939 research outputs found
Computing the common zeros of two bivariate functions via Bezout resultants
The common zeros of two bivariate functions can be computed by finding the common zeros of their polynomial interpolants expressed in a tensor Chebyshev basis. From here we develop a bivariate rootfinding algorithm based on the hidden variable resultant method and B�ezout matrices with polynomial entries. Using techniques including domain subdivision, B�ezoutian regularization and local refinement we are able to reliably and accurately compute the simple common zeros of two smooth functions with polynomial interpolants of very high degree (� 1000). We analyze the resultant method and its conditioning by noting that the B�ezout matrices are matrix polynomials. Our robust algorithm is implemented in the roots command in Chebfun2, a software package written in object-oriented MATLAB for computing with bivariate functions
The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions
Fundamental global similarity solutions of the standard form
u_\g(x,t)=t^{-\a_\g} f_\g(y), with the rescaled variable y= x/{t^{\b_\g}},
\b_\g= \frac {1-n \a_\g}{10}, where \a_\g>0 are real nonlinear eigenvalues (\g
is a multiindex in R^N) of the tenth-order thin film equation (TFE-10) u_{t} =
\nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, n>0, are studied. The
present paper continues the study began by the authors in the previous paper
P. Alvarez-Caudevilla, J.D.Evans, and V.A. Galaktionov, The Cauchy problem
for a tenth-order thin film equation I. Bifurcation of self-similar oscillatory
fundamental solutions, Mediterranean Journal of Mathematics, No. 4, Vol. 10
(2013), 1759-1790.
Thus, the following questions are also under scrutiny:
(I) Further study of the limit n \to 0, where the behaviour of finite
interfaces and solutions as y \to infinity are described. In particular, for
N=1, the interfaces are shown to diverge as follows: |x_0(t)| \sim 10 \left(
\frac{1}{n}\sec\left( \frac{4\pi}{9} \right) \right)^{\frac 9{10}} t^{\frac
1{10}} \to \infty as n \to 0^+.
(II) For a fixed n \in (0, \frac 98), oscillatory structures of solutions
near interfaces.
(III) Again, for a fixed n \in (0, \frac 98), global structures of some
nonlinear eigenfunctions \{f_\g\}_{|\g| \ge 0} by a combination of numerical
and analytical methods
Homogeneous Second-Order Descent Framework: A Fast Alternative to Newton-Type Methods
This paper proposes a homogeneous second-order descent framework (HSODF) for
nonconvex and convex optimization based on the generalized homogeneous model
(GHM). In comparison to the Newton steps, the GHM can be solved by extremal
symmetric eigenvalue procedures and thus grant an advantage in ill-conditioned
problems. Moreover, GHM extends the ordinary homogeneous model (OHM) to allow
adaptiveness in the construction of the aggregated matrix. Consequently, HSODF
is able to recover some well-known second-order methods, such as trust-region
methods and gradient regularized methods, while maintaining comparable
iteration complexity bounds. We also study two specific realizations of HSODF.
One is adaptive HSODM, which has a parameter-free global
complexity bound for nonconvex second-order Lipschitz continuous objective
functions. The other one is homotopy HSODM, which is proven to have a global
linear rate of convergence without strong convexity. The efficiency of our
approach to ill-conditioned and high-dimensional problems is justified by some
preliminary numerical results.Comment: improved writin
Spectral Invariants of Operators of Dirac Type on Partitioned Manifolds
We review the concepts of the index of a Fredholm operator, the spectral flow
of a curve of self-adjoint Fredholm operators, the Maslov index of a curve of
Lagrangian subspaces in symplectic Hilbert space, and the eta invariant of
operators of Dirac type on closed manifolds and manifolds with boundary. We
emphasize various (occasionally overlooked) aspects of rigorous definitions and
explain the quite different stability properties. Moreover, we utilize the heat
equation approach in various settings and show how these topological and
spectral invariants are mutually related in the study of additivity and
nonadditivity properties on partitioned manifolds.Comment: 131 pages, 9 figure
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