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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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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.
The Hahn Banach theorem in the vector space over the field of complex numbers
 Journal of Formalized Mathematics
"... [14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) · (x2 + y2i) = (x1 · x2 − y1 · y2) + (x1 · y2 + x2 · y1)i. ..."
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Cited by 8 (0 self)
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[14], [5], [10], and [9] provide the notation and terminology for this paper. The following propositions are true: 1. PRELIMINARIES (1) For every element z of C holds z  = z. (2) For all real numbers x1, y1, x2, y2 holds (x1 + y1i) · (x2 + y2i) = (x1 · x2 − y1 · y2) + (x1 · y2 + x2 · y1)i. (3) For every real number r holds (r + 0i) · i = 0 + ri. (4) For every real number r holds r + 0i  = r. (5) For every element z of C such that z  � = 0 holds z  + 0i = z z+0i · z. 2. SOME FACTS ON THE FIELD OF COMPLEX NUMBERS Let x, y be real numbers. The functor x + yiCF yielding an element of CF is defined as follows: (Def. 1) x + yiCF = x + yi. The element iCF of CF is defined as follows: (Def. 2) iCF = i. We now state several propositions: (6) iCF = 0 + 1i and iCF = 0 + 1iCF.
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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Cited by 4 (1 self)
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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.
Trigonometric form of complex numbers
 Journal of Formalized Mathematics
"... The scheme Regr without 0 concerns a unary predicateP, and states that: P[1] provided the parameters meet the following requirements: • There exists a non empty natural number k such thatP[k], and • For every non empty natural number k such that k � = 1 andP[k] there exists a non empty natural numbe ..."
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Cited by 3 (0 self)
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The scheme Regr without 0 concerns a unary predicateP, and states that: P[1] provided the parameters meet the following requirements: • There exists a non empty natural number k such thatP[k], and • For every non empty natural number k such that k � = 1 andP[k] there exists a non empty natural number n such that n < k andP[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.
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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Cited by 1 (0 self)
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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.
GünterM.Ziegler,2nded.,Springer1999.
"... and q a = 1holds a = 0. (2) Forallnaturalnumbers a, k, randforeveryrealnumber xsuchthat 1 < xand 0 < kholds x a·k+r = x a · x a·(k− ′ 1)+r (3) Forallnaturalnumbers q, a, bsuchthat 0 < aand 1 < qand q a − ′ 1 q b − ′ 1holds ab. (4) Forallnaturalnumbers n, qsuchthat 0 < qholds q n = q n. 1 Thiswor ..."
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and q a = 1holds a = 0. (2) Forallnaturalnumbers a, k, randforeveryrealnumber xsuchthat 1 < xand 0 < kholds x a·k+r = x a · x a·(k− ′ 1)+r (3) Forallnaturalnumbers q, a, bsuchthat 0 < aand 1 < qand q a − ′ 1 q b − ′ 1holds ab. (4) Forallnaturalnumbers n, qsuchthat 0 < qholds q n = q n. 1 ThisworkhasbeensupportedbyNSERCGrantOGP9207.
versita.com/fm/ A Test for the Stability of Networks
"... Summary. A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an alldominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwit ..."
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Summary. A complex polynomial is called a Hurwitz polynomial, if all its roots have a real part smaller than zero. This kind of polynomial plays an alldominant role in stability checks of electrical (analog or digital) networks. In this article we prove that a polynomial p can be shown to be Hurwitz by checking whether the rational function e(p)/o(p) can be realized as a reactance one port, that is as an electrical impedance or admittance consisting of inductors and capacitors. Here e(p) and o(p) denote the even and the odd part of p [25].
University of Białystok Sorting Operators for Finite Sequences Yatsuka Nakamura
"... Summary. Two kinds of sorting operators, descendent one and ascendent one are introduced for finite sequences of reals. They are also called rearrangement of finite sequences of reals. Maximum and minimum values of finite sequences of reals are also defined. We also discuss relations between these c ..."
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Summary. Two kinds of sorting operators, descendent one and ascendent one are introduced for finite sequences of reals. They are also called rearrangement of finite sequences of reals. Maximum and minimum values of finite sequences of reals are also defined. We also discuss relations between these concepts. MML Identifier: RFINSEQ2.