Asymptotics of lattice walks via analytic combinatorics in several variables

We consider the enumeration of walks on the two dimensional non-negative integer lattice with short steps. Up to isomorphism there are 79 unique two dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pad\'e-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.Comment: 10 pages, 3 tables, as accepted to proceedings of FPSAC 2016 (without conference formatting

Asymptotics of lattice walks via analytic combinatorics in several variables

We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech

Asymptotics of lattice walks via analytic combinatorics in several variables

We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech

Różnice między płciami w odniesieniu do aktywności fizycznej wśród uczniów szkół średnich w krajach Grupy Wyszehradzkiej (V4)

Background. Physical inactivity is also a significant problem in East-Central Europe and Hungary where 20% of the population does sports regularly while 53% of them never, in turn, 24% of the people does sports few times monthly. Insufficient physical activity is an increasing public health problem among young people and adolescents. Several types of research examined the quality and extent of physical activity related to different factors (social demographic, psychosocial, and lifestyle, etc.). Material and methods. Interviewing was carried out from April to June 2015 at the same time in each Visegrad country. IPAQ extended physical activity questionnaire and a self-edited questionnaire were used to assess nutritional and activity habits (In order to evaluate data, INDARES software, and paper-based questionnaires were used for 2145 persons from different secondary schools). Results. In low PA level category, male students were rather found while in high PA level category, male students showed higher ratios than females. From 56.7% to 77.8% of male students occurred in high PA level category opposite to female students where this rate was from 42.4% to 67.4%. We found significant gender differences in total MET/week values (p<0.001) in the V4 countries. Conclusions. We found significant differences in the Visegrad countries and between sexes. These differences draw the attention to improving deficiencies in physical activity of secondary school students with well-defined risk group interventions.Wprowadzenie. Brak aktywności fizycznej to poważny problem występujący w Europie Wschodniej i Środkowej, a także na Węgrzech, gdzie 20% ludności nie uprawia regularnie żadnego sportu, podczas gdy 53% ludzi nie uprawia sportu wcale, natomiast 24% populacji uprawia sport kilka razy w miesiącu. Zbyt niska aktywność fizyczna to rosnący problem z zakresu zdrowia publicznego wśród młodych ludzi i nastolatków. Przeprowadzono już szereg różnego rodzaju badań związanych z jakością i zakresem aktywności fizycznej w odniesieniu do wielu czynników (socjodemograficznych, psychospołecznych oraz związanych ze stylem życia itd.). Materiał i metody. W okresie od kwietnia do czerwca 2015 przeprowadzono wywiady równolegle w każdym z krajów grupy Wyszehradzkiej. W badaniu wykorzystano kwestionariusz IPAQ z rozszerzonym zakresem aktywności fizycznej oraz stworzony przez autorów kwestionariusz w celu dokonania oceny nawyków żywieniowych oraz tych związanych z aktywnością fizyczną (w celu oceny danych wykorzystano oprogramowanie INDARES oraz wydrukowane kwestionariusze w przypadku 2145 uczniów z różnych szkół średnich). Wyniki. W zakresie niskiej aktywności fizycznej prym wiedli uczniowie płci męskiej, podczas gdy w kategorii wysokiego poziomu aktywności fizycznej, uczniowie płci męskiej wykazywali wyższy poziom niż uczennice. Między 56,7% a 77,8% uczniów płci męskiej znalazło się w kategorii wysokiej aktywności fizycznej, w przeciwieństwie do uczennic, w przypadku których ten wskaźnik odnotowano na poziomie między 42,4% a 67,4%. Autorzy badania wykazali także znaczne różnice pod względem płci w całkowitych wartościach MET/tydzień (p<0,001) w krajach V4. Wnioski. Wykazano znaczne różnice w krajach grupy Wyszehradzkiej w odniesieniu do płci. Różnice te kierują naszą uwagę na konieczność ulepszenia braków związanych z aktywnością fizyczną uczniów w szkołach średnich, w obliczu istnienia dobrze zdefiniowanych działań wobec konkretnych grup ryzyka