Old finds of Roman coins in the archives and collection of the National Museum in Krakow III : Dr Wacław Pancerzyński and discoveries of Roman coins in Volhynia

Before World War II, some Roman coins discovered in Volhynia were added to the numismatic col- lection of the National Museum in Krakow (NMK). Some of them were donated by Dr Wacław Pancerzyński, a physician and a major in the Polish Army. He obtained the coins during his service in the Border Protection Corps in Mizocz in what was then Zdołbunowski poviat (today Mizoch, zdolbunivs’kii raion, rivnens’ka oblast’, Ukraine). The three coins, donated in 1927, were most likely part of a hoard of Roman denarii discovered that year in the vicinity of Mizocz. In turn, a bronze coin of Constantine the Great, type “Urbs Roma”, found in Buszcza (today Bushcha, zdolbunivs’kii rayon, rivnens’ka oblast’, Ukraine), entered the NMK collection in 1929. It remains to this day the only discovery of a Roman coin recorded in this town. Unfortunately, due to confusion and the loss of some archives during World War II, it is not possible to accurately identify the coins in the current NMK collection. The same holds true regarding the denarii of Trajan and Antoninus Pius found in another locality in Volhynia, Międzyrzecz Korecki (today Velyki Mezhyrychi, koretskyi raion, rivnens’ka oblast’, Ukraine), donated by Dr Stanisław Tomkowicz. This find is well known in the literature. Unfortunately, we do not know how the donor, an outstanding figure known for his work for the protection of monuments in Kraków and Western Galicia, came into possession of these coins

Eigenvalue Spectra of Functional Networks in fMRI Data and Artificial Models

Full paper available at Springerlink: http://link.springer.com/chapter/10.1007%2F978-3-642-38658-9_19In this work we provide a spectral comparison of functional networks in fMRI data of brain activity and artificial energy-based neural model. The spectra (set of eigenvalues of the graph adjacency matrix) of both networks turn out to obey similar decay rate and characteristic power-law scaling in their middle parts. This extends the set of statistics, which are already confirmed to be similar for both neural models and medical data, by the graph spectrum

Forecasting Cinema Attendance at the Movie Show Level: Evidence from Poland

Background: Cinema programmes are set in advance (usually with a weekly frequency), which motivates us to investigate the short-term forecasting of attendance. In the literature on the cinema industry, the issue of attendance forecasting has gained less research attention compared to modelling the aggregate performance of movies. Furthermore, unlike most existing studies, we use data on attendance at the individual show level (179,103 shows) rather than aggregate box office sales. Objectives: In the paper, we evaluate short-term forecasting models of cinema attendance. The main purpose of the study is to find the factors that are useful in forecasting cinema attendance at the individual show level (i.e., the number of tickets sold for a particular movie, time and cinema). Methods/Approach: We apply several linear regression models, estimated for each recursive sample, to produce one-week ahead forecasts of the attendance. We then rank the models based on the out-of-sample fit. Results: The results show that the best performing models are those that include cinema- and region-specific variables, in addition to movie parameters (e.g., genre, age classification) or title popularity. Conclusions: Regression models using a wide set of variables (cinema- and region-specific variables, movie features, title popularity) may be successfully applied for predicting individual cinema shows attendance in Poland

Studies on Roman coin finds from the Central European Barbaricum in the Institute of Archaeology of the Jagiellonian University : an overview

