Pub Date : 2024-08-01DOI: 10.1016/S0034-4877(24)00057-0
Muhtorjon Makhammadaliev
In this paper, we study the weakly periodic (nonperiodic) Gibbs measures for the Hard Core (HC) model with a countable set ℤ of spin values and with a countable set of parameters λi > 0, i ∈ ℤ, on a Cayley tree of order k ≥ 2. For the considered model in the case ∑i λi < +∞, a complete description of weakly periodic Gibbs measures is obtained for any normal divisor of index two and in the case ∑i; λi = +∞, it is shown that there is no weakly periodic Gibbs measure. Moreover, in the case of a normal divisor of index four the uniqueness conditions for weakly periodic Gibbs measures are found. Further, under certain conditions an exact critical value is found that ensures the existence of weakly periodic Gibbs measures.
{"title":"Weakly periodic gibbs measures for the HC model with a countable set of spin values","authors":"Muhtorjon Makhammadaliev","doi":"10.1016/S0034-4877(24)00057-0","DOIUrl":"10.1016/S0034-4877(24)00057-0","url":null,"abstract":"<div><div>In this paper, we study the weakly periodic (nonperiodic) Gibbs measures for the Hard Core (HC) model with a countable set ℤ of spin values and with a countable set of parameters <em>λ<sub>i</sub> ></em> 0, <em>i</em> ∈ ℤ, on a Cayley tree of order <em>k</em> ≥ 2. For the considered model in the case ∑<em><sub>i</sub></em> λ<em><sub>i</sub></em> < +∞, a complete description of weakly periodic Gibbs measures is obtained for any normal divisor of index two and in the case ∑<em><sub>i</sub></em>; λ<em><sub>i</sub></em> = +∞, it is shown that there is no weakly periodic Gibbs measure. Moreover, in the case of a normal divisor of index four the uniqueness conditions for weakly periodic Gibbs measures are found. Further, under certain conditions an exact critical value is found that ensures the existence of weakly periodic Gibbs measures.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 83-103"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311092","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we deal with the construction of quadratic invariants in a complex phase space under the transformation z = (x + iy) and for various time-dependent systems. For this purpose, Struckmeier and Riedel (SR) approach [1, 2] is used. The constructed invariants include an unknown function f2(t) that is a solution of a third-order differential equation and its coefficients can be determined by the trajectories of the particle. The invariants play an important role in the study of a dynamical system, to access the accuracy in numerical simulations and to investigate the classical and quantum integrability of a system.
{"title":"Construction of quadratic invariants for time-dependent systems in complex phase space using Struckmeier and Riedel approach","authors":"Vipin Kumar, S.B. Bhardwaj, Ram Mehar Singh, Shalini Gupta, Fakir Chand","doi":"10.1016/S0034-4877(24)00052-1","DOIUrl":"10.1016/S0034-4877(24)00052-1","url":null,"abstract":"<div><div>In this paper, we deal with the construction of quadratic invariants in a complex phase space under the transformation <em>z</em> = (<em>x</em> + <em>iy</em>) and\u0000<span><math><mrow><mover><mi>z</mi><mo>¯</mo></mover><mo>=</mo><mrow><mo>(</mo><mrow><mi>x</mi><mo>-</mo><mi>i</mi><mi>y</mi></mrow><mo>)</mo></mrow></mrow></math></span> for various time-dependent systems. For this purpose, Struckmeier and Riedel (SR) approach [<span><span>1</span></span>, <span><span>2</span></span>] is used. The constructed invariants include an unknown function <em>f</em><sub>2</sub>(<em>t</em>) that is a solution of a third-order differential equation and its coefficients can be determined by the trajectories of the particle. The invariants play an important role in the study of a dynamical system, to access the accuracy in numerical simulations and to investigate the classical and quantum integrability of a system.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 1-10"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1016/S0034-4877(24)00056-9
Pankaj Kumar, Rakesh Kumar
We study the Hong–Mandel higher-order squeezing of both quadrature components for an arbitrary 2nth-order (n ≠1) considering the most general Hermitian operator, Xθ = X1 cos θ + iX2 sin θ, in the superposed state, |Ψ〉 = K [|Ψ0) + reiϕ |0〉] of the orthogonal even coherent state and vacuum state. Here | Ψ0〉 = K[|α, +〉 + |iα, +〉] is the orthogonal coherent state, |α, +〉 = K′[|α) + | – α〉] and |iα, +〉 = K″ [|iα, +〉 + | – iα, +〉] are even coherent states, operators X1,2 are defined by X1 + iX2 = a, a is the annihilation operator, α, θ, r and ϕ are arbitrary parameters and the only restriction on these is the normalization condition of the superposed state |Ψ〉. We find that maximum simultaneous 2nth-order Hong–Mandel squeezing of both quadrature components Xθ and Xθ+π/2 exhibited by the orthogonal even coherent state enhances in its superposition with vacuum state. We conclude that the values of higher-order momenta in the superposed state become much closer to the best minimum values of the corresponding values of higher-order momenta explored numerically so far than that obtained in orthogonal even coherent state. Variations of 2nth-order squeezing for n = 2,3 and 4, i.e. fourth, sixth and eighth-order squeezing with different parameters have also been discussed.
