853 results
Search Results
2. Paper pop-ups demonstrating 3D vectors in Cartesian coordinates
- Author
-
Umporn Wutchana, Karnpitcha Bunrangsri, and Narumon Emarat
- Subjects
Algebra ,law ,General Physics and Astronomy ,Cartesian coordinate system ,Education ,law.invention - Published
- 2019
3. Shafarevich's paper 'A general reciprocity law'
- Author
-
Sergei V. Vostokov
- Subjects
Algebra ,symbols.namesake ,Algebra and Number Theory ,Pairing ,symbols ,Hilbert space ,Hilbert's fourteenth problem ,Projective Hilbert space ,Field (mathematics) ,Rigged Hilbert space ,Reciprocity law ,Mathematics ,Hilbert–Poincaré series - Abstract
A new method for calculating an explicit form of the Hilbert pairing is proposed. It is used to calculate the Hilbert pairing in a classical local field and in a complete higher-dimensional field. Bibliography: 25 titles.
- Published
- 2013
4. Errata to the paper by Yu.L. Ershov 'Abstract class field theory (a finitary approach)' (194:2 (2003), 199-223)
- Author
-
Yu. L. Ershov
- Subjects
Algebra ,Discrete mathematics ,Algebra and Number Theory ,Class field theory ,Finitary ,Mathematics - Published
- 2004
5. Errata to the paper by O.N.Ageev 'On the genericity of some non-asymptotic dynamical properties'.Russian Math. Surveys58:1 (2003), 173–174
- Author
-
O. N. Ageev
- Subjects
Algebra ,General Mathematics ,Mathematics education ,Mathematics - Published
- 2003
6. Comment on 'The operational foundations of PT-symmetric and quasi-Hermitian quantum theory'.
- Author
-
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
7. Comment on 'Twisted bialgebroids versus bialgebroids from a Drinfeld twist'.
- Author
-
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
- Full Text
- View/download PDF
8. Reply to Comment on 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras'.
- Author
-
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
9. REMARKS ON MY PAPER 'THE GALOIS GROUP OF THE ALGEBRAIC CLOSURE OF A LOCAL FIELD'
- Author
-
A V Jakovlev
- Subjects
Algebra ,Embedding problem ,symbols.namesake ,Galois cohomology ,Fundamental theorem of Galois theory ,Galois group ,symbols ,General Medicine ,Galois extension ,Galois module ,Algebraic closure ,Field norm ,Mathematics - Published
- 1978
10. CORRECTION TO MY PAPER 'SEMISIMPLE AUTOMORPHISM GROUPS OF FOUR-DIMENSIONAL AFFINE SPACE'
- Author
-
D I Panyushev
- Subjects
Algebra ,Affine coordinate system ,Complex space ,Affine representation ,Inner automorphism ,Affine hull ,Affine group ,Affine space ,General Medicine ,Affine plane ,Mathematics - Published
- 1986
11. CORRECTIONS TO THE PAPER 'DESCRIPTION OF FILTERED LIE ALGEBRAS WITH WHICH GRADED LIE ALGEBRAS OF CARTAN TYPE ARE ASSOCIATED'
- Author
-
Victor G. Kac
- Subjects
Algebra ,Adjoint representation of a Lie algebra ,Cartan matrix ,Real form ,General Medicine ,Killing form ,Kac–Moody algebra ,Affine Lie algebra ,Graded Lie algebra ,Mathematics ,Lie conformal algebra - Published
- 1976
12. CORRECTIONS TO THE PAPER 'THE GEOMETRY OF THE POINCARÉ THETA-DIVISOR OF A PRYM VARIETY'
- Author
-
A N Tjurin
- Subjects
Algebra ,symbols.namesake ,Pure mathematics ,Poincaré conjecture ,symbols ,Geometry ,General Medicine ,Theta divisor ,Prym variety ,Mathematics - Published
- 1978
13. Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces.
