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118
AN ELEMENTARY PROOF OF BORSUK THEOREM
"... In 1933, Borsuk conjectured that any bounded ddimensional set of nonzero diameter can be broken into d + 1 parts of smaller diameter[1]. This conjecture was disproved for large enough d[2, 3, 4, 5, 6, 7], though it is true for low dimensional cases. The paper provides an alternative proof for d = 2 ..."
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= 2 case. Theorem 1 (Borsuk). Any bounded plane figure can be divided into three pieces of smaller diameters. This theorem has a standard proof by first proving the plane figure (assume its diameter to be 1) can be bounded by a hexagon, with opposite sides parallel and separated by a distance
A Borsuk theorem on homotopy types
 Journal of Formalized Mathematics
, 1991
"... Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions, retrac ..."
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Cited by 108 (6 self)
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Summary. We present a Borsuk’s theorem published first in [1] (compare also [2, pages 119–120]). It is slightly generalized, the assumption of the metrizability is omitted. We introduce concepts needed for the formulation and the proofs of the theorems on upper semicontinuous decompositions
EXTENSION OF THE BORSUK THEOREM ON NONEMBEDDABILITY OF SPHERES
, 906
"... Abstract. It is proved that the suspension P M of a closed ndimensional manifold M, n ≥ 1, does not embed in a product of n + 1 curves. In fact, the ultimate result will be proved in a much more general setting. This is a farreaching generalization the Borsuk theorem on nonembeddability of the sp ..."
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Abstract. It is proved that the suspension P M of a closed ndimensional manifold M, n ≥ 1, does not embed in a product of n + 1 curves. In fact, the ultimate result will be proved in a much more general setting. This is a farreaching generalization the Borsuk theorem on non
Combinatorial consequences of relatives of the LusternikShnirelmanBorsuk theorem
, 2005
"... Call a set of 2n + k elements Knesercolored when its nsubsets are put into classes such that disjoint nsubsets are in different classes. Kneser showed that k + 2 classes are sufficient to Knesercolor the nsubsets of a 2n + k element set. There are several proofs that this same number is necessa ..."
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Cited by 3 (0 self)
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is necessary which rely on fixedpoint theorems related to the LusternikSchnirelmannBorsuk (LSB) theorem. By employing generalizations of these theorems we expand the proofs mentioned to obtain proofs of an original result we call the Subcoloring theorem. The Subcoloring theorem asserts the existence of a
A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs
"... For any undirected graph G, let µ(G) be the graph parameter introduced by Colin de Verdière. In this paper we show that (G) 4 if and only if G is linklessly embeddable (in R 3 ). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and ..."
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Cited by 36 (8 self)
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, and Thomas. A key ingredient is a Borsuktype theorem on the existence of a pair of antipodal linked (k \Gamma 1) spheres in certain mappings OE : S 2k ! R 2k\Gamma1 . This result might be of interest in its own right. We also derive that (G) 4 for each linklessly embeddable graph G = (V; E), where
Extensions of the BorsukUlam Theorem
, 2002
"... One of the more well known results from topology is the BorsukUlam Theorem. It states that any continuous function from an ndimensional sphere to ndimensional Euclidean space must map some pair of opposite points on the sphere to the same point in Euclidean space. This is often stated colloquiall ..."
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One of the more well known results from topology is the BorsukUlam Theorem. It states that any continuous function from an ndimensional sphere to ndimensional Euclidean space must map some pair of opposite points on the sphere to the same point in Euclidean space. This is often stated
BorsukUlam theorem and applications
, 2005
"... The BorsukUlam theorem is one of the most applied theorems in topology. It was conjectured by Ulam at the Scottish Café in Lvov. Applications range from combinatorics to differential equations and even economics. The theorem proven in one form by Borsuk in 1933 has many equivalent formulations. One ..."
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The BorsukUlam theorem is one of the most applied theorems in topology. It was conjectured by Ulam at the Scottish Café in Lvov. Applications range from combinatorics to differential equations and even economics. The theorem proven in one form by Borsuk in 1933 has many equivalent formulations
Tverberg partitions and Borsuk–Ulam theorems
 Pacific Jour. of Math
"... An Ndimensional real representation E of a finite group G is said to have the “Borsuk–Ulam Property ” if any continuous Gmap from the (N + 1)fold join of G (an Ncomplex equipped with the diagonal Gaction) to E has a zero. This happens iff the “Van Kampen characteristic class ” of E is nonzero, ..."
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Cited by 13 (0 self)
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An Ndimensional real representation E of a finite group G is said to have the “Borsuk–Ulam Property ” if any continuous Gmap from the (N + 1)fold join of G (an Ncomplex equipped with the diagonal Gaction) to E has a zero. This happens iff the “Van Kampen characteristic class ” of E is nonzero
ON THE EXISTENCE THEOREMS OF KANTOROVICH, MIRANDA AND BORSUK
"... Abstract. The theorems of Kantorovich, Miranda and Borsuk all give conditions on the existence of a zero of a nonlinear mapping. In this paper we are concerned with relations between these theorems in terms of generality in the case that the mapping is finitedimensional. To this purpose we formulat ..."
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Abstract. The theorems of Kantorovich, Miranda and Borsuk all give conditions on the existence of a zero of a nonlinear mapping. In this paper we are concerned with relations between these theorems in terms of generality in the case that the mapping is finitedimensional. To this purpose we
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