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Measuring the Relative Complexity of Mathematical Constructions and Theorems

2019
*
Bulletin of Symbolic Logic
*

prepared by Nebojša Ikodinović

doi:10.1017/bsl.2019.34
fatcat:xw74gnwgfrfgnlt4cuhm3gnk7m
*and*Marija Boričić. ... Abstract We investigate*the**relative**complexity**of**mathematical**constructions**and**theorems*using*the*frameworks*of*computable reducibilities*and*reverse*mathematics*. ... First, we study*the*computational content*of*various*theorems*with reverse*mathematical*strength around Arithmetical Transfinite Recursion (ATR 0 ) from*the*point*of*view*of*computable reducibilities, ...##
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A theorem on linear point sets

1919
*
Bulletin of the American Mathematical Society
*

Since A

doi:10.1090/s0002-9904-1919-03201-7
fatcat:xsyt442lrjct7eqikmedskdpli
*and*H differ by a set*of*zero*measure*,*the**relative*exterior*measure**of**the*one is*the*same at every point as that*of**the*other. ... ,*and**of**mathematical*physics which can be solved by*the*aid*of*that theory. ... We obtain a particular case*of*our*theorem*if we assume A to be a*measurable*set. Exterior*measure*will then be replaced by*measure*,*and**relative*exterior*measure*by "*relative**measure*." ...##
###
Transversal theory, an account of some aspects of combinatorial mathematics

1972
*
Discrete Mathematics
*

Hall's

doi:10.1016/0012-365x(72)90119-7
fatcat:7obg7hlcxbetnjkslosv3pnrm4
*theorem**and**the*notion*of*duality.*The*method*of*'elementary*constructions*'. Rado's selection rinciple. Variants, refinemer: ts,*and*applications*of*all's*theorem*. ndependent transversals. ... Definitions*and*examples*of**complex*manifolds. Sheaves*and*coho- molog;:. Geometry*of**complex*manifolds. Applications for elliptic partial differential equations to deformations. Index. E.D. ...##
###
Page 5 of Mathematical Reviews Vol. 53, Issue 4
[page]

1977
*
Mathematical Reviews
*

A recursive relatedness

*theorem*is proved, showing that any two*relative**complexity**measures*are related by a fixed recursive function. ... This*theorem*allows proofs*of*results for all*measures*to be obtained from proofs for a particular*measure*. ““*Complexity*-determined reducibilities are studied. ...##
###
The Digital and the Real Universe. Foundations of Natural Philosophy and Computational Physics

2019
*
Philosophies
*

,

doi:10.3390/philosophies4010003
fatcat:mgv373e4sbhgreenikab6am5qa
*measuring*,*and*computing. ...*The*following article refers to*the*author's book, "*The*Digital*and**the*Real World. Computational Foundations*of**Mathematics*, Science, Technology,*and*Philosophy." ... Reverse*mathematics*enable us to determine*the*proof-theoretic strength resp.*complexity**of**theorems*by classifying them with respect to equivalent*theorems**and*proofs. ...##
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Sentence complexity of theorems in Mizar
[article]

2013
*
arXiv
*
pre-print

Friedman, we use Mizar to take on

arXiv:1311.1915v1
fatcat:saxhyqkr3jbvvdaxcbgjiceyuy
*the*question*of*surveying*the*sentence*complexity*(*measured*by quantifier alternation)*of**mathematical**theorems*. ... We find, as Friedman suggests, that*the*sentence*complexity**of*most Mizar*theorems*is universal (Π_1, or ∀),*and*as one goes higher in*the*sentence*complexity*hierarchy*the*number*of*Mizar*theorems*having ... We looked at*the*definitions*and**theorems*across*the*MML*and*used prenex normal form as a*measure**of**the**complexity**of*these items. ...##
###
Proof as a Mathematical Object - Proposals for a Research Program

2020
*
Baltic Journal of Modern Computing
*

*The*need to develop a theory for

*measuring*value

*and*

*complexity*

*of*

*mathematical*implications

*and*proofs is discussed including motivations, benefits

*and*implementation problems. ...

*The*problem

*of*representing logical implications

*and*proofs by

*mathematical*objects is considered. ... ,

*measure*

*and*quantify

*mathematical*results such as

*theorems*

*and*lemmas. ...

