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Determinant of Some Matrices of Field Elements
, 2006
"... Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matr ..."
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Here, we present determinants of some square matrices of field elements. First, the determinat of 2 ∗ 2 matrix is shown. Secondly, the determinants of zero matrix and unit matrix are shown, which are equal to 0 in the field and 1 in the field respectively. Thirdly, the determinant of diagonal matrix is shown, which is a product of all diagonal elements of the matrix. At the end, we prove that the determinant of a matrix is the same as the determinant of its transpose.
Properties of First and Second Order Cutting of Binary Relations
, 2005
"... This paper introduces some notions concerning binary relations according to [9]. It is also an attempt to complement the knowledge contained in the Mizar Mathematical Library regarding binary relations. We define here an image and inverse image of element of set A under binary relation of two sets ..."
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This paper introduces some notions concerning binary relations according to [9]. It is also an attempt to complement the knowledge contained in the Mizar Mathematical Library regarding binary relations. We define here an image and inverse image of element of set A under binary relation of two sets A, B as image and inverse image of singleton of the element under this relation, respectively. Next, we define “The First Order Cutting Relation of two sets A, B under a subset of the set A ” as the union of images of elements of this subset under the relation. We also define “The Second Order Cutting Subset of the Cartesian Product of two sets A, B under a subset of the set A” as an intersection of images of elements of this subset under the subset of the Cartesian Product. The paper also defines first and second projection of binary relations. The main goal of the article is to prove properties and collocations of definitions introduced in this paper.
Sperner's Lemma
, 2010
"... Summary. In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function f , which for an arbitrary vertex v of the barycentric subdivision B of simplex K assigns some vertex ..."
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Summary. In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner's lemma. The lemma states that for a function f , which for an arbitrary vertex v of the barycentric subdivision B of simplex K assigns some vertex from a face of K which contains v, we can find a simplex S of B which satisfies f (S) = K (see [10]). MML identifier: SIMPLEX1, version: 7.11.0 4.1 .1 The notation and terminology used in this paper have been introduced in the following papers: [2], [11], [19], [9], [6], [7], [1], [5], [3], [4], [13], [15], [12], [22], Preliminaries We follow the rules: x, y, X denote sets and n, k denote natural numbers. The following two propositions are true: (1) Let R be a binary relation and C be a cardinal number. If for every x such that x ∈ X holds Card(R • x) = C, then Card R = Card(R (dom R \ X)) + C · Card X. (2) Let Y be a non empty finite set. Suppose Card X = Y + 1. Let f be a function from X into Y . Suppose f is onto. Then there exists y such that y ∈ Y and Card(f −1 ({y})) = 2 and for every x such that x ∈ Y and x = y holds Card(f −1 ({x})) = 1. Let X be a 1sorted structure. A simplicial complex structure of X is a simplicial complex structure of the carrier of X. A simplicial complex of X is a simplicial complex of the carrier of X. Let X be a 1sorted structure, let K be a simplicial complex structure of X, and let A be a subset of K. The functor @ A yielding a subset of X is defined by: Let X be a 1sorted structure, let K be a simplicial complex structure of X, and let A be a family of subsets of K. The functor @ A yielding a family of subsets of X is defined by: We now state the proposition (3) Let X be a 1sorted structure and K be a subsetclosed simplicial complex structure of X. Suppose K is total. Let S be a finite subset of K. Suppose S is simplexlike. Then the complex of { @ S} is a subsimplicial complex of K. The Area of an Abstract Simplicial Complex For simplicity, we adopt the following rules: R 1 denotes a non empty RLS