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Connected Choice and the Brouwer Fixed Point Theorem
[article]

2018
*
arXiv
*
pre-print

We study

arXiv:1206.4809v3
fatcat:ybktamzncbeoxk2agabqiylziq
*the*computational content of*the**Brouwer**Fixed**Point**Theorem*in*the*Weihrauch lattice. ... One of our main results is that for any*fixed*dimension*the**Brouwer**Fixed**Point**Theorem*of that dimension is computably equivalent to*connected**choice*of*the*Euclidean unit cube of*the*same dimension. ... Brouwer's*Fixed**Point**Theorem**and**Connected**Choice*In this section, we want to prove that*the**Brouwer**Fixed**Point**Theorem*is computably equivalent to*connected**choice*for any*fixed*dimension. ...##
###
Connected choice and the Brouwer fixed point theorem

2019
*
Journal of Mathematical Logic
*

Definition (

doi:10.1142/s0219061319500041
fatcat:oxsofyejjbc63oxd5qennl46o4
*Choice*)*The**choice*statement (∀ closed A ⊆ X )(∃x ∈ X )(A = ∅ =⇒ x ∈ A) translates into*the**choice*operation C X :⊆ A − (X ) ⇒ X , A → A Realizing*Theorems*Definition Any*theorem*T of form ... By*Fix*n : C n → A n , f → {x ∈ [0, 1] n : f (x) = x} we denote*the**fixed**point*map of dimension n. We note that CC n • Con n •*Fix*n (f ) ⊆ BFT n (f ). ...##
###
On the Computational Content of the Brouwer Fixed Point Theorem
[chapter]

2012
*
Lecture Notes in Computer Science
*

Brouwer's

doi:10.1007/978-3-642-30870-3_7
fatcat:sw75b4h5lbe5ddhto3skhgixgm
*Fixed**Point**Theorem**and**Connected**Choice*In this section we want to show that*the**Brouwer**Fixed**Point**Theorem*is computably equivalent to*connected**choice*for any*fixed*dimension. ... It is easy to see that in general*the**Brouwer**Fixed**Point**Theorem**and**connected**choice*are not independent of*the*dimension. ...##
###
The Brouwer Fixed Point Theorem Revisited
[chapter]

2016
*
Lecture Notes in Computer Science
*

In this paper we attempt to classify

doi:10.1007/978-3-319-40189-8_6
fatcat:yny2hizfwzfivfoyhpuao3budi
*the**Brouwer**Fixed**Point**Theorem*. ... One of our main results, presented in Section 3, is that*the**Brouwer**Fixed**Point**Theorem*is equivalent to*connected**choice*for each*fixed*dimension n, i.e. BFT n ≡ W CC n . ...##
###
The open mapping theorem and the fundamental theorem of algebra
[article]

2011
*
arXiv
*
pre-print

In particular, this shows that one can obtain a proof of

arXiv:0710.5648v2
fatcat:gaylblzq4ffu5cg2uobmsaoure
*the*FTA (albeit indirect) which is based on*the**Brouwer**fixed**point**theorem*, an aim which was not reached in*the*past*and*later*the*possibility ... In addition, it is also independent of*the*fact that*the*roots of an analytic function are isolated.*The*proof is based on either*the*Banach or*Brouwer**fixed**point**theorems*. ... This cast doubt on*the*possibility of proving*the*FTA by applying*the**Brouwer**fixed**point**theorem*. ...##
###
On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem

2006
*
Applied Mathematics and Computation
*

functions that there exists no social welfare function which satisfies transitivity, Pareto principle, independence of irrelevant alternatives,

doi:10.1016/j.amc.2005.02.054
fatcat:op6jqk6tgrhixprox727pke6lu
*and*has no dictator is equivalent to*the**Brouwer**fixed**point*... We will show that in*the*case where there are two individuals*and*three alternatives (or under*the*assumption of free-triple property)*the*Arrow impossibility*theorem*(Arrow (1963)) for social welfare ... to*the**Brouwer**fixed**point**theorem*. ...##
###
On the Invariance of Openness

1958
*
Proceedings of the National Academy of Sciences of the United States of America
*

In

doi:10.1073/pnas.44.5.464
pmid:16590221
pmcid:PMC335445
fatcat:fahn4qnliza35a3bpmhqynggtm
*the*present*theorem*we keep*the*integers a*and*b*fixed*, but ki, k2, k3 are arbitrary integers with d any integer satisfying d 0 0 (mod p). ...*and**fixed*integral exponents with x*and*y unknowns. ... In*the*present*theorem*we keep*the*integers a*and*b*fixed*, but ki, k2, k3 are arbitrary integers with d any integer satisfying d 0 0 (mod p). ...##
###
Intersection properties of Helly families

2001
*
Topology and its Applications
*

*The*Helly convex-set

*theorem*is extended onto topological spaces. ... From our results it follows that if there are given m + 2 convex subsets of an m-dimensional contractible Hausdorff space

*and*

*the*intersection of each collection of m + 1

*the*subsets is a nonempty contractible ... Our main tool is

*the*

*Brouwer*

*fixed*

*point*

*theorem*(1912) which appears here in an equivalent form as lemma on indexed covering (some applications of

*the*lemma to

*the*

*fixed*

*point*theory are given in [6] ...

