Dilated floor functions having nonnegative commutator II. Negative dilations

This paper completes the classification of the set $S$ of all real parameter pairs $(\alpha,\beta)$ such that the dilated floor functions $f_\alpha(x) = \lfloor{\alpha x}\rfloor$, $f_\beta(x) = \lfloor{\beta x}\rfloor$ have a nonnegative commutator, i.e. $[ f_{\alpha}, f_{\beta}](x) = \lfloor{\alpha \lfloor{\beta x}\rfloor}\rfloor - \lfloor{\beta \lfloor{\alpha x}\rfloor}\rfloor \geq 0$ for all real $x$. This paper treats the case where both dilation parameters $\alpha, \beta$ are negative. This result is equivalent to classifying all positive $\alpha, \beta$ satisfying $\lfloor{\alpha \lceil{\beta x}\rceil}\rfloor - \lfloor{\beta \lceil{\alpha x}\rceil}\rfloor \geq 0$ for all real $x$. The classification analysis is connected with the theory of Beatty sequences and with the Diophantine Frobenius problem in two generators.Comment: 18 pages, 8 figure

Relaxed Wythoff has all Beatty Solutions

We find conditions under which the P-positions of three subtraction games arise as pairs of complementary Beatty sequences. The first game is due to Fraenkel and the second is an extension of the first game to non-monotone settings. We show that the P-positions of the second game can be inferred from the recurrence of Fraenkel\u27s paper if a certain inequality is satisfied. This inequality is shown to be necessary if the P-positions are known to be pairs of complementary Beatty sequences, and the family of irrationals for which this inequality holds is explicitly given. We highlight several games in the literature that have P-positions as pairs of complementary Beatty sequences with slope in this family. The third game we present is novel, and we show that the P-positions can be inferred from the same recurrence in any setting. It is shown that any pair of complementary Beatty sequences arises as the P-positions of some game in this family. We also provide background on some inverse problems which have appeared in the field over the last several years, in particular the Duchêne-Rigo conjecture. This paper presents a solution to the Fraenkel problem posed at the 2011 BIRS workshop, a modification of the Duchêne-Rigo conjecture

Wythoff Wisdom

International audienceSix authors tell their stories from their encounters with the famous combinatorial game Wythoff Nim and its sequences, including a short survey on exactly covering systems

Számelmélet és kombinatorikus vonatkozásai = Number Theory and its Interactions with Combinatorics

A kutatók számos érdekes eredményt értek el a kombinatorikus számelmélet és geometria, gráfelmélet, diofantikus approximáció területén, itt csak néhányat említünk. Elekes és Ruzsa a Freiman, Balog-Szemerédi és Laczkovich-Ruzsa tételek közös általánosítását adják, ezzel a témakört egységesítik, és számos kombinatorikus geometriai tételt fejlesztenek tovább. Elekes Szabó E.-vel áttörést ért el a sok szabályosságot tartalmazó konfigurációk karakterizációjának általános problémájában, néhány korábbi eredményt jelentősen továbbfejlesztve. Szemerédi A. Khalfalah-val igazolja Sárközy, Roth és T. Sós azon sejtését, hogy: ha beosztjuk az egész számokat véges sok osztályba, akkor valamely osztályban van két olyan szám, amelyek összege négyzetszám, V. Vu-val közösen pedig Folkman egy sejtését bizonyítja. Biró javítja Ruzsa és Kolountzakis egész számok parkettázására vonatkozó eredményét. Erősíti és általánosítja a "karakterizáló sorozatok" témakör korábbi eredményeit. Ruzsa és B. Green meghatározzák tetszőleges véges kommutatív csoportban a legnagyobb összegmentes halmaz elemszámát. T. Sós Lovász L.-val megmutatja, hogy ha gráfok egy sorozatában a kis részgráfoknak ugyanaz az eloszlása, mint egy általánosított G véletlen gráfban, akkor ezen gráfoknak aszimptotikusan olyan struktúrája van, mint G-nek. T. Sós társszerzőkkel azt az alapkérdést vizsgálja, mikor van közel egymáshoz két gráf. | The participants obtaind several interesting results in combinatorial number theory and geometry, graph theory, diophantine approximation, we list just a few of these results.. Elekes and Ruzsa give a common generalization of the Freiman, Balog-Szemerédi and Laczkovich-Ruzsa theorems, unifying in this way the subject and improving a lot of earlier results. Elekes with E. Szabó achieved a breakthrough in the general problem of characterizing configurations having a lot of reguarity, improving some earlier results. Szemerédi with A. Khalfalah proves the follwing conjecture of Sárközy, Roth and T. Sós: if we divide the set of integers into finitely many classes, then in one of the classes we can find two numbers such that their sum is a square, and with V. Vu he proves a conjecture of Folkman. Biró improves a result of Ruzsa and Kolountzakis on tilings of the integers, and, he proves generalizations and strengthenings of some results in the subject 'characterizing sequences'. Ruzsa and B. Green determine the size of the largest sumfree set in an arbitrary finite Abelian group. L. Lovász and T. Sós showed that generalized quasirandom sequences (whose subgraph densities match those of a fixed finite weighted graph) have a finite structure. T. Sós with co-authors defines the distance of two graphs that reflects the similarity , the closeness of both local and global properties

Tracing Evolution of Gene Transfer Agents Using Comparative Genomics

The accumulating evidence suggest that viruses and their components can be domesticated by their hosts, equipping them with convenient molecular toolkits for various functions. One of such domesticated system is Gene Transfer Agents (GTAs) that are produced by some bacteria and archaea. GTAs morphologically resemble small phage-like particles and contain random fragments of their host genome. They are produced only by a small fraction of the microbial population and are released through a lysis of the host cell. Bioinformatic analyses suggest that GTAs are especially abundant in the taxonomic class of Alphaproteobacteria, where they are vertically inherited and evolve as a part of their host genomes. In this work, we extensively analyze evolutionary patterns of alphaproteobacterial GTAs using comparative genomics, phylogenomics and machine learning methods. We initially develop an algorithm that validate the wide presence of GTA elements in alphaproteobacterial genomes, where they are generally mistaken for prophages due to their homology. Furthermore, we demonstrate that GTAs evolve under the selection that reduces the energetic cost of their production, indicating their importance for the conditions of the nutrient depletion. The genome-wide screenings of translational selection and coevolution signatures highlight the significance of GTAs as a stress-response adaptation for the horizontal gene transfer, revealing a set of previously unknown genes that could play a role in the GTA cycle. As production of GTAs leads to the host death, their maintenance is likely to be under a kin or group level selection. By combining our findings with accumulated body of knowledge, this work proposes a conceptual model illustrating the role of GTAs in bacterial populations and their persistence for hundreds of millions of years of evolution