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Martin Klazar

Publications and source records attributed to Martin Klazar.

At least 19 recordsLinked to original sources
arXiv2026

Elementary Real Analysis

Preliminary version of a book on univariate real analysis, with 14 chapters and 2 appendices. 1. Five numeric domains; 2. Limits of real sequences; 3. Series; 4. Limits of real functions. 5. Elementary functions; 6. Continuous functions; 7. Derivatives; 8. Mean value theorems; 9. Taylor polynomials; 10. Real analytic functions; 11. Newton integral; 12. Riemann integral; 13. Henstock-Kurzweil integral; 14. Applications of integrals; A. Auxiliary notions and notation; B. Solutions to exercises

math.HO↗
arXiv2026

The transcendence of $\mathrm{e}$ via formal power series

We review Hilbert's classical analytical proof of the transcendence of the number $\mathrm{e}$. Then, we show how this result can be obtained algebraically by means of formal power series (FPS). We give two proofs of the transcendence of $\mathrm{e}$ based on FPS. The first of them is a specialization of the 1990 proof by Beukers, Bézivin and Robba of the Lindemann-Weierstrass theorem. The second proof is due to this author and is an adaptation of Hilbert's argument to FPS.

math.NT↗
arXiv2026

Cardinalities in proofreading

In 1975, G. Pólya suggested that if two proofreaders found $a$ and $b$ errors in a text, of which $c$ errors were found by both of them, then a reasonable approximation of the unknown number $e$ of all errors is $e\approx ab/c$. We justify this formula by constructing a realistic model of proofreaders and estimating the efficiency of this model with the help of the bound on large deviations in the Probabilistic Method. In conclusion we discuss the distinction between realistic and probabilistic models of problems.

math.PR↗
arXiv2026

Catalan's conjecture is Mihăilescu's theorem

This text evolves from the lecture notes for my course on Catalan's conjecture in winter term 2025/26. The ultimate goal is to give full details of Mihăilescu's proof. Current chapters: 1. Euler's theorem: $x^2-y^3=1$; 2. V. Lebesgue's theorem: $x^m-y^2=1$; 3. Chao Ko's theorem: $x^2-y^q=1$ with $q\ge5$; 4. Two relations of Cassels: $p\,|\,y$ and $q\,|\,x$; 5. Mihăilescu's theorem: $x^p-y^q=1$ with $p>q>2$; 6. An obstruction group; 7. Super-Cassels relations: $p^2\,|\,y$ and $q^2\,|\,x$; 8. Theorem M4: $p=3,5$ or $q=3,5$; A Results from mathematical anlysis; and B Results from algebra.

math.HO↗
arXiv2025

Cantor digraphs and abbreviations of formulas

A digraph $D=\langle V,E\rangle$ ($E\subset V\times V$) is Cantor if Cantor's theorem - for no set there is a surjection from it to its power set - holds in $D$, in the sense we explain. We construct a ZF formula $φ$ with length $494$ such that $D\modelsφ$ iff $D$ is Cantor. In order to obtain $φ$, which is a word over the alphabet $$ \{x_1,\,x_2,\,\dots\}\cup \{\in,\,=,\,\neg, \,\to,\,\leftrightarrow,\,\wedge,\, \vee,\,\exists,\,\forall,\,(,\,)\}\,, $$ we devise abbreviation schemes of ZF formulas. We introduce extensive and strongly extensive digraphs and show, by the standard argument, that they are Cantor. We construct a countable strongly extensive digraph with arbitrarily large finite in-degrees.

math.LO↗
arXiv2025

Arnol'd's limit and the Lagrange inversion

We show how to prove by means of the Lagrange inversion the limit of Arnol'd that $$ \lim_{x\to0}\frac{\sin(\tan x)-\tan(\sin x)}{\arcsin(\arctan x)-\arctan(\arcsin x)}=1\,. $$ In fact, we obtain a more general result in terms of formal power series.

math.CA↗
arXiv2025

Extending Pólya's random walker beyond probability I. Complex weights

Working in combinatorial model $\mathrm{W_{co}}(d)$, $d=1,2,\dots$, of Pólya's random walker in $\mathbb{Z}^d$, we prove two theorems on recurrence to a vertex. We obtain an effective version of the first theorem if $d=2$. Using a semi-formal approach to generating functions, we extend both theorems beyond probability to a more general model $\mathrm{W_{\mathbb{C}}}$ with complex weights. We relate models $\mathrm{W_{co}}(d)$ to standard models $\mathrm{W_{Ma}}(d)$ based on Markov chains. The follow-up article will treat non-Archimedean models $\mathrm{W_{fo}}(k)$ in which weights are formal power series in $\mathbb{C}[[x_1,x_2,\dots,x_k]]$.

