Here you may download a collection of my non-research papers popularising mathematics or focusing on some classical topics in mathematics (mostly in Polish). A part of them appeared in the students journal [MACIERZATOR] published by the Students' Mathematical Society of the University of Silesia. Some others are based on several talks I gave at seminars and lectures intended mainly for high school and academic students. Still more to upload...

[MACIERZATOR] papers

• Nigel Kalton (1946-2010), Macierzator 41, November 2011 (in Polish).
A biography of the British Mathematician Nigel J. Kalton.

• Joram Lindenstrauss (1936-2012), Macierzator 50, November 2012 (in Polish). A biography of the Israeli Mathematician Joram Lindenstrauss.

• The Olympic impressions series consists of short articles whose goal is to present and analyse selected problems which appeared at various mathematical competitions, as well as some classical results strongly related to olympiad-type problems.
• Macierze dodatnio określone vs. wypukłość (1), Impresje olimpijskie 5, Macierzator 46, April 2012 (in Polish). This article is inspired by the inequality $$\frac{x}{(2a)^t}+\frac{y}{(2b)^t}+\frac{z}{(2c)^t}\geq\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}\, ,$$claimed for all positive numbers $$a,b,c,x,y,z,t$$ satisfying $$1\leq x,y,z\leq 4$$. It was proposed by Géza Kós for the Vojtěch Jarník International Math. Competition, 2012. Although it has a seemingly simple proof, which involves Euler's $$\Gamma$$-function, the trick turned out to be very difficult to spot for the contestants. We try to give a deeper insight into this problem and show that it is, in fact, all about proving positive definiteness of a certain matrix corresponding to the above inequality.
• Lemat Steinitza, Impresje olimpijskie 4, Macierzator 43, January 2012 (in Polish). We analyse the proof of the Steinitz lemma, mentioned in the last part of this series, in the version by Bárány and Grinberg: If $$\{x_j\}_{j=1}^n\subset\mathbb{R}^d$$ and $$\|x_j\|\leq 1$$ (the Euclidean norm) for $$1\leq j\leq n$$, then one may chose signs $$\{\varepsilon_j\}_{j=1}^n\subset\{-1,1\}$$ so that $$\Biggl\|\sum_{j=1}^n\varepsilon_jx_j\Biggr\|\leq\sqrt{d}.$$ We try to underline that the key idea is to consider the vector space $$V$$ of all vanishing linear combinations of $$x_j$$'s and to look for extreme points of the polyhedron $$V\cap [-1,1]^n$$.
• Potęga liniowej zależności, Impresje olimpijskie 3, Macierzator 40, October 2011 (in Polish). We discuss a nice combinatorial-algebraic problem from the International Mathematics Competition for University Students, 2011. The key idea is to fight against a huge number of linearly dependent vectors by considering the linear space of all their vanishing linear combinations and looking for some special properties of this space. This idea is crucial for proving a beautiful geometrical lemma due to Steinitz which concerns minimazing the norm $$\|\sum_{j=1}^n\varepsilon_jx_j\|$$, where vectors $$x_j$$ are given, whereas $$\varepsilon_j=\pm 1$$ are to be chosen.
• Pewne wzmocnienie nierówności Erdösa-Mordella, Impresje olimpijskie 2, Macierzator 36, March 2011 (in Polish). The Erdös-Mordell inequality asserts that if $$P$$ is any point lying in the interior of a given triangle $$XYZ$$ then $$x+y+z\geq 2(p+q+r),$$ where $$x,y,z$$ are the distances between $$P$$ and $$X,Y,Z$$, respectively, and $$p,q,r$$ are the distances between $$P$$ and the lines $$YZ,ZX,XY$$, respectively. We discuss a certain generalisation of this inequality for hexagons which appeared as a problem at the International Mathematical Olympiad in 1996.
• Prawdopodobnie najtrudniejsze zadanie w historii USAMO, Impresje olimpijskie 1, Macierzator 34, December 2010 (in Polish). This is on the remarkable inequality $$\sum_{i,j=1}^n\min\{x_ix_j,y_iy_j\}\leq\sum_{i,j=1}^n\min\{x_iy_j,x_jy_i\}$$ (for positive $$x_i$$ and $$y_j$$) which appeared at the USA Mathematical Olympiad in 2000 and turned out to be extremely difficult to prove.

Other notes

• Combinatorial Nullstellensatz (in Polish). These are the notes for two talks I gave to students in December 2011 and January 2012. We discuss Alon's version of a combinatorial counterpart of Hilbert's zeros theorem. Several typical applications are given in the form of problems with hints.

• O metodzie probabilistycznej Paula Erdösa, Matematyka-Społeczeństwo-Nauczanie 43 (2009), 29-31 (in Polish).

• Rachunek operatorowy Mikusińskiego i jego zastosowanie do równań różniczkowych (in Polish). Here you may find a complete proof of the celebrated Titchmarsh theorem which says that for any continuous functions $$f,g\colon [0,\infty)\to\mathbb{C}$$ we have: $$\forall_{t\geq 0}\,\,\int_0^t f(t-u)g(u)\, du=0\,\,\,\Rightarrow\,\,\, f(t)\equiv 0\,\,\mbox{ or }\,\, g(t)\equiv 0.$$In other words, $$(C[0,\infty),+,\ast)$$ is a non-unital commutative ring with no non-trivial zero divisors.