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Some measures for asymmetry of distributions
Get PDFWe propose several measures, functional and scalar, for asymmetry of distributions by comparing the behavior of probability densities to the right and left of the mode(s) and show how to generate classes of equivalent distributions from a given distribution, allowing for varying asymmetry but retaining some information theoretic properties of the original distribution, such as the entropy
THE EQUATION [x,u]+[y,v]=0 IN FREE LIE ALGEBRAS
No full textWe investigate equations of the form [x,u]+[y,v]=0 over a free Lie algebra
L. In the case where u and v are free generators of L, we
exhibit two series of solutions, we
work out the dimensions of the homogeneous components of the
solution space, and we determine its radical. In
the general case we show that the results on free generator
coefficients are sufficient to obtain the solution space up to
finite codimension. As an application we determine the radical of
the bilinear equation [x_1,x_2]+[x_3,x_4]=0
Backward Error of Polynomial Eigenproblems Solved by Linearization
Get PDFThe most widely used approach for solving the polynomial eigenvalue problem
P(\lambda)x = \bigl(\sum_{i=0}^m \l^i A_i\bigr) x = 0
in matrices is to
linearize to produce a larger order pencil ,
whose eigensystem is then found by any method for generalized eigenproblems.
For a given polynomial , infinitely many linearizations exist
and approximate eigenpairs of computed via
linearization can have widely varying backward errors.
Two main factors affect the backward error.
First, because is usually highly structured, perturbations
to cannot directly be interpreted as equivalent perturbations to .
Second, the ``short'' eigenvectors of can be recovered from the
``long'' eigenvectors of in potentially many ways,
with differing implications on the backward error for .
We show that if a certain one-sided factorization relating to can be
found then a simple formula permits recovery of right eigenvectors of from
those of ,
and the backward error of an approximate eigenpair of can be bounded in
terms of
the backward error for the corresponding approximate eigenpair of .
A similar factorization has the same implications for left eigenvectors.
We use this technique to derive backward error bounds
depending only on the norms of the for the companion pencils
and for the vector space \DL(P) of pencils recently identified by
Mackey, Mackey, Mehl, and Mehrmann.
In all cases, sufficient conditions
are identified for an optimal backward error
for .
These results are shown to be entirely consistent with those of
Higham, Mackey, and Tisseur on the conditioning of linearizations of .
Other contributions of this work are a block scaling of the companion pencils
that yields improved backward error bounds;
a demonstration that the bounds are applicable to certain structured
linearizations of structured polynomials;
and backward error bounds specialized to the quadratic case,
including analysis of the benefits of a scaling recently proposed
by Fan, Lin, and Van Dooren.
The results herein make no assumptions on the stability of the method applied to
or whether the method is direct or iterative
Quantum mushroom billiards
Get PDFWe report the first large-scale statistical study of very high-lying eigenmodes (quantum states) of the mushroom billiard proposed by L. A. Bunimovich [Chaos 11, 802 (2001)]. The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy (1.7%). We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly regular modes. Our model explains our observed density of such superpositions dying as E−1/3 (E is the eigenvalue). We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16 000 odd modes (an order of magnitude more than any existing study). We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers (~105) orders of magnitude faster than with competing methods
Definite Matrix Polynomials and their Linearization by Definite Pencils
Get PDFHyperbolic matrix polynomials
are an important class of Hermitian matrix polynomials
that contain overdamped quadratics as a special case.
They share with definite pencils the spectral property that their eigenvalues
are real and semisimple.
We extend the definition of hyperbolic matrix polynomial
in a way that relaxes the requirement of definiteness of
the leading coefficient matrix,
yielding what we call definite polynomials.
We show that this class of polynomials has an elegant characterization in terms
of definiteness intervals on the extended real line,
and that it includes definite pencils as a special case.
A fundamental question is whether a definite
matrix polynomial can be linearized
in a structure-preserving way.
We show that the answer to this question is affirmative:
is definite if and only if it has a
definite linearization in ,
a certain vector space of Hermitian pencils;
and for definite we give a complete characterization of all the
linearizations in that are definite. For the important
special case of quadratics, we show how a definite quadratic
polynomial can be transformed into a definite linearization with a
positive definite leading coefficient matrix---a form that is
particularly attractive numerically
Calculating the -Norm of Large Sparse Systems via Chandrasekhar Iterations and Extrapolations
Get PDFWe describe an algorithm for estimating the -norm of a large linear time invariant dynamical system described by a discrete time state-space model. The algorithm uses Chandrasekhar iterations to obtain an estimate of the -norm and then uses extrapolation to improve these estimates
Investigation of Properties of Some Inference Processes
Get PDFThe spectrum of Renyi inference processes in the discrete case is found to have
limits of Minimax at one end and CM∞ at the other. Another sequence of processes
is found to have the limit Maximin. Although Maximin is the dual of Minimax,
it is seen to have better characteristics when compared with Maximum Entropy
(ME) than those possessed by Minimax. The comparison of inference processes is
made using a list of desiderata which were shown by Paris/Vencovska to uniquely
characterise ME.
Algorithms are described for calculating Minimax and Maximin, which have the
advantage over ME of inferring belief values which are rational numbers when the
agent’s knowledge is itself expressed purely in terms of rational numbers.
Then Minimax and Maximin are viewed as examples of Partly Linear, or PL
inference processes. This yields a unique characterisation of Maximin. Another
inference process, Meanimax, compares well with Minimax and is a counterexample
of some plausible conjectures about certain properties of inference processes
An overview of the milestones in development of the analytic number theory from antiquity to the present day (in Russian)
Get PDFA brief review of the key results in the analytic number theory, based on Marcus du Sautoy's bestseller "The music of the primes" (2003). Provides a concise and clear description of the key formulas which link the prime counting function with Riemann's zeta function, focusing both on their mathematical and philosophical interpretation
Deflecting dams and the formation of oblique shocks in snow avalanches at Flateyri, Iceland.
No full textSnow avalanches are a threat in many populated mountainous regions, and deflecting dams are often built to divert them away from people, and infrastructure, into less harmful areas. When an avalanche is deflected by a dam or wedge, it often generates rapid changes in the flow thickness and velocity, which can be modeled as an oblique shock wave. This paper reviews classical oblique shock theory, which was originally developed for shallow water flows, and uses it to make predictions of the maximum runup height on a deflecting dam, the downstream flow velocity, and the width of the channelized stream. The theory is used to investigate field observations of snow avalanches at Flateyri in Iceland, where a dam has deflected two avalanches away from the town and produced a channelized stream that flowed parallel to the dam. The results indicate that there is no one single set of upstream flow conditions that parameterizes the flow behavior, but the solution evolves as the avalanche propagates along the dam in response to the deceleration imposed by the slope. Fully time-dependent shock capturing numerical simulations of the Skollahvilft avalanche, which hit the dam on 21 February 1999, are used to show how the channelized stream widens as the avalanche slows down and thickens toward the end of the runout zone. The oblique shock relations nevertheless provide useful local order of magnitude estimates for the flow conditions immediately upstream of the shock
Complex dynamics of shear banded flows
No full textMany complex fluids undergo a flow induced transition to a state of coexisting bands of differing viscosities and internal structuring. This effect, which is called shear banding, is widely observed in wormlike micellar surfactants, onion surfactants, colloidal suspensions and polymer solutions. According to a rapidly accumulating body of experimental evidence, shear bands often exhibit complex dynamics, which can be either oscillatory or chaotic in nature. This can be seen in the unsteady response of the bulk rheological signals, and in the motion of the interface between the bands. After giving a brief overview of this experimental evidence, we review in some detail recent efforts to address it theoretically