On the number of lambda terms with prescribed size of their De Bruijn representation

John Tromp introduced the so-called 'binary lambda calculus' as a way to encode lambda terms in terms of binary words. Later, Grygiel and Lescanne conjectured that the number of binary lambda terms with $m$ free indices and of size $n$ (encoded as binary words of length $n$) is $o(n^{-3/2} \tau^{-n})$ for $\tau \approx 1.963448\ldots$. We generalize the proposed notion of size and show that for several classes of lambda terms, including binary lambda terms with $m$ free indices, the number of terms of size $n$ is $\Theta(n^{-3/2} \rho^{-n})$ with some class dependent constant $\rho$, which in particular disproves the above mentioned conjecture. A way to obtain lower and upper bounds for the constant near the leading term is presented and numerical results for a few previously introduced classes of lambda terms are given

On Bernoulli Sums and Bernstein Polynomials

In the paper we discuss a technology based on Bernstein polynomials of asymptotic analysis of a class of binomial sums that arise in information theory. Our method gives a quick derivation of required sums and can be generalized to multinomial distributions. As an example we derive a formula for the entropy of multinomial distributions. Our method simplifies previous work of Jacquet, Szpankowski and Flajolet from 1999

On Greedy Trie Execution

In the paper "How to select a looser'' Prodinger was analyzing an algorithm where $n$ participants are selecting a leader by flipping fair coins, where recursively, the 0-party (those who i.e. have tossed heads) continues until the leader is chosen. We give an answer to the question stated in the Prodinger's paper – what happens if not a 0-party is recursively looking for a leader but always a party with a smaller cardinality. We show the lower bound on the number of rounds of the greedy algorithm (for fair coin)

Counting embeddings of rooted trees into families of rooted trees

The number of embeddings of a partially ordered set $S$ in a partially ordered set $T$ is the number of subposets of $T$ isomorphic to $S$. If both, $S$ and $T$, have only one unique maximal element, we define good embeddings as those in which the maximal elements of $S$ and $T$ overlap. We investigate the number of good and all embeddings of a rooted poset $S$ in the family of all binary trees on $n$ elements considering two cases: plane (when the order of descendants matters) and non-plane. Furthermore, we study the number of embeddings of a rooted poset $S$ in the family of all planted plane trees of size $n$. We derive the asymptotic behaviour of good and all embeddings in all cases and we prove that the ratio of good embeddings to all is of the order $\Theta(1/\sqrt{n})$ in all cases, where we provide the exact constants. Furthermore, we show that this ratio is non-decreasing with $S$ in the plane binary case and asymptotically non-decreasing with $S$ in the non-plane binary case and in the planted plane case. Finally, we comment on the case when $S$ is disconnected.Comment: 20 pages, 6 figure

PERSPEKTYWY WPISU ELEMENTÓW KRAJOBRAZU KULTUROWEGO SZCZECINA NA LISTĘ POMNIKÓW HISTORII

This article is an attempt to pay attention to the rich heritage of the capital of Western Pomerania and to contribute to the discussion on the need to raise the monument’s history of its most valuable elements. The cultural landscape of Szczecin, which is a multicultural heritage of the historic Pomerania cultural area, is, by virtue of its origin, a carrier of original, authentic values that are absent in other Polish cities and regions.According to the authors it is possible to distinguish of three areas, that together reflect the groundbreaking process of spatial development in the history of the city, having authenticity and originality in the country. The first part of the entry should include the historic urban layout of Szczecin’s Śródmieście (Downtown) at the turn of the nineteenth and twentieth century with the form of star and triangular development quarters, as well as urban and architectural and landscape complexes, which are important components of it: Wały Chrobrego (Hakenterrasse), Jasne Błonia (Quistorpaue) and Cmentarz Centralny (Hauptfriedhof).The spatial and aesthetic qualities, the historical significance and the uniqueness of the particular elements of the aforementioned teams predestine them in our opinion to be included in the list of Historical Monuments.This article is an attempt to pay attention to the rich heritage of the capital of Western Pomerania and to contribute to the discussion on the need to raise the monument’s history of its most valuable elements. The cultural landscape of Szczecin, which is a multicultural heritage of the historic Pomerania cultural area, is, by virtue of its origin, a carrier of original, authentic values that are absent in other Polish cities and regions.According to the authors it is possible to distinguish of three areas, that together reflect the groundbreaking process of spatial development in the history of the city, having authenticity and originality in the country. The first part of the entry should include the historic urban layout of Szczecin’s Śródmieście (Downtown) at the turn of the nineteenth and twentieth century with the form of star and triangular development quarters, as well as urban and architectural and landscape complexes, which are important components of it: Wały Chrobrego (Hakenterrasse), Jasne Błonia (Quistorpaue) and Cmentarz Centralny (Hauptfriedhof).The spatial and aesthetic qualities, the historical significance and the uniqueness of the particular elements of the aforementioned teams predestine them in our opinion to be included in the list of Historical Monuments

Ciepło systemowe z odpadów komunalnych

Białystok jest jednym z nielicznych miast w Polsce, w którym prąd i ciepło systemowe trafiające do mieszkańców, produkowane są dzięki spalaniu odpadów komunalnych. Energię z tego paliwa od 2016 r. wytwarza Zakład Unieszkodliwiania Odpadów Komunalnych należący do miejskiej spółki PUHP LECH. W Polsce istnieje obecnie sześć spalarni, które dzięki termicznemu przekształcaniu odpadów komunalnych mogą produkować energię elektryczną i cieplną. Takie obiekty znajdują się w Bydgoszczy, Koninie, Krakowie, Poznaniu, Warszawie i właśnie w Białymstoku

Energia z odpadów komunalnych w Białymstoku

Prawie 107 tys. ton odpadów komunalnych przetworzył termicznie w pierwszym roku działalności Zakład Unieszkodliwiania Odpadów Komunalnych w Białymstoku. Dzięki temu procesowi ZUOK produkuje energię cieplną i elektryczną, która trafia do mieszkańców miasta. Spalarnia jest bardzo ważnych elementem kompleksowego systemu gospodarki odpadami aglomeracji, który w branży wskazywany jest jako wzorcowy w skali kraju