Relevant bibliographies by topics / NON COMMUTATIVITY

Contents

  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'NON COMMUTATIVITY'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 April 2023

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Journal articles on the topic "NON COMMUTATIVITY"

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Balibrea, Francisco, Jose Salvador Cánovas Peña, and Víctor Jiménez López. "Commutativity and non-commutativity of topological sequence entropy." Annales de l’institut Fourier 49, no. 5 (1999): 1693–709. http://dx.doi.org/10.5802/aif.1735.

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Paban, Sonia, Savdeep Sethi, and Mark Stern. "Non-commutativity and Supersymmetry." Journal of High Energy Physics 2002, no. 03 (March 6, 2002): 012. http://dx.doi.org/10.1088/1126-6708/2002/03/012.

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Banerjee, Rabin, Biswajit Chakraborty, and Kuldeep Kumar. "Membrane and non-commutativity." Nuclear Physics B 668, no. 1-2 (September 2003): 179–206. http://dx.doi.org/10.1016/j.nuclphysb.2003.07.009.

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Fortin, S., M. Gadella, F. Holik, and M. Losada. "Evolution of quantum observables: from non-commutativity to commutativity." Soft Computing 24, no. 14 (December 2, 2019): 10265–76. http://dx.doi.org/10.1007/s00500-019-04546-7.

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Haouam, Ilyas. "The Non-Relativistic Limit of the DKP Equation in Non-Commutative Phase-Space." Symmetry 11, no. 2 (February 14, 2019): 223. http://dx.doi.org/10.3390/sym11020223.

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The non-relativistic limit of the relativistic DKP equation for both of zero and unity spin particles is studied through the canonical transformation known as the Foldy–Wouthuysen transformation, similar to that of the case of the Dirac equation for spin-1/2 particles. By considering only the non-commutativity in phases with a non-interacting fields case leads to the non-commutative Schrödinger equation; thereafter, considering the non-commutativity in phase and space with an external electromagnetic field thus leads to extract a phase-space non-commutative Schrödinger–Pauli equation; there, we examined the effect of the non-commutativity in phase-space on the non-relativistic limit of the DKP equation. However, with both Bopp–Shift linear transformation through the Heisenberg-like commutation relations, and the Moyal–Weyl product, we introduced the non-commutativity in phase and space.
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Liang, Jin, and Chengwei Zhang. "Study on Non-Commutativity Measure of Quantum Discord." Mathematics 7, no. 6 (June 14, 2019): 543. http://dx.doi.org/10.3390/math7060543.

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In this paper, we are concerned with the non-commutativity measure of quantum discord. We first present an explicit expression of the non-commutativity measure of quantum discord in the two-qubit case. Then we compare the geometric quantum discords for two dynamic models with their non-commutativity measure of quantum discords. Furthermore, we show that the results conducted by the non-commutativity measure of quantum discord are different from those conducted by both or one of the Hilbert-Schmidt distance discord and trace distance discord. These intrinsic differences indicate that the non-commutativity measure of quantum discord is incompatible with at least one of the well-known geometric quantum discords in the quantitative and qualitative representation of quantum correlations.
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Chung, Won Sang. "Hall effect on non-commutative plane with space-space non-commutativity and momentum-momentum non-commutativity." Advanced Studies in Theoretical Physics 11 (2017): 357–64. http://dx.doi.org/10.12988/astp.2017.614.

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Tweed, Douglas B., Thomas P. Haslwanter, Vera Happe, and Michael Fetter. "Non-commutativity in the brain." Nature 399, no. 6733 (May 1999): 261–63. http://dx.doi.org/10.1038/20441.

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Sidharth, B. G. "Non-commutativity, fluctuations and unification." Chaos, Solitons & Fractals 13, no. 9 (July 2002): 1763–66. http://dx.doi.org/10.1016/s0960-0779(01)00188-6.

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Guttenberg, S., M. Herbst, M. Kreuzer, and R. Rashkov. "Non-topological non-commutativity in string theory." Fortschritte der Physik 56, no. 4-5 (April 18, 2008): 440–51. http://dx.doi.org/10.1002/prop.200710517.

