A geometric formula for the zeros of the partition function of the inhomogeneous 2d Ising model was recently proposed in terms of the angles of 2d triangulations embedded in the flat 3d space. Here we proceed to an analytical check of this formula on the cubic graph, dual to a double pyramid, and provide a thorough numerical check by generating random 2d planar triangulations. Our method is to generate Delaunay triangulations of the 2-sphere then performing random local rescalings. For every 2d triangulations, we compute the corresponding Ising couplings from the triangle angles and the dihedral angles, and check directly that the Ising partition function vanishes for these couplings (and grows in modulus in their neighborhood). In particular, we lift an ambiguity of the original formula on the sign of the dihedral angles and establish a convention in terms of convexity/concavity. Finally, we extend our numerical analysis to 2d toroidal triangulations and show that the geometric formula does not work and will need to be generalized, as originally expected, in order to accommodate for non-trivial topologies.
{"title":"Geometric Formula for 2d Ising Zeros: Examples & Numerics","authors":"Iñaki Garay, Etera R. Livine","doi":"arxiv-2409.11109","DOIUrl":"https://doi.org/arxiv-2409.11109","url":null,"abstract":"A geometric formula for the zeros of the partition function of the\u0000inhomogeneous 2d Ising model was recently proposed in terms of the angles of 2d\u0000triangulations embedded in the flat 3d space. Here we proceed to an analytical\u0000check of this formula on the cubic graph, dual to a double pyramid, and provide\u0000a thorough numerical check by generating random 2d planar triangulations. Our\u0000method is to generate Delaunay triangulations of the 2-sphere then performing\u0000random local rescalings. For every 2d triangulations, we compute the\u0000corresponding Ising couplings from the triangle angles and the dihedral angles,\u0000and check directly that the Ising partition function vanishes for these\u0000couplings (and grows in modulus in their neighborhood). In particular, we lift\u0000an ambiguity of the original formula on the sign of the dihedral angles and\u0000establish a convention in terms of convexity/concavity. Finally, we extend our\u0000numerical analysis to 2d toroidal triangulations and show that the geometric\u0000formula does not work and will need to be generalized, as originally expected,\u0000in order to accommodate for non-trivial topologies.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider tight-binding models on Bravais lattices with anisotropic onsite potentials that vary along a given direction and are constant along the transverse one. Inspired by our previous work on flatbands in anti-$mathcal{PT}$ symmetric Hamiltonians [Phys. Rev. A 105, L021305 (2022)], we construct an anti-$mathcal{PT}$ symmetric Hamiltonians with an $E=0$ flatband by tuning the hoppings and the shapes of potentials. This construction is illustrated for the square lattice with bounded and unbounded potentials. Unlike flatbands in short-ranged translationally invariant Hamiltonians, we conjecture that the considered $E=0$ flatbands do not host compact localized states. Instead the flatband eigenstates exhibit a localization transition along the potential direction upon increasing the potential strength for bounded potentials. For unbounded potentials flatband eigenstates are always localized irrespective of the potential strength.
