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On the symmetry of epsilon factors for GL_n
Get PDFWe show how the epsilon factors for GL_n factor, as finite morphisms of algebraic varieties, through the corresponding extended quotients. The finite morphisms are, up to a constant, rational characters of complex tori
On finite soluble groups with almost fixed-point-free automorphisms of non-coprime order
Get PDFIt is proved that if a finite -soluble group admits an automorphism of order having at most fixed points on every \f-invariant elementary abelian -section of , then the -length of is bounded above in terms of and ; if in addition the group is soluble, then the Fitting height of is bounded above in terms of and . It is also proved that if a finite soluble group admits an automorphism of order for some primes , then the Fitting height of is bounded above in terms of and
Computing the Action of Trigonometric and Hyperbolic Matrix Functions
Get PDFWe derive a new algorithm for computing the action of the cosine,
sine, hyperbolic cosine, and
hyperbolic sine of a matrix on a matrix ,
without first computing .
The algorithm can compute
and simultaneously, and likewise for
and ,
and it uses only real arithmetic when is real.
The algorithm exploits an existing algorithm \texttt{expmv}
of Al-Mohy and Higham for
and its underlying backward error analysis.
Our experiments show that the new algorithm performs in
a forward stable manner and is generally significantly faster than
alternatives based on multiple invocations of \texttt{expmv}
through formulas such as
A formula for the Frechet derivative of a generalized matrix function
Get PDFWe state and prove an extension of the Daleckii-Krein theorem, thus obtaining an
explicit formula for the Frechet derivative of generalized matrix functions. Moreover,
we prove the differentiability of generalized matrix functions of real matrices under very
mild assumptions. For complex matrices, we argue that generalized matrix functions
are real differentiable but generally not complex differentiable. Finally, we discuss the
application of our result to the study of the condition number of generalized matrix
functions. Along our way, we also derive generalized matrix functional analogues of
a few classical theorems on polynomial interpolation of classical matrix functions and
their derivatives
The Semisimple Elements of E_8(2)
Get PDFIn this paper we determine detailed information on the conjugacy classes and centralizers of semisimple elements in the exceptional Lie-type group E_8(2)
Characterization of objects by electrosensing fish based on the first order polarization tensor
Get PDFWeakly electric fish generate electric current and use hundreds of voltage
sensors on the surface of their body to navigate and locate food. Experiments [G.
von der Emde and S. Fetz, J. Exp Biol, 210, 3082�3095, 2007] show that they can
discriminate between differently shaped conducting or insulating objects by using
electrosensing. One approach to electrically identify and characterize the object with a
lower computational cost rather than full shape reconstruction is to use the first order
Polarization Tensor (PT) of the object.
In this paper, by considering experimental work on Peters� elephantnose fish
Gnathonemus petersii, we investigate the possible role of the first order PT in the
ability of the fish to discriminate between objects of different shape. We also suggest
some experiments that might be performed to further investigate the role of the first
order PT in electrosensing fish. Finally, we speculate on the possibility of electrical
cloaking or camouflage in prey of electrosensing fish and what might be learnt from
the fish in human remote sensing
Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms
Get PDFTheoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and principal values \acos, \asin, \acosh, and \asinh are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. New results include a ``round trip'' formula that relates to and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Pad\'e approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be superior to computing these functions via logarithmic formulas
Matching Exponential-Based and Resolvent-Based Centrality Measures
Get PDFThe relative importance of nodes in a network can be quantified via functions of the adjacency matrix. Two popular choices of function are the exponential, which is parameter-free, and the resolvent function, which yields the Katz centrality measure. Katz centrality can be the more computationally efficient, especially for large directed networks, and has the benefit of generalizing naturally to time-dependent network sequences, but it depends on a parameter. We give a prescription for selecting the Katz parameter based on the objective of matching the centralities of the exponential counterpart. For our new choice of parameter the resolvent can be very ill conditioned, but we argue that the centralities computed in floating point arithmetic can nevertheless reliably be used for ranking. Experiments on \revised{six} real networks show that the new choice of Katz parameter leads to rankings of nodes that \revised{generally} match those from the exponential centralities well in practice
An Equivalent of Language Invariance
Get PDFWe give a simpler equivalent of the Principle of Language Invariance within the framework of Pure Inductive Logic which is more evidently rational
Introduction to the dynamics of piecewise smooth maps
Get PDFThese are preliminary notes for the Advanced Course on Piecewise Smooth Dynamical Systems
11--15 April 2016 at the CRM, Barcelona
as part of the CRM Intensive Research Program in Nonsmooth Dynamics