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Dissertations / Theses on the topic "NON COMMUTATIVITY"

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López, Armand Idárraga. "Position dependent non-commutativity in two dimensions." reponame:Repositório Institucional da UFABC, 2015.

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Orientador: Prof. Dr. Vladislav Kupriyanov
Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2015.
No presente trabalho estudamos as consequências físicas da não-comutatividade dependente da posição e rotacionalmente invariante em duas dimensões [x, y] = iq f (x2 + y2), usando a teoria de perturbações em mecânica quântica e considerando os modelos exatamente solúveis como o oscilador harmônico isotrópico e o problema de Landau. Nós demonstramos a consistência da abordagem proposta, em particular, derivamos a versão não-comutativa da equação de continuidade e mostramos que a probabilidade é conservada na nossa abordagem. Pesquisamos três formas gerais diferentes para a f (r): constante, monomial de r2 e exponencial Gaussiana. Obtendo resultados diversos de acordo com as características específicas de cada f (e. g. a potência do monomio, largura da Gaussiana). Para a maior parte das escolhas da f , temos encontrado quebra da degenerescência.
In the present work we study the physical consequences of the position dependent rotationally invariant noncommutativity in two dimensions [x, y] = iq f (x2 + y2), using the perturbation theory in quantum mechanics and considering the exactly solvable models in standard quantum mechanics: isotropic harmonic oscillator and Landau problem. We demonstrate the consistency of the proposed approach, in particular, we derive the noncommutative continuity equation and show that the probability is conserved in our approach. We investigate three different general forms of f (r): constant, monomial of r2 and Gaussian exponential. Obtaining diverse results according to specific characteristics of each f (e. g. monomial power and Gaussian width). Degeneracy breaking is found in most of the cases.
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Bouatta, Nazim. "String field theory, non-commutativity and higher spins." Doctoral thesis, Universite Libre de Bruxelles, 2008. http://hdl.handle.net/2013/ULB-DIPOT:oai:dipot.ulb.ac.be:2013/210481.

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In Chapter 1, we give an introduction to the topic of open string field theory. The concepts presented include gauge invariance, tachyon condensation, as well as the star product.

In Chapter 2, we give a brief review of vacuum string field theory (VSFT), an approach to open string field theory around the stable vacuum of the tachyon. We discuss the sliver state explaining its role as projector in the space of half-string basis. We review the construction of D-brane solutions in vacuum string field theory. We show that in the sliver basis the star product correspond to a matrix product.

Using the material introduced in the previous chapters, in Chapter 3 we establish a translation dictionary between open and closed strings, starting from open string field theory. Under this correspondence, we show that (off--shell) level--matched closed string states are represented by star algebra projectors in open string field theory. As an outcome of our identification, we show that boundary states, which in closed string theory represent D-branes, correspond to the identity string field in the open string side.

We then turn to noncommutative field theories. In Chapter 4, we introduce the framework in which we will work. The tools introduced are solitons, projectors, and partial isometries.

The ideas of Chapter 4 are applied to specific examples in Chapter 5, where we present new solutions of noncommutative gauge theories in which coincident vortices expand into circular shells. As the theories are noncommutative, the naive definition of the locations of the vortices and shells is gauge-dependent, and so we define and calculate the profiles of these solutions using the gauge-invariant noncommutative Wilson lines introduced by Gross and Nekrasov. We find that charge 2 vortex solutions are characterized by two positions and a single nonnegative real number, which we demonstrate is the radius of the shell. We find that the radius is identically zero in all 2-dimensional solutions. If one considers solutions that depend on an additional commutative direction, then there are time-dependent solutions in which the radius oscillates, resembling a braneworld description of a cyclic universe. There are also smooth BIon-like space-dependent solutions in which the shell expands to infinity, describing a vortex ending on a domain wall.

In Chapter 6, we review the Fronsdal models for free high-spin fields that exhibit peculiar properties. We discuss the triplet structure of totally symmetric tensors of the free String Field Theory and their generalization to AdS background.

