170 results
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2. Comment on 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'.
- Author
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Znojil, Miloslav
- Subjects
QUANTUM theory ,HILBERT space ,OPEN-ended questions ,ALGEBRA - Abstract
In Alase et al (2022 J. Phys. A: Math. Theor. 55 244003), Alase et al wrote that 'the constraint of quasi-Hermiticity on observables' is not 'sufficient to extend the standard quantum theory' because 'such a system is equivalent to a standard quantum system.' Three addenda elucidating the current state of the art are found necessary. The first one concerns the project: in the related literature the original 'aim of extending standard quantum theory' has already been abandoned shortly after its formulation. The second comment concerns the method, viz., the study in 'the framework of general probabilistic theories' (GPT). It is noticed that a few other, mathematically consistent GPT-like theories are available. The authors do not mention, in particular, the progress achieved, under the quasi-Hermiticity constraint, in the approach using the effect algebras. We add that this approach already found its advanced realistic applications in the quasi-Hermitian models using the unbounded operators of observables acting in the infinite-dimensional Hilbert spaces. Thirdly, the 'intriguing open question' about 'what possible constraints, if any, could lead to such a meaningful extension' (in the future) is given an immediate tentative answer: the possibility is advocated that the desirable constraint could really be just the quasi-Hermiticity of the observables, provided only that one has in mind its recently developed non-stationary version. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
3. Comment on 'Twisted bialgebroids versus bialgebroids from a Drinfeld twist'.
- Author
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Škoda, Zoran and Stojić, Martina
- Subjects
NONCOMMUTATIVE algebras ,HOPF algebras ,ALGEBRA ,MODEL theory ,MATHEMATICS - Abstract
A class of left bialgebroids whose underlying algebra A ♯ H is a smash product of a bialgebra H with a braided commutative Yetter–Drinfeld H -algebra A has recently been studied in relation to models of field theories on noncommutative spaces. In Borowiec and Pachoł (2017 J. Phys. A: Math. Theor. 50 055205) a proof has been presented that the bialgebroid A F ♯ H F where H
F and AF are the twists of H and A by a Drinfeld 2-cocycle F = ∑ F 1 ⊗ F 2 is isomorphic to the twist of bialgebroid A ♯ H by the bialgebroid 2-cocycle ∑ 1 ♯ F 1 ⊗ 1 ♯ F 2 induced by F. They assume H is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter–Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with a general coaction and the correct prebraiding, and even without the assumption of quasitriangularity. [ABSTRACT FROM AUTHOR]- Published
- 2024
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4. Reply to Comment on 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras'.
- Author
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Jacobsen, Jesper Lykke
- Subjects
EIGENVALUES ,ALGEBRA ,MATHEMATICS - Abstract
The authors replies to the comment made by Yang and Zhou (2024 J. Phys. A: Math. Theor.) on his 2015 paper entitled 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras' (Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003). [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces.
- Author
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Marquette, Ian, Zhang, Junze, and Zhang, Yao-Zhong
- Subjects
ALGEBRA - Abstract
Superintegrable systems in two-dimensional (2D) Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
6. Orthosymplectic Z2×Z2Z2×Z2 -graded Lie superalgebras and parastatistics.
- Author
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Stoilova, N I and der Jeugt, J Van
- Subjects
LIE superalgebras ,ALGEBRA ,PARASOCIAL relationships ,YANG-Baxter equation - Abstract
A Z 2 × Z 2 -graded Lie superalgebra g is a Z 2 × Z 2 -graded algebra with a bracket [ [ ⋅ , ⋅ ] ] that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie superalgebra. We construct the most general orthosymplectic Z 2 × Z 2 -graded Lie superalgebra o s p (2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2) in terms of defining matrices. A special case of this algebra appeared already in work of Tolstoy in 2014. Our construction is based on the notion of graded supertranspose for a Z 2 × Z 2 -graded matrix. Since the orthosymplectic Lie superalgebra o s p (2 m + 1 | 2 n) is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from o s p (2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2) . Some special cases are of particular interest, even when one is dealing with parabosons only. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
7. Integrable boundary conditions for staggered vertex models.
- Author
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Frahm, Holger and Gehrmann, Sascha
- Subjects
HAMILTONIAN systems ,ALGEBRA ,TRANSFER matrix - Abstract
Yang–Baxter integrable vertex models with a generic Z 2 -staggering can be expressed in terms of composite R -matrices given in terms of the elementary R -matrices. Similarly, integrable open boundary conditions can be constructed through generalized reflection algebras based on these objects and their representations in terms of composite boundary matrices K ± . We show that only two types of staggering yield a local Hamiltonian with integrable open boundary conditions in this approach. The staggering in the underlying model allows for a second hierarchy of commuting integrals of motion (in addition to the one including the Hamiltonian obtained from the usual transfer matrix), starting with the so-called quasi momentum operator. In this paper, we show that this quasi momentum operator can be obtained together with the Hamiltonian for both periodic and open models in a unified way from enlarged Yang–Baxter or reflection algebras in the composite picture. For the special case of the staggered six-vertex model, this allows constructing an integrable spectral flow between the two local cases. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