structure, K 1 , K 2 , K 3 denote simplicial complex structures of R 1 , V denotes a real linear space, and K 4 denotes a non void simplicial complex of V . Let us consider R 1 , K 1 . The functor K 1  yields a subset of R 1 and is defined by: (Def. 3) x ∈ K 1  iff there exists a subset A of K 1 such that A is simplexlike and x ∈ conv @ A. One can prove the following propositions: (6) Let K be a subsetclosed simplicial complex structure of V . Then x ∈ K if and only if there exists a subset A of K such that A is simplexlike and x ∈ Int( @ A). (8) For every subset A of R 1 holds the complex of {A} = conv A. (ii) for every subset A of it such that A is simplexlike there exists a subset B of K 1 such that B is simplexlike and conv @ A ⊆ conv @ B. The following proposition is true (10) For every subdivision structure P of K 1 holds K 1  = P . Let us consider R 1 and let K 1 be a simplicial complex structure of R 1 with a nonempty element. Observe that every subdivision structure of K 1 has a nonempty element. We now state four propositions: (11) K 1 is a subdivision structure of K 1 . Let us consider V and let K be a simplicial complex structure of V . A subdivision of K is a finitemembered subsetclosed subdivision structure of K. We now state the proposition (15) Let K be a simplicial complex of V with empty element. Suppose K ⊆ Ω K . Let B be a function from 2 the carrier of V + into the carrier of V . Suppose that for every simplex S of K such that S is non empty holds B(S) ∈ conv @ S. Then subdivision(B, K) is a subdivision structure of K. Let us consider V , K 4 . One can verify that there exists a subdivision of K 4 which is non void. The Barycentric Subdivision Let us consider V , K 4 . Let us assume that K 4  ⊆ Ω (K 4 ) . The functor BCS K 4 yields a non void subdivision of K 4 and is defined by: (Def. 5) BCS K 4 = subdivision(the center of mass of V , K 4 ). Let us consider n and let us consider V , K 4 . Let us assume that K 4  ⊆ Ω (K 4 ) . The functor BCS(n, K 4 ) yields a non void subdivision of K 4 and is defined by: 192 karol pąk (Def. 6) BCS(n, K 4 ) = subdivision(n, the center of mass of V , K 4 ). Next we state several propositions: Let us consider n, V and let K be a non void total simplicial complex of V . Note that BCS(n, K) is total. Let us consider n, V and let K be a non void finitevertices total simplicial complex of V . Note that BCS(n, K) is finitevertices. Selected Properties of Simplicial Complexes Let us consider V and let K be a simplicial complex structure of V . We say that K is affinelyindependent if and only if: (Def. 7) For every subset A of K such that A is simplexlike holds @ A is affinelyindependent. Let us consider R 1 , K 1 . We say that K 1 is simplexjoinclosed if and only if: (Def. 8) For all subsets A, B of K 1 such that A is simplexlike and B is simplexlike holds conv @ A ∩ conv @ B = conv @ A ∩ B. Let us consider V . Note that every simplicial complex structure of V which is emptymembered is also affinelyindependent. Let F be an affinelyindependent family of subsets of V . Observe that the complex of F is affinelyindependent. Let us consider R 1 . One can verify that every simplicial complex structure of R 1 which is emptymembered is also simplexjoinclosed. Let us consider V and let I be an affinelyindependent subset of V . One can check that the complex of {I} is simplexjoinclosed. Let us consider V . One can check that there exists a subset of V which is non empty, trivial, and affinelyindependent. Let us consider V . One can check that there exists a simplicial complex of V which is finitevertices, affinelyindependent, simplexjoinclosed, and total and has a nonempty element. Sperner's lemma 193 Let us consider V and let K be an affinelyindependent simplicial complex structure of V . One can verify that every subsimplicial complex of K is affinelyindependent. Let us consider V and let K be a simplexjoinclosed simplicial complex structure of V . One can check that every subsimplicial complex of K is simplexjoinclosed. Next we state the proposition (25) Let K be a subsetclosed simplicial complex structure of V . Then K is simplexjoinclosed if and only if for all subsets A, B of K such that A is simplexlike and B is simplexlike and Int For simplicity, we follow the rules: K 5 is a simplexjoinclosed simplicial complex of V , A 1 , B 1 are subsets of K 5 , K 6 is a non void affinelyindependent simplicial complex of V , K 7 is a non void affinelyindependent simplexjoinclosed simplicial complex of V , and K is a non void affinelyindependent simplexjoinclosed total simplicial complex of V . Let us consider V , K 6 and let S be a simplex of K 6 . Note that @ S is affinelyindependent. One can prove the following propositions: (26) If A 1 is simplexlike and B 1 is simplexlike and Int (27) If A 1 is simplexlike and @ A 1 is affinelyindependent and B 1 is simplexlike, then Int( Let us consider V and let K 6 be a non void affinelyindependent total simplicial complex of V . Observe that BCS K 6 is affinelyindependent. Let us consider n. Observe that BCS(n, K 6 ) is affinelyindependent. Let us consider V , K 7 . One can verify that (the center of mass of V ) the topology of K 7 is onetoone. We now state the proposition Let us consider V , K. Note that BCS K is simplexjoinclosed. Let us consider n. Observe that BCS(n, K) is simplexjoinclosed. The following four propositions are true: and for every n such that n ≤ degree(K 4 ) there exists a simplex S of K 4 such that S = n + 1 and @ S is affinelyindependent. Then degree(K 4 ) = degree(BCS K 4 ). (32) If K 6  ⊆ Ω (K 6 ) , then degree(K 6 ) = degree(BCS(n, K 6 )). 194 karol pąk (33) Let S be a simplexlike family of subsets of K 7 . If S has non empty elements, then Card S = Card((the center of mass of V ) • S). For simplicity, we adopt the following convention: A 2 denotes a finite affinelyindependent subset of V , A 3 , B 2 denote finite subsets of V , B denotes a subset of V , S, T denote finite families of subsets of V , S 3 denotes a ⊆linear finite finitemembered family of subsets of V , S 4 , T 1 denote finite simplexlike families of subsets of K, and A 4 denotes a simplex of K. The following propositions are true: (34) Let S 6 , S 5 be simplexlike families of subsets of K 7 . Suppose that (38) Let given S 3 . Suppose S 3 has non empty elements and S 3 = S 3 . Let v be an element of V . Suppose v / ∈ S 3 and S 3 ∪ {v} is affinelyindependent. Then {S 6 ; S 6 ranges over simplexes of S 3 and BCS (the complex of { S 3 ∪ {v}}): (the center of mass of (39) Let given S 3 . Suppose S 3 has non empty elements and S 3 +1 = S 3 and S 3 is affinelyindependent. Then Card{S 6 ; S 6 ranges over simplexes of S 3 and BCS (the complex of { S 3 }): (the center of mass of V ) • S 3 ⊆ S 6 } = 2. (41) Suppose S 4 has non empty elements and S 4 + n ≤ degree(K). Then the following statements are equivalent (i) A 3 is a simplex of n + S 4 and BCS K and (the center of mass of V ) • S 4 ⊆ A 3 , (ii) there exists T 1 such that T 1 misses S 4 and T 1 ∪ S 4 is ⊆linear and has non empty elements and T 1 = n + 1 and A 3 = (the center of mass of (42) Suppose S 4 is ⊆linear and has non empty elements and S 4 = S 4 and S 4 ⊆ A 4 and A 4 = S 4 + 1. Then {S 6 ; S 6 ranges over simplexes of S 4 and BCS K : (the center of mass of (43) Suppose S 4 is ⊆linear and has non empty elements and S 4 + 1 = S 4 . Then Card{S 6 ; S 6 ranges over simplexes of S 4 and BCS K : (the center of and (iv) for every simplex S of n − 1 and K and for every X such that X = {S 6 ; S 6 ranges over simplexes of n and K: S ⊆ S 6 } holds if conv @ S meets Int A 3 , then Card X = 2 and if conv @ S misses Int A 3 , then Card X = 1. Let S be a simplex of n−1 and BCS K and given X such that X = {S 6 ; S 6 ranges over simplexes of n and BCS K : S ⊆ S 6 }. Then (v) if conv @ S meets Int A 3 , then Card X = 2, and (vi) if conv @ S misses Int A 3 , then Card X = 1. (45) Let S be a simplex of n − 1 and BCS(k, the complex of {A 2 }) such that A 2 = n + 1 and X = {S 6 ; S 6 ranges over simplexes of n and BCS(k, the complex of {A 2 }): S ⊆ S 6 }. Then (i) if conv @ S meets Int A 2 , then Card X = 2, and (ii) if conv @ S misses Int A 2 , then Card X = 1. The Main Theorem In the sequel v is a vertex of BCS(k, the complex of {A 2 }) and F is a function from Vertices BCS(k, the complex of {A 2 }) into A 2 . The following two propositions are true:
The Sum and Product of Finite Sequences of Complex Numbers
, 2010
"... Summary. This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11]. MML identifier: RVSUM 2, version: 7.11.07 4.156.1112 The notation and terminology used in this paper have been introduced in the ..."
