highly resembling the ones about Hausdorff dimension of level sets, the latter exhibits a different behaviour. Instead of "finding narrow-cross sections", results related to upper box-counting dimension try to "measure" how much level sets can spread out on the fractal, how widely the generic function can "oscillate" on it. Key differences are illustrated by giving estimates concerning the Sierpiński triangle.

math.CA↗

arXiv2022

Generic Hölder level sets and fractal conductivity

Hausdorff dimensions of level sets of generic continuous functions defined on fractals can give information about the "thickness/narrow cross-sections'' of a "network" corresponding to a fractal set, $F$. This lead to the definition of the topological Hausdorff dimension of fractals. In this paper we continue our study of the level sets of generic $1$-Hölder-$α$ functions. While in a previous paper we gave the initial definitions and established some properties of these generic level sets, in this paper we provide numerical estimates in the case of the Sierpiński triangle. These calculations give better insight and illustrate why can one think of these generic $1$-Hölder-$α$ level sets as something measuring "thickness/narrow cross-sections/conductivity'' of a fractal "network". We also give an example for the phenomenon which we call phase transition for $D_{*}(α, F)$. This roughly means that for a certain lower range { of $α$s } only the geometry of $F$ determines $D_{*}(α, F)$ while for larger values the Hölder exponent, $α$ also matters.

math.CA↗

arXiv2022

Generic Hölder level sets on Fractals

Hausdorff dimensions of level sets of generic continuous functions defined on fractals were considered in two papers by R. Balka, Z. Buczolich and M. Elekes. In those papers the topological Hausdorff dimension of fractals was defined. In this paper we start to study level sets of generic $1$-Hölder-$α$ functions defined on fractals. This is related to some sort of "thickness", "conductivity" properties of fractals. The main concept of our paper is $D_{*}(α, F)$ which is the essential supremum of the Hausdorff dimensions of the level sets of a generic $1$-Hölder-$α$ function defined on the fractal $F$. We prove some basic properties of $D_{*}(α, F)$, we calculate its value for an example of a "thick fractal sponge", we show that for connected self similar sets $D_{*}(α, F)$ it equals the Hausdorff dimension of almost every level in the range of a generic $1$-Hölder-$α$ function.

math.CA↗

arXiv2021

Generic power series on subsets of the unit disk

In this paper, we examine the boundary behaviour of the generic power series $f$ with coefficients chosen from a fixed bounded set $Λ$ in the sense of Baire category. Notably, we prove that for any open subset $U$ of the unit disk $D$ with a non-real boundary point on the unit circle, $f(U)$ is a dense set of $\CC$. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.

math.CV↗

arXiv2021

Strong one-sided density without uniform density

In this paper we give an example of a closed, strongly one-sided dense set which is not of uniform density type. We also show that there is a set of uniform density type which is not of strong uniform density type.

math.CA↗

arXiv2021

Big and little Lipschitz one sets

Given a continuous function $f: {\mathbb R}\to {\mathbb R}$ we denote the so-called "big Lip" and "little lip" functions by $ {\mathrm {Lip}} f$ and $ {\mathrm {lip}} f$ respectively}. In this paper we are interested in the following question. Given a set $E {\subset} {\mathbb R}$ is it possible to find a continuous function $f$ such that $ {\mathrm {lip}} f=\mathbf{1}_E$ or $ {\mathrm {Lip}} f=\mathbf{1}_E$? For monotone continuous functions we provide the rather straightforward answer. For arbitrary continuous functions the answer is much more difficult to find. We introduce the concept of uniform density type (UDT) and show that if $E$ is $G_δ$ and UDT then there exists a continuous function $f$ satisfying $ {\mathrm {Lip}} f =\mathbf{1}_E$, that is, $E$ is a $ {\mathrm {Lip}} 1$ set. In the other direction we show that every ${\mathrm {Lip}} 1$ set is $G_δ$ and weakly dense. We also show that the converse of this statement is not true, namely that there exist weakly dense $G_{δ}$ sets which are not $ {\mathrm {Lip}} 1$. We say that a set $E\subset \mathbb{R}$ is ${\mathrm {lip}} 1$ if there is a continuous function $f$ such that ${\mathrm {lip}} f=\mathbf{1}_E$. We introduce the concept of strongly one-sided density and show that every ${\mathrm {lip}} 1$ set is a strongly one-sided dense $F_σ$ set.

math.CA↗

arXiv2021

Chest X-ray lung and heart segmentation based on minimal training sets

As the COVID-19 pandemic aggravated the excessive workload of doctors globally, the demand for computer aided methods in medical imaging analysis increased even further. Such tools can result in more robust diagnostic pipelines which are less prone to human errors. In our paper, we present a deep neural network to which we refer to as Attention BCDU-Net, and apply it to the task of lung and heart segmentation from chest X-ray (CXR) images, a basic but ardous step in the diagnostic pipeline, for instance for the detection of cardiomegaly. We show that the fine-tuned model exceeds previous state-of-the-art results, reaching $98.1\pm 0.1\%$ Dice score and $95.2\pm 0.1\%$ IoU score on the dataset of Japanese Society of Radiological Technology (JSRT). Besides that, we demonstrate the relative simplicity of the task by attaining surprisingly strong results with training sets of size 10 and 20: in terms of Dice score, $97.0\pm 0.8\%$ and $97.3\pm 0.5$, respectively, while in terms of IoU score, $92.2\pm 1.2\%$ and $93.3\pm 0.4\%$, respectively. To achieve these scores, we capitalize on the mixup augmentation technique, which yields a remarkable gain above $4\%$ IoU score in the size 10 setup.

eess.IV↗