Pierre Pellat-Finet · SPIRE - Sciences Po Institutional REpository · 2024
Quaternions have been used to represent polarization states and polarization operators. But so far, only polarizers, dichroic or non-depolarizing devices have been represented in that way. We propose a quaternionic representation of perfect as well as partial depolarizers. It leads us to design actual setups for producing natural (unpolarized) light from an arbitrary partially or completely polarized wave or for reducing the degree of polarization of a wave. We conclude by making the link with Azzam's polarization orthogonalization problem.
Cited by 2Open access
C. Manuel Carlevaro, Ramiro M. Irastorza, Fernando Vericat · arXiv (Cornell University) · 2015
A heuristic diagram of the evolution of the standard genetic code is presented. It incorporates, in a way that resembles the energy levels of an atom, the physical notion of broken symmetry and it is consistent with original ideas by Crick on the origin and evolution of the code as well as with the chronological order of appearence of the amino acids along the evolution as inferred from work that mixtures known experimental results with theoretical speculations. Suggested by the diagram we propose a Hamilton quaternions based mathematical representation of the code as it stands now-a-days. The central object in the description is a codon function that assigns to each amino acid an integer quaternion in such a way that the observed code degeneration is preserved. We emphasize the advantages of a quaternionic representation of amino acids taking as an example the folding of proteins. With this aim we propose an algorithm to go from the quaternions sequence to the protein three dimensional structure which can be compared with the corresponding experimental one stored at the Protein Data Bank. In our criterion the mathematical representation of the genetic code in terms of quaternions merits to be taken into account because it describes not only most of the known properties of the genetic code but also opens new perspectives that are mainly derived from the close relationship between quaternions and rotations.
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Igor Frenkel, Matvei Libine · Advances in Mathematics · 2008
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C. Manuel Carlevaro, Ramiro M. Irastorza, Fernando Vericat · Biosystems · 2016
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Igor Frenkel, Matvei Libine · Advances in Theoretical and Mathematical Physics · 2021
We develop further quaternionic analysis introducing left and right doubly regular functions. We derive Cauchy-Fueter type formulas for these doubly regular functions that can be regarded as another counterpart of Cauchy's integral formula for the second order pole, in addition to the one studied in the first paper with the same title. We also realize the doubly regular functions as a subspace of the quaternionic-valued functions satisfying a Euclidean version of Maxwell's equations for the electromagnetic field.
Then we return to the study of the original quaternionic analogue of Cauchy's second order pole formula and its relation to the polarization of vacuum. We find the decomposition of the space of quaternionic-valued functions into irreducible components that include the spaces of doubly left and right regular functions. Using this decomposition, we show that a regularization of the vacuum polarization diagram is achieved by subtracting the component corresponding to the one-dimensional subrepresentation of the conformal group. After the regularization, the vacuum polarization diagram is identified with a certain second order differential operator which yields a quaternionic version of Maxwell equations.
Next, we introduce two types of quaternionic algebras consisting of spaces of scalar-valued and quaternionic-valued functions. We emphasize that these algebra structures are invariant under the action of the conformal Lie algebra. This uses techniques from our study of the vacuum polarization diagram. These algebras are not associative, but we can define an infinite family of n-multiplications, and we conjecture that they have structures of weak cyclic A-infinity algebras. We also conjecture the relation between the multiplication operations of the scalar and non-scalar quaternionic algebras with the n-photon Feynman diagrams in the scalar and ordinary conformal QED.
Cited by 18
Luis Arenas-Carmona · Archiv der Mathematik · 2010
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R. Deepika, K. Muthunagai · 2024
The introduction of complex numbers marked a significant leap in mathematics, introducing the imaginary unit i to represent the square root of -1. This innovative concept proved invaluable in solving equations involving square roots of negative numbers. The extension to quaternions involved introducing additional imaginary units denoted as j and k . A quaternion in an interesting concept that extends complex numbers to 4-D.The manuscript is about using quaternions to calculate wave packets in one-dimension, and anti-hermitian operators to obtain the results in quaternionic form including expectation values of position, momentum and energy. The results are compared to the exisiting results on complex wave packets in one-dimension.
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Gerhard Hiß, Rafał Lutowski, Andrzej Szczepański · Communications in Algebra · 2020
We are interested in the question of the existence of flat manifolds for which all R-irreducible components of the holonomy representation are either absolutely irreducible, of complex or of quaternionic type. In the first two cases such examples are well known. But the existence of the third type of flat manifolds was unknown to the authors. In this article, we construct such an example. Moreover, we present a list of finite groups for which a construction of manifolds of quaternionic type is impossible.
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Niki-Lina Matiadou, Perikles Papadopoulos, D Finkelstein et al. · Journal of Pure and Applied Mathematics Advances and Applications · 2015
The fundamental theorem of algebra over the quaternion skew field has gained little attention. It is of less importance than that over the complex number field, though some problems may reduce to it.
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Graziano Gentili, Caterina Stoppato · Proceedings of the American Mathematical Society · 2020
After their introduction in 2006, quaternionic slice regular functions have mostly been studied over domains that are symmetric with respect to the real axis. This choice was motivated by some foundational results published in 2009, such as the Representation Formula for axially symmetric domains. The present work studies slice regular functions over domains that are not axially symmetric, partly correcting the hypotheses of some previously published results. In particular, this work includes a Local Representation Formula valid without the symmetry hypothesis. Moreover, it determines a class of domains, called simple, having the following property: every slice regular function on a simple domain can be uniquely extended to the symmetric completion of its domain.
