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**1 - 4**of**4**### 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 1-sorted 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 1-sorted 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 1-sorted 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 1-sorted structure and K be a subset-closed simplicial complex structure of X. Suppose K is total. Let S be a finite subset of K. Suppose S is simplex-like. 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 simplex-like and x ∈ conv @ A. One can prove the following propositions: (6) Let K be a subset-closed simplicial complex structure of V . Then x ∈ |K| if and only if there exists a subset A of K such that A is simplex-like 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 simplex-like there exists a subset B of K 1 such that B is simplex-like 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 non-empty element. Observe that every subdivision structure of K 1 has a non-empty 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 finite-membered subset-closed 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 finite-vertices total simplicial complex of V . Note that BCS(n, K) is finite-vertices. Selected Properties of Simplicial Complexes Let us consider V and let K be a simplicial complex structure of V . We say that K is affinely-independent if and only if: (Def. 7) For every subset A of K such that A is simplex-like holds @ A is affinelyindependent. Let us consider R 1 , K 1 . We say that K 1 is simplex-join-closed if and only if: (Def. 8) For all subsets A, B of K 1 such that A is simplex-like 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 empty-membered is also affinely-independent. Let F be an affinely-independent family of subsets of V . Observe that the complex of F is affinely-independent. Let us consider R 1 . One can verify that every simplicial complex structure of R 1 which is empty-membered is also simplex-join-closed. Let us consider V and let I be an affinely-independent subset of V . One can check that the complex of {I} is simplex-join-closed. Let us consider V . One can check that there exists a subset of V which is non empty, trivial, and affinely-independent. Let us consider V . One can check that there exists a simplicial complex of V which is finite-vertices, affinely-independent, simplex-join-closed, and total and has a non-empty element. Sperner's lemma 193 Let us consider V and let K be an affinely-independent 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 simplex-join-closed simplicial complex structure of V . One can check that every subsimplicial complex of K is simplexjoin-closed. Next we state the proposition (25) Let K be a subset-closed simplicial complex structure of V . Then K is simplex-join-closed if and only if for all subsets A, B of K such that A is simplex-like and B is simplex-like and Int For simplicity, we follow the rules: K 5 is a simplex-join-closed simplicial complex of V , A 1 , B 1 are subsets of K 5 , K 6 is a non void affinely-independent simplicial complex of V , K 7 is a non void affinely-independent simplex-joinclosed simplicial complex of V , and K is a non void affinely-independent simplexjoin-closed 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 simplex-like and B 1 is simplex-like and Int (27) If A 1 is simplex-like and @ A 1 is affinely-independent and B 1 is simplexlike, then Int( Let us consider V and let K 6 be a non void affinely-independent total simplicial complex of V . Observe that BCS K 6 is affinely-independent. Let us consider n. Observe that BCS(n, K 6 ) is affinely-independent. Let us consider V , K 7 . One can verify that (the center of mass of V ) the topology of K 7 is one-to-one. We now state the proposition Let us consider V , K. Note that BCS K is simplex-join-closed. Let us consider n. Observe that BCS(n, K) is simplex-join-closed. 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 simplex-like 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 finite-membered family of subsets of V , S 4 , T 1 denote finite simplex-like 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 simplex-like 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 affinely-independent. 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:

### Partial Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

"... Summary. In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]). ..."

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Summary. In this article, we define and develop partial differentiation of vector-valued functions on n-dimensional real normed linear spaces (refer to [19] and [20]).

### DOI: 10.2478/v10037-010-0025-7 Differentiation of Vector-Valued Functions on n-Dimensional Real Normed Linear Spaces

"... Summary. In this article, we define and develop differentiation of vectorvalued functions on n-dimensional real normed linear spaces (refer to [16] and [17]). ..."

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Summary. In this article, we define and develop differentiation of vectorvalued functions on n-dimensional real normed linear spaces (refer to [16] and [17]).

### Planes and Spheres as Topological Manifolds. Stereographic Projection

"... Summary. The goal of this article is to show some examples of topolo-gical manifolds: planes and spheres in Euclidean space. In doing it, the article introduces the stereographic projection [25]. ..."

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Summary. The goal of this article is to show some examples of topolo-gical manifolds: planes and spheres in Euclidean space. In doing it, the article introduces the stereographic projection [25].