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142
Factorial and Newton coefficients
- Journal of Formalized Mathematics
, 1990
"... Summary. We define the following functions: exponential function (for natural exponent), factorial function and Newton coefficients. We prove some basic properties of notions introduced. There is also a proof of binominal formula. We prove also that ∑ n �n � k=0 k = 2n. ..."
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Summary. We define the following functions: exponential function (for natural exponent), factorial function and Newton coefficients. We prove some basic properties of notions introduced. There is also a proof of binominal formula. We prove also that ∑ n �n � k=0 k = 2n.
The Euclidean Space
, 1991
"... this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R ..."
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Cited by 83 (0 self)
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this paper. In this paper k, n are natural numbers and r is a real number. Let us consider n. The functor R
Sum and product of finite sequences of elements of a field
- Journal of Formalized Mathematics
, 1992
"... Summary. This article is concerned with a generalization of concepts introduced in [11], i.e., there are introduced the sum and the product of finite number of elements of any field. Moreover, the product of vectors which yields a vector is introduced. According to [11], some operations on i-tuples ..."
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Cited by 21 (0 self)
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Summary. This article is concerned with a generalization of concepts introduced in [11], i.e., there are introduced the sum and the product of finite number of elements of any field. Moreover, the product of vectors which yields a vector is introduced. According to [11], some operations on i-tuples of elements of field are introduced: addition, subtraction, and complement. Some properties of the sum and the product of finite number of elements of a field are present.
Little Bezout theorem (factor theorem)
- FORMALIZED MATHEMATICS
, 2004
"... We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x − r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero po ..."
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Cited by 12 (3 self)
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We present a formalization of the factor theorem for univariate polynomials, also called the (little) Bezout theorem: Let r belong to a commutative ring L and p(x) be a polynomial over L. Then x − r divides p(x) iff p(r) = 0. We also prove some consequences of this theorem like that any non zero polynomial of degree n over an algebraically closed integral domain has n (non necessarily distinct) roots.
Basic Properties of Rough Sets and Rough Membership Function
- Journal of Formalized Mathematics
"... Summary. We present basic concepts concerning rough set theory. We define tolerance and approximation spaces and rough membership function. Different rough inclusions as well as the predicate of rough equality of sets are also introduced. ..."
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Cited by 9 (7 self)
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Summary. We present basic concepts concerning rough set theory. We define tolerance and approximation spaces and rough membership function. Different rough inclusions as well as the predicate of rough equality of sets are also introduced.