我们考虑的是布拉维网格上的紧约束模型，它的各向异性原位势沿给定方向变化，沿横向不变。受我们之前关于反$mathcal{PT}$对称哈密顿的平带的工作[Phys. Rev. A 105, L021305 (2022)]的启发，我们通过调整电势的跳跃和形状，构造了一个具有$E=0$平带的反（anti-$mathcal{PT}$）对称哈密顿。与短程平移不变哈密顿中的平带不同，我们猜想所考虑的$E=0$平带并不承载紧凑的局部化态。相反，当有界电势的电势强度增大时，平带特征状态会沿着电势方向出现局部过渡。对于无约束电势，无论电势强度如何，平带特征状态始终是局域化的。
{"title":"Flatbands in tight-binding lattices with anisotropic potentials","authors":"Arindam Mallick, Alexei Andreanov","doi":"arxiv-2409.11336","DOIUrl":"https://doi.org/arxiv-2409.11336","url":null,"abstract":"We consider tight-binding models on Bravais lattices with anisotropic onsite\u0000potentials that vary along a given direction and are constant along the\u0000transverse one. Inspired by our previous work on flatbands in\u0000anti-$mathcal{PT}$ symmetric Hamiltonians [Phys. Rev. A 105, L021305 (2022)],\u0000we construct an anti-$mathcal{PT}$ symmetric Hamiltonians with an $E=0$\u0000flatband by tuning the hoppings and the shapes of potentials. This construction\u0000is illustrated for the square lattice with bounded and unbounded potentials.\u0000Unlike flatbands in short-ranged translationally invariant Hamiltonians, we\u0000conjecture that the considered $E=0$ flatbands do not host compact localized\u0000states. Instead the flatband eigenstates exhibit a localization transition\u0000along the potential direction upon increasing the potential strength for\u0000bounded potentials. For unbounded potentials flatband eigenstates are always\u0000localized irrespective of the potential strength.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260448","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We perform an asymptotic evaluation of the Hankel transform, $int_0^{infty}J_{nu}(lambda x) f(x)mathrm{d}x$, for arbitrarily large $lambda$ of an entire exponential type function, $f(x)$, of type $tau$ by shifting the contour of integration in the complex plane. Under the situation that $J_{nu}(lambda x)f(x)$ has an odd parity with respect to $x$ and the condition that the asymptotic parameter $lambda$ is greater than the type $tau$, we obtain an exactly terminating Poincar{'e} expansion without any trailing subdominant exponential terms. That is the Hankel transform evaluates exactly into a polynomial in inverse $lambda$ as $lambda$ approaches infinity.
{"title":"Terminating Poincare asymptotic expansion of the Hankel transform of entire exponential type functions","authors":"Nathalie Liezel R. Rojas, Eric A. Galapon","doi":"arxiv-2409.10948","DOIUrl":"https://doi.org/arxiv-2409.10948","url":null,"abstract":"We perform an asymptotic evaluation of the Hankel transform,\u0000$int_0^{infty}J_{nu}(lambda x) f(x)mathrm{d}x$, for arbitrarily large\u0000$lambda$ of an entire exponential type function, $f(x)$, of type $tau$ by\u0000shifting the contour of integration in the complex plane. Under the situation\u0000that $J_{nu}(lambda x)f(x)$ has an odd parity with respect to $x$ and the\u0000condition that the asymptotic parameter $lambda$ is greater than the type\u0000$tau$, we obtain an exactly terminating Poincar{'e} expansion without any\u0000trailing subdominant exponential terms. That is the Hankel transform evaluates\u0000exactly into a polynomial in inverse $lambda$ as $lambda$ approaches\u0000infinity.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"29 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An important result in the theory of quantum control is the "universality" of $2$-local unitary gates, i.e. the fact that any global unitary evolution of a system of $L$ qudits can be implemented by composition of $2$-local unitary gates. Surprisingly, recent results have shown that universality can break down in the presence of symmetries: in general, not all globally symmetric unitaries can be constructed using $k$-local symmetric unitary gates. This also restricts the dynamics that can be implemented by symmetric local Hamiltonians. In this paper, we show that obstructions to universality in such settings can in general be understood in terms of superoperator symmetries associated with unitary evolution by restricted sets of gates. These superoperator symmetries lead to block decompositions of the operator Hilbert space, which dictate the connectivity of operator space, and hence the structure of the dynamical Lie algebra. We demonstrate this explicitly in several examples by systematically deriving the superoperator symmetries from the gate structure using the framework of commutant algebras, which has been used to systematically derive symmetries in other quantum many-body systems. We clearly delineate two different types of non-universality, which stem from different structures of the superoperator symmetries, and discuss its signatures in physical observables. In all, our work establishes a comprehensive framework to explore the universality of unitary circuits and derive physical consequences of its absence.