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C. Manuel Carlevaro, Ramiro M. Irastorza, Fernando Vericat · Biosystems · 2016
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Neal Wadhwa, Michael Rubinstein, Frédo Durand et al. · DSpace@MIT (Massachusetts Institute of Technology) · 2014
Recently, we presented a new image pyramid, called the Riesz pyramid, that uses the Riesz transform to manipulate the phase in non-oriented sub-bands of an image sequence to produce real-time motion-magnified videos. In this report we give a quaternionic formulation of the Riesz pyramid, and show how several seemingly heuristic choices in how to use the Riesz transform for phase-based video magnifica-tion fall out of this formulation in a natural and principled way. We intend this report to accompany the original paper on the Riesz pyramid for video magnification. 1.
Cited by 23Open access
Neal Wadhwa, Michael Rubinstein, Fredo Dur et al. · 2014
Recently, we presented a new image pyramid, called the Riesz pyramid, that uses the Riesz transform to manipulate the phase in non-oriented sub-bands of an image sequence to produce real-time motion-magnified videos. In this report we give a quaternionic formulation of the Riesz pyramid, and show how several seemingly heuristic choices in how to use the Riesz transform for phase-based video magnifica-tion fall out of this formulation in a natural and principled way. We intend this report to accompany the original paper on the Riesz pyramid for video magnification. 1.
Cited by 0
Nevena Jurčević Peček · Repository of the Faculty of Science, University of Zagreb · 2016
U ovoj disertaciji proučavamo problem reducibilnosti reprezentacija p-adskih hermitskih kvaternionskih grupa koje su parabolički inducirane iz kuspidalnih i (esencijalno) kvadratno integrabilnih reprezentacija Levijevih faktora standardnih paraboličkih podgrupa. Glavni rezultati su generalizacija Tadićevih kriterija reducibilnosti za rascjepive simplektičke i neparne specijalne ortogonalne grupe na slučaj proizvoljnih hermitskih kvaternionskih grupa. Dokazi se baziraju na tehnikama Jacquetovih modula te koriste strukturnu formulu i teoriju R-grupa. Pritom je pokazano da za hermitske kvaternionske grupe vrijedi strukturna formula te su određene R-grupe.
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PETIT, Frédéric, FERNANDEZ-MALOIGNE, Christine, CAPELLE-LAIZE, Anne-Sophie et al. · OpenGrey (Institut de l'Information Scientifique et Technique) · 2010
Les océans représentent la majeure partie de la surface de la planète. Couvrant plus de 70% de la surface de la Terre, l'exploration de l'environnement marin est un thème de recherche essentiel. De nombreuses technologies ont été développées afin de surveiller et de suivre l' évolution de l'environnement marin. La vision par ordinateur et le traitement d' images ont été étudiés dans ce cadre ces dernières années afin de développer de nouveaux outils pour l'océanographie. Les images acquises sous l'eau souffrent de nombreuses dégradations dues à l'environnement aquatique. Nous nous sommes intéréssé au développement de traitements prenant en compte la nature physique des phénomènes qui interfèrent lors de l'acquisition. L'exploitation d'un modèle bio-optique a permis d'établir une corrélation entre des propriétés bio-optiques et des attributs colorimétriques des images. L'inversion de la propagation de la lumière a alors été réalisée en établissant un modèle d'estimation des paramètres physiques impliqués à partir des informations colorimétriques de l'image et de mettre en oeuvre une correction complètement automatique d' images sous-marines. Dans un deuxième temps, dans un but de détection d'objets d'intérêts, nous nous sommes interessé à un critère quaternionique permettant une extraction ciblée de zones/objets possèdant une ou plusieurs caractéristiques colorométriques communes par le biais d'une définition hypercomplexe de sous-espaces couleurs basée sur la teinte. Cette méthode a été testée sur des vidéos sous-marines en réalisant une extraction de poissons de couleur unie.
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George Apostolopoulos, Athanasios Koutras, Ioanna Christoyianni et al. · World Congress on Medical Physics and Biomedical Engineering, September 7 - 12, 2009, Munich, Germany · 2016
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Eugene B. Postnikov, Olga Stepanova · Advances in Applied Clifford Algebras · 2005
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Cherng-tiao Perng · Contemporary mathematics - American Mathematical Society · 2014
Based on factorization of Lipschitz quaternions, the author gave a proof of Jacobi’s formula for the number of representations of a positive integer as a sum of four integer squares. In this note, we generalize the method to the Lipschitz type quaternions associated with two other quadratic forms. We obtain factorization theory of these Lipschitz type quaternions, and formulas for the number of representations of a positive integer in terms of these quadratic forms.
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Junliang Wu, Pingping Zhang · Advances in Pure Mathematics · 2011
In this paper, a series of bicomplex representation methods of quaternion division algebra is introduced. We present a new multiplication concept of quaternion matrices, a new determinant concept, a new inverse concept of quaternion matrix and a new similar matrix concept. Under the new concept system, many quaternion algebra problems can be transformed into complex algebra problems to express and study. These concepts can perfect the theory of [J.L. Wu, A new representation theory and some methods on quaternion division algebra, JP Journal of Algebra, 2009, 14(2): 121-140] and unify the complex algebra and quaternion division algebra.
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Jesse Ira Deutsch · Journal of Number Theory · 2005
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