##
###
A FIXED-POINT THEOREM FOR PLANAR HOMEOMORPHISMS

1999
*
Topology
*

Acknowledgement-I am grateful to

doi:10.1016/s0040-9383(98)00001-9
fatcat:nhvcthlddrfadepf7yineys53e
*the*referee for a very careful reading of*the*manuscript*and*for many helpful suggestions that improved*the*exposition. ... INTRODUCTION*The*main result of this paper is*the*following*fixed**point**theorem*.*THEOREM*2.3 . ... As before, fI "S is*fixed**point*free*and**the*Lefschetz*Theorem*implies that fI ,*and*hence f, has a*fixed**point*of positive index. This contradiction completes*the*proof of (a). ...##
###
Monotone principle of forked points and its consequences

2015
*
Mathematica Moravica
*

*Brouwer*

*and*Schauder

*theorems*are two direct censequences of

*the*monotone principle od forked

*points*. ... Annalen, 65 (1908), 107-128; translated in van Heijenoort 1967, 183-198],

*and*by M. Ta s k o v i ć [

*The*axiom of

*choice*,

*fixed*

*point*

*theorems*,

*and*inductive ordered sets, Proc. Amer. Math. ... We notice that this statement is a direct translation of

*the*

*Brouwer*

*fixed*

*point*

*theorem*to Banach spaces. Proof. ...

##
###
THE HEX GAME THEOREM AND THE ARROW IMPOSSIBILITY THEOREM: THE CASE OF WEAK ORDERS

2009
*
Metroeconomica
*

Because Gale showed that

doi:10.1111/j.1467-999x.2008.00332.x
fatcat:ldjg3g7klrcvbobegbisieedjq
*the**Brouwer**fixed**point**theorem*is equivalent to*the*HEX game*theorem*, this paper indirectly shows*the*equivalence of*the**Brouwer**fixed**point**theorem**and**the*Arrow impossibility ... Chichilnisky showed*the*equivalence of her impossibility*theorem**and**the**Brouwer**fixed**point**theorem*,*and*Baryshnikov showed that*the*impossibility*theorem*by Chichilnisky*and**the*Arrow impossibility*theorem*... CONCLUDING REMARKS We have considered*the*relationship between*the*HEX game*theorem**and**the*Arrow impossibility*theorem*when individual preferences are weak orders,*and*have shown their equivalence. ...##
###
The Game of Hex and the Brouwer Fixed-Point Theorem

1979
*
The American mathematical monthly
*

This research has been partially supported by

doi:10.1080/00029890.1979.11994922
fatcat:iymfcjvexfc6dia5qoc4aytfti
*the*National Science Foundation under Grant MCS77-16054*and*SOC78-05196 with*the*University of California, Berkeley. ... Reproduction in whole or in part is permitted for any purpose of*the*United States Government. ... We have thus obtained a*fixed*-*point*-free mapping, which contradicts*the**Brouwer**Theorem*. . 4 . 4*The*n-dimensional Hex*Theorem*. ...##
###
The Game of Hex and the Brouwer Fixed-Point Theorem

1979
*
The American mathematical monthly
*

This research has been partially supported by

doi:10.2307/2320146
fatcat:wo5ghy7dkfg3nl6n6eucjzztja
*the*National Science Foundation under Grant MCS77-16054*and*SOC78-05196 with*the*University of California, Berkeley. ... Reproduction in whole or in part is permitted for any purpose of*the*United States Government. ... We have thus obtained a*fixed*-*point*-free mapping, which contradicts*the**Brouwer**Theorem*. . 4 . 4*The*n-dimensional Hex*Theorem*. ...##
###
On Schauder's 54th problem in scottish book revisited

2002
*
Mathematica Moravica
*

On

doi:10.5937/matmor0206119t
fatcat:oxmhmt5v5bardnc62z2hlurvji
*the*other hand, in this sense, we extend*and**connected*former results of*Brouwer*, ... does every continuous mapping f : C → C has a*fixed**point*? ... In*connection*with this, proofs are*the*analogous to*the*proofs of*the*preceding statements of*Brouwer*, Schauder*and*Banach. ...##
###
Direction Preserving Zero Point Computing and Applications
[chapter]

2009
*
Lecture Notes in Computer Science
*

We study

doi:10.1007/978-3-642-10841-9_37
fatcat:gjiubueigzft5l47expzkforba
*the**connection*between*the*direction preserving zero*point**and**the*discrete*Brouwer**fixed**point*in terms of their computational complexity. ... Building upon*the**connection*between*the*two types of combinatorial structures for continuous*fixed**point**theorems*, we derive an immediate proof that TUCKER is PPAD-complete for all constant dimensions ... There is however a*choice*to make when we discretize*the*original concept of*fixed**point*from*the*continuous domain. ...
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