量子控制理论中的一个重要结果是 2 美元局部单元门的 "普遍性"，即由 L 美元量子单元组成的系统的任何全局单元演化都可以通过组成 2 美元局部单元门来实现。令人惊讶的是，最近的研究结果表明，在存在对称性的情况下，普遍性可能会被打破：一般来说，并非所有的全局对称单元都能用 $k$ 局部对称单元门来构造。这也限制了对称局部汉密尔顿所能实现的动力学。在本文中，我们表明在这种情况下，普遍性的障碍一般可以从与受限门集的单元演化相关的超算子对称性来理解。这些超算子对称性导致了算子希尔伯特空间的块分解，决定了算子空间的连通性，进而决定了动态李代数的结构。我们利用换元代数框架从门结构中系统地推导出超算子对称性，在几个例子中明确地证明了这一点，换元代数框架已被用于系统地推导其他量子多体系统中的对称性。我们清楚地划分了源于超算子对称性不同结构的两种不同类型的非普遍性，并讨论了其在物理观测中的特征。总之，我们的工作建立了一个全面的框架来探索单元电路的普遍性，并推导出其不存在的物理后果。
{"title":"Non-Universality from Conserved Superoperators in Unitary Circuits","authors":"Marco Lastres, Frank Pollmann, Sanjay Moudgalya","doi":"arxiv-2409.11407","DOIUrl":"https://doi.org/arxiv-2409.11407","url":null,"abstract":"An important result in the theory of quantum control is the \"universality\" of\u0000$2$-local unitary gates, i.e. the fact that any global unitary evolution of a\u0000system of $L$ qudits can be implemented by composition of $2$-local unitary\u0000gates. Surprisingly, recent results have shown that universality can break down\u0000in the presence of symmetries: in general, not all globally symmetric unitaries\u0000can be constructed using $k$-local symmetric unitary gates. This also restricts\u0000the dynamics that can be implemented by symmetric local Hamiltonians. In this\u0000paper, we show that obstructions to universality in such settings can in\u0000general be understood in terms of superoperator symmetries associated with\u0000unitary evolution by restricted sets of gates. These superoperator symmetries\u0000lead to block decompositions of the operator Hilbert space, which dictate the\u0000connectivity of operator space, and hence the structure of the dynamical Lie\u0000algebra. We demonstrate this explicitly in several examples by systematically\u0000deriving the superoperator symmetries from the gate structure using the\u0000framework of commutant algebras, which has been used to systematically derive\u0000symmetries in other quantum many-body systems. We clearly delineate two\u0000different types of non-universality, which stem from different structures of\u0000the superoperator symmetries, and discuss its signatures in physical\u0000observables. In all, our work establishes a comprehensive framework to explore\u0000the universality of unitary circuits and derive physical consequences of its\u0000absence.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"54 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
J. M. Hoff da Silva, R. T. Cavalcanti, G. M. Caires da Rocha
This paper delves into the deformation of spinor structures within nontrivial topologies and their physical implications. The deformation is modeled by introducing real functions that modify the standard spinor dynamics, leading to distinct physical regions characterized by varying degrees of Lorentz symmetry violation. It allows us to investigate the effects in the dynamical equation and a geometrized nonlinear sigma model. The findings suggest significant implications for the spinor fields in regions with nontrivial topologies, providing a robust mathematical approach to studying exotic spinor behavior.
{"title":"Deformations in spinor bundles: Lorentz violation and further physical implications","authors":"J. M. Hoff da Silva, R. T. Cavalcanti, G. M. Caires da Rocha","doi":"arxiv-2409.11168","DOIUrl":"https://doi.org/arxiv-2409.11168","url":null,"abstract":"This paper delves into the deformation of spinor structures within nontrivial\u0000topologies and their physical implications. The deformation is modeled by\u0000introducing real functions that modify the standard spinor dynamics, leading to\u0000distinct physical regions characterized by varying degrees of Lorentz symmetry\u0000violation. It allows us to investigate the effects in the dynamical equation\u0000and a geometrized nonlinear sigma model. The findings suggest significant\u0000implications for the spinor fields in regions with nontrivial topologies,\u0000providing a robust mathematical approach to studying exotic spinor behavior.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We provide a geometric perspective on the kinetic interaction of matter and radiation, based on a metriplectic approach. We discuss the interaction of kinetic theories via dissipative brackets, with our fundamental example being the coupling of matter, described by the Boltzmann equation, and radiation, described by the radiation transport equation. We explore the transition from kinetic systems to their corresponding moment systems, provide a Hamiltonian description of such moment systems, and give a geometric interpretation of the moment closure problem for kinetic theories. As applications, we discuss in detail diffusion radiation hydrodynamics as an example of a geometric moment closure of kinetic matter-radiation interaction and additionally, we apply the variable moment closure framework of Burby (2023) to derive novel Hamiltonian moment closures for pure radiation transport and discuss an interesting connection to the Hamiltonian fluid closures derived by Burby (2023).
