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The fundamental properties of natural numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1.h ..."
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Cited by 682 (76 self)
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Summary. Some fundamental properties of addition, multiplication, order relations, exact division, the remainder, divisibility, the least common multiple, the greatest common divisor are presented. A proof of Euclid algorithm is also given. MML Identifier:NAT_1. WWW:http://mizar.org/JFM/Vol1/nat_1
Partial Functions
"... this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following rules ..."
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Cited by 494 (10 self)
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this article we prove some auxiliary theorems and schemes related to the articles: [1] and [2]. MML Identifier: PARTFUN1. WWW: http://mizar.org/JFM/Vol1/partfun1.html The articles [4], [6], [3], [5], [7], [8], and [1] provide the notation and terminology for this paper. We adopt the following
Relations defined on sets
 Journal of Formalized Mathematics
, 1989
"... Summary. The article includes theorems concerning properties of relations defined as a subset of the Cartesian product of two sets (mode Relation of X,Y where X,Y are sets). Some notions, introduced in [4] such as domain, codomain, field of a relation, composition of relations, image and inverse ima ..."
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Cited by 517 (0 self)
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Summary. The article includes theorems concerning properties of relations defined as a subset of the Cartesian product of two sets (mode Relation of X,Y where X,Y are sets). Some notions, introduced in [4] such as domain, codomain, field of a relation, composition of relations, image and inverse image of a set under a relation are redefined.
Basic Properties of Real Numbers
 Journal of Formalized Mathematics
, 1989
"... this paper. A real number is an element of R ..."
The ordinal numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and oth ..."
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Cited by 722 (70 self)
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Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and other properties of order relation of cardinals. In the second section we show that every set has cardinal number equipotence to it. We introduce notion of alephs and we deal with the concept of finite set. At the end of the article we show two schemes of cardinal induction. Some definitions are based on [9] and [10].
Relations and their basic properties
 Journal of Formalized Mathematics
, 1989
"... Summary. We define here: mode Relation as a set of pairs, the domain, the codomain, and the field of relation; the empty and the identity relations, the composition of relations, the image and the inverse image of a set under a relation. Two predicates, = and ⊆, and three functions, ∪, ∩ and \ are ..."
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Cited by 1069 (6 self)
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Summary. We define here: mode Relation as a set of pairs, the domain, the codomain, and the field of relation; the empty and the identity relations, the composition of relations, the image and the inverse image of a set under a relation. Two predicates, = and ⊆, and three functions, ∪, ∩ and \ are redefined. Basic facts about the above mentioned notions are presented.
Functions from a set to a set
 Journal of Formalized Mathematics
, 1989
"... function from a set X into a set Y, denoted by “Function of X,Y ”, the set of all functions from a set X into a set Y, denoted by Funcs(X,Y), and the permutation of a set (mode Permutation of X, where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1] ..."
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Cited by 1094 (23 self)
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function from a set X into a set Y, denoted by “Function of X,Y ”, the set of all functions from a set X into a set Y, denoted by Funcs(X,Y), and the permutation of a set (mode Permutation of X, where X is a set). Theorems and schemes included in the article are reformulations of the theorems of [1
2003: Global analyses of sea surface temperature, sea ice, and night marine air temperature since the late Nineteenth Century
 J. Geophysical Research
"... data set, HadISST1, and the nighttime marine air temperature (NMAT) data set, HadMAT1. HadISST1 replaces the global sea ice and sea surface temperature (GISST) data sets and is a unique combination of monthly globally complete fields of SST and sea ice concentration on a 1 ° latitudelongitude grid ..."
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Cited by 517 (3 self)
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data set, HadISST1, and the nighttime marine air temperature (NMAT) data set, HadMAT1. HadISST1 replaces the global sea ice and sea surface temperature (GISST) data sets and is a unique combination of monthly globally complete fields of SST and sea ice concentration on a 1 ° latitudelongitude grid
Mega: molecular evolutionary genetic analysis software for microcomputers
 CABIOS
, 1994
"... A computer program package called MEGA has been developed for estimating evolutionary distances, reconstructing phylogenetic trees and computing basic statistical quantities from molecular data. It is written in C+ + and is intended to be used on IBM and IBMcompatible personal computers. In this pr ..."
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Cited by 471 (6 self)
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A computer program package called MEGA has been developed for estimating evolutionary distances, reconstructing phylogenetic trees and computing basic statistical quantities from molecular data. It is written in C+ + and is intended to be used on IBM and IBMcompatible personal computers. In this program, various methods for estimating evolutionary distances from nucleotide and amino acid sequence data, three different methods of phylogenetic inference (UPGMA, neighborjoining and maximum parsimony) and two statistical tests of topological differences are included. For the maximum parsimony method, new algorithms of branchandbound and heuristic searches are implemented. In addition, MEGA computes statistical quantities such as nucleotide and amino acid frequencies, transition/transversion biases, codon frequencies (codon usage tables), and the number of variable sites in specified segments in nucleotide and amino acid sequences. Advanced onscreen sequence data and phylogenetictree editors facilitate publicationquality outputs with a wide range of printers. Integrated and interactive designs, online contextsensitive helps, and a textfile editor make MEGA easy to use.
Results 1  10
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