272 research outputs found
Integrability of the hyperbolic reduced Maxwell-Bloch equations for strongly correlated Bose-Einstein condensates
We derive and study the hyperbolic reduced Maxwell-Bloch equations (HRMB) which acts as a simplified model for the dynamics of strongly correlated Bose-Einstein condensates. A proof of their integrability is found by the derivation of a Lax pair which is valid for both the hyperbolic and standard cases of the reduced Maxwell-Bloch equations. The origin of the latter lies in quantum optics. We derive explicit solutions of the HRMB equations that correspond to kinks propagating on the Bose-Einstein condensate (BEC). These solutions are different from Gross-Pitaevskii solitons because the nonlinearity of the HRMB equations arises from the interaction of the BEC and excited atoms
On the R-matrix realization of Yangians and their representations
We study the Yangians Y(a) associated with the simple Lie algebras a of type
B, C or D. The algebra Y(a) can be regarded as a quotient of the extended
Yangian X(a) whose defining relations are written in an R-matrix form. In this
paper we are concerned with the algebraic structure and representations of the
algebra X(a). We prove an analog of the Poincare-Birkhoff-Witt theorem for X(a)
and show that the Yangian Y(a) can be realized as a subalgebra of X(a).
Furthermore, we give an independent proof of the classification theorem for the
finite-dimensional irreducible representations of X(a) which implies the
corresponding theorem of Drinfeld for the Yangians Y(a). We also give explicit
constructions for all fundamental representation of the Yangians.Comment: 65 page
A stochastic large deformation model for computational anatomy
In the study of shapes of human organs using computational anatomy, variations are found to arise from inter-subject anatomical differences, disease-specific effects, and measurement noise. This paper introduces a stochastic model for incorporating random variations into the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework. By accounting for randomness in a particular setup which is crafted to fit the geometrical properties of LDDMM, we formulate the template estimation problem for landmarks with noise and give two methods for efficiently estimating the parameters of the noise fields from a prescribed data set. One method directly approximates the time evolution of the variance of each landmark by a finite set of differential equations, and the other is based on an Expectation-Maximisation algorithm. In the second method, the evaluation of the data likelihood is achieved without registering the landmarks, by applying bridge sampling using a stochastically perturbed version of the large deformation gradient flow algorithm. The method and the estimation algorithms are experimentally validated on synthetic examples and shape data of human corpora callosa
Generalization of the U_q(gl(N)) algebra and staggered models
We develop a technique of construction of integrable models with a Z_2
grading of both the auxiliary (chain) and quantum (time) spaces. These models
have a staggered disposition of the anisotropy parameter. The corresponding
Yang-Baxter Equations are written down and their solution for the gl(N) case
are found. We analyze in details the N=2 case and find the corresponding
quantum group behind this solution. It can be regarded as quantum
U_{q,B}(gl(2)) group with a matrix deformation parameter qB with (qB)^2=q^2.
The symmetry behind these models can also be interpreted as the tensor product
of the (-1)-Weyl algebra by an extension of U_q(gl(N)) with a Cartan generator
related to deformation parameter -1.Comment: 12 pages ; Latex2
Integrable XYZ Model with Staggered Anisotropy Parameter
We apply to the XYZ model the technique of construction of integrable models
with staggered parameters, presented recently for the XXZ case. The solution of
modified Yang-Baxter equations is found and the corresponding integrable
zig-zag ladder Hamiltonian is calculated. The result is coinciding with the XXZ
case in the appropriate limit.Comment: 8 pages ; epic packag
Exotic Bialgebra S03: Representations, Baxterisation and Applications
The exotic bialgebra S03, defined by a solution of the Yang-Baxter equation,
which is not a deformation of the trivial, is considered. Its FRT dual algebra
is studied. The Baxterisation of the dual algebra is given in two
different parametrisations. The finite-dimensional representations of
are considered. Diagonalisations of the braid matrices are used to yield
remarkable insights concerning representations of the L-algebra and to
formulate the fusion of finite-dimensional representations. Possible
applications are considered, in particular, an exotic eight-vertex model and an
integrable spin-chain model.Comment: 24 pages, Latex; V2: revised subsection 4.1, added 9 references, to
appear in Annales Henri Poincare in the volume dedicated to D. Arnaudo
A new class of statistical models: Transfer matrix eigenstates, chain Hamiltonians, factorizable -matrix
Statistical models corresponding to a new class of braid matrices
() presented in a previous paper are studied. Indices
labeling states spanning the dimensional base space of ,
the -th order transfer matrix are so chosen that the operators (the sum
of the state labels) and (CP) (the circular permutation of state labels)
commute with . This drastically simplifies the construction of
eigenstates, reducing it to solutions of relatively small number of
simultaneous linear equations. Roots of unity play a crucial role. Thus for
diagonalizing the 81 dimensional space for N=3, , one has to solve a
maximal set of 5 linear equations. A supplementary symmetry relates invariant
subspaces pairwise ( and so on) so that only one of each pair needs
study. The case N=3 is studied fully for . Basic aspects for all
are discussed. Full exploitation of such symmetries lead to a formalism
quite different from, possibly generalized, algebraic Bethe ansatz. Chain
Hamiltonians are studied. The specific types of spin flips they induce and
propagate are pointed out. The inverse Cayley transform of the YB matrix giving
the potential leading to factorizable -matrix is constructed explicitly for
N=3 as also the full set of relations. Perspectives are discussed
in a final section.Comment: 27 page
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