8. Reduced qKZ equation and genuine qKZ equation.
- Author
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Razumov, Alexander V
- Subjects
EQUATIONS ,ALGEBRA ,DENSITY ,TEMPERATURE - Abstract
The work is devoted to the study of quantum integrable systems associated with quantum loop algebras. The recently obtained equation for the zero temperature inhomogeneous reduced density operator is analyzed. It is demonstrated that any solution of the corresponding qKZ equation generates a solution to this equation. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
9. Classification of the reducible Verma modules over the Jacobi algebra G2.
- Author
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Aizawa, N, Dobrev, V K, and Doi, S
- Subjects
ALGEBRA ,REPRESENTATIONS of algebras ,LIE algebras ,CLASSIFICATION ,INDECOMPOSABLE modules ,JACOBI polynomials - Abstract
In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra G 2 . We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
10. Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere.
- Author
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Correa, F, del Olmo, M A, Marquette, I, and Negro, J
- Subjects
ALGEBRA ,POLYNOMIALS ,LIE algebras ,DIFFERENTIAL operators ,SPHERES - Abstract
Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. These algebras often display structure constants which depend on the central elements and in particular on the Hamiltonian. In this paper, we develop a new approach reexamining the quadratically superintegrable system on the two-sphere for which a symmetry algebra is known to be the Racah algebra R(3). Such a model is related to the 59 two dimensional quadratically superintegrable systems on conformally flat spaces via contractions and limits. We demonstrate that using further polynomials of degree 2, 3 and 4 in the enveloping algebra of su(3) one can generate an algebra based only on abstract commutation relations of su(3) Lie algebra without explicit constraints on the representations or realisations. This construction relies on the maximal Abelian subalgebra, also called MASA, which are the Cartan generators and their commutant. We obtain a new six-dimensional cubic algebra where the structure constant are integer numbers which reduce from a quartic algebra for which the structure constant depend on the Cartan generator and the Casimir invariant. We also present other form of the symmetry algebra using the quadratic and cubic Casimir invariants of su(3). It reduces as the known quadratic Racah algebra R(3) only when using an explicit realization. This algebraic structure describes the symmetry of the quadratically superintegrable systems on the 2 sphere. We also present a contraction to another six-dimensional cubic algebra which would corresponding to the symmetry algebra of a Smorodinsky–Winternitz model. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
11. A fourth-order superintegrable system with a rational potential related to Painlevé VI.
- Author
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Marquette, Ian, Post, Sarah, and Ritter, Lisa
- Subjects
PAINLEVE equations ,REPRESENTATIONS of algebras ,HARMONIC oscillators ,WAVE functions ,JACOBI polynomials ,ALGEBRA ,DUFFING oscillators ,DARBOUX transformations - Abstract
In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order integral of motion and use the classification of such systems to show that the potential gives a rational solution associated with the sixth Painlevé equation. Additionally, we show that the integrals of the motion close to form a cubic algebra and describe briefly deformed oscillator representations of this algebra. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
12. Deformed Heisenberg algebra and minimal length.
- Author
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Masłowski, T,, Nowicki, A., and Tkachuk, V. M.
- Subjects
ALGEBRA ,MATHEMATICAL functions ,MATHEMATICAL analysis ,NUMERICAL analysis ,HEISENBERG uncertainty principle ,MATHEMATICS - Abstract
A one-dimensional deformed Heisenberg algebra [X, P] = i f (P) is studied. We answer the question: for what function of deformation f (P) does there exist a nonzero minimal uncertainty in position (minimal length)? We also find an explicit expression for the minimal length in the case of an arbitrary function of deformation. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
13. Dynamical symmetry algebras of two superintegrable two-dimensional systems.
- Author
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Marquette, I and Quesne, C
- Subjects
ALGEBRA ,SYMMETRY ,POLYNOMIALS ,INTEGRALS - Abstract
A complete classification of 2D quadratically superintegrable systems with scalar potential on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems E
8 and E10 from such a classification by a new approach based on extra sets of ladder operators. They correspond in fact to two of those equivalence classes. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the underlying dynamical symmetry algebra allows to write the integrals in terms of its generators and therefore explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states which are beyond the well-known eigenstates of diagonalizable Hamiltonians and satisfy more complicated polynomial identities. Our approach provides a natural framework for such states. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF
14. Comment on 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras'.
- Author
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Yang, Yi and Zhou, Shuigeng
- Subjects
ALGEBRA ,TIME complexity ,EIGENVALUES ,TRANSFER matrix ,EXTRAPOLATION ,RANDOM graphs - Abstract
We present an algorithm to compute the exact critical probability h (n) for an n × ∞ helical square lattice with random and independent site occupancy. The algorithm has time complexity O (n 2 c n) and space complexity O (c n) with c = 2.7459... and allows us to compute h (n) up to n = 24. Since the extrapolation result of h (n) is inconsistent with the current best estimation of p
c , we also compute and extend the exact critical probability p c (n) for an n × ∞ cylindrical square lattice to n = 24. Our calculation shows that the current best result of p c = 0.592 746 050 792 10 (2) by Jacobsen (2015 J. Phys. A: Math. Theor. 48 454003) is incorrect and the corrected value should be 0.592 746 050 7896 (1) . [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
15. Triangular solutions to the reflection equation for Uq(sln^).
- Author
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Kolyaskin, Dmitry and Mangazeev, Vladimir V
- Subjects
YANG-Baxter equation ,AFFINE algebraic groups ,EQUATIONS ,ALGEBRA - Abstract
We study solutions of the reflection equation related to the quantum affine algebra U q ( s l n ^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of U q ( s l n ^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
16. Superintegrability and deformed oscillator realizations of quantum TTW Hamiltonians on constant-curvature manifolds and with reflections in a plane.
- Author
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Marquette, Ian and Parr, Anthony
- Subjects
SPACES of constant curvature ,FINITE integration technique ,ALGEBRA ,MATHEMATICS ,DIFFERENTIAL operators - Abstract
We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins et al (2010 J. Phys. A: Math. Theor. 43 265205). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (Kalnins et al 2011 SIGMA 7 031). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay et al (2009 J. Phys. A: Math. Theor. 42 242001) as well as a deformation discovered by Post et al (2011 J. Phys. A: Math. Theor. 44 505201) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realizations with finite-dimensional irreps which fill a gap in the literature. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. Bianchi spaces and their three-dimensional isometries as S-expansions of two-dimensional isometries.
- Author
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Caroca, Ricardo, Kondrashuk, Igor, Merino, Nelson, and Nadal, Felip
- Subjects
BIANCHI groups ,ISOCHORIC processes ,ALGEBRA ,ABELIAN semigroups ,COMPUTER software ,LIE algebras - Abstract
In this paper we show that certain three-dimensional isometry algebras, specifically those of type I, II, III and V (according to Bianchi's classification), can be obtained as expansions of the isometries in two dimensions. In particular, we use the so-called S-expansionmethod, whichmakes use of the finite Abelian semigroups, because it is the most general procedure known until now. Also, it is explicitly shown why it is impossible to obtain the algebras of type IV, VI-IX as expansions from the isometry algebras in two dimensions. All the results are checked with computer programs. This procedure shows that the problem of how to relate, by an expansion, two Lie algebras of different dimensions can be entirely solved. In particular, the procedure can be generalized to higher dimensions, which could be useful for diverse physical applications, as we discuss in our conclusions. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
18. Tetrahedron equation and quantum cluster algebras.
- Author
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Inoue, Rei, Kuniba, Atsuo, and Terashima, Yuji
- Subjects
CLUSTER algebras ,YANG-Baxter equation ,EQUATIONS ,TETRAHEDRA ,ALGEBRA - Abstract
We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y -variables in terms q -Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's R matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when q is specialized to a root of unity. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes.
- Author
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Bernard, Pierre-Antoine, Mann, Zachary, Parez, Gilles, and Vinet, Luc
- Subjects
FERMIONS ,CUBES ,VERTEX operator algebras ,REPRESENTATION theory ,QUANTUM entanglement ,ALGEBRA - Abstract
This study investigates the scaling behavior of the ground-state entanglement entropy in a model of free fermions on folded cubes. An analytical expression is derived in the large-diameter limit, revealing a strict adherence to the area law. The absence of the logarithmic enhancement expected for free fermions is explained using a decomposition of folded cubes in chains based on its Terwilliger algebra and s o (3) − 1 . The entanglement Hamiltonian and its relation to Heun operators are also investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. Generic triangular solutions of the reflection equation: case.