- Author
-
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
14. Gravitation in terms of observables 2: the algebra of fundamental observables.
- Author
-
Gambini, Rodolfo, Rastgoo, Saeed, and Pullin, Jorge
- Subjects
POISSON algebras ,ALGEBRA ,GRAVITY - Abstract
In a previous paper, we showed how to use the techniques of the group of loops to formulate the loop approach to gravity proposed by Mandelstam in the 1960's. Those techniques allow to overcome some of the difficulties that had been encountered in the earlier treatment. In this approach, gravity is formulated entirely in terms of Dirac observables without constraints, opening attractive new possibilities for quantization. In this paper we discuss the Poisson algebra of the resulting Dirac observables, associated with the intrinsic components of the Riemann tensor. This provides an explicit realization of the non-local algebra of observables for gravity that several authors have conjectured. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
15. Orthosymplectic Z2×Z2Z2×Z2 -graded Lie superalgebras and parastatistics.
- Author
-
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
16. On higher-order Riccati equations as Bäcklund transformations
- Author
-
Decio Levi, Am Grundland, Grundland, Am, and Levi, Decio
- Subjects
Algebra ,Bäcklund transform ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Integrable system ,Short paper ,Riccati equation ,Mathematics::Mathematical Physics ,General Physics and Astronomy ,Order (group theory) ,Statistical and Nonlinear Physics ,Mathematical Physics ,Mathematics ,Algebraic Riccati equation - Abstract
In this short paper we illustrate by few examples the special role played by higher-order Riccati equations in the construction of Backlund transformations for integrable systems.
- Published
- 1999
17. Integrable boundary conditions for staggered vertex models.
- Author
-
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
18. THE FUNDAMENTAL GROUP OF THE COMPLEMENT OF A HYPERSURFACE IN $ \mathbf C^n$
- Author
-
Vik S Kulikov
- Subjects
Algebra ,Fundamental group ,Pure mathematics ,Hypersurface ,Conjecture ,General Medicine ,Paper based ,Algebraic number ,Commutative property ,Mathematics ,Knot (mathematics) - Abstract
Let be a complex algebraic hypersurface in not passing through the point . The generators of the fundamental group and the relations among them are described in terms of the real cone over with apex at . This description is a generalization to the algebraic case of Wirtinger's corepresentation of the fundamental group of a knot in . A new proof of Zariski's conjecture about commutativity of the fundamental group for a projective nodal curve is given in the second part of the paper based on the description of the generators and the relations in the group obtained in the first part.
- Published
- 1992
19. Reduced qKZ equation and genuine qKZ equation.
- Author
-
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
20. Classification of the reducible Verma modules over the Jacobi algebra G2.
- Author
-
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
21. Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere.
- Author
-
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
22. Recursion and Hamiltonian operators for integrable nonabelian difference equations.
- Author
-
Casati, Matteo and Wang, Jing Ping
- Subjects
HAMILTONIAN operator ,DIFFERENCE equations ,ALGEBRA ,DARBOUX transformations ,DIFFERENTIAL-difference equations ,NONABELIAN groups ,POLYNOMIALS ,EQUATIONS - Abstract
In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we study a family of integrable equations: the nonabelian Narita–Itoh–Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove the locality of infinitely many commuting symmetries generated from their highly nonlocal recursion operators. Finally, we discuss the nonabelian version of several integrable difference systems, including the relativistic Toda chain and Ablowitz–Ladik lattice. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