A direct successor of the oldest tradition of academic archaeology in Poland, the Institute of Archaeology of the Jagiellonian University ranks among the leading research centres with respect to studies on the influx of Roman coins into European Barbaricum. The interest in Roman coinage at the Jagiellonian University pre-dates archaeology and can be traced back to the 16th century and the professors of the Kraków Academy (the name of the university at that time) Maciej of Miechów (1457-1523) and Stanisław Grzebski (1524-1570). In the 19th century, Roman coins discovered in the vicinity of Kraków attracted the interest of Jerzy Samuel Bandtke (1768-1835). However, the time when this area of research enjoyed particular development falls to the last years before WWII and the post-war period. A significant role in this respect was played by researchers either representing the JU Institute of Archaeology, like Professors Rudolf Jamka (1906-1972), Kazimierz Godłowski (1934-1995), and Piotr Kaczanowski (1944-2015), or those cooper-ating with the Institute like Professor Stefan Skowronek (1928-2019). Their activity laid the foundations for today’s research on the finds of Roman coins and their inflow into the territories of the Roman Period Barbaricum. Currently, this area of studies is within the focus of two of the departments of the Institute of Archaeology: the Department of Iron Age Archaeology and the Department of Classical Archaeology. The intensification of research on the inflow of Roman coins owes much to the Finds of Roman coins in Poland and lands connected historically with PL project, carried out in 2014–2018 under the leadership of Professor Aleksander Burshe, with important contributions provided by a group of scholars from the JU Institute of Archaeology. Despite the conclusion of the project, studies on the inflow of Roman coins will continue.1

Graph Polynomials and Group Coloring of Graphs

Let $\Gamma$ be an Abelian group and let $G$ be a simple graph. We say that $G$ is $\Gamma$-colorable if for some fixed orientation of $G$ and every edge labeling $\ell:E(G)\rightarrow \Gamma$, there exists a vertex coloring $c$ by the elements of $\Gamma$ such that $c(y)-c(x)\neq \ell(e)$, for every edge $e=xy$ (oriented from $x$ to $y$). Langhede and Thomassen proved recently that every planar graph on $n$ vertices has at least $2^{n/9}$ different $\mathbb{Z}_5$-colorings. By using a different approach based on graph polynomials, we extend this result to $K_5$-minor-free graphs in the more general setting of field coloring. More specifically, we prove that every such graph on $n$ vertices is $\mathbb{F}$-$5$-choosable, whenever $\mathbb{F}$ is an arbitrary field with at least $5$ elements. Moreover, the number of colorings (for every list assignment) is at least $5^{n/4}$.Comment: 14 page

On a Problem of Steinhaus

Let $N$ be a positive integer. A sequence $X=(x_1,x_2,\ldots,x_N)$ of points in the unit interval $[0,1)$ is piercing if $\{x_1,x_2,\ldots,x_n\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. In 1958 Steinhaus asked whether piercing sequences can be arbitrarily long. A negative answer was provided by Schinzel, who proved that any such sequence may have at most $74$ elements. This was later improved to the best possible value of $17$ by Warmus, and independently by Berlekamp and Graham. In this paper we study a more general variant of piercing sequences. Let $f(n)\geq n$ be an infinite nondecreasing sequence of positive integers. A sequence $X=(x_1,x_2,\ldots,x_{f(N)})$ is $f$-piercing if $\{x_1,x_2,\ldots,x_{f(n)}\}\cap \left[\frac{i}{n},\frac{i+1}{n} \right) \neq\emptyset$ holds for every $n=1,2,\ldots, N$ and every $i=0,1,\ldots,n-1$. A special case of $f(n)=n+d$, with $d$ a fixed nonnegative integer, was studied by Berlekamp and Graham. They noticed that for each $d\geq 0$, the maximum length of any $(n+d)$-piercing sequence is finite. Expressing this maximum length as $s(d)+d$, they obtained an exponential upper bound on the function $s(d)$, which was later improved to $s(d)=O(d^3)$ by Graham and Levy. Recently, Konyagin proved that $2d\leqslant s(d)< 200d$ holds for all sufficiently big $d$. Using a different technique based on the Farey fractions and stick-breaking games, we prove here that the function $s(d)$ satisfies $\left\lfloor{}c_1d\right\rfloor{}\leqslant s(d)\leqslant c_2d+o(d)$, where $c_1=\frac{\ln 2}{1-\ln 2}\approx2.25$ and $c_2=\frac{1+\ln2}{1-\ln2}\approx5.52$. We also prove that there exists an infinite $f$-piercing sequence with $f(n)= \gamma n+o(n)$ if and only if $\gamma\geq\frac{1}{\ln 2}\approx 1.44$.Comment: 16 page