- Author
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Tsuboi, Zengo
- Subjects
REFLECTIONS ,EQUATIONS ,ALGEBRA ,TRIANGULAR norms - Abstract
We consider intertwining relations of the triangular q-Onsager algebra, and obtain generic triangular boundary K-operators in terms of the Borel subalgebras of U
q (sl2 ). These K-operators solve the reflection equation. [ABSTRACT FROM AUTHOR]- Published
- 2020
- Full Text
- View/download PDF
21. On n-ary Hom-Lie H-pseudoalgebras.
- Author
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Qinxiu Sun
- Subjects
LIE algebras ,GENERALIZATION ,MATHEMATICAL analysis ,NUMERICAL analysis ,LIE groups ,ALGEBRA - Abstract
The aim of this paper is to study n-ary Hom-LieH-pseudoalgebras generalizing both n-ary Hom-Nambu-Lie algebras and n-Lie H-pseudoalgebras. We list examples of the new structure and present some properties and the construction theorem. Furthermore, we provide a way to construct an n-ary Hom-Nambu- Lie algebra from an n-ary Hom-LieH-pseudoalgebra. The equivalent definition of n-ary Hom-Lie H-pseudoalgebra is also described. Finally, we discuss the prospects of n-ary Hom-Lie H-pseudoalgebras in physics. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
22. Integrable models from twisted half-loop algebras.
- Author
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N Cramp, é and, and C A S Young
- Subjects
QUANTUM theory ,MAGNETIC materials ,MATHEMATICAL symmetry ,ALGEBRA ,MATHEMATICAL analysis - Abstract
This paper is devoted to the construction of new integrable quantum-mechanical models based on certain subalgebras of the half-loop algebra of . Various results about these subalgebras are proven by presenting them in the notation of the St Petersburg school. These results are then used to demonstrate the integrability, and find the symmetries, of two types of physical system: twisted Gaudin magnets and Calogero-type models of particles on several half lines meeting at a point. [ABSTRACT FROM AUTHOR]
- Published
- 2007
23. Integral preserving discretization of 2D Toda lattices.
- Author
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Smirnov, Sergey V
- Subjects
GENERALIZED integrals ,LIE algebras ,CHARACTERISTIC functions ,INTEGRALS ,BANACH lattices ,ALGEBRA - Abstract
There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case (that is, they admit complete families of characteristic integrals in both directions). We prove that semi-discrete analogs of Toda lattices associated to the Cartan matrices of all simple Lie algebras are Darboux integrable. By examining the properties of Habibullin's discretization we show that if a function is a characteristic integral for a generalized Toda lattice in the continuous case, then the same function is a characteristic integral in the semi-discrete case as well. We consider characteristic algebras of such integral-preserving discretizations of Toda lattices to prove the existence of complete families of characteristic integrals in the second direction. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
24. On S -expansions and other transformations of Lie algebras.
- Author
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Alvarez, M A and Rosales-Gómez, J
- Subjects
LIE algebras ,ALGEBRA - Abstract
The aim of this work is to study the relation between S -expansions and other transformations of Lie algebras. In particular, we prove that contractions, deformations and central extensions of Lie algebras are preserved by S -expansions. We also provide several examples and give conditions so transformations of reduced subalgebras of S -expanded algebras are preserved by the S -expansion procedure. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
25. Runge–Lenz vector as a 3d projection of SO(4) moment map in R4×R4 phase space.
- Author
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Ikemori, Hitoshi, Kitakado, Shinsaku, Matsui, Yoshimitsu, and Sato, Toshiro
- Subjects
GRAPHICAL projection ,KEPLER problem ,PARTICLE motion ,SYMPLECTIC geometry ,ALGEBRA ,PHASE space - Abstract
We show, using the methods of geometric algebra, that Runge–Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, Runge–Lenz vector is a consequence of geometric symmetry of R 4 × R 4 phase space. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
26. Limit shape for infinite rank limit of tensor power decomposition for Lie algebras of series so2n+1.
- Author
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Nazarov, Anton, Nikitin, Pavel, and Postnova, Olga
- Subjects
LIE algebras ,CENTRAL limit theorem ,PROBABILITY measures ,TENSOR products ,ALGEBRA - Abstract
We consider the Plancherel measure on irreducible components of tensor powers of the spinor representation of s o 2 n + 1 . The irreducible representations correspond to the generalized Young diagrams. With respect to this measure the probability of an irreducible representation is the product of its multiplicity and dimension, divided by the total dimension of the tensor product. We study the limit shape of the generalized Young diagram when the tensor power N and the rank n of the algebra tend to infinity with N / n fixed. We derive an explicit formula for the limit shape and prove convergence to it in probability. We prove central limit theorem for global fluctuations around the limit shape. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
