131,636 research outputs found
Blow-up of generalized complex 4-manifolds
We introduce blow-up and blow-down operations for generalized complex
4-manifolds. Combining these with a surgery analogous to the logarithmic
transform, we then construct generalized complex structures on nCP2 # m
\bar{CP2} for n odd, a family of 4-manifolds which admit neither complex nor
symplectic structures unless n=1. We also extend the notion of a symplectic
elliptic Lefschetz fibration, so that it expresses a generalized complex
4-manifold as a fibration over a two-dimensional manifold with boundary.Comment: 25 pages, 15 figures. This is the final version, which was published
in J. To
New Exact and Numerical Solutions of the (Convection-)Diffusion Kernels on SE(3)
We consider hypo-elliptic diffusion and convection-diffusion on , the quotient of the Lie group of rigid body motions SE(3) in
which group elements are equivalent if they are equal up to a rotation around
the reference axis. We show that we can derive expressions for the convolution
kernels in terms of eigenfunctions of the PDE, by extending the approach for
the SE(2) case. This goes via application of the Fourier transform of the PDE
in the spatial variables, yielding a second order differential operator. We
show that the eigenfunctions of this operator can be expressed as (generalized)
spheroidal wave functions. The same exact formulas are derived via the Fourier
transform on SE(3). We solve both the evolution itself, as well as the
time-integrated process that corresponds to the resolvent operator.
Furthermore, we have extended a standard numerical procedure from SE(2) to
SE(3) for the computation of the solution kernels that is directly related to
the exact solutions. Finally, we provide a novel analytic approximation of the
kernels that we briefly compare to the exact kernels.Comment: Revised and restructure
Equivalences of twisted K3 surfaces
We prove that two derived equivalent twisted K3 surfaces have isomorphic
periods. The converse is shown for K3 surfaces with large Picard number. It is
also shown that all possible twisted derived equivalences between arbitrary
twisted K3 surfaces form a subgroup of the group of all orthogonal
transformations of the cohomology of a K3 surface.
The passage from twisted derived equivalences to an action on the cohomology
is made possible by twisted Chern characters that will be introduced for
arbitrary smooth projective varieties.Comment: Final version. 35 pages. to appear in Math. An
Fully Complex Magnetoencephalography
Complex numbers appear naturally in biology whenever a system can be analyzed
in the frequency domain, such as physiological data from magnetoencephalography
(MEG). For example, the MEG steady state response to a modulated auditory
stimulus generates a complex magnetic field for each MEG channel, equal to the
Fourier transform at the stimulus modulation frequency. The complex nature of
these data sets, often not taken advantage of, is fully exploited here with new
methods. Whole-head, complex magnetic data can be used to estimate complex
neural current sources, and standard methods of source estimation naturally
generalize for complex sources. We show that a general complex neural vector
source is described by its location, magnitude, and direction, but also by a
phase and by an additional perpendicular component. We give natural
interpretations of all the parameters for the complex equivalent-current dipole
by linking them to the underlying neurophysiology. We demonstrate complex
magnetic fields, and their equivalent fully complex current sources, with both
simulations and experimental data.Comment: 23 pages, 1 table, 5 figures; to appear in Journal of Neuroscience
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Topological dualities in the Ising model
We relate two classical dualities in low-dimensional quantum field theory:
Kramers-Wannier duality of the Ising and related lattice models in
dimensions, with electromagnetic duality for finite gauge theories in
dimensions. The relation is mediated by the notion of boundary field theory:
Ising models are boundary theories for pure gauge theory in one dimension
higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t
Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects
the multiplicity of topological boundary states. In the process we describe
lattice theories as (extended) topological field theories with boundaries and
domain walls. This allows us to generalize the duality to non-abelian groups;
finite, semi-simple Hopf algebras; and, in a different direction, to finite
homotopy theories in arbitrary dimension
Zariski -plets via dessins d'enfants
We construct exponentially large collections of pairwise distinct
equisingular deformation families of irreducible plane curves sharing the same
sets of singularities. The fundamental groups of all curves constructed are
abelian.Comment: Final version accepted for publicatio
Topological dualities in the Ising model
We relate two classical dualities in low-dimensional quantum field theory:
Kramers-Wannier duality of the Ising and related lattice models in
dimensions, with electromagnetic duality for finite gauge theories in
dimensions. The relation is mediated by the notion of boundary field theory:
Ising models are boundary theories for pure gauge theory in one dimension
higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t
Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects
the multiplicity of topological boundary states. In the process we describe
lattice theories as (extended) topological field theories with boundaries and
domain walls. This allows us to generalize the duality to non-abelian groups;
finite, semi-simple Hopf algebras; and, in a different direction, to finite
homotopy theories in arbitrary dimension.Comment: 62 pages, 22 figures; v2 adds important reference [S]; v2 has
reworked introduction, additional reference [KS], and minor changes; v4 for
publication in Geometry and Topology has all new figures and a few minor
changes and additional reference
A New Class of Group Field Theories for 1st Order Discrete Quantum Gravity
Group Field Theories, a generalization of matrix models for 2d gravity,
represent a 2nd quantization of both loop quantum gravity and simplicial
quantum gravity. In this paper, we construct a new class of Group Field Theory
models, for any choice of spacetime dimension and signature, whose Feynman
amplitudes are given by path integrals for clearly identified discrete gravity
actions, in 1st order variables. In the 3-dimensional case, the corresponding
discrete action is that of 1st order Regge calculus for gravity (generalized to
include higher order corrections), while in higher dimensions, they correspond
to a discrete BF theory (again, generalized to higher order) with an imposed
orientation restriction on hinge volumes, similar to that characterizing
discrete gravity. The new models shed also light on the large distance or
semi-classical approximation of spin foam models. This new class of group field
theories may represent a concrete unifying framework for loop quantum gravity
and simplicial quantum gravity approaches.Comment: 48 pages, 4 figures, RevTeX, one reference adde
Monotonically convergent optimal control theory of quantum systems under a nonlinear interaction with the control field
We consider the optimal control of quantum systems interacting non-linearly
with an electromagnetic field. We propose new monotonically convergent
algorithms to solve the optimal equations. The monotonic behavior of the
algorithm is ensured by a non-standard choice of the cost which is not
quadratic in the field. These algorithms can be constructed for pure and
mixed-state quantum systems. The efficiency of the method is shown numerically
on molecular orientation with a non-linearity of order 3 in the field.
Discretizing the amplitude and the phase of the Fourier transform of the
optimal field, we show that the optimal solution can be well-approximated by
pulses that could be implemented experimentally.Comment: 24 pages, 11 figure
Non-commutative flux representation for loop quantum gravity
The Hilbert space of loop quantum gravity is usually described in terms of
cylindrical functionals of the gauge connection, the electric fluxes acting as
non-commuting derivation operators. It has long been believed that this
non-commutativity prevents a dual flux (or triad) representation of loop
quantum gravity to exist. We show here, instead, that such a representation can
be explicitly defined, by means of a non-commutative Fourier transform defined
on the loop gravity state space. In this dual representation, flux operators
act by *-multiplication and holonomy operators act by translation. We describe
the gauge invariant dual states and discuss their geometrical meaning. Finally,
we apply the construction to the simpler case of a U(1) gauge group and compare
the resulting flux representation with the triad representation used in loop
quantum cosmology.Comment: 12 pages, matches published versio
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