Representation of Lie Groups and Special Functions

Klimyk, A. U.; Vilenkin, N. Ya.

doi:10.1007/978-94-017-2881-2

Cited by 285 publications

(134 citation statements)

References 93 publications

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“…4). Therefore, in general, denoting f(β h ) as the probability distribution of the bending angle β h at the hinge, the PDF of rigid-body motion on SE(3) at this hinge can be obtained as a convolution (15) where α ∈ [0,2π], β ∈ [0, π], and γ ∈ [0,2π], and Then the Fourier transform of f j (a,R) on SE(3) can be obtained as (18) where (19) (20) where δ i,j is the Kronecker delta function, and is the generalized Legendre function 51,62 .…”

Section: The Hinge Casementioning

confidence: 99%

Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints

Chirikjian

2006

Macromolecules

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Fluctuations in the bending angles at internal irregularities of DNA and RNA (such as symmetric loops, bulges, and nicks/gaps) have been observed from various experiments. However, little effort has been made to computationally predict and explain the statistical behavior of semiflexible chains with internal defects. In this paper, we describe the general structure of these macromolecular chains as inextensible elastic chains with one or more internal joints which have limited ranges of rotation, and propose a method to compute the probability density functions of the end-to-end pose of these macromolecular chains. Our method takes advantage of the operational properties of the non-commutative Fourier transform for the group of rigid-body motions in three-dimensional space, SE(3). Two representative types of joints, the hinge for planar rotation and the ball joint for spatial rotation, are discussed in detail. The proposed method applies to various stiffness models of semi-flexible chain-like macromolecules. Examples are calculated using the Kratky-Porod model with specified stiffness, angular fluctuation, and joint locations. Entropic effects associated with internal angular fluctuations of semi-flexible macromolecular chains with internal joints can be computed using this formulation. Our method also provides a potential tool to detect the existence of internal irregularities.

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Section: The Hinge Casementioning

confidence: 99%

Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints

Chirikjian

2006

Macromolecules

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“…In the above definition, the rotational part, are matrix elements of the irreducible unitary representations for SO (3), which are defined as [37,38] (2)…”

Section: Matrix Elements Of the Irreducible Unitary Representations Omentioning

confidence: 99%

“…One can also use the following series form to calculate the translational part of the matrix elements of IURs for SE(3): (6) where C(k, 0; l′, s|l, s), C(k, m−m′; l′, m′|l, m) are Clebsch-Gordan coefficients, are spherical harmonic functions, and j k (pr) is the k th spherical Bessel function. According to [37], Clebsch-Gordan coefficients are defined as (7) where Finally, we are at the stage of defining the Fourier transform for SE (3). Based on the above formulae, the matrix elements of the Fourier transform of a function F(g), wherein g = (r, R) ∈ SE (3), is obtained by the following relation (8) where dg = dRdr with dR = (1/8π 2 ) sinβ dα dβ dγ and dr = r 2 sinθ da dθ dφ.…”

Section: Matrix Elements Of the Irreducible Unitary Representations Omentioning

confidence: 99%

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A unified approach to conformational statistics of classical polymer and polypeptide models

Kim

Chirikjian

2005

Polymer

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We present a unified method to generate conformational statistics which can be applied to any of the classical discrete-chain polymer models. The proposed method employs the concepts of Fourier transform and generalized convolution for the group of rigid-body motions in order to obtain probability density functions of chain end-to-end distance. In this paper, we demonstrate the proposed method with three different cases: the freely-rotating model, independent energy model, and interdependent pairwise energy model (the last two are also well-known as the Rotational Isomeric State model). As for numerical examples, for simplicity, we assume homogeneous polymer chains. For the freely-rotating model, we verify the proposed method by comparing with well-known closedform results for mean-squared end-to-end distance. In the interdependent pairwise energy case, we take polypeptide chains such as polyalanine and polyvaline as examples.

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“…Being a modification of monomial symmetric functions, they are directly related to the theory of symmetric (Laurent) polynomials [14,15,16,17] (see Section 11 in [7]). …”

Section: Introductionmentioning

confidence: 99%

E-Orbit Functions

Klimyk¹,

Patera

2008

SIGMA

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Abstract. We review and further develop the theory of E-orbit functions. They are functions on the Euclidean space E n obtained from the multivariate exponential function by symmetrization by means of an even part W e of a Weyl group W , corresponding to a CoxeterDynkin diagram. Properties of such functions are described. They are closely related to symmetric and antisymmetric orbit functions which are received from exponential functions by symmetrization and antisymmetrization procedure by means of a Weyl group W . The E-orbit functions, determined by integral parameters, are invariant with respect to even part W aff e of the affine Weyl group corresponding to W . The E-orbit functions determine a symmetrized Fourier transform, where these functions serve as a kernel of the transform. They also determine a transform on a finite set of points of the fundamental domain F e of the group W aff e (the discrete E-orbit function transform).

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Representation of Lie Groups and Special Functions

Cited by 285 publications

References 93 publications

Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints

Conformational Statistics of Semiflexible Macromolecular Chains with Internal Joints

A unified approach to conformational statistics of classical polymer and polypeptide models

E-Orbit Functions

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