Collective Operations on Number-Membered Sets

The article starts with definitions of sets of opposite and inverse numbers of a given number membered set. Next, collective addition, subtraction, multiplication and division of two sets are defined. Complex numbers cases and extended real numbers ones are introduced separately and unified for reals. Shortcuts for singletons cases are also defined.Institute of Computer Science, University of Białystok, Sosnowa 64, 15-887 Białystok PolandGrzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4):341-347, 2003.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Some Properties of p-Groups and Commutative p-Groups

This article describes some properties of p-groups and some properties of commutative p-groups.Liang Xiquan - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaGrzegorz 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.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Marco Riccardi. The Sylow theorems. Formalized Mathematics, 15(3):159-165, 2007, doi:10.2478/v10037-007-0018-3.Dariusz Surowik. Cyclic groups and some of their properties - part I. Formalized Mathematics, 2(5):623-627, 1991.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Commutator and center of a group. Formalized Mathematics, 2(4):461-466, 1991.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Formalization of Integral Linear Space

In this article, we formalize integral linear spaces, that is a linear space with integer coefficients. Integral linear spaces are necessary for lattice problems, LLL (Lenstra-Lenstra-Lovász) base reduction algorithm that outputs short lattice base and cryptographic systems with lattice [8].Futa Yuichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz 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 and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Dimension of real unitary space. Formalized Mathematics, 11(1):23-28, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Daniele Micciancio and Shafi Goldwasser. Complexity of lattice problems: A cryptographic perspective (the international series in engineering and computer science). 2002.Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329-334, 1990.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Basis of real linear space. Formalized Mathematics, 1(5):847-850, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992

Occurrence and degree of iliopsoas muscle contracture in regular male squash players

Introduction: Sprints combined with changes in direction and repeated lunges are the most frequent movements during a squash game. These motions overload the iliopsoas muscle which may cause a lot of microinjuries. Accumulating microinjuries combined with a lack of stretching exercises may lead to iliopsoas contracture. Aim of the study: Assessment of the frequency and degree of iliopsoas contracture in regular squash players. Material and methods: The experimental group comprised 25 regular squash players (minimum 2 years of playing at least twice a week) and 21 non-players (control group). A modified Thomas Test was used to assess iliopsoas contracture using goniometric and linear measurements. Results: Iliopsoas contracture was observed in 96% of the squash players and 66.7% of the non-players (p = 0.0089). The degree of muscle contracture in the goniometric measurement was greater in squash players than in the non-players in both the left (p = 0.0303) and right (p = 0.0007) iliopsoas muscles. There were no statistically significant differences in the linear measurement. Conclusions: There is a positive relationship between regularly playing squash and the frequency of iliopsoas contracture occurrence being significantly greater in squash players than in non-players

The Geometric Interior in Real Linear Spaces

We introduce the notions of the geometric interior and the centre of mass for subsets of real linear spaces. We prove a number of theorems concerning these notions which are used in the theory of abstract simplicial complexes.Institute of Informatics, University of Białystok, PolandGrzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 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.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Noboru Endou, Takashi Mitsuishi, and Yasunari Shidama. Convex sets and convex combinations. Formalized Mathematics, 11(1):53-58, 2003.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.Karol Pąk. Affine independence in vector spaces. Formalized Mathematics, 18(1):87-93, 2010, doi: 10.2478/v10037-010-0012-z.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Wojciech A. Trybulec. Linear combinations in real linear space. Formalized Mathematics, 1(3):581-588, 1990.Wojciech A. Trybulec. Partially ordered sets. Formalized Mathematics, 1(2):313-319, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990

On Rough Subgroup of a Group

This article describes a rough subgroup with respect to a normal subgroup of a group, and some properties of the lower and the upper approximations in a group.Liang Xiquan - Qingdao University of Science and Technology, ChinaLi Dailu - Qingdao University of Science and Technology, ChinaWojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990

Injuries and factors determining their occurrence in paratroopers of airborne forces

The purpose of this study was to determine the type and incidence of injuries among airborne forces paratroopers, and also to analyze the factors that determine the probability of suffering injuries while parachuting. 165 soldiers in active service, from the 6th Airborne Brigade in Cracow, participated in the study. The survey was carried out via the author’s questionnaire. 32.72% of the examined soldiers were injured during the parachute jump. Crude injury incidence was calculated as 27.86/10,000 jumps. In terms of types of injuries, the frequency of their occurrence was as follows: sprains (34%), fractures (17%), muscle strains (13%), complete muscle ruptures (8%), partial muscle ruptures (8%), dislocations (6%), and others. The most common locations of the injuries were: the ankle joint (31%), the knee joint (24%) and the spine (18%). The most injuries (83%) happened during the landing phase of the parachute jump. The factors that increase the risk of injury during parachute jumps were as follows: higher body weight, older age, longer time of parachuting and serving in the Airborne Forces, greater number of parachute jumps. The most common reason for injury during parachuting was parachutist error

Set of Points on Elliptic Curve in Projective Coordinates

In this article, we formalize a set of points on an elliptic curve over GF(p). Elliptic curve cryptography [10], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.Futa Yuichi - Shinshu University, Nagano, JapanOkazaki Hiroyuki - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz 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 and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Józef Białas. Group and field definitions. Formalized Mathematics, 1(3):433-439, 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.Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35-40, 1990.G. Seroussi I. Blake and N. Smart. Elliptic Curves in Cryptography. Number 265 in London Mathematical Society Lecture Note Series. Cambridge University Press, 1999.Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.Christoph Schwarzweller. The ring of integers, euclidean rings and modulo integers. Formalized Mathematics, 8(1):29-34, 1999.Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97-105, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 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.Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized Mathematics, 1(1):85-89, 1990

The Real Vector Spaces of Finite Sequences are Finite Dimensional

In this paper we show the finite dimensionality of real linear spaces with their carriers equal Rn. We also give the standard basis of such spaces. For the set Rn we introduce the concepts of linear manifold subsets and orthogonal subsets. The cardinality of orthonormal basis of discussed spaces is proved to equal n.Yatsuka Nakamura - Shinshu University Nagano, JapanNagato Oya - Shinshu University Nagano, JapanYasunari Shidama - Shinshu University Nagano, JapanArtur Korniłowicz - Institute of Computer Science, University of Białystok, Sosnowa 64, 15-887 Białystok, Polan