5,812 research outputs found
A possible combinatorial point for XYZ-spin chain
We formulate and discuss a number of conjectures on the ground state vectors
of the XYZ-spin chains of odd length with periodic boundary conditions and a
special choice of the Hamiltonian parameters. In particular, arguments for the
validity of a sum rule for the components, which describes in a sense the
degree of antiferromagneticity of the chain, are given.Comment: AMSLaTeX, 15 page
On kernel engineering via PaleyâWiener
A radial basis function approximation takes the form
where the coefficients a 1,âŠ,a n are real numbers, the centres b 1,âŠ,b n are distinct points in â d , and the function Ï:â d ââ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which Ï is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure ÎŒ for which the convolution Ï=ÎŒ Ï is a function of compact support, and when Ï is polyharmonic. The novelty of this construction is its use of the PaleyâWiener theorem to identify compact support via analysis of the Fourier transform of the new kernel Ï, so providing a new form of kernel engineering
Reply to Comment on "Exact analytic solution for the generalized Lyapunov exponent of the 2-dimensional Anderson localization"
We reply to comments by P.Marko, L.Schweitzer and M.Weyrauch
[preceding paper] on our recent paper [J. Phys.: Condens. Matter 63, 13777
(2002)]. We demonstrate that our quite different viewpoints stem for the
different physical assumptions made prior to the choice of the mathematical
formalism. The authors of the Comment expect \emph{a priori} to see a single
thermodynamic phase while our approach is capable of detecting co-existence of
distinct pure phases. The limitations of the transfer matrix techniques for the
multi-dimensional Anderson localization problem are discussed.Comment: 4 pages, accepted for publication in J.Phys.: Condens. Mat
Critical and Tricritical Hard Objects on Bicolorable Random Lattices: Exact Solutions
We address the general problem of hard objects on random lattices, and
emphasize the crucial role played by the colorability of the lattices to ensure
the existence of a crystallization transition. We first solve explicitly the
naive (colorless) random-lattice version of the hard-square model and find that
the only matter critical point is the non-unitary Lee-Yang edge singularity. We
then show how to restore the crystallization transition of the hard-square
model by considering the same model on bicolored random lattices. Solving this
model exactly, we show moreover that the crystallization transition point lies
in the universality class of the Ising model coupled to 2D quantum gravity. We
finally extend our analysis to a new two-particle exclusion model, whose
regular lattice version involves hard squares of two different sizes. The exact
solution of this model on bicolorable random lattices displays a phase diagram
with two (continuous and discontinuous) crystallization transition lines
meeting at a higher order critical point, in the universality class of the
tricritical Ising model coupled to 2D quantum gravity.Comment: 48 pages, 13 figures, tex, harvmac, eps
Three-Dimensional Vertex Model in Statistical Mechanics, from Baxter-Bazhanov Model
We find that the Boltzmann weight of the three-dimensional Baxter-Bazhanov
model is dependent on four spin variables which are the linear combinations of
the spins on the corner sites of the cube and the Wu-Kadanoff duality between
the cube and vertex type tetrahedron equations is obtained explicitly for the
Baxter-Bazhanov model. Then a three-dimensional vertex model is obtained by
considering the symmetry property of the weight function, which is
corresponding to the three-dimensional Baxter-Bazhanov model. The vertex type
weight function is parametrized as the dihedral angles between the rapidity
planes connected with the cube. And we write down the symmetry relations of the
weight functions under the actions of the symmetry group of the cube. The
six angles with a constrained condition, appeared in the tetrahedron equation,
can be regarded as the six spectrums connected with the six spaces in which the
vertex type tetrahedron equation is defined.Comment: 29 pages, latex, 8 pasted figures (Page:22-29
Three-coloring statistical model with domain wall boundary conditions. I. Functional equations
In 1970 Baxter considered the statistical three-coloring lattice model for
the case of toroidal boundary conditions. He used the Bethe ansatz and found
the partition function of the model in the thermodynamic limit. We consider the
same model but use other boundary conditions for which one can prove that the
partition function satisfies some functional equations similar to the
functional equations satisfied by the partition function of the six-vertex
model for a special value of the crossing parameter.Comment: 16 pages, notations changed for consistency with the next part,
appendix adde
On spherical averages of radial basis functions
A radial basis function (RBF) has the general form
where the coefficients a 1,âŠ,a n are real numbers, the points, or centres, b 1,âŠ,b n lie in â d , and Ï:â d ââ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when Ï is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm Ï(x)=âxâ when d is an odd positive integer, the thin plate spline Ï(x)=âxâ2log ââxâ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243â264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the PaleyâWiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserlesâs study of geometric integration
sl(N) Onsager's Algebra and Integrability
We define an analog of Onsager's Algebra through a finite set of
relations that generalize the Dolan Grady defining relations for the original
Onsager's Algebra. This infinite-dimensional Lie Algebra is shown to be
isomorphic to a fixed point subalgebra of Loop Algebra with respect
to a certain involution. As the consequence of the generalized Dolan Grady
relations a Hamiltonian linear in the generators of Onsager's Algebra
is shown to posses an infinite number of mutually commuting integrals of
motion
A new representation for the partition function of the six vertex model with domain wall boundaries
We obtain a new representation for the partition function of the six vertex
model with domain wall boundaries using a functional equation recently derived
by the author. This new representation is given in terms of a sum over the
permutation group where the partial homogeneous limit can be taken trivially.
We also show by construction that this partition function satisfies a linear
partial differential equation.Comment: 14 pages, v2: added references, accepted for publication in J. Stat.
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