378 research outputs found
On discrete integrable equations with convex variational principles
We investigate the variational structure of discrete Laplace-type equations
that are motivated by discrete integrable quad-equations. In particular, we
explain why the reality conditions we consider should be all that are
reasonable, and we derive sufficient conditions (that are often necessary) on
the labeling of the edges under which the corresponding generalized discrete
action functional is convex. Convexity is an essential tool to discuss
existence and uniqueness of solutions to Dirichlet boundary value problems.
Furthermore, we study which combinatorial data allow convex action functionals
of discrete Laplace-type equations that are actually induced by discrete
integrable quad-equations, and we present how the equations and functionals
corresponding to (Q3) are related to circle patterns.Comment: 39 pages, 8 figures. Revision of the whole manuscript, reorder of
sections. Major changes due to additional reality conditions for (Q3) and
(Q4): new Section 2.3; Theorem 1 and Sections 3.5, 3.6, 3.7 update
Integrable discrete nets in Grassmannians
We consider discrete nets in Grassmannians which generalize
Q-nets (maps with planar elementary
quadrilaterals) and Darboux nets (-valued maps defined on the
edges of such that quadruples of points corresponding to
elementary squares are all collinear). We give a geometric proof of
integrability (multidimensional consistency) of these novel nets, and show that
they are analytically described by the noncommutative discrete Darboux system.Comment: 10 p
A constructive approach to the soliton solutions of integrable quadrilateral lattice equations
Scalar multidimensionally consistent quadrilateral lattice equations are
studied. We explore a confluence between the superposition principle for
solutions related by the Backlund transformation, and the method of solving a
Riccati map by exploiting two kn own particular solutions. This leads to an
expression for the N-soliton-type solutions of a generic equation within this
class. As a particular instance we give an explicit N-soliton solution for the
primary model, which is Adler's lattice equation (or Q4).Comment: 22 page
Deuteron Photodissociation in Ultraperipheral Relativistic Heavy-Ion on Deuteron Collisions
In ultraperipheral relativistic deuteron on heavy-ion collisions, a photon
emitted from the heavy nucleus may dissociate the deuterium ion. We find
deuterium breakup cross sections of 1.38 barns for deuterium-gold collisions at
a center of mass energy of 200 GeV per nucleon, as studied at the Relativistic
Heavy Ion Collider, and 2.49 barns for deuterium-lead collisions at a center of
mass energy of 6.2 TeV, as proposed for the Large Hadron Collider. This cross
section includes an energy-independent 140 mb contribution from hadronic
diffractive dissociation. At the LHC, the cross section is as large as that of
hadronic interactions. The estimated error is 5%. Deuteron dissociation could
be used as a luminosity monitor and a `tag' for moderate impact parameter
collisions.Comment: Final version, to appear in Phys. Rev. C. Diffractive dissociation
included 10 pages with 3 figure
The curious nonexistence of Gaussian 2-designs
2-designs -- ensembles of quantum pure states whose 2nd moments equal those
of the uniform Haar ensemble -- are optimal solutions for several tasks in
quantum information science, especially state and process tomography. We show
that Gaussian states cannot form a 2-design for the continuous-variable
(quantum optical) Hilbert space L2(R). This is surprising because the affine
symplectic group HWSp (the natural symmetry group of Gaussian states) is
irreducible on the symmetric subspace of two copies. In finite dimensional
Hilbert spaces, irreducibility guarantees that HWSp-covariant ensembles (such
as mutually unbiased bases in prime dimensions) are always 2-designs. This
property is violated by continuous variables, for a subtle reason: the
(well-defined) HWSp-invariant ensemble of Gaussian states does not have an
average state because the averaging integral does not converge. In fact, no
Gaussian ensemble is even close (in a precise sense) to being a 2-design. This
surprising difference between discrete and continuous quantum mechanics has
important implications for optical state and process tomography.Comment: 9 pages, no pretty figures (sorry!
On the pion electroproduction amplitude
We analyze amplitudes for the pion electroproduction on proton derived from
Lagrangians based on the local chiral SU(2) x SU(2) symmetries. We show that
such amplitudes do contain information on the nucleon axial form factor F_A in
both soft and hard pion regimes. This result invalidates recent Haberzettl's
claim that the pion electroproduction at threshold cannot be used to extract
any information regarding F_A.Comment: 14 pages, 6 figures, revised version, accepted for publication in
Phys. Rev.
Systems of Hess-Appel'rot type
We construct higher-dimensional generalizations of the classical
Hess-Appel'rot rigid body system. We give a Lax pair with a spectral parameter
leading to an algebro-geometric integration of this new class of systems, which
is closely related to the integration of the Lagrange bitop performed by us
recently and uses Mumford relation for theta divisors of double unramified
coverings. Based on the basic properties satisfied by such a class of systems
related to bi-Poisson structure, quasi-homogeneity, and conditions on the
Kowalevski exponents, we suggest an axiomatic approach leading to what we call
the "class of systems of Hess-Appel'rot type".Comment: 40 pages. Comm. Math. Phys. (to appear
Diluting Cosmological Constant In Infinite Volume Extra Dimensions
We argue that the cosmological constant problem can be solved in a braneworld
model with infinite-volume extra dimensions, avoiding no-go arguments
applicable to theories that are four-dimensional in the infrared. Gravity on
the brane becomes higher-dimensional at super-Hubble distances, which entails
that the relation between the acceleration rate and vacuum energy density flips
upside down compared to the conventional one. The acceleration rate decreases
with increasing the energy density. The experimentally acceptable rate is
obtained for the energy density larger than (1 TeV). The results are stable
under quantum corrections because supersymmetry is broken only on the brane and
stays exact in the bulk of infinite volume extra space. Consistency of 4D
gravity and cosmology on the brane requires the quantum gravity scale to be
around eV. Testable predictions emerging within this approach are:
(i) simultaneous modifications of gravity at sub-millimeter and the Hubble
scales; (ii) Hagedorn-type saturation in TeV energy collisions due to the Regge
spectrum with the spacing equal to eV.Comment: 36 pages, 1 eps fig; 4 refs and comment adde
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