Aerodynamic characteristics of the modified 40- by 80-foot wind tunnel as measured in a 1/50th-scale model

The aerodynamic characteristics of the 40- by 80-Foot Wind Tunnel at Ames Research Center were measured by using a 1/50th-scale facility. The model was configured to closely simulate the features of the full-scale facility when it became operational in 1986. The items measured include the aerodynamic effects due to changes in the total-pressure-loss characteristics of the intake and exhaust openings of the air-exchange system, total-pressure distributions in the flow field at locations around the wind tunnel circuit, the locations of the maximum total-pressure contours, and the aerodynamic changes caused by the installation of the acoustic barrier in the southwest corner of the wind tunnel. The model tests reveal the changes in the aerodynamic performance of the 1986 version of the 40- by 80-Foot Wind Tunnel compared with the performance of the 1982 configuration

Hip Hop and the Huxtables: Identity, Hip Hop, and the Cosby Effect in Colson Whitehead\u27s \u3cem\u3eSag Harbor\u3c/em\u3e

Identity is a tricky thing for anyone in the formative years of adolescence, a thing made much more complex when you don’t fit the mold of any preexisting social group. For a black American in the 1980s, the formulation of identity was a remarkably unique challenge. The rise of hip hop as a major element of American culture gave a far-reaching voice to the challenges faced many black Americans, but its roots in and content about impoverished, usually violent urban areas offered a decidedly limited and negative view of black Americans. In Sag Harbor, Colson Whitehead delves into this complicated identity problem through Benji, a black prep-school New York high school student “catching up on months” of black culture he has missed out on in his upper-middle class world (Whitehead 37). Benji fits the role of the “black geek” during his school year, playing Dungeons & Dragons, obsessing over comic books and Star Wars, yet he is drawn to the cultural world of hip hop in search of a more authentically black experience. Benji is caught in the midst of swirling social identities: too black to fully assimilate into his prep-school world and too white to be part of the hip hop world. Sag Harbor is the place where he tries to negotiate this tension and reinvent himself

Some remarks to the legal status of platform workers in the light of the latest European jurisprudence

The COVID-19 pandemic has had a negative impact on the working conditions of so-called platform workers that have faced the lack of labour and social protection deriving from their formal status of independent contractors. Seeking for protection, numerous claims have been filed by the riders and drivers of digital platforms that were asking for a recognition of the subordinate work. The aim of the present article is to give a critical and brief overview of the latest European jurisprudence regarding the legal status of platform workers. The article focuses on the methodology and criteria applied by the judges in order to examine the particularities of the new forms of work and new forms of surveillance as well. Special attention is paid to on-location work performance by low-skilled individuals conducting services for the digital platforms operating in food delivery and transportation sectors

More on Divisibility Criteria for Selected Primes

This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].Naumowicz Adam - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandPiliszek Radosław - Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.Grzegorz Bancerek. Veblen hierarchy. Formalized Mathematics, 19(2):83-92, 2011. doi:10.2478/v10037-011-0014-5.C.C. Briggs. Simple divisibility rules for the 1st 1000 prime numbers. arXiv preprint arXiv:math/0001012, 2000.Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Recursive definitions. Formalized Mathematics, 1(2):321-328, 1990.Magdalena Jastrz¸ebska and Adam Grabowski. Some properties of Fibonacci numbers. Formalized Mathematics, 12(3):307-313, 2004.Artur Korniłowicz. On the real valued functions. Formalized Mathematics, 13(1):181-187, 2005.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Yatsuka Nakamura and Hisashi Ito. Basic properties and concept of selected subsequence of zero based finite sequences. Formalized Mathematics, 16(3):283-288, 2008. doi:10.2478/v10037-008-0034-y.Adam Naumowicz. On the representation of natural numbers in positional numeral systems. Formalized Mathematics, 14(4):221-223, 2006. doi:10.2478/v10037-006-0025-9.Karol Pak. Stirling numbers of the second kind. Formalized Mathematics, 13(2):337-345, 2005.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

