178,600 research outputs found
Path Gain Algebraic Formulation for the Scalar Linear Network Coding Problem
In the algebraic view, the solution to a network coding problem is seen as a
variety specified by a system of polynomial equations typically derived by
using edge-to-edge gains as variables. The output from each sink is equated to
its demand to obtain polynomial equations. In this work, we propose a method to
derive the polynomial equations using source-to-sink path gains as the
variables. In the path gain formulation, we show that linear and quadratic
equations suffice; therefore, network coding becomes equivalent to a system of
polynomial equations of maximum degree 2. We present algorithms for generating
the equations in the path gains and for converting path gain solutions to
edge-to-edge gain solutions. Because of the low degree, simplification is
readily possible for the system of equations obtained using path gains. Using
small-sized network coding problems, we show that the path gain approach
results in simpler equations and determines solvability of the problem in
certain cases. On a larger network (with 87 nodes and 161 edges), we show how
the path gain approach continues to provide deterministic solutions to some
network coding problems.Comment: 12 pages, 6 figures. Accepted for publication in IEEE Transactions on
Information Theory (May 2010
Equivariant Schr\"odinger Maps in two spatial dimensions
We consider equivariant solutions for the Schr\"odinger map problem from
to with energy less than and show that
they are global in time and scatter
Construction of a Lax Pair for the -Painlev\'e System
We construct a Lax pair for the -Painlev\'e system from first
principles by employing the general theory of semi-classical orthogonal
polynomial systems characterised by divided-difference operators on discrete,
quadratic lattices [arXiv:1204.2328]. Our study treats one special case of such
lattices - the -linear lattice - through a natural generalisation of the big
-Jacobi weight. As a by-product of our construction we derive the coupled
first-order -difference equations for the -Painlev\'e
system, thus verifying our identification. Finally we establish the
correspondences of our result with the Lax pairs given earlier and separately
by Sakai and Yamada, through explicit transformations
Exact boundary observability for nonautonomous quasilinear wave equations
By means of a direct and constructive method based on the theory of
semiglobal solution, the local exact boundary observability is shown for
nonautonomous 1-D quasilinear wave equations. The essential difference between
nonautonomous wave equations and autonomous ones is also revealed.Comment: 18 pages, 5 figure
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