Results 21 to 30 of about 104 (81)
The New Notation for Hyperoperation of a Sequence
, 2022 For a sequence \(a_1, a_2, \ldots, a_n\), we define the exponent, tetration, and pentation of a sequence \(a_n\) as \(\overset{n}{\underset{k = 1}{\textrm{E}}} (a_k) = a_1[3]a_2[3]\cdots[3]a_n\), \(\overset{n}{\underset{k = 1}{\textrm{T}}} (a_k) = a_1[4 ...
Kyumin Nam
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Emergent tetratic order in crowded systems of rotationally asymmetric hard kite particles [PDF]
Nature Communications, 2020 AbstractConsidering multi-body systems of monodisperse hard Brownian particles, it remains challenging to predict the forms of order that can emerge in their dense assembled structures. Surprisingly, here, using Monte Carlo simulations, we show that tetratic-ordered phases emerge in a dense two-dimensional system of hard kites that are rotationally ...
Zhanglin Hou +5 more
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, 2021 19 pages, 2 tables.International audienceWe solve a few open problems related to a peculiar property of the integer tetration ${^{b}a}$, which is the constancy of its congruence speed for any sufficiently large $b=b(a)$.
Ripà, Marco
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, 2023 Research data for the kind reviewers of JNT about the smallest perfect powers of degree c=3,4,5,...,32, congruent to 5 modulo 20, having a constant congruence speed of c (please, note that this result has not been included in the submitted manuscript ...
Marco Ripà (16426323)
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Asymptotic Solutions of the Tetration Equation
, 2022 100 pages, about 30 ...
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Tetratic phase in 2D crystals of squares
Soft MatterWe report the tetratic phase as an intermediate phase between an isotropic fluid and a square crystal composed of micrometer sized squares in two dimensions. The squares are manufactured by direct laser writing in a photoresist.
Robert Löffler +2 more
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Extension of tetration to real and complex heights
, 2021 The continuous tetrational function ${^x}r=τ(r,x)$, the unique solution of equation $τ(r,x)=r^{τ(r,x-1)}$ and its differential equation $τ'(r,x) =q τ(r,x) τ'(r,x-1)$, is given explicitly as ${^x}r=\exp_{r}^{\lfloor x \rfloor+1}[\{x\}]_q$, where $x$ is a real variable called height, $r$ is a real constant called base, $\{x\}=x-\lfloor x \rfloor$ is the ...
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Nonequilibrium steady states in a vibrated-rod monolayer: tetratic, nematic, and smectic correlations [PDF]
Journal of Statistical Mechanics: Theory and Experiment, 2006 We study experimentally the nonequilibrium phase behaviour of a horizontal monolayer of macroscopic rods. The motion of the rods in two dimensions is driven by vibrations in the vertical direction. In addition to varying packing fraction and aspect ratio as in most studies on hard-particle systems, we take advantage of our ability to vary the precise ...
Narayan, Vijay +2 more
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Epsilon Numbers and Cantor Normal Form
, 2009 An epsilon number is a transfinite number which is a fixed point of an exponential map: ωϵ = ϵ. The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, ↑). Namely, the ordinal indexing of epsilon numbers is
Grzegorz Bancerek, Bancerek, Grzegorz
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THE PROBLEM OF NEW PROPERTIES OF NUMBERS
Надежность и качество сложных системBackground. The proposed work is devoted to research to clarify the axiomatization of arithmetic in connection with the presence of special assumptions in it.
L.G. Sushkov
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