In Chapter 7, in the context of massless higher spin gauge fields in constant curvature spaces discussed in chapter 6, we compute the surface charges which generalize the electric charge for spin one, the color charges in Yang-Mills theories and the energy-momentum and the angular momentum for asymptotically flat gravitational fields. We show that there is a one-to-one map from surface charges onto divergence free Killing tensors. These Killing tensors are computed by relating them to a cohomology group of the first quantized BRST model underlying the Fronsdal action.


Doctorat en Sciences
info:eu-repo/semantics/nonPublished

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Tumwesigye, Alex Behakanira. "On one-dimensional dynamical systems and commuting elements in non-commutative algebras." Licentiate thesis, Mälardalens högskola, Utbildningsvetenskap och Matematik, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-31437.

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This thesis work is about commutativity which is a very important topic in mathematics, physics, engineering and many other fields. Two processes are said to be commutative if the order of "operation" of these processes does not matter. A typical example of two processes in real life that are not commutative is the process of opening the door and the process of going through the door. In mathematics, it is well known that matrix multiplication is not always commutative. Commutating operators play an essential role in mathematics, physics engineering and many other fields. A typical example of the importance of commutativity comes from signal processing. Signals pass through filters (often called operators on a Hilbert space by mathematicians) and commutativity of two operators corresponds to having the same result even when filters are interchanged. Many important relations in mathematics, physics and engineering are represented by operators satisfying a number of commutation relations. In chapter two of this thesis we treat commutativity of monomials of operatos satisfying certain commutation relations in relation to one-dimensional dynamical systems. We derive explicit conditions for commutativity of the said monomials in relation to the existence of periodic points of certain one-dimensional dynamical systems. In chapter three, we treat the crossed product algebra for the algebra of piecewise constant functions on given set, describe the commutant of this algebra of functions which happens to be the maximal commutative subalgebra of the crossed product containing this algebra. In chapter four, we give a characterization of the commutant for the algebra of piecewise constant functions on the real line, by comparing commutants for a non decreasing sequence of algebras.
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Gawell, Elin. "Centra of Quiver Algebras." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-106734.

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A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by monomials and (anti-)commutativity relations. We give a combinatorial description of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiveralgebra and state necessary and sufficient conditions for the center to be finitely genteratedas a K-algebra.Examples are provided of partly (anti-)commutative quiver algebras that are Koszul algebras. Necessary and sufficient conditions for finite generation of the Hochschild cohomology ring modulo nilpotent elements for a partly (anti-)commutative Koszul quiver algebra are given.
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Ipia, Carlos Andrés Palechor. "Deformações e invariâncias em modelos supersimétricos em três e quatro dimensões espaçotemporais." reponame:Repositório Institucional da UFABC, 2017.