{"title":"Higher-order squeezing of both quadrature components in superposition of orthogonal even coherent state and vacuum state","authors":"Pankaj Kumar, Rakesh Kumar","doi":"10.1016/S0034-4877(24)00056-9","DOIUrl":"10.1016/S0034-4877(24)00056-9","url":null,"abstract":"<div><div>We study the Hong–Mandel higher-order squeezing of both quadrature components for an arbitrary 2<em>n</em><sup>th</sup>-order (<em>n</em> ≠1) considering the most general Hermitian operator, <em>X<sub>θ</sub> = X</em><sub>1</sub> cos <em>θ + iX</em><sub>2</sub> sin <em>θ</em>, in the superposed state, |<em>Ψ</em>〉 = <em><strong>K</strong></em> [|Ψ<sub>0</sub>) + <em>re</em><sup><em>i</em>ϕ</sup> |0〉] of the orthogonal even coherent state and vacuum state. Here | Ψ<sub>0</sub>〉 = <strong><em>K</em></strong>[|α, +〉 + |iα, +〉] is the orthogonal coherent state, |α, +〉 = <strong><em>K</em></strong>′[|α) + | – α〉] and |<em>i</em>α, +〉 = <strong><em>K</em></strong><em>″</em> [|<em>i</em>α, +〉 + | – <em>i</em>α, +〉] are even coherent states, operators <em>X</em><sub>1,2</sub> are defined by <em>X</em><sub>1</sub> + <em>iX</em><sub>2</sub> = <em>a, a</em> is the annihilation operator, α, <em>θ</em>, <em>r</em> and ϕ are arbitrary parameters and the only restriction on these is the normalization condition of the superposed state |<em>Ψ</em>〉. We find that maximum simultaneous 2<em>n</em><sup>th</sup>-order Hong–Mandel squeezing of both quadrature components <em>X<sub>θ</sub></em> and <em>X<sub>θ+π</sub></em>/2 exhibited by the orthogonal even coherent state enhances in its superposition with vacuum state. We conclude that the values of higher-order momenta in the superposed state become much closer to the best minimum values of the corresponding values of higher-order momenta explored numerically so far than that obtained in orthogonal even coherent state. Variations of 2<em>n</em><sup>th</sup>-order squeezing for <em>n</em> = 2,3 and 4, i.e. fourth, sixth and eighth-order squeezing with different parameters have also been discussed.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 73-82"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1016/S0034-4877(24)00058-2
Cheng Da , Hony-Yi Fan
By introducing appropriate electron's coordinate eigenstate and momentum eigenstate we propose new realization of angular momentum operators for describing electron's motion in uniform magnetic field. The coordinate eigenstates make up a representation and embody quantum entanglement between magnetic field and electron. The eigenstate of angular momentum's lowering-ascending operator L± are derived, the way we tackle this problem is to make an analogue between the transform to the squeezing mechanism. All these discussions reveal that the quantum theory for charged particles' motion in magnetic field needs to be developed.
