65,024 research outputs found
Continued Fractions: A New Form
While the traditional form of continued fractions is well-documented, a new form, designed to approximate real numbers between 1 and 2, is less well-studied. This report first describes prior research into the new form, describing the form and giving an algorithm for generating approximations for a given real number. It then describes a rational function giving the rational number represented by the continued fraction made from a given tuple of integers and shows that no real number has a unique continued fraction. Next, it describes the set of real numbers that are hardest to approximate; that is, given a positive integer , it describes the real number that maximizes the value , where is the closest continued fraction to generated from a tuple of length . Finally, it lays out plans for future work
Preconditioned Locally Harmonic Residual Method for Computing Interior Eigenpairs of Certain Classes of Hermitian Matrices
We propose a Preconditioned Locally Harmonic Residual (PLHR) method for
computing several interior eigenpairs of a generalized Hermitian eigenvalue
problem, without traditional spectral transformations, matrix factorizations,
or inversions. PLHR is based on a short-term recurrence, easily extended to a
block form, computing eigenpairs simultaneously. PLHR can take advantage of
Hermitian positive definite preconditioning, e.g., based on an approximate
inverse of an absolute value of a shifted matrix, introduced in [SISC, 35
(2013), pp. A696-A718]. Our numerical experiments demonstrate that PLHR is
efficient and robust for certain classes of large-scale interior eigenvalue
problems, involving Laplacian and Hamiltonian operators, especially if memory
requirements are tight
Improved approximation of arbitrary shapes in dem simulations with multi-spheres
DEM simulations are originally made for spherical particles only. But most of real particles are anything but not spherical. Due to this problem, the multi-sphere method was invented. It provides the possibility to clump several spheres together to create complex shape structures. The proposed algorithm offers a novel method to create multi-sphere clumps for the given arbitrary shapes. Especially the use of modern clustering algorithms, from the field of computational intelligence, achieve satisfactory results. The clustering is embedded into an optimisation algorithm which uses a pre-defined criterion. A mostly unaided algorithm with only a few input and hyperparameters is able to approximate arbitrary shapes
Interpolation and Approximation of Polynomials in Finite Fields over a Short Interval from Noisy Values
Motivated by a recently introduced HIMMO key distribution scheme, we consider
a modification of the noisy polynomial interpolation problem of recovering an
unknown polynomial from approximate values of the residues of
modulo a prime at polynomially many points taken from a short
interval
Bremsstrahlung in the gravitational field of a cosmic string
In the framework of QED we investigate the bremsstrahlung process for an
electron passing by a straight static cosmic string. This process is precluded
in empty Minkowski space-time by energy and momentum conservation laws. It
happens in the presence of the cosmic string as a consequence of the conical
structure of space, in spite of the flatness of the metric. The cross section
and emitted electromagnetic energy are computed and analytic expressions are
found for different energies of the incoming electron. The energy interval is
divided in three parts depending on whether the energy is just above electron
rest mass , much larger than , or exceeds , with the
string mass per unit length in Planck units. We compare our results with those
of scalar QED and classical electrodynamics and also with conic pair production
process computed earlier.Comment: 21 pages, to appear in Phys. Rev. D., KONS-RGKU-94-0
The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry
The closest tensors of higher symmetry classes are derived in explicit form
for a given elasticity tensor of arbitrary symmetry. The mathematical problem
is to minimize the elastic length or distance between the given tensor and the
closest elasticity tensor of the specified symmetry. Solutions are presented
for three distance functions, with particular attention to the Riemannian and
log-Euclidean distances. These yield solutions that are invariant under
inversion, i.e., the same whether elastic stiffness or compliance are
considered. The Frobenius distance function, which corresponds to common
notions of Euclidean length, is not invariant although it is simple to apply
using projection operators. A complete description of the Euclidean projection
method is presented. The three metrics are considered at a level of detail far
greater than heretofore, as we develop the general framework to best fit a
given set of moduli onto higher elastic symmetries. The procedures for finding
the closest elasticity tensor are illustrated by application to a set of 21
moduli with no underlying symmetry.Comment: 48 pages, 1 figur
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