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Orientador: Prof. Dr. Alysson Fábio Ferrari
Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Física, 2017.
As deformações do espaço-tempo têm sido bastante estudadas desde diferentes abordagens tais como a não comutatividade canônica e deformações via álgebras de Hopf, com a motivação de que estas deformações podem aparecer a escalas de altas energias, como por exemplo a escala de Planck. De igual forma, pode-se buscar estender deformações para a estrutura do superespaço e a supersimetria, e assim estudar o comportamento clássico e quântico, como a invariância supersimétrica e renormalizabilidade, em modelos definidos sobre estas estruturas. Dois tipos de deformações possíveis da supersimetria foram estudadas neste trabalho. O primeiro deles envolve a introdução de um produto não comutativo em (3+1) dimensões, que embora seja um produto não associativo e que quebra a álgebra da supersimetria, permite construir um modelo de Wess-Zumino com correções de derivadas de ordem superior do tipo Lee-wick, e que resultam ser invariante sob as transformações da SUSY usual. O segundo tipo de deformação estudado utiliza o conceito de álgebras de Hopf, através de um twist de Drinfel¿d. No caso do modelo de Wess-Zumino em (2 + 1) dimensões, veremos que apesar de que as estruturas sejam construídas de forma consistente e seja possível preservar a álgebra da SUSY usando geradores deformados, o modelo resulta não ser invariante sob esta última e não renormalizável. Também foi usado o formalismo de twist para um modelo de Chern-simons com SUSY N = 2 em (2 + 1) dimensões, que permite construir um modelo invariante de calibre, no entanto a invariância da SUSY não seja evidente. Neste modelo, embora em principio a álgebra da SUSY pode ser preservada pelo uso de geradores deformados, estes tornam-se bastante complicados, dificultando a prova da invariância supersimétrica. Pode-se concluir que existem diferentes formas de deformar as estruturas algébricas da supersimetria e que devido aos vínculos de cada modelo em específico torna-se difícil a construção de modelos que preservem algumas das propriedades importantes de modelos supersimétricos que se estudam, tais como a invariância e renormalização.
The space-time deformations have been well studied using different approaches, like as canonical commutativity and deformations via Hopf algebras, with the motivation of such deformations can appear in high scale energies, for example, planck scales. The same way, they can extend deformations to superspace and supersymmetry structures, and thus, study the quantum and classical behavior, like as the supersymmetry invariance and renormalizability, in models defined on these structures. Two classes of possible transformation of supersymmetry were studied in this work. The first one involves the introduction of one non commutative product in (3 + 1) dimensions, although it is not associative and breaks the supersymmetry algebra. It allows the construction of a Wess- Zumino model with higher order derivatives corrections like as Lee-Wick models, and it is invariant under usual SUSY transformations. The second deformation class studied utilizes the Hopf algebra concept, through Drinfel¿d twist. In the Wess-Zumino case in (2 + 1) dimensions, we can observe, although, the construction of the algebraic structure is consistent and it is possible preserve the SUSY algebra using deformed generators, the model is not invariant under this last and non renormalizable, also the twist formalism was used to Chern-Simons model N = 2 in (2 + 1) dimensions, it allows to construct an invariant gauge model, however the SUSY invariance is not evident. In this model, although the SUSY algebra can be preserved using the deformed generators, they become complicated, making it difficult to prove the supersymmetric invariance. It is possible to conclude that there are different ways to deform the algebraic structures of supersymmetry and because of the constraints of each specific model, it is difficult the construction of models which preserve some important properties of supersymmetry models studies, like as invariance and renormalizability.
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Baudot, Rémi. "Programmation logique : non-commutativité et polarisation." Paris 13, 2000. http://www.theses.fr/2000PA132024.

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Dans cette thèse, je m'intéresse aux langages de programmation logique, fondes sur l'isomorphisme de Curry-Howard (programmation - recherche de preuve dans une logique). Afin d'accroitre le pouvoir d'expression de tels langages, je propose de définir un langage de programmation logique base sur la logique non-commutative (nl) de Ruet. Dans ce cadre, j'ai réalisé le prouveur de théorèmes sigma 1 3 qui exploite les propriétés de focalisation et de réversibilité des formules logiques, et possède une implémentation séquentielle des phénomènes d'extraction de formules et de séparation de ressources. Son évolution a donné lieu au langage de programmation noclog décliné en deux versions, basées sur l'alternance des comportements synchrones et asynchrones des connecteurs logiques de nl. Ce travail a permis d'exhiber le pouvoir de représentation spatiale de la logique non-commutative et les limites du focusing, entre autres dans la représentation des décompositions partielles. Me replaçant dans un formalisme commutatif, mes travaux se sont orientés vers l'étude de la logique linéaire polarisée afin d'étudier la nature fondamentale des langages de programmation logique vis-a-vis du paradigme de la focalisation. De ces travaux sont issus une nouvelle division en quatre classes des formules logiques et les deux calculs des séquents llpn et llpp. Ces derniers, éliminant par le fait la frontière entre logique et programmation logique, sont deux langages de programmation polarises. Ces dernières avancées permettent de représenter les phénomènes de décompositions partielles et de représentation du temps logique. On peut ainsi modéliser la sequentialité ou la non-commutativité temporelle grâce à l'utilisation de doubles décalages, contrairement
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Ballanti, Federico. "Lo spettro di alcuni oscillatori non commutativi." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2015. http://amslaurea.unibo.it/8978/.

