9,696 research outputs found
Continuous and discrete Clebsch variational principles
The Clebsch method provides a unifying approach for deriving variational
principles for continuous and discrete dynamical systems where elements of a
vector space are used to control dynamics on the cotangent bundle of a Lie
group \emph{via} a velocity map. This paper proves a reduction theorem which
states that the canonical variables on the Lie group can be eliminated, if and
only if the velocity map is a Lie algebra action, thereby producing the
Euler-Poincar\'e (EP) equation for the vector space variables. In this case,
the map from the canonical variables on the Lie group to the vector space is
the standard momentum map defined using the diamond operator. We apply the
Clebsch method in examples of the rotating rigid body and the incompressible
Euler equations. Along the way, we explain how singular solutions of the EP
equation for the diffeomorphism group (EPDiff) arise as momentum maps in the
Clebsch approach. In the case of finite dimensional Lie groups, the Clebsch
variational principle is discretised to produce a variational integrator for
the dynamical system. We obtain a discrete map from which the variables on the
cotangent bundle of a Lie group may be eliminated to produce a discrete EP
equation for elements of the vector space. We give an integrator for the
rotating rigid body as an example. We also briefly discuss how to discretise
infinite-dimensional Clebsch systems, so as to produce conservative numerical
methods for fluid dynamics
Variational Principles for Lagrangian Averaged Fluid Dynamics
The Lagrangian average (LA) of the ideal fluid equations preserves their
transport structure. This transport structure is responsible for the Kelvin
circulation theorem of the LA flow and, hence, for its convection of potential
vorticity and its conservation of helicity.
Lagrangian averaging also preserves the Euler-Poincar\'e (EP) variational
framework that implies the LA fluid equations. This is expressed in the
Lagrangian-averaged Euler-Poincar\'e (LAEP) theorem proven here and illustrated
for the Lagrangian average Euler (LAE) equations.Comment: 23 pages, 3 figure
An integrable shallow water equation with peaked solitons
We derive a new completely integrable dispersive shallow water equation that
is biHamiltonian and thus possesses an infinite number of conservation laws in
involution. The equation is obtained by using an asymptotic expansion directly
in the Hamiltonian for Euler's equations in the shallow water regime. The
soliton solution for this equation has a limiting form that has a discontinuity
in the first derivative at its peak.Comment: LaTeX file. Figure available from authors upon reques
Finite Size Polyelectrolyte Bundles at Thermodynamic Equilibrium
We present the results of extensive computer simulations performed on
solutions of monodisperse charged rod-like polyelectrolytes in the presence of
trivalent counterions. To overcome energy barriers we used a combination of
parallel tempering and hybrid Monte Carlo techniques. Our results show that for
small values of the electrostatic interaction the solution mostly consists of
dispersed single rods. The potential of mean force between the polyelectrolyte
monomers yields an attractive interaction at short distances. For a range of
larger values of the Bjerrum length, we find finite size polyelectrolyte
bundles at thermodynamic equilibrium. Further increase of the Bjerrum length
eventually leads to phase separation and precipitation. We discuss the origin
of the observed thermodynamic stability of the finite size aggregates
Leray and LANS- modeling of turbulent mixing
Mathematical regularisation of the nonlinear terms in the Navier-Stokes
equations provides a systematic approach to deriving subgrid closures for
numerical simulations of turbulent flow. By construction, these subgrid
closures imply existence and uniqueness of strong solutions to the
corresponding modelled system of equations. We will consider the large eddy
interpretation of two such mathematical regularisation principles, i.e., Leray
and LANS regularisation. The Leray principle introduces a {\bfi
smoothed transport velocity} as part of the regularised convective
nonlinearity. The LANS principle extends the Leray formulation in a
natural way in which a {\bfi filtered Kelvin circulation theorem},
incorporating the smoothed transport velocity, is explicitly satisfied. These
regularisation principles give rise to implied subgrid closures which will be
applied in large eddy simulation of turbulent mixing. Comparison with filtered
direct numerical simulation data, and with predictions obtained from popular
dynamic eddy-viscosity modelling, shows that these mathematical regularisation
models are considerably more accurate, at a lower computational cost.Comment: 42 pages, 12 figure
On Berenstein-Douglas-Seiberg Duality
I review the proposal of Berenstein-Douglas for a completely general
definition of Seiberg duality. To give evidence for their conjecture I present
the first example of a physical dual pair and explicitly check that it
satisfies the requirements. Then I explicitly show that a pair of toric dual
quivers is also dual according to their proposal. All these computations go
beyond tilting modules, and really work in the derived category. I introduce
all necessary mathematics where needed.Comment: 22 pages, LaTe
Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications
We develop the necessary tools, including a notion of logarithmic derivative
for curves in homogeneous spaces, for deriving a general class of equations
including Euler-Poincar\'e equations on Lie groups and homogeneous spaces.
Orbit invariants play an important role in this context and we use these
invariants to prove global existence and uniqueness results for a class of PDE.
This class includes Euler-Poincar\'e equations that have not yet been
considered in the literature as well as integrable equations like Camassa-Holm,
Degasperis-Procesi, CH and DP equations, and the geodesic equations
with respect to right invariant Sobolev metrics on the group of diffeomorphisms
of the circle
Signalment risk factors for cutaneous and renal glomerular vasculopathy (Alabama rot) in dogs in the UK
Seasonal outbreaks of cutaneous and renal glomerular vasculopathy (CRGV) have been reported annually in UK dogs since 2012, yet the aetiology of the disease remains unknown. The objectives of this study were to explore whether any breeds had an increased or decreased risk of being diagnosed with CRGV, and to report on age and sex distributions of CRGV cases occurring in the UK. Multivariable logistic regression was used to compare 101 dogs diagnosed with CRGV between November 2012 and May 2017 with a denominator population of 446,453 dogs from the VetCompass database. Two Kennel Club breed groups—hounds (odds ratio (OR) 10.68) and gun dogs (OR 9.69)—had the highest risk of being diagnosed with CRGV compared with terriers, while toy dogs were absent from among CRGV cases. Females were more likely to be diagnosed with CRGV (OR 1.51) as were neutered dogs (OR 3.36). As well as helping veterinarians develop an index of suspicion for the disease, better understanding of the signalment risk factors may assist in the development of causal models for CRGV and help identify the aetiology of the disease
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