Simple Fractal Calculus from Fractal Arithmetic

Aerts, Diederik; Czachor, Marek; Kuna, Maciej

doi:10.1016/s0034-4877(18)30053-3

Cited by 13 publications

(18 citation statements)

References 32 publications

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“…For all that, treated just as a mathematical trick, the idea has found concrete applications in fractal theory [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

“…From the Burgin perspective it is essential that one can also encounter nonlinear functions f (Fechner's logarithms, Cantor-type functions for Cantor sets [1][2][3], Peano-type space-filling curves in Sierpiński-set cases [4]...). Whenever one tries to apply a Burgin-type generalization to a physical system, one immediately encounters the problem that nontrivial f s typically require dimensionless arguments in f (x).…”

Section: Introductionmentioning

confidence: 99%

“…We will consider two spaces, 4 , which become Minkowski spacetimes when appropriate arithmetic is selected. We will then consider the problem of electromagnetic fields produced by pointlike sources, but the Maxwell field will be formulated in terms of this concrete arithmetic ⊕, , , , which will make the space Minkowskian.…”

Section: Introductionmentioning

confidence: 99%

“…In Section III, the two types of reals are employed in construction of two types of Minkowski spaces. Sections IV and V discuss in detail the two concrete examples of Minkowski spaces: 4 . Section VI discusses electromagnetic fields produced by a pointlike charge, with Maxwell equations formulated in terms of a non-Diophantine formalism defined by some f .…”

Section: Introductionmentioning

confidence: 99%

“…The examples of four-dimensional spaces, R 4 4 , are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms of the corresponding calculus.…”

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confidence: 99%

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If Gravity is Geometry, is Dark Energy just Arithmetic?

Czachor

2017

Int J Theor Phys

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Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, R 4 4 , are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms of the corresponding calculus. However, when one switches to the 'natural' Diophantine arithmetic and calculus, the Minkowskian character of the space is lost and what one effectively obtains is a Lorentzian manifold. I discuss in more detail the problem of electromagnetic fields produced by a pointlike charge. The solution has the standard form when expressed in terms of the non-Diophantine formalism. When the 'natural' formalsm is used, the same solution looks as if the fields were created by a charge located in an expanding universe, with nontrivially accelerating expansion. The effect is clearly visible also in solutions of the Friedman equation with vanishing cosmological constant. All of this suggests that phenomena attributed to dark energy may be a manifestation of a miss-match between the arithmetic employed in mathematical modeling, and the one occurring at the level of natural laws. Arithmetic is as physical as geometry.

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“…For all that, treated just as a mathematical trick, the idea has found concrete applications in fractal theory [1][2][3][4].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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If Gravity is Geometry, is Dark Energy just Arithmetic?

Czachor

2017

Int J Theor Phys

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A Loophole of All ‘Loophole-Free’ Bell-Type Theorems

Czachor

2020

Found Sci

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Bell's theorem cannot be proved if complementary measurements have to be represented by random variables which cannot be added or multiplied. One such case occurs if their domains are not identical. The case more directly related to the Einstein-Rosen-Podolsky argument occurs if there exists an 'element of reality' but nevertheless addition of complementary results is impossible because they are represented by elements from different arithmetics. A naive mixing of arithmetics leads to contradictions at a much more elementary level than the Clauser-Horne-Shimony-Holt inequality.

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Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics

Czachor¹

2020

Found Sci

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Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ (MOND) change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition $$\beta _1\oplus \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)+\tanh ^{-1}(\beta _2)\big )$$ β 1 ⊕ β 2 = tanh ( tanh - 1 ( β 1 ) + tanh - 1 ( β 2 ) ) , although multiplication $$\beta _1\odot \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)\cdot \tanh ^{-1}(\beta _2)\big )$$ β 1 ⊙ β 2 = tanh ( tanh - 1 ( β 1 ) · tanh - 1 ( β 2 ) ) , and division $$\beta _1\oslash \beta _2=\tanh \big (\tanh ^{-1}(\beta _1)/\tanh ^{-1}(\beta _2)\big )$$ β 1 ⊘ β 2 = tanh ( tanh - 1 ( β 1 ) / tanh - 1 ( β 2 ) ) do not seem to appear in the literature. The map $$f_{\mathbb{X}}(\beta )=\tanh ^{-1}(\beta )$$ f X ( β ) = tanh - 1 ( β ) defines an isomorphism of the arithmetic in $${\mathbb{X}}=(-1,1)$$ X = ( - 1 , 1 ) with the standard one in $${\mathbb{R}}$$ R . The new arithmetic is projective and non-Diophantine in the sense of Burgin (Uspekhi Matematicheskich Nauk 32:209–210 (in Russian), 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov (Technika Molodezhi 10:16–19 (11:30–33 in Russian), 1961). Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz (Non-Newtonian calculus, Lee Press, Pigeon Cove, 1972). Treating the above example as a paradigm, we ask what can be said about the set $${\mathbb{X}}$$ X of ‘real numbers’, and the isomorphism $$f_{{\mathbb{X}}}:{\mathbb{X}}\rightarrow {\mathbb{R}}$$ f X : X → R , if we assume the standard form of Newtonian mechanics and general relativity (formulated by means of the new calculus) but demand agreement with astrophysical observations. It turns out that the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if $$f_{\mathbb{X}}(t/t_H)\approx 0.8\sinh (t-t_1)/(0.8\, t_H)$$ f X ( t / t H ) ≈ 0.8 sinh ( t - t 1 ) / ( 0.8 t H ) . The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with $$\Omega _\Lambda =0.7$$ Ω Λ = 0.7 , $$\Omega _M=0.3$$ Ω M = 0.3 . Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element $$0_{{\mathbb{X}}}$$ 0 X of addition, is nonzero from the point of view of the standard arithmetic of $${\mathbb{R}}$$ R . This implies $$f^{-1}_{{\mathbb{X}}}(0)=0_{{\mathbb{X}}}>0$$ f X - 1 ( 0 ) = 0 X > 0 . The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters. Physics of the dark universe seems to be both geometry and arithmetic.

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Simple Fractal Calculus from Fractal Arithmetic

Cited by 13 publications

References 32 publications

If Gravity is Geometry, is Dark Energy just Arithmetic?

If Gravity is Geometry, is Dark Energy just Arithmetic?

A Loophole of All ‘Loophole-Free’ Bell-Type Theorems

Non-Newtonian Mathematics Instead of Non-Newtonian Physics: Dark Matter and Dark Energy from a Mismatch of Arithmetics

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