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Primitive roots of unity and cyclotomic polynomials
 Journal of Formalized Mathematics
"... Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials. ..."
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Summary. We present a formalization of roots of unity, define cyclotomic polynomials and demonstrate the relationship between cyclotomic polynomials and unital polynomials.
Hermitan functionals. Canonical construction of scalar product in quotient vector space
 Formalized Mathematics
"... Summary. In the article we present antilinear functionals, sesquilinear and hermitan forms. We prove Schwarz and Minkowski inequalities, and Parallelogram Law for non negative hermitan form. The proof of Schwarz inequality is based on [16]. The incorrect proof of this fact can be found in [13]. The ..."
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Summary. In the article we present antilinear functionals, sesquilinear and hermitan forms. We prove Schwarz and Minkowski inequalities, and Parallelogram Law for non negative hermitan form. The proof of Schwarz inequality is based on [16]. The incorrect proof of this fact can be found in [13]. The construction of scalar product in quotient vector space from non negative hermitan functions is the main result of the article.
Yatsuka Nakamura
"... Summary. Concepts of the inner product and conjugate of matrix of complex numbers are defined here. Operations such as addition, subtraction, scalar multiplication and inner product are introduced using correspondent definitions of the conjugate of a matrix of a complex field. Many equations for suc ..."
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Summary. Concepts of the inner product and conjugate of matrix of complex numbers are defined here. Operations such as addition, subtraction, scalar multiplication and inner product are introduced using correspondent definitions of the conjugate of a matrix of a complex field. Many equations for such operations consist like a case of the conjugate of matrix of a field and some operations on the set of sum of complex numbers are introduced.
Fundamental Theorem of Algebra 1 MML Identifier: POLYNOM5.
"... provide the notation and terminology for this paper. One can prove the following propositions: 1. PRELIMINARIES (1) For all natural numbers n, m such that n � = 0 and m � = 0 holds (n · m − n − m) + 1 ≥ 0. (2) For all real numbers x, y such that y> 0 holds min(x,y) max(x,y) ≤ 1. (3) For all real ..."
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provide the notation and terminology for this paper. One can prove the following propositions: 1. PRELIMINARIES (1) For all natural numbers n, m such that n � = 0 and m � = 0 holds (n · m − n − m) + 1 ≥ 0. (2) For all real numbers x, y such that y> 0 holds min(x,y) max(x,y) ≤ 1. (3) For all real numbers x, y such that for every real number c such that c> 0 and c < 1 holds c · x ≥ y holds y ≤ 0. (4) Let p be a finite sequence of elements of R. Suppose that for every natural number n such that n ∈ dom p holds p(n) ≥ 0. Let i be a natural number. If i ∈ dom p, then ∑ p ≥ p(i). (5) For all real numbers x, y holds −(x + yiCF) = −x + (−y)iCF. (6) For all real numbers x1, y1, x2, y2 holds (x1 + y1iCF) − (x2 + y2iCF) = (x1 − x2) + (y1 − y2)iCF. In this article we present several logical schemes. The scheme ExDHGrStrSeq deals with a non empty groupoid A and a unary functor F yielding an element of A, and states that: There exists a sequence S of A such that for every natural number n holds S(n) = F (n) for all values of the parameters. The scheme ExDdoubleLoopStrSeq deals with a non empty double loop structure A and a unary functor F yielding an element of A, and states that: There exists a sequence S of A such that for every natural number n holds S(n) = F (n) for all values of the parameters. The following proposition is true (8) 1 For every element z of CF such that z � = 0CF power CF (z, n)  = zn.
Trigonometric Form of Complex Numbers MML Identifier: COMPTRIG.
"... The scheme Regr without 0 concerns a unary predicate P, and states that: P [1] provided the parameters meet the following requirements: • There exists a non empty natural number k such that P [k], and • For every non empty natural number k such that k � = 1 and P [k] there exists a non empty natural ..."
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The scheme Regr without 0 concerns a unary predicate P, and states that: P [1] provided the parameters meet the following requirements: • There exists a non empty natural number k such that P [k], and • For every non empty natural number k such that k � = 1 and P [k] there exists a non empty natural number n such that n < k and P [n]. The following propositions are true: (3) 1 For every element z of C holds ℜ(z) ≥ −z. (4) For every element z of C holds ℑ(z) ≥ −z. (5) For every element z of CF holds ℜ(z) ≥ −z. (6) For every element z of CF holds ℑ(z) ≥ −z. (7) For every element z of CF holds z  2 = ℜ(z) 2 + ℑ(z) 2. (8) For all real numbers x1, x2, y1, y2 such that x1 + x2iCF = y1 + y2iCF holds x1 = y1 and x2 = y2. (9) For every element z of CF holds z = ℜ(z) + ℑ(z)iCF. (10) 0CF = 0 + 0iCF. (12) 2 For every unital non empty groupoid L and for every element x of L holds power L (x, 1) = x. (13) For every unital non empty groupoid L and for every element x of L holds power L (x, 2) = x · x. (14) Let L be an addassociative right zeroed right complementable right distributive unital non empty double loop structure and n be a natural number. If n> 0, then power L (0L, n) = 0L. 1 The propositions (1) and (2) have been removed. 2 The proposition (11) has been removed.
Bilinear Functionals in Vector Spaces 1
"... Summary. The main goal of the article is the presentation of the theory of bilinear functionals in vector spaces. It introduces standard operations on bilinear functionals and proves their classical properties. It is shown that quotient functionals are non degenerated on the left and the right. In t ..."
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Summary. The main goal of the article is the presentation of the theory of bilinear functionals in vector spaces. It introduces standard operations on bilinear functionals and proves their classical properties. It is shown that quotient functionals are non degenerated on the left and the right. In the case of symmetric and alternating bilinear functionals it is shown that the left and right kernels are equal.
Quotient Vector Spaces and Functionals 1
"... Summary. The article presents well known facts about quotient vector spaces and functionals (see [8]). There are repeated theorems and constructions with either weaker assumptions or in more general situations (see [11], [7], [10]). The construction of coefficient functionals and non degenerated fun ..."
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Summary. The article presents well known facts about quotient vector spaces and functionals (see [8]). There are repeated theorems and constructions with either weaker assumptions or in more general situations (see [11], [7], [10]). The construction of coefficient functionals and non degenerated functional in quotient vector space generated by functional in the given vector space are the only new things which are done.