1,391 research outputs found
On the perturbation of an -orthogonal projection
The -orthogonal projection onto a subspace is an important mathematical
tool, which has been widely applied in many fields such as linear least squares
problems, eigenvalue problems, ill-posed problems, and randomized algorithms.
In some numerical applications, the entries of a matrix will seldom be known
exactly, so it is necessary to develop some bounds to characterize the effects
of the uncertainties caused by matrix perturbation. In this paper, we establish
new perturbation bounds for the -orthogonal projection onto the column
space of a matrix, which involve upper (lower) bounds and combined upper
(lower) bounds. The new bounds contain some sharper counterparts of the
existing ones. Numerical examples are also given to illustrate our theoretical
results
An improved upper bound for the number of distinct eigenvalues of a matrix after perturbation
An upper bound for the number of distinct eigenvalues of a perturbed matrix
has been recently established by P. E. Farrell [1, Theorem 1.3]. The estimate
is the central result in Farrell's work and can be applied to estimate the
number of Krylov iterations required for solving a perturbed linear system. In
this paper, we present an improved upper bound for the number of distinct
eigenvalues of a matrix after perturbation. Furthermore, some results based on
the improved estimate are presented
New upper bounds for the spectral variation of a general matrix
Let be a normal matrix with spectrum
, and let be a perturbed matrix with spectrum
. If is still normal,
the celebrated Hoffman--Wielandt theorem states that there exists a permutation
of such that
,
where denotes the Frobenius norm of a matrix. This theorem
reveals the strong stability of the spectrum of a normal matrix. However, if
or is non-normal, the Hoffman--Wielandt theorem does not
hold in general. In this paper, we present new upper bounds for
,
provided that both and are general matrices. Some of our
estimates improve or generalize the existing ones
A combined field approach for the two-way coupling problem in the liquid evaporation
During liquid evaporation, the temperature of the liquid determines the
saturated vapor pressure above it, which controls the evaporation rate and thus
determines the liquid temperature through latent heat. Therefore, the equations
for the vapor concentration in the atmosphere and for the temperature in the
liquid are coupled and must be solved in an iterative manner. In the present
paper, a combined field approach which unifies the coupled fields into one
single field and thus makes the iteration unnecessary is proposed. The present
work will be useful in scientific and industrial processes involving liquid
evaporation and may also have more general applications to coupled field
problems in which all the fields have the same governing equation.Comment: 9 pages, 1 figur
Generalization of the Sherman-Morrison-Woodbury formula involving the Schur complement
Let and be
nonsingular matrices, and let . Explicit
expressions for the Moore-Penrose inverses of and a two-by-two block
matrix, under appropriate conditions, have been established by
Castro-Gonz\'{a}lez et al. [Linear Algebra Appl. 471 (2015) 353-368]. Based on
these results, we derive a novel expression for the Moore-Penrose inverse of
under suitable conditions, where ,
, and . In
particular, if both and are nonsingular matrices, our
expression reduces to the celebrated Sherman-Morrison-Woodbury formula.
Moreover, we extend our results to the bounded linear operators case
Effects of Sliding Speed on the Intensity of Triboluminescence in Slide contact: Experimental Measurements and Theoretical Analyses
Triboluminescence (TL) is the emission of light produced by rubbing or
striking two materials together. Here, the light emission has been observed
from the sliding contact between two disks under dry condition. The effects of
the sliding speed on the intensity of TL have been experimentally investigated.
The results show that the intensity of the emission light increases
significantly with the sliding speed. A theoretical model is also proposed and
an analytical expression is deduced for the intensity of TL in the slide
contact. The theoretical prediction is found consistent with the experimental
results. The present work may be helpful to the understanding of the mechanism
of light emission when friction
A new estimate for a quantity involving the Chebyshev polynomials of the first kind
In this paper, we establish a new estimate (including lower and upper bounds)
for an important quantity involved in the convergence analysis of smoothed
aggregation algebraic multigrid methods. The new upper bound improves the
existing ones. And our upper bound is optimal
Convergence analysis of a two-grid method for nonsymmetric positive definite problems
Multigrid is a powerful solver for large-scale linear systems arising from
discretized partial differential equations. The convergence theory of multigrid
methods for symmetric positive definite problems has been well developed over
the past decades, while, for nonsymmetric problems, such theory is still not
mature. As a foundation for multigrid analysis, two-grid convergence theory
plays an important role in motivating multigrid algorithms. Regarding two-grid
methods for nonsymmetric problems, most previous works focus on the spectral
radius of iteration matrix or rely on convergence measures that are typically
difficult to compute in practice. Moreover, the existing results are confined
to two-grid methods with exact solution of the coarse-grid system. In this
paper, we analyze the convergence of a two-grid method for nonsymmetric
positive definite problems (e.g., linear systems arising from the
discretizations of convection-diffusion equations). In the case of exact coarse
solver, we establish an elegant identity for characterizing two-grid
convergence factor, which is measured by a smoother-induced norm. The identity
can be conveniently used to derive a class of optimal restriction operators and
analyze how the convergence factor is influenced by restriction. More
generally, we present some convergence estimates for an inexact variant of the
two-grid method, in which both linear and nonlinear coarse solvers are
considered
On the perturbation of the Moore-Penrose inverse of a matrix
The Moore-Penrose inverse of a matrix has been extensively investigated and
widely applied in many fields over the past decades. One reason for the
interest is that the Moore-Penrose inverse can succinctly express some
important geometric constructions in finite-dimensional spaces, such as the
orthogonal projection onto a subspace and the linear least squares problem. In
this paper, we establish new perturbation bounds for the Moore-Penrose inverse
under the Frobenius norm, some of which are sharper than the existing ones
Effects of the Position Reversal of Friction Pairs on the Strength of Tribocharging and Tribodischarging
The friction-induced charging (i.e., tribocharging) and the following
discharging (referred here as tribodischarging) are always believed to have
negative effects on the daily life and on the industrial production. Thus, how
to inhibit the tribocharging and the tribodischarging has caused wide public
concern. Because the discharge caused by the electrical breakdown of the
ambient gas is generally accompanied with the generation of light, we
investigated here the tribocharging and the tribodischarging by observing the
light emitted during friction. We found that the position reversal of the
friction pair has a dramatic impact on the intensity of the tribo-induced
light. Experimental results show that an intense light is produced when a
stationary Al2O3 disk is sliding on a rotating SiO2 disk, but only a weak light
is observed for the case of a stationary SiO2 disk and a rotating Al2O3 disk.
This means that the process of the tribocharging and the tribodischarging can
be significantly influenced owing to the change in the relative position of the
friction couple. The experimentally measured polarities of the tribo-induced
charge on the friction surfaces further indicated that the strong discharging
occurs when the rotating surface is negatively charged. The reason for the
difference in the intensity of the tribocharging and tribodischarging can be
attributed to the combined effects of the contact potential difference and the
temperature gradient between the contacting surfaces on the charge transfer
when friction. Finally, a simple, low cost, yet effective approach, i.e., just
keep the friction partner whose surface is tribo-induced negatively charged as
the stationary one, can be utilized to suppress the intensity of the
tribocharging and the tribodischarging. This work may provide potential
applications in numerous areas of science and engineering and also in the
everyday life.Comment: 15 pages, 6 figure
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