1,225 research outputs found
A direct proof of Kim's identities
As a by-product of a finite-size Bethe Ansatz calculation in statistical
mechanics, Doochul Kim has established, by an indirect route, three
mathematical identities rather similar to the conjugate modulus relations
satisfied by the elliptic theta constants. However, they contain factors like
and , instead of . We show here that
there is a fourth relation that naturally completes the set, in much the same
way that there are four relations for the four elliptic theta functions. We
derive all of them directly by proving and using a specialization of
Weierstrass' factorization theorem in complex variable theory.Comment: Latex, 6 pages, accepted by J. Physics
Linear and multiplicative 2-forms
We study the relationship between multiplicative 2-forms on Lie groupoids and
linear 2-forms on Lie algebroids, which leads to a new approach to the
infinitesimal description of multiplicative 2-forms and to the integration of
twisted Dirac manifolds.Comment: to appear in Letters in Mathematical Physic
A sigma model field theoretic realization of Hitchin's generalized complex geometry
We present a sigma model field theoretic realization of Hitchin's generalized
complex geometry, which recently has been shown to be relevant in
compactifications of superstring theory with fluxes. Hitchin sigma model is
closely related to the well known Poisson sigma model, of which it has the same
field content. The construction shows a remarkable correspondence between the
(twisted) integrability conditions of generalized almost complex structures and
the restrictions on target space geometry implied by the Batalin--Vilkovisky
classical master equation. Further, the (twisted) classical Batalin--Vilkovisky
cohomology is related non trivially to a generalized Dolbeault cohomology.Comment: 28 pages, Plain TeX, no figures, requires AMS font files AMSSYM.DEF
and amssym.tex. Typos in eq. 6.19 and some spelling correcte
Energy minimization using Sobolev gradients: application to phase separation and ordering
A common problem in physics and engineering is the calculation of the minima
of energy functionals. The theory of Sobolev gradients provides an efficient
method for seeking the critical points of such a functional. We apply the
method to functionals describing coarse-grained Ginzburg-Landau models commonly
used in pattern formation and ordering processes.Comment: To appear J. Computational Physic
Nodal domains on quantum graphs
We consider the real eigenfunctions of the Schr\"odinger operator on graphs,
and count their nodal domains. The number of nodal domains fluctuates within an
interval whose size equals the number of bonds . For well connected graphs,
with incommensurate bond lengths, the distribution of the number of nodal
domains in the interval mentioned above approaches a Gaussian distribution in
the limit when the number of vertices is large. The approach to this limit is
not simple, and we discuss it in detail. At the same time we define a random
wave model for graphs, and compare the predictions of this model with analytic
and numerical computations.Comment: 19 pages, uses IOP journal style file
Exact solution for random walks on the triangular lattice with absorbing boundaries
The problem of a random walk on a finite triangular lattice with a single
interior source point and zig-zag absorbing boundaries is solved exactly. This
problem has been previously considered intractable.Comment: 10 pages, Latex, IOP macro
Connections and dynamical trajectories in generalised Newton-Cartan gravity I. An intrinsic view
The "metric" structure of nonrelativistic spacetimes consists of a one-form
(the absolute clock) whose kernel is endowed with a positive-definite metric.
Contrarily to the relativistic case, the metric structure and the torsion do
not determine a unique Galilean (i.e. compatible) connection. This subtlety is
intimately related to the fact that the timelike part of the torsion is
proportional to the exterior derivative of the absolute clock. When the latter
is not closed, torsionfreeness and metric-compatibility are thus mutually
exclusive. We will explore generalisations of Galilean connections along the
two corresponding alternative roads in a series of papers. In the present one,
we focus on compatible connections and investigate the equivalence problem
(i.e. the search for the necessary data allowing to uniquely determine
connections) in the torsionfree and torsional cases. More precisely, we
characterise the affine structure of the spaces of such connections and display
the associated model vector spaces. In contrast with the relativistic case, the
metric structure does not single out a privileged origin for the space of
metric-compatible connections. In our construction, the role of the Levi-Civita
connection is played by a whole class of privileged origins, the so-called
torsional Newton-Cartan (TNC) geometries recently investigated in the
literature. Finally, we discuss a generalisation of Newtonian connections to
the torsional case.Comment: 79 pages, 7 figures; v2: added material on affine structure of
connection space, former Section 4 postponed to 3rd paper of the serie
Converting Classical Theories to Quantum Theories by Solutions of the Hamilton-Jacobi Equation
By employing special solutions of the Hamilton-Jacobi equation and tools from
lattice theories, we suggest an approach to convert classical theories to
quantum theories for mechanics and field theories. Some nontrivial results are
obtained for a gauge field and a fermion field. For a topologically massive
gauge theory, we can obtain a first order Lagrangian with mass term. For the
fermion field, in order to make our approach feasible, we supplement the
conventional Lagrangian with a surface term. This surface term can also produce
the massive term for the fermion.Comment: 30 pages, no figures, v2: discussions and references added, published
version matche
Optical Flow on Evolving Surfaces with an Application to the Analysis of 4D Microscopy Data
We extend the concept of optical flow to a dynamic non-Euclidean setting.
Optical flow is traditionally computed from a sequence of flat images. It is
the purpose of this paper to introduce variational motion estimation for images
that are defined on an evolving surface. Volumetric microscopy images depicting
a live zebrafish embryo serve as both biological motivation and test data.Comment: The final publication is available at link.springer.co
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