132,571 research outputs found
On tree form-factors in (supersymmetric) Yang-Mills theory
{\it Perturbiner}, that is, the solution of field equations which is a
generating function for tree form-factors in N=3 supersymmetric
Yang-Mills theory, is studied in the framework of twistor formulation of the
N=3 superfield equations. In the case, when all one-particle asymptotic states
belong to the same type of N=3 supermultiplets (without any restriction on
kinematics), the solution is described very explicitly. It happens to be a
natural supersymmetrization of the self-dual perturbiner in non-supersymmetric
Yang-Mills theory, designed to describe the Parke-Taylor amplitudes. In the
general case, we reduce the problem to a neatly formulated algebraic geometry
problem (see Eqs(\ref{5.15i}),(\ref{5.15ii}),(\ref{5.15iii})) and propose an
iterative algorithm for solving it, however we have not been able to find a
closed-form solution. Solution of this problem would, of course, produce a
description of all tree form-factors in non-supersymmetric Yang-Mills theory as
well. In this context, the N=3 superfield formalism may be considered as a
convenient way to describe a solution of the non-supersymmetric Yang-Mills
theory, very much in the spirit of works by E.Witten \cite{Witten} and by
J.Isenberg, P.B.Yasskin and P.S.Green \cite{2}.Comment: 17 pages, Latex, the form of citation in the abstract have been
corrected by xxx.lanl.gov reques
A two-scale Stefan problem arising in a model for tree sap exudation
The study of tree sap exudation, in which a (leafless) tree generates
elevated stem pressure in response to repeated daily freeze-thaw cycles, gives
rise to an interesting multi-scale problem involving heat and multiphase
liquid/gas transport. The pressure generation mechanism is a cellular-level
process that is governed by differential equations for sap transport through
porous cell membranes, phase change, heat transport, and generation of osmotic
pressure. By assuming a periodic cellular structure based on an appropriate
reference cell, we derive an homogenized heat equation governing the global
temperature on the scale of the tree stem, with all the remaining physics
relegated to equations defined on the reference cell. We derive a corresponding
strong formulation of the limit problem and use it to design an efficient
numerical solution algorithm. Numerical simulations are then performed to
validate the results and draw conclusions regarding the phenomenon of sap
exudation, which is of great importance in trees such as sugar maple and a few
other related species. The particular form of our homogenized temperature
equation is obtained using periodic homogenization techniques with two-scale
convergence, which we investigate theoretically in the context of a simpler
two-phase Stefan-type problem corresponding to a periodic array of melting
cylindrical ice bars with a constant thermal diffusion coefficient. For this
reduced model, we prove results on existence, uniqueness and convergence of the
two-scale limit solution in the weak form, clearly identifying the missing
pieces required to extend the proofs to the fully nonlinear sap exudation
model. Numerical simulations of the reduced equations are then compared with
results from the complete sap exudation model.Comment: 35 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1411.303
On the performance of a cavity method based algorithm for the Prize-Collecting Steiner Tree Problem on graphs
We study the behavior of an algorithm derived from the cavity method for the
Prize-Collecting Steiner Tree (PCST) problem on graphs. The algorithm is based
on the zero temperature limit of the cavity equations and as such is formally
simple (a fixed point equation resolved by iteration) and distributed
(parallelizable). We provide a detailed comparison with state-of-the-art
algorithms on a wide range of existing benchmarks networks and random graphs.
Specifically, we consider an enhanced derivative of the Goemans-Williamson
heuristics and the DHEA solver, a Branch and Cut Linear/Integer Programming
based approach. The comparison shows that the cavity algorithm outperforms the
two algorithms in most large instances both in running time and quality of the
solution. Finally we prove a few optimality properties of the solutions
provided by our algorithm, including optimality under the two post-processing
procedures defined in the Goemans-Williamson derivative and global optimality
in some limit cases
A Fourier interpolation method for numerical solution of FBSDEs: Global convergence, stability, and higher order discretizations
The implementation of the convolution method for the numerical solution of
backward stochastic differential equations (BSDEs) introduced in [19] uses a
uniform space grid. Locally, this approach produces a truncation error, a space
discretization error, and an additional extrapolation error. Even if the
extrapolation error is convergent in time, the resulting absolute error may be
high at the boundaries of the uniform space grid. In order to solve this
problem, we propose a tree-like grid for the space discretization which
suppresses the extrapolation error leading to a globally convergent numerical
solution for the (F)BSDE. On this alternative grid the conditional expectations
involved in the BSDE time discretization are computed using Fourier analysis
and the fast Fourier transform (FFT) algorithm as in the initial
implementation. The method is then extended to higher-order time
discretizations of FBSDEs. Numerical results demonstrating convergence are also
presented.Comment: 28 pages, 8 figures; Previously titled 'Global convergence and
stability of a convolution method for numerical solution of BSDEs'
(1410.8595v1
A contribution to the optimal numbering of tree structures
This paper develops an automatic procedure for the optimal numbering of members and nodes in tree structures. With it the stiffness matrix is optimally conditioned either if a direct solution algorithm or a frontal one is used to solve the system of equations. In spite of its effectiveness, the procedure is strikingly simple and so is the computer program shown below
EM's Convergence in Gaussian Latent Tree Models
We study the optimization landscape of the log-likelihood function and the
convergence of the Expectation-Maximization (EM) algorithm in latent Gaussian
tree models, i.e. tree-structured Gaussian graphical models whose leaf nodes
are observable and non-leaf nodes are unobservable. We show that the unique
non-trivial stationary point of the population log-likelihood is its global
maximum, and establish that the expectation-maximization algorithm is
guaranteed to converge to it in the single latent variable case. Our results
for the landscape of the log-likelihood function in general latent tree models
provide support for the extensive practical use of maximum likelihood
based-methods in this setting. Our results for the EM algorithm extend an
emerging line of work on obtaining global convergence guarantees for this
celebrated algorithm. We show our results for the non-trivial stationary points
of the log-likelihood by arguing that a certain system of polynomial equations
obtained from the EM updates has a unique non-trivial solution. The global
convergence of the EM algorithm follows by arguing that all trivial fixed
points are higher-order saddle points
Kinetic Solvers with Adaptive Mesh in Phase Space
An Adaptive Mesh in Phase Space (AMPS) methodology has been developed for
solving multi-dimensional kinetic equations by the discrete velocity method. A
Cartesian mesh for both configuration (r) and velocity (v) spaces is produced
using a tree of trees data structure. The mesh in r-space is automatically
generated around embedded boundaries and dynamically adapted to local solution
properties. The mesh in v-space is created on-the-fly for each cell in r-space.
Mappings between neighboring v-space trees implemented for the advection
operator in configuration space. We have developed new algorithms for solving
the full Boltzmann and linear Boltzmann equations with AMPS. Several recent
innovations were used to calculate the discrete Boltzmann collision integral
with dynamically adaptive mesh in velocity space: importance sampling,
multi-point projection method, and the variance reduction method. We have
developed an efficient algorithm for calculating the linear Boltzmann collision
integral for elastic and inelastic collisions in a Lorentz gas. New AMPS
technique has been demonstrated for simulations of hypersonic rarefied gas
flows, ion and electron kinetics in weakly ionized plasma, radiation and light
particle transport through thin films, and electron streaming in
semiconductors. We have shown that AMPS allows minimizing the number of cells
in phase space to reduce computational cost and memory usage for solving
challenging kinetic problems
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