Lucas' theorem: its generalizations, extensions and applications (1878--2014)

In 1878 \'E. Lucas proved a remarkable result which provides a simple way to compute the binomial coefficient ${n\choose m}$ modulo a prime $p$ in terms of the binomial coefficients of the base-$p$ digits of $n$ and $m$: {\it If $p$ is a prime, $n=n_0+n_1p+\cdots +n_sp^s$ and $m=m_0+m_1p+\cdots +m_sp^s$ are the $p$-adic expansions of nonnegative integers $n$ and $m$, then \begin{equation*} {n\choose m}\equiv \prod_{i=0}^{s}{n_i\choose m_i}\pmod{p}. \end{equation*}} The above congruence, the so-called {\it Lucas' theorem} (or {\it Theorem of Lucas}), plays an important role in Number Theory and Combinatorics. In this article, consisting of six sections, we provide a historical survey of Lucas type congruences, generalizations of Lucas' theorem modulo prime powers, Lucas like theorems for some generalized binomial coefficients, and some their applications. In Section 1 we present the fundamental congruences modulo a prime including the famous Lucas' theorem. In Section 2 we mention several known proofs and some consequences of Lucas' theorem. In Section 3 we present a number of extensions and variations of Lucas' theorem modulo prime powers. In Section 4 we consider the notions of the Lucas property and the double Lucas property, where we also present numerous integer sequences satisfying one of these properties or a certain Lucas type congruence. In Section 5 we collect several known Lucas type congruences for some generalized binomial coefficients. In particular, this concerns the Fibonomial coefficients, the Lucas $u$-nomial coefficients, the Gaussian $q$-nomial coefficients and their generalizations. Finally, some applications of Lucas' theorem in Number Theory and Combinatorics are given in Section 6.Comment: 51 pages; survey article on Lucas type congruences closely related to Lucas' theore

Teacher roles during amusement park visits – insights from observations, interviews and questionnaires

Amusement parks offer rich possibilities for physics learning, through observations and experiments that illustrate important physical principles and often involve the whole body. Amusement parks are also among the most popular school excursions, but very often the learning possibilities are underused. In this work we have studied different teacher roles and discuss how universities, parks or event managers can encourage and support teachers and schools in their efforts to make amusement park visits true learning experiences for their students

The Ecological Role of Salamanders as Predators and Prey

Salamanders are relevant components of many terrestrial and aquatic ecosystems. However, despite the importance of salamanders in many resource–consumer networks, their functional role remains remarkably understudied. Therefore, this volume, entitled The Ecological Role of Salamanders as Prey and Predators, provides an opportunity for researchers to highlight the new research on the ecological role of salamanders and newts in prey–predator systems, their trophic behavior, and the variability of their trophic niche in space and time. Various innovative methods, such as COI metabarcoding and network analysis, are applied in the present study to test both the classical and new hypotheses concerning the trophic ecology of salamanders and their interactions with their prey. The present volume is composed of one review and seven research papers, all of which are published after undergoing a complete and impartial peer-review process

Advances in farm animal genomic resources

The history of livestock started with the domestication of their wild ancestors: a restricted number of species allowed to be tamed and entered a symbiotic relationship with humans. In exchange for food, shelter and protection, they provided us with meat, eggs, hides, wool and draught power, thus contributing considerably to our economic and cultural development. Depending on the species, domestication took place in different areas and periods. After domestication, livestock spread over all inhabited regions of the earth, accompanying human migrations and becoming also trade objects. This required an adaptation to different climates and varying styles of husbandry and resulted in an enormous phenotypic diversity. Approximately 200 years ago, the situation started to change with the rise of the concept of breed. Animals were selected for the same visible characteristics, and crossing with different phenotypes was reduced. This resulted in the formation of different breeds, mostly genetically isolated from other populations. A few decades ago, selection pressure was increased again with intensive production focusing on a limited range of types and a subsequent loss of genetic diversity. For short-term economic reasons, farmers have abandoned traditional breeds. As a consequence, during the 20th century, at least 28% of farm animal breeds became extinct, rare or endangered. The situation is alarming in developing countries, where native breeds adapted to local environments and diseases are being replaced by industrial breeds. In the most marginal areas, farm animals are considered to be essential for viable land use and, in the developing world, a major pathway out of poverty. Historic documentation from the period before the breed formation is scarce. Thus, reconstruction of the history of livestock populations depends on archaeological, archeo-zoological and DNA analysis of extant populations. Scientific research into genetic diversity takes advantage of the rapid advances in molecular genetics. Studies of mitochondrial DNA, microsatellite DNA profiling and Y-chromosomes have revealed details on the process of domestication, on the diversity retained by breeds and on relationships between breeds. However, we only see a small part of the genetic information and the advent of new technologies is most timely in order to answer many essential questions. High-throughput single-nucleotide polymorphism genotyping is about to be available for all major farm animal species. The recent development of sequencing techniques calls for new methods of data management and analysis and for new ideas for the extraction of information. To make sense of this information in practical conditions, integration of geo-environmental and socio-economic data are key elements. The study and management of farm animal genomic resources (FAnGR) is indeed a major multidisciplinary issue. The goal of the present Research Topic was to collect contributions of high scientific quality relevant to biodiversity management, and applying new methods to either new genomic and bioinformatics approaches for characterization of FAnGR, to the development of FAnGR conservation methods applied ex-situ and in-situ, to socio-economic aspects of FAnGR conservation, to transfer of lessons between wildlife and livestock biodiversity conservation, and to the contribution of FAnGR to a transition in agriculture (FAnGR and agro-ecology)