All Liouville Numbers are Transcendental

In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real number x is a Liouville number iff for every positive integer n, there exist integers p and q such that q > 1 and 0 < x − p q < 1 q n . It is easy to show that all Liouville numbers are irrational. The definition and basic notions are contained in [10], [1], and [12]. Liouvile constant, which is defined formally in [12], is the first explicit transcendental (not algebraic) number, another notable examples are e and π [5], [11], and [4]. Algebraic numbers were formalized with the help of the Mizar system [13] very recently, by Yasushige Watase in [23] and now we expand these techniques into the area of not only pure algebraic domains (as fields, rings and formal polynomials), but also for more settheoretic fields. Finally we show that all Liouville numbers are transcendental, based on Liouville’s theorem on Diophantine approximation.Korniłowicz Artur - Institute of Informatics, University of Białystok, Białystok, PolandNaumowicz Adam - Institute of Informatics, University of Białystok, Białystok, PolandGrabowski Adam - Institute of Informatics, University of Białystok, Białystok, PolandTom M. Apostol. Modular Functions and Dirichlet Series in Number Theory. Springer- Verlag, 2nd edition, 1997.Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Sophie Bernard, Yves Bertot, Laurence Rideau, and Pierre-Yves Strub. Formal proofs of transcendence for e and _ as an application of multivariate and symmetric polynomials. In Jeremy Avigad and Adam Chlipala, editors, Proceedings of the 5th ACM SIGPLAN Conference on Certified Programs and Proofs, pages 76-87. ACM, 2016.Jesse Bingham. Formalizing a proof that e is transcendental. Journal of Formalized Reasoning, 4:71-84, 2011.Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.J.H. Conway and R.K. Guy. The Book of Numbers. Springer-Verlag, 1996.Manuel Eberl. Liouville numbers. Archive of Formal Proofs, December 2015. http://isa-afp.org/entries/Liouville_Numbers.shtml, Formal proof development.Adam Grabowski and Artur Korniłowicz. Introduction to Liouville numbers. Formalized Mathematics, 25(1):39-48, 2017.Adam Grabowski, Artur Korniłowicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191-198, 2015.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.Joseph Liouville. Nouvelle démonstration d’un théorème sur les irrationnelles algébriques, inséré dans le Compte Rendu de la dernière séance. Compte Rendu Acad. Sci. Paris, Sér.A (18):910–911, 1844.Anna Justyna Milewska. The field of complex numbers. Formalized Mathematics, 9(2): 265-269, 2001.Robert Milewski. The ring of polynomials. Formalized Mathematics, 9(2):339-346, 2001.Robert Milewski. The evaluation of polynomials. Formalized Mathematics, 9(2):391-395, 2001.Robert Milewski. Fundamental theorem of algebra. Formalized Mathematics, 9(3):461-470, 2001.Michał Muzalewski and Lesław W. Szczerba. Construction of finite sequences over ring and left-, right-, and bi-modules over a ring. Formalized Mathematics, 2(1):97-104, 1991.Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1 (3):495-500, 1990.Wojciech A. Trybulec. Non-contiguous substrings and one-to-one finite sequences. Formalized Mathematics, 1(3):569-573, 1990.Yasushige Watase. Algebraic numbers. Formalized Mathematics, 24(4):291-299, 2016.Katarzyna Zawadzka. The sum and product of finite sequences of elements of a field. Formalized Mathematics, 3(2):205-211, 1992

Eko-teologia jako forma chrześcijańskiej diakonii wobec stworzenia

Jürgen Moltmann, one of the world’s leading Protestant theologians, is widely recognized as a key figure in the growing concern for eco-theology. For Moltmann, the present ecological crisis can be overcome by a new understanding of God and creation. The concept of a triune God helps to dispel the myth of our unilateral relationship of dominion over nature. Moltmann sees also some positive aspects of Gaia theory. He champions a commitment to justice for all creatures, not just human. He moves beyond anthropocentrism and tries to balance the traditional emphasis on God’s creative work by insisting that a doctrine of creation should culminate in the Sabbath celebration

A Note on the Seven Bridges of Königsberg Problem

In this paper we account for the formalization of the seven bridges of Königsberg puzzle. The problem originally posed and solved by Euler in 1735 is historically notable for having laid the foundations of graph theory, cf. [7]. Our formalization utilizes a simple set-theoretical graph representation with four distinct sets for the graph’s vertices and another seven sets that represent the edges (bridges). The work appends the article by Nakamura and Rudnicki [10] by introducing the classic example of a graph that does not contain an Eulerian path. This theorem is item #54 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.Institute of Informatics University of Białystok Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1): 55–65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.Czesław Byliński and Piotr Rudnicki. The correspondence between monotonic many sorted signatures and well-founded graphs. Part I. Formalized Mathematics, 5(4):577– 582, 1996.Gary Chartrand. Introductory Graph Theory. New York: Dover, 1985.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.Krzysztof Hryniewiecki. Graphs. Formalized Mathematics, 2(3):365–370, 1991.Yatsuka Nakamura and Piotr Rudnicki. Euler circuits and paths. Formalized Mathematics, 6(3):417–425, 1997.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73–83, 1990. Received June 16, 201

Elementary Number Theory Problems. Part I

In this paper we demonstrate the feasibility of formalizing recreational mathematics in Mizar ([1], [2]) drawing examples from W. Sierpinski’s book “250 Problems in Elementary Number Theory” [4]. The current work contains proofs of initial ten problems from the chapter devoted to the divisibility of numbers. Included are problems on several levels of difficulty.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Marco Riccardi. The perfect number theorem and Wilson’s theorem. Formalized Mathematics, 17(2):123–128, 2009. doi:10.2478/v10037-009-0013-y.Wacław Sierpinski. 250 Problems in Elementary Number Theory. Elsevier, 1970.Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825–829, 2001.Li Yan, Xiquan Liang, and Junjie Zhao. Gauss lemma and law of quadratic reciprocity. Formalized Mathematics, 16(1):23–28, 2008. doi:10.2478/v10037-008-0004-4.11512

Decoding the Christian Era of Dionysius Exiguus

Dotychczas nie znaleziono satysfakcjonującego wyjaśnienia, w jaki sposób rzymski mnich Dionysius Exiguus ustalił rok narodzenia Jezusa w swoim systemie rachuby lat, który wprowadził w 525 roku. W badaniach dominują bowiem dwa bezzasadne założenia: pierwsze, że twórca tej rachuby nie podał jej uzasadnienia, i drugie, że rozwiązania problemu należy szukać jedynie w pracach tego autora dotyczących obliczania dat Wielkanocy. W tym artykule przyjąłem następującą hipotezę badawczą: wyjaśnienie nowej rachuby lat znajduje się w pismach Dionizego, do których należą jego prace komputystyczne, ale także teologiczne. Analiza tych pism pozwala odkryć historyczne i teologiczne racje, które stały się faktyczną podstawą ustalenia daty narodzenia Jezusa na rok odpowiadający dzisiejszemu AD 1