27. On the classification of rational K-matrices.
- Author
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Tamás Gombor
- Subjects
YANG-Baxter equation ,ASYMPTOTIC expansions ,ALGEBRA ,INFINITY (Mathematics) ,CLASSIFICATION ,SYMMETRY - Abstract
This paper presents a derivation of the possible residual symmetries of rational K-matrices which are invertible in the ‘classical limit’ (the spectral parameter goes to infinity). This derivation uses only the boundary Yang–Baxter equation and the asymptotic expansions of the R-matrices. The result proves the previous assumption of the literature: if the original and the residual symmetry algebras are and then there exists a Lie-algebra involution of for which the invariant sub-algebra is . In addition, we study a K-matrix which is not invertible in the ‘classical limit’. It is shown that its symmetry algebra is not reductive but a semi-direct sum of reductive and solvable Lie-algebras. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
28. Regularization of δ ′ potential in general case of deformed space with minimal length.
- Author
-
Samar, M I and Tkachuk, V M
- Subjects
FUNCTION spaces ,POTENTIAL energy ,ALGEBRA ,EIGENFUNCTIONS - Abstract
In the general case of deformed Heisenberg algebra leading to the minimal length, we present a definition of the δ ′(x) potential as a linear kernel of potential energy operator in momentum representation. We find exactly the energy level and corresponding eigenfunction for δ ′(x) and δ (x) â' δ ′(x) potentials in deformed space with arbitrary function of deformation. The energy spectrum for different partial cases of deformation function is analysed. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
29. Recurrence relations for symplectic realization of (quasi)-Poisson structures.
- Author
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Vladislav G Kupriyanov
- Subjects
CAYLEY numbers (Algebra) ,ALGEBRA ,COMMUTATION (Electricity) ,GEOMETRIC quantization ,MANIFOLDS (Mathematics) ,MICROPLATES ,SYMPLECTIC manifolds ,COORDINATES - Abstract
It is known that any Poisson manifold can be embedded into a bigger space which admits a description in terms of a global symplectic structure. Such a procedure is known as a symplectic realization and has a number of important applications like the quantization of the original Poisson manifold. In the present paper we extend the above idea to the case of quasi-Poisson structures which should not necessarily satisfy the Jacobi identity. For any given quasi-Poisson structure we provide a closed recursive formula for local embedding functions and Darboux coordinates. Our construction is illustrated for the examples of the constant R-flux algebra, quasi-Poisson structure isomorphic to the commutator algebra of imaginary octonions and the non-geometric M-theory R-flux background. In all cases we derive explicit formulas for the symplectic realization and the corresponding expression for Darboux coordinates. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
30. AGT correspondence: Ding–Iohara algebra at roots of unity and Lepowsky–Wilson construction.
- Author
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Lev Spodyneiko
- Subjects
CONFORMAL field theory ,STRING theory ,INSTANTONS ,FIELD theory (Physics) ,ALGEBRA - Abstract
It was recently conjectured that the AGT correspondence between the —instanton counting on and the two-dimensional field theories with the conformal symmetry algebra can be considered as a root of unity limit of its K-theoretic analogue. From this point of view, the algebra and a special basis in its representation are limits of the Ding–Iohara algebra and the Macdonald polynomials respectively. In this paper we confirm this conjecture for the special case r = 1. We uncover the implicit symmetry in this limit. We also found that the vertex operators in the special basis have factorized AFLT form. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
31. Integrable dispersive chains and energy dependent Schrödinger operator.
- Author
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Pavlov, Maxim V
- Subjects
EQUATIONS ,ALGEBRA ,NUMERICAL analysis ,SCHRODINGER operator ,DIFFERENTIAL operators - Abstract
In this paper we consider integrable dispersive chains associated with the so-called ‘energy dependent’ Schrödinger operator. In a general case multi-component reductions of these dispersive chains are new integrable systems, which are characterized by two arbitrary natural numbers. Also we show that integrable three-dimensional linearly degenerate quasilinear equations of a second order possess infinitely many differential constraints. Corresponding dispersive reductions are integrable systems associated with the ‘energy dependent’ Schrödinger operator. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
32. Bethe vectors of quantum integrable models based on.
- Author
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Pakuliak, S, Ragoucy, E, and Slavnov, N A
- Subjects
QUANTUM theory ,ALGEBRA ,MATHEMATICAL formulas ,VECTORS (Calculus) ,BOREL subgroups - Abstract
We study quantum integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the right and left universal off-shell nested Bethe vectors. It is shown that these formulas can be related by certain morphisms of the positive Borel subalgebra in into analogous subalgebra in . [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