{"title":"A Geometric Perspective on Kinetic Matter-Radiation Interaction and Moment Systems","authors":"Brian K. Tran, Joshua W. Burby, Ben S. Southworth","doi":"arxiv-2409.11495","DOIUrl":"https://doi.org/arxiv-2409.11495","url":null,"abstract":"We provide a geometric perspective on the kinetic interaction of matter and\u0000radiation, based on a metriplectic approach. We discuss the interaction of\u0000kinetic theories via dissipative brackets, with our fundamental example being\u0000the coupling of matter, described by the Boltzmann equation, and radiation,\u0000described by the radiation transport equation. We explore the transition from\u0000kinetic systems to their corresponding moment systems, provide a Hamiltonian\u0000description of such moment systems, and give a geometric interpretation of the\u0000moment closure problem for kinetic theories. As applications, we discuss in\u0000detail diffusion radiation hydrodynamics as an example of a geometric moment\u0000closure of kinetic matter-radiation interaction and additionally, we apply the\u0000variable moment closure framework of Burby (2023) to derive novel Hamiltonian\u0000moment closures for pure radiation transport and discuss an interesting\u0000connection to the Hamiltonian fluid closures derived by Burby (2023).","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"207 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260381","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the spectrum of meson masses in large $N_c$ QCD$_2$ governed by celebrated 't Hooft's integral equation. We generalize analytical methods proposed by Fateev, Lukyanov and Zamolodchikov to the case of arbitrary, but equal quark masses $m_1=m_2.$ Our results include analytical expressions for spectral sums and systematic large-$n$ expansion.
{"title":"Meson mass spectrum in QCD$_2$ 't Hooft's model","authors":"Alexey Litvinov, Pavel Meshcheriakov","doi":"arxiv-2409.11324","DOIUrl":"https://doi.org/arxiv-2409.11324","url":null,"abstract":"We study the spectrum of meson masses in large $N_c$ QCD$_2$ governed by\u0000celebrated 't Hooft's integral equation. We generalize analytical methods\u0000proposed by Fateev, Lukyanov and Zamolodchikov to the case of arbitrary, but\u0000equal quark masses $m_1=m_2.$ Our results include analytical expressions for\u0000spectral sums and systematic large-$n$ expansion.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Influence network of events is a view of the universe based on events that may be related to one another via influence. The network of events form a partially-ordered set which, when quantified consistently via a technique called chain projection, results in the emergence of spacetime and the Minkowski metric as well as the Lorentz transformation through changing an observer from one frame to another. Interestingly, using this approach, the motion of a free electron as well as the Dirac equation can be described. Indeed, the same approach can be employed to show how a discrete version of some of the features of Euclidean geometry, including directions, dimensions, subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network formalism, we build on some of our previous works to further develop aspects of Euclidean geometry. Specifically, we present the emergence of geometric shapes, a discrete version of the Parallel postulate, the dot product, and the outer (wedge product) in 2+1 dimensions. Finally, we show that the scalar quantification of two concatenated orthogonal intervals exhibits features that are similar to those of the well-known concept of geometric product in geometric Clifford algebras.
