Dedekind’s η-function and Euler’s theorem on pentagonal numbers

The Paving Stone Theory of World Tectonics

10.1093/oso/9780198717867.003.0003 ◽

2018 ◽

Author(s):

Roy Livermore

Keyword(s):

Indian Ocean ◽

Plate Tectonics ◽

Earth Science ◽

Transform Faults ◽

Plate Motions ◽

Euler’S Theorem ◽

History Of ◽

New Ideas ◽

Entire History ◽

The Indian Ocean

Tuzo Wilson introduces the concept of transform faults, which has the effect of transforming Earth Science forever. Resistance to the new ideas is finally overcome in the late 1960s, as the theory of moving plates is established. Two scientists play a major role in quantifying the embryonic theory that is eventually dubbed ‘plate tectonics’. Dan McKenzie applies Euler’s theorem, used previously by Teddy Bullard to reconstruct the continents around the Atlantic, to the problem of plate rotations on a sphere and uses it to unravel the entire history of the Indian Ocean. Jason Morgan also wraps plate tectonics around a sphere. Tuzo Wilson introduces the idea of a fixed hotspot beneath Hawaii, an idea taken up by Jason Morgan to create an absolute reference frame for plate motions.

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A theorized new class of polyhedral hydrocarbons of molecular formula CnHn and their bottom-up scaffold expansions into hyperstructures

Scientific Reports ◽

10.1038/s41598-021-84562-6 ◽

2021 ◽

Vol 11 (1) ◽

Author(s):

Camila M. B. Machado ◽

Nathalia B. D. Lima ◽

Sóstenes L. S. Lins ◽

Alfredo M. Simas

Keyword(s):

General Formula ◽

Chemical Structure ◽

Molecular Formula ◽

Fusion Reactions ◽

3D Scaffold ◽

3D Scaffolds ◽

Bottom Up ◽

New Class ◽

Euler’S Theorem ◽

2D And 3D

AbstractWe address the use of Euler's theorem and topological algorithms to design 18 polyhedral hydrocarbons of general formula CnHn that exist up to 28 vertexes containing four- and six-membered rings only; compounds we call “nuggets”. Subsequently, we evaluated their energies to verify the likelihood of their chemical existence. Among these compounds, 13 are novel systems, of which 3 exhibit chirality. Further, the ability of all nuggets to perform fusion reactions either through their square faces, or through their hexagonal faces was evaluated. Indeed, they are potentially able to form bottom-up derived molecular hyperstructures with great potential for several applications. By considering these fusion abilities, the growth of the nuggets into 1D, 2D, and 3D-scaffolds was studied. The results indicate that nugget24a (C24H24) is predicted to be capable of carrying out fusion reactions. From nugget24a, we then designed 1D, 2D, and 3D-scaffolds that are predicted to be formed by favorable fusion reactions. Finally, a 3D-scaffold generated from nugget24a exhibited potential to be employed as a voxel with a chemical structure remarkably similar to that of MOF ZIF-8. And, such a voxel, could in principle be employed to generate any 3D sculpture with nugget24a as its level of finest granularity.

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Metaplectic flavor symmetries from magnetized tori

Journal of High Energy Physics ◽

10.1007/jhep05(2021)078 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Yahya Almumin ◽

Mu-Chun Chen ◽

Víctor Knapp-Pérez ◽

Saúl Ramos-Sánchez ◽

Michael Ratz ◽

...

Keyword(s):

Zero Mode ◽

Metaplectic Groups ◽

Common Multiple ◽

Least Common Multiple ◽

Yukawa Couplings ◽

Flavor Symmetries ◽

Euler’S Theorem ◽

Modular Transformations ◽

Analytic Expressions ◽

Geometric Picture

Abstract We revisit the flavor symmetries arising from compactifications on tori with magnetic background fluxes. Using Euler’s Theorem, we derive closed form analytic expressions for the Yukawa couplings that are valid for arbitrary flux parameters. We discuss the modular transformations for even and odd units of magnetic flux, M, and show that they give rise to finite metaplectic groups the order of which is determined by the least common multiple of the number of zero-mode flavors involved. Unlike in models in which modular flavor symmetries are postulated, in this approach they derive from an underlying torus. This allows us to retain control over parameters, such as those governing the kinetic terms, that are free in the bottom-up approach, thus leading to an increased predictivity. In addition, the geometric picture allows us to understand the relative suppression of Yukawa couplings from their localization properties in the compact space. We also comment on the role supersymmetry plays in these constructions, and outline a path towards non-supersymmetric models with modular flavor symmetries.

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Extension of Euler's theorem to n-dimensional spaces

IEEE Transactions on Aerospace and Electronic Systems ◽

10.1109/7.40731 ◽

1989 ◽

Vol 25 (6) ◽

pp. 903-909 ◽

Cited By ~ 14

Author(s):

I.Y. Bar-Itzhack

Keyword(s):

Euler’S Theorem

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COMPARISON STUDY OF FERMAT, SOLOVAY-STRASSEN AND MILLER-RABIN PRIMALITY TEST USING MATHEMATICA 6.0

Visipena Journal ◽

10.46244/visipena.v3i1.48 ◽

2012 ◽

Vol 3 (1) ◽

pp. 1-10

Author(s):

Ega Gradini

Keyword(s):

Quantitative Comparison ◽

Comparison Study ◽

Source Codes ◽

Powerful Test ◽

Euler’S Theorem ◽

Primality Test ◽

Miller Test ◽

Mathematica Software ◽

The Given

This paper presents three primality tests; Fermat test, Solovay-Strassen test, and Rabin-Miller test. Mathematica software is used to carry out the primality tests. The application of Fermat’s Litle Theorem as well as Euler’s Theorem on the tests was also discussed and this leads to the concept of pseudoprime. This paper is also discussed some results on pseudoprimes with certain range and do quantitative comparison. Those primality tests need to be evaluated in terms of its ability to compute as well as correctness in determining primality of given numbers. The answer to this is to create a source codes for those tests and evaluate them by using Mathematica 6.0. Those are Miller-Rabin test, Solovay-Strassen test, Fermat test and Lucas-Lehmer test. Each test was coded using an algorithm derived from number theoretic theorems and coded using the Mathematica version 6.0. Miller-Rabin test, SolovayStrassen test, and Fermat test are probabilistic tests since they cannot certainly identify the given number is prime, sometimes they fail. Using Mathematica 6.0, comparison study of primality test has been made and given the Miller- Rabin test as the most powerful test than other.

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