##
###
Page 350 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 25, Issue
[page]

1919
*
American Mathematical Society. Bulletin of the American Mathematical Society
*

We then define

*the*“*relative*exterior*measure**of*A at*the*point x” as * Cf. Denjoy, Journal de Mathématiques, ser. 7, vol. 1 (1915), p. 130. ... ,*and**of**mathematical*physics which can be solved by*the*aid*of*that theory. ...##
###
On coordinatization of mathematics
[article]

2018
*
arXiv
*
pre-print

*The*need to develop a theory for

*measuring*value

*and*

*complexity*

*of*

*mathematical*implications

*and*proofs is discussed including motivations, benefits

*and*implementation problems. ...

*The*problem

*of*advancing coordinatization

*of*

*mathematics*is considered. ... problems, study

*the*network

*of*polynomial reductions, 6) build a map relating

*theorems*

*and*reductions

*of*computational classes for a given domain, 7) introduce

*measures*

*of*cognitive

*complexity*

*of*

*mathematical*...

##
###
Capacitability for Co-Analytic Sets

2022
*
New Zealand Journal of Mathematics
*

It follows from a

doi:10.53733/170
fatcat:5kuatjaxu5ffhh6lvujxna4t6q
*theorem**of*Davies (1952) that if A is an analytic subset*of**the*Cantor middle third set, λ is positive*and**the*Hausdorff s-*measure**of*A is greater than λ, then there is a compact subset ... C*of*A such that*the*Hausdorff s-*measure**of*C is greater than λ. ... For any B,*the*set*of*X such that X is Martin-Löf random*relative*to B has full*measure*. ...##
###
Page 3706 of Mathematical Reviews Vol. , Issue 92g
[page]

1992
*
Mathematical Reviews
*

In §3, we solve

*the*Henson*and*Moore problem by introducing a*construction**of*countably additive*measures*from internal Banach space-valued*measures*. ... For example,*the*Hadamard gap*theorem*, Bloch’s theo- rem,*and**the*two Picard*theorems*all are proved in*the*first half*of**the*book,*and**the*prime number*theorem*is covered near*the*three-quarter point ...##
###
Page 546 of Mathematical Reviews Vol. , Issue 86b
[page]

1986
*
Mathematical Reviews
*

This

*construction*has since been used by a number*of*researchers to obtain new results in many areas including*measure**and*probability theory, potential theory, math- ematical physics*and**mathematical*... {For*the*entire collection see MR 85d:00025.}*MEASURE**AND*INTEGRATION 30 FUNCTIONS*OF*A*COMPLEX*VARIABLE See also 01021. Dettman, John W. (1-OAKL) * Applied*complex*variables. ...##
###
Quantum Model Theory (QMod): Modeling Contextual Emergent Entangled Interfering Entities
[chapter]

2012
*
Lecture Notes in Computer Science
*

In this sense, QMod is a generalization

doi:10.1007/978-3-642-35659-9_12
fatcat:si22hvr3rzawjplpmheo7rntrq
*of*quantum mechanics, similar to how*the*general*relativity*manifold*mathematical*formalism is a generalization*of*special*relativity*. ... . for each context corresponding to a*measurement*,*and*for its real version it does not need*the*property*of*'linearity*of**the*set*of*states' to model*the*quantum effect. ... Let us now work out a*mathematical**construction*put forward in*the*representation*theorem*proven in Sec. 2. ...##
###
Page 5416 of Mathematical Reviews Vol. , Issue 93j
[page]

1993
*
Mathematical Reviews
*

Summary: “A unified treatment is given

*of**the**relative*position*of*L? ... /G,*and*unlike*the*usual convex hull*construction*where*the*convex hulls*of*two subgroups may be disjoint,*the*hull for any subgroup*of*G contains*the*base point. ...##
###
Guest Editorial: Tenth International Conference on Computability, Complexity and Randomness (CCR 2015)

2017
*
Theory of Computing Systems
*

This article is part

doi:10.1007/s00224-017-9809-2
fatcat:gtfithndffdulndutrqnek3qiq
*of**the*Topical Collection on Special Issue on Computability,*Complexity**and*Randomness (CCR 2015) London School*of*Economics Department*of**Mathematics*, London School*of*Economics*and*... In "Exact*constructive**and*computable dimensions" by Ludwig Staiger,*the*author provides a systematic development*of*exact*constructive*dimension, which quantitatively refines*constructive*dimension just ...
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