math.PR↗
arXiv2025

Countable real analysis

HMC sets are hereditarily at most countable sets. We rework a substantial part of univariate real analysis in a form in which only HMC real functions are used. In such countable real analysis we carry out Hilbert's proof of transcendence of the number $\mathrm{e}$. We also construct a uniformly continuous function $f:[0,1]\cap\mathbb{Q}\to\mathbb{R}$ such that $f'=1$ on $[0,1]\cap\mathbb{Q}$ and $\lim_{\substack{a\to1/\sqrt{2}\\a\in\mathbb{Q}}}f(a)=\frac{1}{\sqrt{2}}>f(b)$ for every $b\in[0,1]\cap\mathbb{Q}$.

math.LO↗
arXiv2023

HMC real numbers in Countable Mathematical Analysis

We develop a theory of real numbers as rational Cauchy sequences, in which any two of them, $(a_n)$ and $(b_n)$, are equal iff $\lim\,(a_n-b_n)=0$. We need such reals in the Countable Mathematical Analysis ([4]) which allows to use only hereditarily at most countable (HMC) sets.

math.LO↗
arXiv2022

Bertrand's paradox on a monitor

We investigate Bertrand's probabilistic paradox through the lens of discrete geometry and old-fashioned but reliable discrete probability. We approximate the plane unit circle with $1/n$ times $1/n$ boxes and count the pairs of boxes separated by distance more than $\sqrt{3}$. For $n\to\infty$ the proportion of such pairs goes to $$ \frac{1+\sqrt{3}}{8}-\frac{π(2-\sqrt{3})}{96}=0.33273\dots\;. $$

math.PR↗
arXiv2021

Rethinking real numbers as infinite decimals

We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that both operations are {\em weakly computable}; we introduce both kinds of computability in greater generality. We determine which additive and multiplicative shifts (restrictions of addition and multiplication to one variable) are computable, and prove that each of these shifts becomes computable after a~permutation of encoding. We ask if it is the case for the bivariate addition and multiplication.

math.LO↗
arXiv2021

Around Hilbert's theorem: the center of a circle is not constructible by straightedge alone

In order to state the theorem in the title formally and to review its rigorous proof, we extend and make more precise the Uspenskiy-Shen-Akopyan-Fedorov model of Euclidean constructions with arbitrary points; we also introduce formalizations for infinite configurations and for the projective plane. We exemplify the proof method by simpler and not so well known results that it is impossible to construct the unit length, or a given point, by compass and straightedge from nothing by means of classical arbitrary points. On the other hand we construct any given point by compass and straightedge from nothing by means of arbitrary points determined by horizontal segments. We quote a "proof" of Hilbert's theorem from the literature and explain why it is problematic. We rigorously prove Hilbert's theorem and present three variants of it, the last one for the projective plane.

math.MG↗
arXiv2020

On growth functions of ordered hypergraphs

For $k,l\ge2$ we consider ideals of edge $l$-colored complete $k$-uniform hypergraphs $(n,χ)$ with vertex sets $[n]=\{1, 2, \dots n\}$ for $n\in\mathbb{N}$. An ideal is a set of such colored hypergraphs that is closed to the relation of induced ordered subhypergraph. We obtain analogues of two results of Klazar [arXiv:0703047] who considered graphs, namely we prove two dichotomies for growth functions of such ideals of colored hypergraphs. The first dichotomy is for any $k,l\ge2$ and says that the growth function is either eventually constant or at least $n-k+2$. The second dichotomy is only for $k=3,l=2$ and says that the growth function of an ideal of edge two-colored complete $3$-uniform hypergraphs grows either at most polynomially, or for $n\ge23$ at least as $G_n$ where $G_n$ is the sequence defined by $G_1=G_2=1$, $G_3=2$ and $G_n = G_{n-1} + G_{n-3}$ for $n\ge4$. The lower bounds in both dichotomies are tight.

math.CO↗
arXiv2020

Effective formulas for linear recurrence sequences of integers

We propose a new definition of effective formulas for problems in enumerative combinatorics. We outline the proof of the fact that every linear recurrence sequence of integers has such a formula. It follows from a lower bound that can be deduced from the Skolem-Mahler-Lech theorem and the Subspace Theorem. We will give details of this deduction that is due to P. Corvaja in the full version of this extended abstract.

math.CO↗
arXiv2019

The Newton integral and the Stirling formula

We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval. We review in its framework in detail two derivations of the Stirling formula. The first approximates log(1)+log(2)+...+log(n) with an integral and the second uses the classical gamma function and a Fubini-type result. We mention two more integral representations of n!.

math.HO↗
arXiv2018

What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I

For enumerative problems, i.e. computable functions f from N to Z, we define the notion of an effective (or closed) formula. It is an algorithm computing f(n) in the number of steps that is polynomial in the combined size of the input n and the output f(n), both written in binary notation. We discuss many examples of enumerative problems for which such closed formulas are, or are not, known. These problems include (i) linear recurrence sequences and holonomic sequences, (ii) integer partitions, (iii) pattern-avoiding permutations, (iv) triangle-free graphs and (v) regular graphs. In part I we discuss problems (i) and (ii) and defer (iii)--(v) to part II. Besides other results, we prove here that every linear recurrence sequence of integers has an effective formula in our sense.

math.CO↗