23. A fourth-order superintegrable system with a rational potential related to Painlevé VI.
- Author
-
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
24. Deformed Heisenberg algebra and minimal length.
- Author
-
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
25. Dynamical symmetry algebras of two superintegrable two-dimensional systems.
- Author
-
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
26. Comment on 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras'.
- Author
-
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
27. Triangular solutions to the reflection equation for Uq(sln^).
- Author
-
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
28. A covariant tapestry of linear GUP, metric-affine gravity, their Poincaré algebra and entropy bound.
- Author
-
Farag Ali, Ahmed and Wojnar, Aneta
- Subjects
ENTROPY ,GRAVITY ,HEISENBERG uncertainty principle ,ALGEBRA ,TAPESTRY ,AFFINE algebraic groups ,PHASE space - Abstract
Motivated by the potential connection between metric-affine gravity and linear generalized uncertainty principle (GUP) in the phase space, we develop a covariant form of linear GUP and an associated modified Poincaré algebra, which exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP. We use three-torus geometry to visually represent linear GUP within a covariant framework. The three-torus area provides an exact geometric representation of Bekenstein's universal bound. We depart from Bousso's approach, which adapts Bekenstein's bound by substituting the Schwarzschild radius ( r s ) with the radius (R) of the smallest sphere enclosing the physical system, thereby basing the covariant entropy bound on the sphere's area. Instead, our revised covariant entropy bound is described by the area of a three-torus, determined by both the inner radius r s and outer radius R where r s ⩽ R due to gravitational stability. This approach results in a more precise geometric representation of Bekenstein's bound, notably for larger systems where Bousso's bound is typically much larger than Bekensetin's universal bound. Furthermore, we derive an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the three-torus covariant entropy bound, suggesting a new avenue of quantum gravity. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Superintegrability and deformed oscillator realizations of quantum TTW Hamiltonians on constant-curvature manifolds and with reflections in a plane.
- Author
-
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
30. Bianchi spaces and their three-dimensional isometries as S-expansions of two-dimensional isometries.
- Author
-
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
31. Two new discrete integrable systems.
- Author
-
Chen Xiao-Hong and Zhang Hong-Qing
- Subjects
DISCRETE systems ,SYSTEMS theory ,ALGEBRA ,LATTICE theory ,HAMILTONIAN systems - Abstract
In this paper, we focus on the construction of new (1+1)-dimensional discrete integrable systems according to a subalgebra of loop algebra Ã
1 . By designing two new (1+1)-dimensional discrete spectral problems, two new discrete integrable systems are obtained, namely, a 2-field lattice hierarchy and a 3-field lattice hierarchy. When deriving the two new discrete integrable systems, we find the generalized relativistic Toda lattice hierarchy and the generalized modi?ed Toda lattice hierarchy. Moreover, we also obtain the Hamiltonian structures of the two lattice hierarchies by means of the discrete trace identity. [ABSTRACT FROM AUTHOR]- Published
- 2013
- Full Text
- View/download PDF
32. Tetrahedron equation and quantum cluster algebras.
- Author
-
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
33. Absence of logarithmic enhancement in the entanglement scaling of free fermions on folded cubes.
- Author
-
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
34. Generic triangular solutions of the reflection equation: case.
- Author
-
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
35. The generalized Plücker–Klein map
- Author
-
Vyacheslav Alekseevich Krasnov
- Subjects
Algebra ,General Mathematics ,Plucker ,Mathematics - Abstract
The intersection of two quadrics is called a biquadric. If we mark a non-singular quadric in the pencil of quadrics through a given biquadric, then the given biquadric is called a marked biquadric. In the classical papers of Plücker and Klein, a Kummer surface was canonically associated with every three-dimensional marked biquadric (that is, with a quadratic line complex provided that the Plücker–Klein quadric is marked). In Reid’s thesis, this correspondence was generalized to odd-dimensional marked biquadrics of arbitrary dimension . In this case, a Kummer variety of dimension corresponds to every biquadric of dimension . Reid only constructed the generalized Plücker–Klein correspondence. This map was not studied later. The present paper is devoted to a partial solution of the problem of creating the corresponding theory.