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Summary. This article extends the [10]. We define the sum and the product of the sequence of complex numbers, and formalize these theorems. Our method refers to the [11]. MML identifier: RVSUM 2, version: 7.11.07 4.156.1112 The notation and terminology used in this paper have been introduced in the following papers: [5], [7], [6], [4], [8], [13], [9], [2], [3], [15], [10], [12], and [14]. Auxiliary Theorems Let F be a complexvalued binary relation. Then rng F is a subset of C. Let D be a non empty set, let F be a function from C into D, and let F 1 be a complexvalued finite sequence. Note that F · F 1 is finite sequencelike. For simplicity, we adopt the following rules: i, j denote natural numbers, x, x 1 denote elements of C, c denotes a complex number, F , F 1 , F 2 denote complexvalued finite sequences, and R, R 1 denote ielement finite sequences of elements of C. The unary operation sqrcomplex on C is defined as follows: (Def. 1) For every c holds (sqrcomplex)(c) = c 2 . Next we state two propositions: Let us observe that the functor F 1 + F 2 is commutative. Let us consider i, R 1 , R 2 . Then R 1 + R 2 is an element of C i . The following propositions are true: Let us consider F . Then −F is a finite sequence of elements of C and it can be characterized by the condition: Let us consider i, R. Then −R is an element of C i . The following propositions are true: Let us consider F 1 , F 2 . Then F 1 − F 2 is a finite sequence of elements of C and it can be characterized by the condition: The following propositions are true: Let us consider F , c. We introduce c · F as a synonym of c F. The sum and product of finite sequences of . . . 109 Let us consider F , c. Then c · F is a finite sequence of elements of C and it can be characterized by the condition: One can prove the following four propositions: is a finite sequence of elements of C and it can be characterized by the condition: Let us note that the functor Next we state four propositions: Finite Sum of Finite Sequence of Complex Numbers One can prove the following propositions: 110 keiichi miyajima and takahiro kato The Product of Finite Sequences of Complex Numbers One can prove the following propositions: Modified Part of [1] We now state several propositions: (50) For every complexvalued finite sequence x holds len(−x) = len x. (51) For all complexvalued finite sequences x 1 , x 2 such that len x 1 = len x 2 holds len(x 1 + x 2 ) = len x 1 . (52) For all complexvalued finite sequences x 1 , x 2 such that len x 1 = len x 2 holds len(x 1 − x 2 ) = len x 1 . (53) For every real number a and for every complexvalued finite sequence x holds len(a · x) = len x. (54) For all complexvalued finite sequences x, y, z such that len x = len y = len z holds (x + y) References [1] Kanchun and Yatsuka Nakamura. The inner product of finite sequences and of points of ndimensional topological space. Formalized Mathematics, 11
Formulas and Identities of Trigonometric Functions
, 2004
"... MML Identifier: SIN COS5. The articles In this paper t 1 , t 2 , t 3 , t 4 are real numbers. One can prove the following propositions: (1) If cos t 1 = 0, then cosec t 1 = sec t 1 tan t 1 . (2) If sin t 1 = 0, then cos t 1 = sin t 1 · cot t 1 . (3) If sin t 2 = 0 and sin t 3 = 0 and sin t 4 = 0, t ..."
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MML Identifier: SIN COS5. The articles In this paper t 1 , t 2 , t 3 , t 4 are real numbers. One can prove the following propositions: (1) If cos t 1 = 0, then cosec t 1 = sec t 1 tan t 1 . (2) If sin t 1 = 0, then cos t 1 = sin t 1 · cot t 1 . (3) If sin t 2 = 0 and sin t 3 = 0 and sin t 4 = 0, then sin( (4) If sin t 2 = 0 and sin t 3 = 0 and sin t 4 = 0, then cos(t 2 + t 3 + t 4 ) = −sin t 2 · sin t 3 · sin t 4 · ((cot t 2 + cot t 3 + cot t 4 ) − cot t 2 · cot t 3 · cot t 4 ). (11) If cos t 1 = 0, then (sec t 1 ) 2 = 1 + (tan t 1 ) 2 . (12) cot t 1 = 1 tan t 1 . 243
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.
Trigonometric Functions on Complex Space
"... Summary. This article describes definitions of sine, cosine, hyperbolic sine and hyperbolic cosine. Some of their basic properties are discussed. ..."
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Summary. This article describes definitions of sine, cosine, hyperbolic sine and hyperbolic cosine. Some of their basic properties are discussed.
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.