Online size Ramsey numbers: Odd cycles vs connected graphs

Given two graph families $\mathcal H_1$ and $\mathcal H_2$, a size Ramsey game is played on the edge set of $K_\mathbb{N}$. In every round, Builder selects an edge and Painter colours it red or blue. Builder is trying to force Painter to create as soon as possible a red copy of a graph from $\mathcal H_1$ or a blue copy of a graph from $\mathcal H_2$. The online (size) Ramsey number $\tilde{r}(\mathcal H_1,\mathcal H_2)$ is the smallest number of rounds in the game provided Builder and Painter play optimally. We prove that if $\mathcal H_1$ is the family of all odd cycles and $\mathcal H_2$ is the family of all connected graphs on $n$ vertices and $m$ edges, then $\tilde{r}(\mathcal H_1,\mathcal H_2)\ge \varphi n + m-2\varphi+1$, where $\varphi$ is the golden ratio, and for $n\ge 3$, $m\le (n-1)^2/4$ we have $\tilde{r}(\mathcal H_1,\mathcal H_2)\le n+2m+O(\sqrt{m-n+1})$. We also show that $\tilde{r}(C_3,P_n)\le 3n-4$ for $n\ge 3$. As a consequence we get $2.6n-3\le \tilde{r}(C_3,P_n)\le 3n-4$ for every $n\ge 3$.Comment: 14 pages, 0 figures; added appendix containing intuition behind the potential function used for lower bound; corrected typos and added a few clarification

Quantum walks on cycles

We consider asymptotic behaviour of a Hadamard walk on a cycle. For a walk which starts with a state in which all the probability is concentrated on one node, we find the explicit formula for the limiting distribution and discuss its asymptotic behaviour when the length of the cycle tends to infinity. We also demonstrate that for a carefully chosen initial state, the limiting distribution of a quantum walk on cycle can lie further away from the uniform distribution than its initial state

Podstawy teoretyczne koncepcji WAZO, cz. 1

W serii Metody Organizacji i Zarządzania (MOiZ) publikowane są cyklicznie opracowania, których celem jest upowszechnienie metod (technik, narzędzi) z zakresu organizacji i zarządzania oraz ułatwienie ich transferu z nauki do biznesu. Seria jest wydawana w ramach Centrum Metod Organizacji i Zarządzania działającego na Wydziale Zarządzania Uniwersytetu Łódzkiego.Metodyka WAZO powstała w ramach projektu finansowanego ze środków Narodowego Centrum Nauki przyznanych na podstawie decyzji numer DEC-2013/09/B/HS4/02722

Long-term spatiotemporal stability and dynamic changes in the haemoparasite community of spiny mice (Acomys dimidiatus) in four montane wadis in the St. Katherine Protectorate, Sinai, Egypt

Background: Long-term field studies of parasite communities are rare but provide a powerful insight into the ecological processes shaping host-parasite interactions. The aim of our study was to monitor long-term trends in the haemoparasite communities of spiny mice (Acomys dimidiatus) and to identify the principal factors responsible for changes over a 12 year period. Methods: To this end we sampled four semi-isolated populations of mice (n= 835) in 2000, 2004, 2008 and 2012 in four dry montane valleys (wadis) located in the Sinai Massif, Egypt. Results: Overall 76.2 % of spiny mice carried at least one of the five haemoparasite genera (Babesia, Bartonella, Haemobartonella, Hepatozoon, Trypanosoma) recorded in the study. Prevalence of haemoparasites varied significantly between the sites with the highest overall prevalence in Wadi Tlah and the lowest in W. El Arbaein, and this changed significantly with time. In the first two surveys there was little change in prevalence, but by 2008, when the first signs of a deepening drought in the region had become apparent, prevalence began to drift downwards, and by 2012 prevalence had fallen to the lowest values recorded from all four sites over the entire 12-year period. The overall mean species richness was 1.2 ± 0.03, which peaked in 2004 and then dropped by more than 50 % by 2012. Species richness was highest among mice from Wadi Tlah and peaked in age class 2 mice (young adults). Site was the most significant factor affecting the prevalence of individual parasite species, with Trypanosoma acomys and Hepatozoon sp. occurring mainly in two wadis (W. Tlah & W. Gharaba). In four of the five genera recorded in the study we observed a significant drop in prevalence or/and abundance since 2004, the exception being Hepatozoon sp. Conclusions: During the 12-year-long period of study in the Sinai, we observed dynamic changes and possibly even cycles of prevalence and abundance of infections which differed depending on parasite species. Although the exact reasons cannot be identified at this time, we hypothesize that the effects of a 15-year-long scarcity of rainfall in the local environment and a fall in host densities over the period of study may have been responsible for a drop in transmission rates, possibly by a negative impact on vector survival