- Published
- 2022
36. Comparison study of typical algorithms for reconstructing time series from the recurrence plot of dynamical systems.
- Author
-
Liu Jie, Shi Shu-Ting, and Zhao Jun-Chan
- Subjects
DYNAMICAL systems ,ALGORITHMS ,TIME series analysis ,STATISTICAL correlation ,ALGEBRA - Abstract
The three most widely used methods for reconstructing the underlying time series via the recurrence plots (RPs) of a dynamical system are compared with each other in this paper. We aim to reconstruct a toy series, a periodical series, a random series, and a chaotic series to compare the effectiveness of the most widely used typical methods in terms of signal correlation analysis. The application of the most effective algorithm to the typical chaotic Lorenz system verifies the correctness of such an effective algorithm. It is verified that, based on the unthresholded RPs, one can reconstruct the original attractor by choosing different RP thresholds based on the Hirata algorithm. It is shown that, in real applications, it is possible to reconstruct the underlying dynamics by using quite little information from observations of real dynamical systems. Moreover, rules of the threshold chosen in the algorithm are also suggested. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
37. ALGEBRA: Algorithm for the heterogeneous dosimetry based on GEANT4 for BRAchytherapy.
- Author
-
Afsharpour, H., Landry, G., D'Amours, M., Enger, S., Reniers, B., Poon, E., Carrier, J-F, Verhaegen, F., and Beaulieu, L.
- Subjects
ALGEBRA ,ALGORITHMS ,RADIATION dosimetry ,RADIOISOTOPE brachytherapy ,NUMERICAL calculations ,MONTE Carlo method ,ARTIFICIAL implants - Abstract
Task group 43 (TG43)-based dosimetry algorithms are efficient for brachytherapy dose calculation inwater. However, human tissues have chemical compositions and densities different thanwater. Moreover, themutual shielding effect of seeds on each other (interseed attenuation) is neglected in the TG43- based dosimetry platforms. The scientific community has expressed the need for an accurate dosimetry platform in brachytherapy. The purpose of this paper is to present ALGEBRA, a Monte Carlo platform for dosimetry in brachytherapy which is sufficiently fast and accurate for clinical and research purposes. ALGEBRA is based on the GEANT4 Monte Carlo code and is capable of handling the DICOM RT standard to recreate a virtual model of the treated site. Here, the performance of ALGEBRA is presented for the special case of LDR brachytherapy in permanent prostate and breast seed implants. However, the algorithm is also capable of handling other treatments such as HDR brachytherapy. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
38. On n-ary Hom-Lie H-pseudoalgebras.
- Author
-
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
39. A generalized linear equation for non-linear diffusion in external fields and non-ideal systems.
- Author
-
Hang Li and Laosheng Wu
- Subjects
EQUATIONS ,DIFFUSION ,FOKKER-Planck equation ,PARTIAL differential equations ,ALGEBRA ,FORCE & energy - Abstract
Based on macroscopic thermodynamic analysis, we have established a generalized linear theory for non-linear diffusion in external fields and non-ideal systems, which was classically described by the Fokker-Planck equation (or the Smoluchowski equation) and the non-linear Fickian equation, respectively. The new theory includes three basic equations expressed in 'apparent variables' as defined in this paper: (i) a generalized linear flux equation for non-linear diffusion; (ii) an apparent mass conservation equation and (iii) a generalized linear non-steady state equation for non-linear diffusion. Our analysis shows that (i) all of the existing linear and non-linear equations are the special cases of the new non-steady state general diffusion equation. It was also demonstrated that the general equation of the non-steady state is equivalent to the Fokker-Planck equation; (ii) coupling diffusion with multiple driving forces can be unified to a single force: the apparent concentration gradient; (iii) the exact relationship between diffusion coefficient and concentration in the nonlinear Fickian equation under non-ideal conditions could be established and (iv) the potential energy is conservative in a diffusion process. An application of the generalized linear equation showed that the solution is simple. For the first time, an analytic solution of the Smoluchowski equation with a time-dependent potential in the algebraic form was obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
40. Integrable models from twisted half-loop algebras.
- Author
-
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
41. The Liouville integrable coupling system of the m-AKNS hierarchy and its Hamiltonian structure.
- Author
-
Yue Chao, Yang Geng, Wen and, and Xu Yue
- Subjects
- *
HAMILTONIAN systems , *LOOPS (Group theory) , *MATHEMATICAL analysis , *ALGEBRA - Abstract
In this paper a type of 9-dimensional vector loop algebra is constructed, which is devoted to establish an isospectral problem. It follows that a Liouville integrable coupling system of the m-AKNS hierarchy is obtained by employing the Tu scheme, whose Hamiltonian structure is worked out by making use of constructed quadratic identity. The method given in the paper can be used to obtain many other integrable couplings and their Hamiltonian structures. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
