Mathematical Proof of Four-Color Theorem

2019 ◽  
Vol 09 (03) ◽  
pp. 410-413
Author(s):  
山中 邹
Keyword(s):  
Mathematical Proof ◽  
Four Color Theorem

scholarly journals From the four-color theorem to a generalizing “four-letter theorem”: A sketch for “human proof” and the philosophical interpretation

2020 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Mathematical Proof ◽  
General Theorem ◽  
Human Capabilities ◽  
Letter Alphabet ◽  
Philosophical Interpretation ◽  
Four Color Theorem ◽  
The Universe

The “four-color” theorem seems to be generalizable as follows. The four-letter alphabet is sufficient to encode unambiguously any set of well-orderings including a geographical map or the “map” of any logic and thus that of all logics or the DNA (RNA) plan(s) of any (all) alive being(s).Then the corresponding maximally generalizing conjecture would state: anything in the universe or mind can be encoded unambiguously by four letters.That admits to be formulated as a “four-letter theorem”, and thus one can search for a properly mathematical proof of the statement.It would imply the “four colour theorem”, the proof of which many philosophers and mathematicians believe not to be entirely satisfactory for it is not a “human proof”, but intermediated by computers unavoidably since the necessary calculations exceed the human capabilities fundamentally. It is furthermore rather unsatisfactory because it consists in enumerating and proving all cases one by one.Sometimes, a more general theorem turns out to be much easier for proving including a general “human” method, and the particular and too difficult for proving theorem to be implied as a corollary in certain simple conditions.The same approach will be followed as to the four colour theorem, i.e. to be deduced more or less trivially from the “four-letter theorem” if the latter is proved. References are only classical and thus very well-known papers: their complete bibliographic description is omitted.

The Four-Color Theorem and Mathematical Proof

1980 ◽  
Vol 77 (12) ◽  
pp. 803 ◽  
Author(s):  
Michael Detlefsen ◽  
Mark Luker
Keyword(s):  
Mathematical Proof ◽  
Four Color Theorem

Mathematical proof that rocked number theory will be published

Nature ◽  
2020 ◽  
Vol 580 (7802) ◽  
pp. 177-177
Author(s):  
Davide Castelvecchi
Keyword(s):  
Number Theory ◽  
Mathematical Proof

How is science to be carried forward, and its conclusions reported?

2018 ◽  
Author(s):  
Andrew Briggs ◽  
Hans Halvorson ◽  
Andrew Steane
Keyword(s):  
Mathematical Proof ◽  
Good Practice ◽  
Natural World ◽  
Component Part ◽  
Journal Articles ◽  
Scientific Publications ◽  
Direct Role ◽  
Intellectual Activity ◽  
Self Identity ◽  
The World

The chapter appraises science as an intellectual activity that is appropriately carried out on its own terms. Consequently, it is not appropriate to introduce references to God as a component part of a mathematical proof, nor of a system of forces in the natural world, nor of a sequence of impersonal processes in the biosphere. This does not mean that it is inappropriate to be thankful to God and to celebrate all these aspects of the world as gifts. They can be employed as opportunities to express appreciation through studying and understanding them better in their own right. Nevertheless, there may be processes, such as those which shape a person’s self-identity, in which it is appropriate to recognize God’s more direct role. Good practice concerning acknowledgements sections in scientific publications such as doctoral theses and journal articles is then discussed.

scholarly journals Nonlinear stability of two-layer shallow water flows with a free surface

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190594
Author(s):  
Francisco de Melo Viríssimo ◽  
Paul A. Milewski
Keyword(s):  
Free Surface ◽  
Initial Data ◽  
Nonlinear Stability ◽  
Mathematical Proof ◽  
Passive Layer ◽  
Physical Parameters ◽  
Immiscible Fluid ◽  
Dimensional System ◽  
The Cauchy Problem ◽  
The Difference

The problem of two layers of immiscible fluid, bordered above by an unbounded layer of passive fluid and below by a flat bed, is formulated and discussed. The resulting equations are given by a first-order, four-dimensional system of PDEs of mixed-type. The relevant physical parameters in the problem are presented and used to write the equations in a non-dimensional form. The conservation laws for the problem, which are known to be only six, are explicitly written and discussed in both non-Boussinesq and Boussinesq cases. Both dynamics and nonlinear stability of the Cauchy problem are discussed, with focus on the case where the upper unbounded passive layer has zero density, also called the free surface case. We prove that the stability of a solution depends only on two ‘baroclinic’ parameters (the shear and the difference of layer thickness, the former being the most important one) and give a precise criterion for the system to be well-posed. It is also numerically shown that the system is nonlinearly unstable, as hyperbolic initial data evolves into the elliptic region before the formation of shocks. We also discuss the use of simple waves as a tool to bound solutions and preventing a hyperbolic initial data to become elliptic and use this idea to give a mathematical proof for the nonlinear instability.

A Relationship Between the Factor Indivisibility and the Output Elasticity of the Indivisible Factor

2021 ◽  
pp. 232102222098516
Author(s):  
Dipankar Das
Keyword(s):  
Production Function ◽  
Limit Point ◽  
Mathematical Proof ◽  
Wage Equation ◽  
Marginal Value ◽  
Output Elasticity ◽  
New Type ◽  
First Time ◽  
Jel Classifications

The paper puts forth a notion and derives a special type of production function where labour is an indivisible factor and is in the integer space. Thus, Newtonian calculus is not an appropriate method of deriving the marginal value because limit point does not exist. This shows that indivisibility determines the output elasticity. In the first part, the paper propounds a notion regarding how indivisibility determines curvature of the production function. In the second part, the paper incorporates the findings within a production function and derives a new type accordingly. Moreover, it formally derives the standard wage equation considering all the entitlements of labour, namely (a) normal wages, (b) interest and (c) rent of ability. So far, no such mathematical proof is there to support this wage composition. This paper, for the first time, derives this wage equation considering indivisibility of labour. JEL Classifications: J23, J24, J31, D24, C61, E24, L8

scholarly journals The Vogt-Russell Theorem, and New Results on an Old Problem

1978 ◽  
Vol 80 ◽  
pp. 303-311
Author(s):  
Helmuth Kähler
Keyword(s):  
Existence And Uniqueness ◽  
Thermal Equilibrium ◽  
Equilibrium Configuration ◽  
Mathematical Proof ◽  
Stellar Structure

Half a centry ago Henry Norris Russell and Heinrich Vogt independently made a conjecture concerning the structure of spherical stars which are in hydrostatic and thermal equilibrium (Russell, 1927; Vogt, 1926). This conjecture has later come to be known as the Vogt-Russell theorem and is usually formulated as follows: The structure of a star is uniquely determined by the mass and the composition. In other words, the statement claims the existence and uniqueness of a stellar equilibrium configuration for given parameters mass and composition, and you may find what is called a mathematical proof in many textbooks on stellar structure.

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