{"title":"New operator realization of angular momentum for description of electron's motion in uniform magnetic field","authors":"Cheng Da , Hony-Yi Fan","doi":"10.1016/S0034-4877(24)00058-2","DOIUrl":"10.1016/S0034-4877(24)00058-2","url":null,"abstract":"<div><div>By introducing appropriate electron's coordinate eigenstate and momentum eigenstate we propose new realization of angular momentum operators for describing electron's motion in uniform magnetic field. The coordinate eigenstates make up a representation and embody quantum entanglement between magnetic field and electron. The eigenstate of angular momentum's lowering-ascending operator <strong><em>L</em><sub>±</sub></strong> are derived, the way we tackle this problem is to make an analogue between the transform\u0000<span><math><mrow><msup><mi>e</mi><mrow><mn>2</mn><mi>f</mi><msub><mi>L</mi><mi>z</mi></msub></mrow></msup><msub><mi>L</mi><mo>±</mo></msub><msup><mi>e</mi><mrow><mo>-</mo><mn>2</mn><mi>f</mi><msub><mi>L</mi><mi>z</mi></msub></mrow></msup></mrow></math></span> to the squeezing mechanism. All these discussions reveal that the quantum theory for charged particles' motion in magnetic field needs to be developed.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 105-115"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311087","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-08-01DOI: 10.1016/S0034-4877(24)00054-5
Uday Chand De, Krishnendu De
This article deals with the characterization of an almost pseudo symmetric spacetime and we illustrate that a conformally flat almost pseudo symmetric spacetime with nonzero constant scalar curvature represents a perfect fluid spacetime. Also, it is established that the chosen spacetime represents a spacetime of quasi constant sectional curvature and generalized Robertson-Walker spacetime. Besides, we find under what condition the spacetime represents radiation era and dust matter fluid. Further it is shown that under certain restriction in an almost pseudo symmetric spacetime, if the Ricci tensor is Killing, then it represents a static spacetime and it is vacuum. Lastly, we study the impact of this spacetime under f(R) gravity scenario and deduce several energy conditions.
{"title":"A note on almost pseudo symmetric spacetimes with certain application to f(R), gravity","authors":"Uday Chand De, Krishnendu De","doi":"10.1016/S0034-4877(24)00054-5","DOIUrl":"10.1016/S0034-4877(24)00054-5","url":null,"abstract":"<div><div>This article deals with the characterization of an almost pseudo symmetric spacetime and we illustrate that a conformally flat almost pseudo symmetric spacetime with nonzero constant scalar curvature represents a perfect fluid spacetime. Also, it is established that the chosen spacetime represents a spacetime of quasi constant sectional curvature and generalized Robertson-Walker spacetime. Besides, we find under what condition the spacetime represents radiation era and dust matter fluid. Further it is shown that under certain restriction in an almost pseudo symmetric spacetime, if the Ricci tensor is Killing, then it represents a static spacetime and it is vacuum. Lastly, we study the impact of this spacetime under <em>f</em>(<em>R)</em> gravity scenario and deduce several energy conditions.</div></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"94 1","pages":"Pages 35-45"},"PeriodicalIF":1.0,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142311095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.1016/S0034-4877(24)00043-0
Muzaffar M. Rahmatullaev, Zulxumor A. Burxonova
In the present paper, we construct new Gibbs measures for the Ising model on the Cayley tree of order two. Moreover, we find the free energy corresponding to the found measures and compare it with the known ones.
{"title":"Constructive Gibbs measures for the Ising model on the Cayley tree","authors":"Muzaffar M. Rahmatullaev, Zulxumor A. Burxonova","doi":"10.1016/S0034-4877(24)00043-0","DOIUrl":"https://doi.org/10.1016/S0034-4877(24)00043-0","url":null,"abstract":"<div><p>In the present paper, we construct new Gibbs measures for the Ising model on the Cayley tree of order two. Moreover, we find the free energy corresponding to the found measures and compare it with the known ones.</p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"93 3","pages":"Pages 361-370"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141479223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.1016/S0034-4877(24)00040-5
Wen-Xiu Ma
This paper aims to study a Kaup-Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct an associated soliton hierarchy of combined derivative nonlinear Schrödinger (NLS) equations, within the zero curvature formulation. The Liouville integrability of the resulting soliton hierarchy is shown by exploring its hereditary recursion operator and bi-Hamiltonian formulation. The first nonlinear example provides an integrable model consisting of combined derivative NLS equations with two arbitrary constants.