42. Algebraically decaying pulses in a Ginzburg-Landau system with a neutrally stable mode.
- Author
-
Arjen Doelman, Geertje Hek, and Nienke Valkhoff
- Subjects
- *
ALGEBRA , *HEAT equation , *DIFFUSION , *CHEMICAL reactions - Abstract
In this paper, we study the existence and stability of pulse solutions in a system with interacting instability mechanisms, which is described by a Ginzburg-Landau equation for an A-mode, coupled to a diffusion equation for a B-mode. Our main question is whether this coupling may stabilize solutions of the Ginzburg-Landau equation that are unstable when the interactions with the neutrally stable B-mode are not included in the model. The spatially homogeneous B-mode is supposed to be neutrally stable. This implies that the pulse solutions cannot decay exponentially, but must decay with an algebraic rate as x? ±?. As a consequence, the methods that exist in the literature by which the stability of pulses in singularly perturbed reaction-diffusion systems can be studied need to be extended. This results in an 'algebraic NLEP approach', which is expected to be relevant beyond the setting of this paper. As in the case of a (weakly) stable B-mode (Doelman et al2004 J. Nonlin. Sci.14237-78) we establish by the application of this approach that the B-mode indeed introduces a mechanism that may stabilize pulses that are unstable when the interactions with the B-mode are not taken into account. [ABSTRACT FROM AUTHOR]
- Published
- 2007
- Full Text
- View/download PDF
43. Integral preserving discretization of 2D Toda lattices.
- Author
-
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
44. On S -expansions and other transformations of Lie algebras.
- Author
-
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
45. Runge–Lenz vector as a 3d projection of SO(4) moment map in R4×R4 phase space.
- Author
-
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
46. Limit shape for infinite rank limit of tensor power decomposition for Lie algebras of series so2n+1.
- Author
-
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
47. Blow-up solutions for L 2 supercritical gKdV equations with exactly k blow-up points.
- Author
-
Yang Lan
- Subjects
EQUATIONS ,NUMERICAL analysis ,MATHEMATICS ,ALGEBRA ,DYNAMICS - Abstract
In this paper we consider the slightly L
2 -supercritical gKdV equations , with the nonlinearity and . In the previous work of the author, we know that there exists a stable self-similar blow-up dynamics for slightly L2 -supercritical gKdV equations. Such solutions can be viewed as solutions with a single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time. [ABSTRACT FROM AUTHOR]- Published
- 2017
- Full Text
- View/download PDF
48. On the classification of rational K-matrices.
- Author
-
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
49. Infants and young children modeling method for numerical dosimetry studies: application to plane wave exposure.
- Author
-
S Dahdouh, N Varsier, M A Nunez Ochoa, J Wiart, A Peyman, and I Bloch
- Subjects
INFANT care ,RADIATION dosimetry ,PLANE wavefronts ,ALGORITHMS ,ALGEBRA - Abstract
Numerical dosimetry studies require the development of accurate numerical 3D models of the human body. This paper proposes a novel method for building 3D heterogeneous young children models combining results obtained from a semi-automatic multi-organ segmentation algorithm and an anatomy deformation method. The data consist of 3D magnetic resonance images, which are first segmented to obtain a set of initial tissues. A deformation procedure guided by the segmentation results is then developed in order to obtain five young children models ranging from the age of 5 to 37 months. By constraining the deformation of an older child model toward a younger one using segmentation results, we assure the anatomical realism of the models. Using the proposed framework, five models, containing thirteen tissues, are built. Three of these models are used in a prospective dosimetry study to analyze young child exposure to radiofrequency electromagnetic fields. The results lean to show the existence of a relationship between age and whole body exposure. The results also highlight the necessity to specifically study and develop measurements of child tissues dielectric properties. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
50. 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
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.