6,321 research outputs found
An Overview of Polynomially Computable Characteristics of Special Interval Matrices
It is well known that many problems in interval computation are intractable,
which restricts our attempts to solve large problems in reasonable time. This
does not mean, however, that all problems are computationally hard. Identifying
polynomially solvable classes thus belongs to important current trends. The
purpose of this paper is to review some of such classes. In particular, we
focus on several special interval matrices and investigate their convenient
properties. We consider tridiagonal matrices, {M,H,P,B}-matrices, inverse
M-matrices, inverse nonnegative matrices, nonnegative matrices, totally
positive matrices and some others. We focus in particular on computing the
range of the determinant, eigenvalues, singular values, and selected norms.
Whenever possible, we state also formulae for determining the inverse matrix
and the hull of the solution set of an interval system of linear equations. We
survey not only the known facts, but we present some new views as well
On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow
In this article, we study the axisymmetric surface diffusion flow (ASD), a
fourth-order geometric evolution law. In particular, we prove that ASD
generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older
regular surfaces of revolution embedded in R^3 and satisfying periodic boundary
conditions. We also give conditions for global existence of solutions and prove
that solutions are real analytic in time and space. Further, we investigate the
geometric properties of solutions to ASD. Utilizing a connection to
axisymmetric surfaces with constant mean curvature, we characterize the
equilibria of ASD. Then, focusing on the family of cylinders, we establish
results regarding stability, instability and bifurcation behavior, with the
radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana
Ultraspinning instability: the missing link
We study linearized perturbations of Myers-Perry black holes in d=7, with two
of the three angular momenta set to be equal, and show that instabilities
always appear before extremality. Analogous results are expected for all higher
odd d. We determine numerically the stationary perturbations that mark the
onset of instability for the modes that preserve the isometries of the
background. The onset is continuously connected between the previously studied
sectors of solutions with a single angular momentum and solutions with all
angular momenta equal. This shows that the near-extremality instabilities are
of the same nature as the ultraspinning instability of d>5 singly-spinning
solutions, for which the angular momentum is unbounded. Our results raise the
question of whether there are any extremal Myers-Perry black holes which are
stable in d>5.Comment: 19 pages. 1 figur
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