33. Classical exchange algebra of the superstring on S5 with AdS-time.
- Author
-
Aoyama, Shogo
- Subjects
YANG-Baxter equation ,DIRAC function ,SUPERSTRING theories ,ALGEBRA ,STRING models (Physics) - Abstract
A classical exchange algebra of the superstring on S
5 with AdS-time is shown on the light-like plane. To this end, we use the geometrical method of which consistency is guaranteed by the classical Yang–Baxter equation. The Dirac method does not work, since there are constraints in which first-class and second-class constraints are mixed and one can hardly disentangle with each other keeping the isometry. [ABSTRACT FROM AUTHOR]- Published
- 2014
- Full Text
- View/download PDF
34. Defining quantumness via the Jordan product.
- Author
-
Facchi, Paolo, Ferro, Leonardo, Marmo, Giuseppe, and Pascazio, Saverio
- Subjects
QUANTUM theory ,ALGEBRA ,QUANTUM states ,MATHEMATICAL models ,COMMUTATORS (Operator theory) ,MATHEMATICAL physics - Abstract
We propose alternative definitions of classical states and quantumnesswitnesses by focusing on the algebra of observables of the system. A central role is assumed by the anticommutator of the observables, namely the Jordan product. This approach turns out to be suitable for generalizations to infinite dimensional systems. We then show that the whole algebra of observables can be generated by three elements by repeated application of the Jordan product. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
35. Vertex operator (super)algebras and LCFT.
- Author
-
Adamović, Dražen and Milas, Antun
- Subjects
OPERATOR theory ,ALGEBRA ,MATHEMATICAL analysis ,LOGARITHMS ,REPRESENTATION theory ,MATHEMATICAL models - Abstract
We reviewsome of the developments in logarithmic conformal field theory from the vertex algebra point of view. Several important examples of vertex operator (super)algebras of the triplet type are discussed, including their representation theory. Particular emphasis is put on C
2 -cofiniteness of these vertex algebras, a description of Zhu's algebras and the construction of logarithmic modules. [ABSTRACT FROM AUTHOR]- Published
- 2013
- Full Text
- View/download PDF
36. Elliptic genus of singular algebraic varieties and quotients.
- Author
-
Anatoly Libgober
- Subjects
ELLIPTIC equations ,ALGEBRA ,NUMERICAL analysis ,PARTIAL differential equations ,MATHEMATICAL analysis - Abstract
This paper discusses the basic properties of various versions of the two-variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the theories regarding the elliptic genera of phases on N = 2 introduced in Witten (1993 Nucl. Phys. B 403 159–222). [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
37. Constraint quantization of a worldline system invariant under reciprocal relativity: II.
- Author
-
P D Jarvis and S O Morgan
- Subjects
GEOMETRIC quantization ,MATHEMATICAL symmetry ,RELATIVITY (Physics) ,WIGNER distribution ,ALGEBRA ,WORLD line (Physics) ,COSMOLOGICAL constant ,MATHEMATICAL physics - Abstract
We consider the worldline quantization of a system invariant under the symmetries of reciprocal relativity. Imposition of the first class constraint, the generator of local time reparametrizations, on physical states enforces identification of the worldline cosmological constant with a fixed value of the quadratic Casimir of the quaplectic symmetry group Q(3, 1) [?] U(3, 1)[?]H(4), the semi-direct product of the pseudo-unitary group with the Weyl-Heisenberg group. In our previous paper, J. Phys. A: Math. Theor. 40 (2007) 12095, the 'spin' degrees of freedom were handled as covariant oscillators, leading to a unique choice of cosmological constant, required for projecting out negative-norm states from the physical gauge-invariant states. In the present paper, the spin degrees of freedom are treated as standard oscillators with positive norm states (wherein Lorentz boosts are not number-conserving in the auxiliary space; reciprocal transformations are of course not spin-conserving in general). As in the covariant approach, the spectrum of the square of the energy-momentum vector is continuous over the entire real line, and thus includes tachyonic (spacelike) and null branches. Adopting standard frames, the Wigner method on each branch is implemented, to decompose the auxiliary space into unitary irreducible representations of the respective little algebras and additional degeneracy algebras. The physical state space is vastly enriched as compared with the covariant approach, and contains towers of integer spin massive states, as well as unconventional massless representations of continuous spin type, with continuous Euclidean momentum and arbitrary integer helicity. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
38. Semi-classical propagation of wavepackets for the phase space Schrödinger equation: interpretation in terms of the Feichtinger algebra.