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Questa tesi si occupa della teoria spettrale di certi sistemi di equazioni ordinarie chiamati oscillatori non commutativi. Dopo avere introdotto i fondamenti necessari per la teoria vengono dimostrati alcuni teoremi qualitativi sullo spettro di tali sistemi.
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El, Khoury Antoine. "Méthodes de vérification de la commutativité des diagrammes dans les catégories symétriques monoïdales fermées libres et non-libres." Toulouse 3, 2010. http://thesesups.ups-tlse.fr/1582/.

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Dans la thèse on s'occupe des méthodes et des algorithmes de vérifications de la commutativité des diagrammes dans les catégories symétriques monoidales fermées (catégories des modules, des semi-modules etc. . . ). On démontre l'infinité de structures de catégories intermédiaires entre la structure des catégories libres et la structure triviale et on étudie l'utilisation des méthodes de la théorie de la démonstration pour la vérification de la commutativité dans ces cas intermédiaires
The subject of this thesis belongs to the field of categorical proof theory, which lies somewhere between category theory and proof theory. It uses proof theoretical methods in solving problems related to some general matters in category theory, which are syntactical nature (for example, question of commuting diagrams in canonical structure of some freely generated category belonging to a particular class of categories). On the other hand, categrorical proof theory uses categories as contexts where comme questions of particular interest fo general proof theory may be correctly formulated and answered (for example, the question whether two dérivations are equal, which is the main question of general proof theory)
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Malaty, George. "Isomorphic Visualization and Understanding of the Commutativity of Multiplication: from multiplication of whole numbers to multiplication of fractions." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-82868.

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Books on the topic "NON COMMUTATIVITY"

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Non-Commutativity, Infinite Dimensionality and Probability at the Crossroads: Proceedings of the Rims Workshop on Infinite-Dimensional Analysis and Quantum ... Quantum Probability & White Noise Analysis). World Scientific Pub Co Inc, 2003.

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Book chapters on the topic "NON COMMUTATIVITY"

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Amblard, Maxime. "Encoding Phases Using Commutativity and Non-commutativity in a Logical Framework." In Logical Aspects of Computational Linguistics, 1–16. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-22221-4_1.

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Blaisdell, Eben, Max Kanovich, Stepan L. Kuznetsov, Elaine Pimentel, and Andre Scedrov. "Non-associative, Non-commutative Multi-modal Linear Logic." In Automated Reasoning, 449–67. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-10769-6_27.

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AbstractAdding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so to allow for commutativity as well. In this work, we close the cycle by considering associativity. We show that the resulting system ($$\mathsf {acLL}_\varSigma $$ acLL Σ ) admits the (multi)cut rule, and we prove two undecidability results for fragments/variations of $$\mathsf {acLL}_\varSigma $$ acLL Σ .
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Krajči, S., R. Lencses, J. Medina, M. Ojeda-Aciego, A. Valverde, and P. Vojtáš. "Non-commutativity and Expressive Deductive Logic Databases." In Logics in Artificial Intelligence, 149–60. Berlin, Heidelberg: Springer Berlin Heidelberg, 2002. http://dx.doi.org/10.1007/3-540-45757-7_13.

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Manolakos, G., and G. Zoupanos. "Non-commutativity in Unified Theories and Gravity." In Springer Proceedings in Mathematics & Statistics, 177–205. Singapore: Springer Singapore, 2018. http://dx.doi.org/10.1007/978-981-13-2715-5_10.

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Medina, Jesús. "Overcoming Non-commutativity in Multi-adjoint Concept Lattices." In Lecture Notes in Computer Science, 278–85. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-02478-8_35.