{"title":"An Observer-Based View of Euclidean Geometry","authors":"Newshaw Bahreyni, Carlo Cafaro, Leonardo Rossetti","doi":"arxiv-2409.10843","DOIUrl":"https://doi.org/arxiv-2409.10843","url":null,"abstract":"Influence network of events is a view of the universe based on events that\u0000may be related to one another via influence. The network of events form a\u0000partially-ordered set which, when quantified consistently via a technique\u0000called chain projection, results in the emergence of spacetime and the\u0000Minkowski metric as well as the Lorentz transformation through changing an\u0000observer from one frame to another. Interestingly, using this approach, the\u0000motion of a free electron as well as the Dirac equation can be described.\u0000Indeed, the same approach can be employed to show how a discrete version of\u0000some of the features of Euclidean geometry, including directions, dimensions,\u0000subspaces, Pythagorean theorem, and geometric shapes can emerge. In this paper, after reviewing the essentials of the influence network\u0000formalism, we build on some of our previous works to further develop aspects of\u0000Euclidean geometry. Specifically, we present the emergence of geometric shapes,\u0000a discrete version of the Parallel postulate, the dot product, and the outer\u0000(wedge product) in 2+1 dimensions. Finally, we show that the scalar\u0000quantification of two concatenated orthogonal intervals exhibits features that\u0000are similar to those of the well-known concept of geometric product in\u0000geometric Clifford algebras.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Non-Markovian dynamics in open quantum systems arise when the system's evolution is influenced by its past interactions with the environment. Here, we present a novel metric for quantifying non-Markovianity based on local quantum Fisher information (LQFI). The proposed metric offers a distinct perspective compared to existing measures, providing a deeper understanding of information flow between the system and its environment. By comparing the LQFI-based measure to the LQU-based measure, we demonstrate its effectiveness in detecting non-Markovianity and its ability to capture the degree of non-Markovian behavior in various quantum channels. Furthermore, we show that a positive time derivative of LQFI signals the flow of information from the environment to the system, providing a clear interpretation of non-Markovian dynamics. Finally, the computational efficiency of the LQFI-based measure makes it a practical tool for characterizing non-Markovianity in diverse physical systems.
{"title":"Quantifying non-Markovianity via local quantum Fisher information","authors":"Yassine Dakir, Abdallah Slaoui, Lalla Btissam Drissi, Rachid Ahl Laamara","doi":"arxiv-2409.10163","DOIUrl":"https://doi.org/arxiv-2409.10163","url":null,"abstract":"Non-Markovian dynamics in open quantum systems arise when the system's\u0000evolution is influenced by its past interactions with the environment. Here, we\u0000present a novel metric for quantifying non-Markovianity based on local quantum\u0000Fisher information (LQFI). The proposed metric offers a distinct perspective\u0000compared to existing measures, providing a deeper understanding of information\u0000flow between the system and its environment. By comparing the LQFI-based\u0000measure to the LQU-based measure, we demonstrate its effectiveness in detecting\u0000non-Markovianity and its ability to capture the degree of non-Markovian\u0000behavior in various quantum channels. Furthermore, we show that a positive time\u0000derivative of LQFI signals the flow of information from the environment to the\u0000system, providing a clear interpretation of non-Markovian dynamics. Finally,\u0000the computational efficiency of the LQFI-based measure makes it a practical\u0000tool for characterizing non-Markovianity in diverse physical systems.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"188 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The theory of composite mixtures consisting of $n$ constituents is framed within the schema provided by the notion of $n$-groupoid. The point of departure is the analysis of $n$-dimensional hypercubes and their skeletons, to each of whose edges an element (an arrow) of one of $n$ given material groupoids is assigned according to the coordinate class to which it belongs. In this way a $GL(3,{mathbb R})$-weighted digraph is obtained. It is shown that if the double groupoid associated with each pair of constituents consists of commuting squares, the resulting $n$-groupoid is conservative. The core of this $n$-groupoid is transitive if, and only if, the mixture is materially uniform.
{"title":"Hypercubes, $n$-groupoids, and mixtures","authors":"Marcelo Epstein","doi":"arxiv-2409.10730","DOIUrl":"https://doi.org/arxiv-2409.10730","url":null,"abstract":"The theory of composite mixtures consisting of $n$ constituents is framed\u0000within the schema provided by the notion of $n$-groupoid. The point of\u0000departure is the analysis of $n$-dimensional hypercubes and their skeletons, to\u0000each of whose edges an element (an arrow) of one of $n$ given material\u0000groupoids is assigned according to the coordinate class to which it belongs. In\u0000this way a $GL(3,{mathbb R})$-weighted digraph is obtained. It is shown that\u0000if the double groupoid associated with each pair of constituents consists of\u0000commuting squares, the resulting $n$-groupoid is conservative. The core of this\u0000$n$-groupoid is transitive if, and only if, the mixture is materially uniform.","PeriodicalId":501312,"journal":{"name":"arXiv - MATH - Mathematical Physics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142260454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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