- Subjects
WAVE packets ,SCHRODINGER equation ,ALGEBRA ,NUMERICAL solutions to partial differential equations ,ORBIT method ,COHERENT states - Abstract
The nearby orbit method is a powerful tool for constructing semi-classical solutions of Schrödinger's equation when the initial datum is a coherent state. In this paper, we first extend this method to arbitrary squeezed states and thereafter apply our results to the Schrödinger equation in phase space. This adaptation requires the phase-space Weyl calculus developed in previous work of ours. We also study the regularity of the semi-classical solutions from the point of view of the Feichtinger algebra familiar from the theory of modulation spaces. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
39. The Q-operator and functional relations of the eight-vertex model at root-of-unity \eta = \frac{2m K}{N} for odd N.
- Subjects
MATHEMATICAL analysis ,ALGEBRA ,VERTEX operator algebras ,OPERATOR algebras ,OPERATOR theory - Abstract
Following Baxter's method of producing Q72-operator, we construct the Q-operator of the root-of-unity eight-vertex model for the crossing parameter \eta = \frac{2m K}{N} with odd N where Q72 does not exist. We use this new Q-operator to study the functional relations in the Fabricius-McCoy comparison between the root-of-unity eight-vertex model and the superintegrable N-state chiral Potts model. By the compatibility of the constructed Q-operator with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we verify the set of functional relations of the root-of-unity eight-vertex model using the explicit form of the Q-operator and fusion weights of the SOS model. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
40. Generalized differential forms.
- Subjects
DIFFERENTIAL forms ,ALGEBRA ,REPRESENTATIONS of algebras ,MATHEMATICAL physics ,MATHEMATICAL analysis - Abstract
The algebra and calculus of generalized differential forms are reviewed and developed. Bases of minus one-forms are studied and used in the investigation of groups of generalized forms and generalized connections. Different representations of generalized forms are discussed. Physical and mathematical applications of generalized forms are presented in a number of examples. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
41. Deformations of N= 2 superconformal algebra and supersymmetric two-component Camassa-Holm equation.
- Author
-
H Aratyn, J F Gomes, and A H Zimerman
- Subjects
- *
DEFORMATIONS (Mechanics) , *SUPERSYMMETRY , *ALGEBRA , *MATHEMATICAL analysis , *MOMENTUM (Mechanics) , *HAMILTONIAN operator - Abstract
This paper is concerned with a link between central extensions of N= 2 superconformal algebra and a supersymmetric two-component generalization of the Camassa-Holm equation. Deformations of superconformal algebra give rise to two compatible bracket structures. One of the bracket structures is derived from the central extension and admits a momentum operator which agrees with the Sobolev norm of a co-adjoint orbit element. The momentum operator induces, via Lenard relations, a chain of conserved Hamiltonians of the resulting supersymmetric Camassa-Holm hierarchy. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
42. The fully packed loop model as a non-rational W 3 conformal field theory.
- Author
-
T Dupic, B Estienne, and Y Ikhlef
- Subjects
ALGEBRA ,CONFORMAL field theory ,LATTICE theory - Abstract
The fully packed loop (FPL) model is a statistical model related to the integrable vertex model. In this paper we study the continuum limit of the FPL. With the appropriate weight of non-contractible loops, we give evidence of an extended W
3 symmetry in the continuum. The partition function on the torus is calculated exactly, yielding new modular invariants of W3 characters. The full conformal field theory spectrum is obtained, and is found to be in excellent agreement with exact diagonalisation. [ABSTRACT FROM AUTHOR]- Published
- 2016
- Full Text
- View/download PDF
43. Quantum deformations and q-boson operators.
- Author
-
P D Jarvis and M A Lohe
- Subjects
BOSONS ,QUANTUM mechanics ,DEFORMATIONS (Mechanics) ,ALGEBRA ,GENERALIZATION - Abstract
The authors discuss various topics to show the impact of the articles by A. J. Macfarlane and L. C. Biedenharn in which q-boson operators play a significant role. Topics include the extensions and generalizations of the q-boson algebra, position and momentum operators and q-deformed quantum mechanics, and representations of quantum algebras and q-functions. Also mentioned are the applications of q-boson operators to various problems in physics.
- Published
- 2016
- Full Text
- View/download PDF
44. Second-order delay ordinary differential equations, their symmetries and application to a traffic problem.
- Author
-
Dorodnitsyn, Vladimir A, Kozlov, Roman, Meleshko, Sergey V, and Winternitz, Pavel
- Subjects
ORDINARY differential equations ,DELAY differential equations ,GROUP theory ,LIE groups ,TRAFFIC flow ,SYMMETRY ,ALGEBRA - Abstract
This article is the third in a series, the aim of which is to use Lie group theory to obtain exact analytic solutions of delay ordinary differential systems (DODSs). Such a system consists of two equations involving one independent variable x and one dependent variable y. As opposed to ordinary differential equations (ODEs) the variable x figures in more than one point (we consider the case of two points, x and x
− ). The dependent variable y and its derivatives figure in both x and x− . Two previous articles were devoted to first-order DODSs, here we concentrate on a large class of second-order ones. We show that within this class the symmetry algebra can be of dimension n with 0 ⩽ n ⩽ 6 for nonlinear DODSs and must be infinite-dimensional for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow. [ABSTRACT FROM AUTHOR]- Published
