Mussardo, Giuseppe · arXiv (Cornell University) · 2010
We discuss the main features of quantum integrable models taking the classes of universality of the Ising model and the repulsive Lieb-Liniger model as paradigmatic examples. We address the breaking of integrability by means of two approaches, the Form Factor Perturbation Theory and semiclassical methods. Each of them has its own advantage. Using the first approach, one can relate the confinement phenomena of topological excitations to the semi-locality of the operator which breaks integrability. Using the second approach, one can control the bound states which arise in each phase of the theory and predict that their number cannot be more than two.
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Interdisciplinary applied mathematics · 2007
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2014
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2005
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Khanh Quoc Ngo · 2011
In this thesis, we explore the following question: what is the accessible set of a distribution H on a manifold M? In particular, when is the accessible set the entire manifold M and when does H give rise to a foliation of M? The first question is answered by a theorem of Chow, the second by a theorem of Frobenius. The intermediate case - when the accessible sets are of dimension between the dimensions of H and that of M- is answered be Sussmann's Orbit theorem. After introducing the necessary concepts, we prove Chow's and Frobenius Theorem ( the former one in a special case) and describe some of their applications
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Jiří Jarník, Jaroslav Kurzweil · Czechoslovak Mathematical Journal · 1994
In [1] the author proved that every A/i-integrable function (see [2]) is Pfeffer integrable (see [3]).The aim of this note is to disprove the converse inclusion by constructing a function that is Pfeffer-integrable but not Mi -integrable.We will do so by modifying our construction from [4] or [5].Let us recall the relevant notation and definitions.We will work in a Euclidean
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Boris Khesin, Fedor Soloviev · Journal of Geometry and Physics · 2014
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Marek Kuś · Journal of Modern Optics · 2002
Abstract The concepts of integrability, non-integrability and chaos in quantum mechanics are examined, and it is indicated that they all are sensibly definable only in connection with the corresponding properties of their classical analogues. The concrete examples concern the quantum and classical properties of dynamic systems on SU3 algebra and classically integrable but quantum nonintegrable system with two degrees of freedom.
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Birkhäuser Basel eBooks · 2007
In this chapter, we consider one of the two main mechanisms which seem to underlie the existence of centers in polynomial vector fields. We only hint at the historical side, which is covered in detail by Schlomiuk [57].
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Carlson, Warren, Koch, Robert de Mello, Lin, Hai · arXiv (Cornell University) · 2011
In this article we study operators with a dimension $Δ\sim O(N)$ and show that simple analytic expressions for the action of the dilatation operator can be found. The operators we consider are restricted Schur polynomials. There are two distinct classes of operators that we consider: operators labeled by Young diagrams with two long columns or two long rows. The main complication in working with restricted Schur polynomials is in building a projector from a given $S_{n+m}$ irreducible representation to an $S_n\times S_m$ irreducible representation (both specified by the labels of the restricted Schur polynomial). We give an explicit construction of these projectors by reducing it to the simple problem of addition of angular momentum in ordinary non-relativistic quantum mechanics. The diagonalizationof the dilatation operator reduces to solving three term recursion relations. The fact that the recursion relations have only three terms is a direct consequence of the weak mixing at one loop of the restricted Schur polynomials. The recursion relations can be solved exactly in terms of symmetric Kravchuk polynomials or in terms of Clebsch-Gordan coefficients. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable.
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K M Tatmzhmani, A Ramani, B Grammaticos et al. · 2019
Physicists and mathematicians have been familiar with differential systems. The fundamental equations used in the modelling of natural phenomena are cast in differential form based on the underlying assumption that spacetime is continuous. While numerical algorithms are often considered as inaccurate approximations of a continuous ‘reality’, there exist domains where discrete systems arise naturally. This chapter deals with a presentation of the various approaches proposed for the detection of integrability, and presents results which will illustrate the parallel existing between integrable discrete and continuous systems. It provides results on systems which either lie between the discrete and continuous ones or go beyond the discrete systems. The chapter discusses a plethora of examples which will serve as a guide for the reader to develop his/her own understanding of the subject. It focuses on two topics which have been among the main themes of the work: integrability detectors and discrete Painleve equations.
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Fels, Mark E. · Digital Commons - USU (Utah State University) · 2006
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Gleb Arutyunov · UNITEXT for physics · 2019
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Takeo Inami, Shie Minakami, Muneto Nitta · 2006
We examine integrability of self-dual Yang-Mills system in the Higgs phase, with taking simpler cases of vortices and domain walls. We show that the vortex equations and the domain-wall equations do not have Painlevé property. This fact suggests that these equations are not integrable.
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Alejandro Salado · Systems Engineering · 2023
Abstract Most systems engineers have likely heard of or pursued integrability when developing systems. It is also likely that systems engineers perceive that, in general, integrability is a desired quality to attain. But what is integrability, really? While there is likely a consensus within the systems engineering community about the general idea of integrability, this paper poses that reasoning about integrability requires a deeper, comprehensive interpretation of what integrability is and entails. Recognizing that the meaning of integrability is subjective, this paper provides a deep dive into understanding and interpreting integrability. The findings presented in this paper could be used as a guide for others to find clarity on what they mean by achieving integrability when developing systems.
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Robert G. Bartle · Graduate studies in mathematics · 2001
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Rafael I. Nepomechie · Journal of Physics A Mathematical and Theoretical · 2016
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PROCESI, Michela, Degasperis A. · IRIS Research product catalog (Sapienza University of Rome) · 1999
We describe a test for Asymptotic integrability and apply it to a class of dispersive PDEs. We identify a new equation which is integrable up to order two.
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Xu, Yujie · ArXiv.org · 2025
In this short note, we prove the equivalence of Grothendieck-Katz $p$-curvature Conjecture with Conjecture F in Ekedahl-Shepherd-Barron-Taylor. More precisely, we show that Conjecture F implies the $p$-curvature conjecture, and that the $p$-curvature Conjecture implies Conjecture F for the foliation attached to a vector bundle with integrable connection.
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Oxford University Press eBooks · 2023
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