{"title":"A combined derivative nonlinear SchrÖdinger soliton hierarchy","authors":"Wen-Xiu Ma","doi":"10.1016/S0034-4877(24)00040-5","DOIUrl":"https://doi.org/10.1016/S0034-4877(24)00040-5","url":null,"abstract":"<div><p>This paper aims to study a Kaup-Newell type matrix eigenvalue problem with four potentials, based on a specific matrix Lie algebra, and construct an associated soliton hierarchy of combined derivative nonlinear Schrödinger (NLS) equations, within the zero curvature formulation. The Liouville integrability of the resulting soliton hierarchy is shown by exploring its hereditary recursion operator and bi-Hamiltonian formulation. The first nonlinear example provides an integrable model consisting of combined derivative NLS equations with two arbitrary constants.</p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"93 3","pages":"Pages 313-325"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.1016/S0034-4877(24)00039-9
G. Dattoli, R. Garra
Logistic equations play a pivotal role in the study of any nonlinear evolution process exhibiting growth and saturation. The interest for the phenomenology they rule goes well beyond physical processes and covers many aspects of ecology, population growth, economy. . . According to such a broad range of applications, there are different forms of functions and distributions which are recognized as generalized logistics. Sometimes they are obtained by fitting procedures. Therefore, criteria might be needed to infer the associated nonlinear differential equations, useful to guess “hidden” evolution mechanisms. In this article we analyze different forms of logistic functions and use simple means to reconstruct the differential equation they satisfy. Our study includes also differential equations containing nonstandard forms of derivative operators, like those of the Laguerre type.
{"title":"A note on differential equations of logistic type","authors":"G. Dattoli, R. Garra","doi":"10.1016/S0034-4877(24)00039-9","DOIUrl":"https://doi.org/10.1016/S0034-4877(24)00039-9","url":null,"abstract":"<div><p>Logistic equations play a pivotal role in the study of any nonlinear evolution process exhibiting growth and saturation. The interest for the phenomenology they rule goes well beyond physical processes and covers many aspects of ecology, population growth, economy. . . According to such a broad range of applications, there are different forms of functions and distributions which are recognized as generalized logistics. Sometimes they are obtained by fitting procedures. Therefore, criteria might be needed to infer the associated nonlinear differential equations, useful to guess “hidden” evolution mechanisms. In this article we analyze different forms of logistic functions and use simple means to reconstruct the differential equation they satisfy. Our study includes also differential equations containing nonstandard forms of derivative operators, like those of the Laguerre type.</p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"93 3","pages":"Pages 301-312"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.1016/S0034-4877(24)00036-3
Gunjan Agrawal, Deepanshi
In the present paper, it has been obtained that the fundamental group of n-dimensional Minkowski space with the time topology contains uncountably many copies of the additive group of integers and is not abelian. The result has been first proved for n = 2. Thereafter, it is extended to n > 2 by proving that loops nonhomotopic in M2 continue to be nonhomotopic in Mn using embedding of M2 in Mn as a retract through the projection map.
本文得出，具有时间拓扑的 n 维闵科夫斯基空间的基群包含不可计数的整数加法群副本，并且不是无边际的。这一结果首先是在 n = 2 时证明的。此后，通过使用 M2 在 Mn 中的嵌入作为通过投影图的回缩，证明 M2 中的非同调环在 Mn 中继续是非同调的，从而将其扩展到 n > 2。
{"title":"Nonabelianness of fundamental group of flat spacetime","authors":"Gunjan Agrawal, Deepanshi","doi":"10.1016/S0034-4877(24)00036-3","DOIUrl":"https://doi.org/10.1016/S0034-4877(24)00036-3","url":null,"abstract":"<div><p>In the present paper, it has been obtained that the fundamental group of <em>n</em>-dimensional Minkowski space with the time topology contains uncountably many copies of the additive group of integers and is not abelian. The result has been first proved for <em>n</em> = 2. Thereafter, it is extended to <em>n</em> > 2 by proving that loops nonhomotopic in <em>M</em><sup>2</sup> continue to be nonhomotopic in <em>M<sup>n</sup></em> using embedding of <em>M</em><sup>2</sup> in <em>M<sup>n</sup></em> as a retract through the projection map.</p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"93 3","pages":"Pages 261-270"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483875","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.1016/S0034-4877(24)00042-9
Hyungjin Huh
We find an explicit solution formula of the time dependent self-dual equations of Chern—Simons—Higgs model. The solution is expressed completely in terms of initial data.
我们找到了切尔恩-西蒙斯-希格斯模型时间相关自偶方程的明确求解公式。解完全用初始数据表示。
{"title":"Complete solvability of the time dependent self-dual equations Of Chern—Simons—Higgs model","authors":"Hyungjin Huh","doi":"10.1016/S0034-4877(24)00042-9","DOIUrl":"https://doi.org/10.1016/S0034-4877(24)00042-9","url":null,"abstract":"<div><p>We find an explicit solution formula of the time dependent self-dual equations of Chern—Simons—Higgs model. The solution is expressed completely in terms of initial data.</p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"93 3","pages":"Pages 353-360"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141487044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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