- 2021
- Full Text
- View/download PDF
45. Nullspaces of entanglement breaking channels and applications.
- Author
-
Kribs, David W, Levick, Jeremy, Olfert, Katrina, Pereira, Rajesh, and Rahaman, Mizanur
- Subjects
QUANTUM information science ,QUANTUM entanglement ,QUANTUM communication ,ALGEBRA - Abstract
Quantum entanglement breaking channels are a fundamental class of quantum operations; originally investigated for quantum information theoretic reasons, their study has since grown to touch on many aspects of quantum information science. Here we investigate the nullspace structures of entanglement breaking channels and we derive a pair of related applications. We show that every operator space of trace zero matrices is the nullspace of an entanglement breaking channel. We derive a test for mixed unitarity of quantum channels based on complementary channel behaviour and entanglement breaking channel nullspaces. We identify conditions that guarantee the existence of private algebras for certain classes of entanglement breaking channels. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
46. On the superconformal index of Argyres–Douglas theories.
- Author
-
Matthew Buican and Takahiro Nishinaka
- Subjects
YANG-Mills theory ,INFINITE series (Mathematics) ,RENORMALIZATION (Physics) ,COULOMB functions ,ALGEBRA - Abstract
We conjecture a closed-form expression for the Schur limit of the superconformal index of two infinite series of Argyres–Douglas (AD) superconformal field theories (SCFTs): the and the theories. While these SCFTs can be realized at special points on the Coulomb branch of certain gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the and indices by using a relation to two-dimensional q-deformed Yang–Mills theory and data from the class construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the and SCFTs. As checks of our conjectures, we study the consistency of our results with an S-duality recently discussed by us in collaboration with Giacomelli and Papageorgakis, we reproduce known Higgs branch relations, we check consistency with a series of renormalization group flows, and we verify that the small S
1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S1 reduction of our indices in a separate paper. [ABSTRACT FROM AUTHOR]- Published
- 2016
- Full Text
- View/download PDF
47. Gauge transformation and symmetries of the commutative multicomponent BKP hierarchy.
- Author
-
Chuanzhong Li
- Subjects
GAUGE invariance ,KADOMTSEV-Petviashvili equation ,EQUATIONS ,ALGEBRA ,POLYNOMIALS - Abstract
In this paper, we defined a new multi-component B type Kadomtsev-Petviashvili (BKP) hierarchy that takes values in a commutative subalgebra of After this, we give the gauge transformation of this commutative multicomponent BKP (CMBKP) hierarchy. Meanwhile, we construct a new constrained CMBKP hierarchy that contains some new integrable systems, including coupled KdV equations under a certain reduction. After this, the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi-component BKP hierarchy will be constructed. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
48. An extension of the Derrida–Lebowitz–Speer–Spohn equation.
- Author
-
Charles Bordenave, Pierre Germain, and Thomas Trogdon
- Subjects
EQUATIONS ,NUMERICAL analysis ,ALGEBRA ,MATHEMATICS ,MATHEMATICAL analysis - Abstract
We show how the derivation of the Derrida–Lebowitz–Speer–Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy–Widom GOE distribution. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
49. Generalized Bäcklund transformations for affine Toda hierarchies.
- Author
-
de Carvalho Ferreira, J M, Gomes, J F, Lobo, G V, and Zimerman, A H
- Subjects
BACKLUND transformations ,AFFINE transformations ,GAUGE invariance ,HIERARCHIES ,ALGEBRA - Abstract
The construction of generalized Bäcklund transformation for the A
n affine Toda hierarchy is proposed in terms of gauge transformation acting on the zero curvature representation. Such construction is based upon the graded structure of the underlying affine algebra which induces a classification of generalized Bäcklund transformations. Moreover, explicit examples for sl(3) and sl(4) lead to uncover interesting composition properties of various types of Bäcklund transformations. The universality character of the gauge-Bäcklund transformation method is extended to all equations of the hierarchy. Such interesting property provides a systematic framework to construct Bäcklund transformations to higher flow equations. Explicit example for the simplest higher flow of the sl(3) hierarchy is presented. [ABSTRACT FROM AUTHOR]- Published
- 2021
- Full Text
- View/download PDF
50. Continuous coexistency preservers on effect algebras.
- Author
-
Mori, Michiya and Šemrl, Peter
- Subjects
ALGEBRA ,HILBERT space - Abstract
Let H be a finite-dimensional Hilbert space, dim H ⩾ 2. We prove that every continuous coexistency preserving map on the effect algebra E(H) is either a standard automorphism of E(H), or a standard automorphism of E(H) composed with the orthocomplementation. We present examples showing the optimality of the result. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
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