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Ghoderao, Pulkit S., Rajiv V. Gavai, and P. Ramadevi. "Scale of Non-commutativity and the Hydrogen Spectrum." In Springer Proceedings in Physics, 429–35. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-33-4408-2_60.

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Guglielmi, Alessio, and Lutz Straßburger. "Non-commutativity and MELL in the Calculus of Structures." In Computer Science Logic, 54–68. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-44802-0_5.

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Chechik, Marsha, Ioanna Stavropoulou, Cynthia Disenfeld, and Julia Rubin. "FPH: Efficient Non-commutativity Analysis of Feature-Based Systems." In Fundamental Approaches to Software Engineering, 319–36. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-89363-1_18.

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Sergyeyev, Artur. "Time Dependence and (Non)commutativity of Symmetries of Evolution Equations." In Noncommutative Structures in Mathematics and Physics, 379–90. Dordrecht: Springer Netherlands, 2001. http://dx.doi.org/10.1007/978-94-010-0836-5_31.

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Faragó, I., and Á. Havasi. "The Mathemathical Background of Operator Splitting and the Effect of Non-Commutativity." In Large-Scale Scientific Computing, 264–71. Berlin, Heidelberg: Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-45346-6_27.

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Conference papers on the topic "NON COMMUTATIVITY"

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Sabido, M., O. Obregón, E. Mena, Alfredo Herrera-Aguilar, Francisco S. Guzmán Murillo, Ulises Nucamendi Gómez, and Israel Quiros. "Non Commutativity and Λ." In GRAVITATION AND COSMOLOGY: Proceedings of the Third International Meeting on Gravitation and Cosmology. AIP, 2008. http://dx.doi.org/10.1063/1.3058570.

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Faggian, Claudia. "Proof construction and non-commutativity." In the 2nd ACM SIGPLAN international conference. New York, New York, USA: ACM Press, 2000. http://dx.doi.org/10.1145/351268.351278.

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Recknagel, Andreas. "Branes, boundary CFT and non-commutativity." In Non-perturbative Quantum Effects 2000. Trieste, Italy: Sissa Medialab, 2000. http://dx.doi.org/10.22323/1.006.0031.

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MOCIOIU, I., M. POSPELOV, and R. ROIBAN. "LIMITS ON THE NON-COMMUTATIVITY SCALE." In Proceedings of the Second Meeting. WORLD SCIENTIFIC, 2002. http://dx.doi.org/10.1142/9789812778123_0044.

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Seridi, M. A., and N. Belaloui. "Parabosonic string and space-time non-commutativity." In THE 8TH INTERNATIONAL CONFERENCE ON PROGRESS IN THEORETICAL PHYSICS (ICPTP 2011). AIP, 2012. http://dx.doi.org/10.1063/1.4715445.

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Zhao, Sumu, Damian Pascual, Gino Brunner, and Roger Wattenhofer. "Of Non-Linearity and Commutativity in BERT." In 2021 International Joint Conference on Neural Networks (IJCNN). IEEE, 2021. http://dx.doi.org/10.1109/ijcnn52387.2021.9533563.

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Pimentel, Luis O. "Non-commutativity in classical and quantum cosmology." In GRAVITATION AND COSMOLOGY: 2nd Mexican Meeting on Mathematical and Experimental Physics. AIP, 2005. http://dx.doi.org/10.1063/1.1900526.

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Disenfeld, Cynthia, Ioanna Stavropoulou, Julia Rubin, and Marsha Chechik. "FPH: Efficient Detection of Feature Interactions through Non-Commutativity." In 2017 IEEE/ACM 39th International Conference on Software Engineering Companion (ICSE-C). IEEE, 2017. http://dx.doi.org/10.1109/icse-c.2017.71.

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Horváthy, P. A. "Exotic Galilean Symmetry, Non-commutativity & the Hall Effect." In Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812772527_0018.

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LUKIERSKI, J., and M. WORONOWICZ. "Twisted Space-Time Symmetry, Non-Commutativity and Particle Dynamics." In Proceedings of the 23rd International Conference of Differential Geometric Methods in Theoretical Physics. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812772527_0028.

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