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More on pseudospectra for polynomial eigenvalue problems and applications in control theory
No full textDefinitions and characterizations of pseudospectra are given for rectangular matrix poly-nomials expressed in homogeneous form: P(α,β)=α^dA_d+α^{d−1}βA_{d−1}+...+β^dA_0. It is shown that problems with infinite (pseudo)eigenvalues are elegantly treated in this framework. For such problems stereographic projection onto the Riemann sphere is shown to provide a convenient way to visualize pseudospectra. Lower bounds for the distance to the nearest nonregular polynomial and the nearest uncontrollable dth order system (with equality for standard state-space systems) are obtained in terms of pseudospectra, showing that pseudospectra are a fundamental tool for reasoning about matrix polynomials in areas such as control theory. How and why to incorporate linear structure into pseudospectra is also discussed by example
Symplectic Amalgams
Get PDFThe aim of this book is the classification of symplectic amalgams - structures which are intimately related to the finite simple groups. In all there sixteen infinite families of symplectic amalgams together with 62 more exotic examples. The classification touches on many important aspects of modern group theory: * p-local analysis * the amalgam method * representation theory over finite fields; and * properties of the finite simple groups. The account is for the most part self-contained and the wealth of detail makes this book an excellent introduction to these recent developments for graduate students, as well as a valuable resource and reference for specialists in the are
Shock capturing and front tracking methods for granular avalanches
No full textShock formations are observed in granular avalanches when supercritical flow merges into a region of subcritical flow. In this paper we employ a shock-capturing numerical scheme for the one-dimensional Savage–Hutter theory of granular flow to describe this phenomenon. A Lagrangian moving mesh scheme applied to the nonconservative form of the equations reproduces smooth solutions of these free boundary problems very well, but fails when shocks are formed. A nonoscillatory central (NOC) difference scheme with TVD limiter or WENO cell reconstruction for the conservative equations is therefore introduced. For the avalanche free boundary problems it must be combined with a front-tracking method, developed here, to properly describe the margin evolution. It is found that this NOC scheme combined with the front-tracking module reproduces both the shock wave and the smooth solution accurately. A piecewise quadratic WENO reconstruction improves the smoothness of the solution near local extrema. The schemes are checked against exact solutions for (1) an upward moving shock wave, (2) the motion of a parabolic cap down an inclined plane, and (3) the motion of a parabolic cap down a curved slope ending in a flat run-out region, where a shock is formed as the avalanche comes to a halt
Rotation numbers for quasi-periodically forced monotone circle maps
Get PDFRotation numbers have played a central role in the study of (unforced) monotone circle maps. In such a case it is possible to obtain a priori bounds of the form
ρ - 1/n ≤ (1/n)(yn-y0) ≤ ρ + 1/n,
where (1/n)(yn-y0) is an estimate of the rotation number obtained from an orbit of length n with initial condition y0, and ρ is the true rotation number. This allows rotation numbers to be computed reliably and efficiently. Although Herman has proved that quasi-periodically forced circle maps also possess a well defined rotation number, independent of initial condition, the analogous bound does not appear to hold. In particular, two of the authors have recently given numerical evidence that there exist quasi-periodically forced circle maps for which yn-y0-ρn is not bounded. This renders the estimation of rotation numbers for quasiperiodically forced circle maps much more problematical. In this paper, we derive a new characterization of the rotation number for quasi-periodically forced circle maps based upon integrating iterates of an arbitrary smooth curve. This satisfies analogous bounds to above and permits us to develop improved numerical techniques for computing the rotation number. Additionally, we consider the boundedness of yn-y0-ρn. We show that if this quantity is bounded (both above and below) for one orbit, then it is bounded for all orbits.
Conversely, if for any orbit yn-y0-ρn is unbounded either above or below, then there is a residual set of orbits for which yn-y0-ρn is unbounded both above and below. In proving these results we also present a min-max characterization of the rotation number. We evaluate the performance of an algorithm based on this, and on the whole find it to be inferior to the integral based method
Multi-companion matrices
No full textIn this paper, we introduce and study the class of multi-companion matrices. They generalize companion matrices in various ways and possess a number of interesting properties. We find explicit expressions for the generalized eigenvectors of multi-companion matrices such that each generalized eigenvector depends on the corresponding eigenvalue and a number of quantities which are functionally independent of the eigenvalues of the matrix and (up to a uniqueness constraint) of each other. Moreover, we obtain a parameterization of a multi-companion matrix through the eigenvalues and these additional quantities. The number of parameters in this parameterization is equal to the number of non-trivial elements of the multi-companion matrix. The results can be applied to statistical estimation, simulation and theoretical studies of periodically correlated and multivariate time series in both discrete- and continuous-time
Weak detonations, their paths and transition to strong detonation
No full textPreviously, a quasi-steady form of the classical Rankine–Hugoniot weak
detonation has been shown to play an integral part in describing certain forms of
detonation initiation, arising during an intermediate stage between the thermal
ignition of the material and the first appearance of a strong detonation with
Zeldovich–von Neumann–D¨oring (ZND) structure. In this paper, we use a
parametric variable integration to calculate numerically the path of the weak
detonation in two important initiation scenarios, shock-induced and initial
disturbance-induced transition to detonation, via a large activation energy
induction domain model. The influence that the nature of the path may have on
the weak detonation structure is also discussed. In each case these calculations
enable us to predict how, where and when the transition to a strong detonation
with ZND structure will occur. Explanations for several phenomena observed
in both experiments and numerical studies on transition to detonation are also
uncovered by these calculations
Fluctuations and pinch-offs observed in viscous fingering
Get PDFOur experiments on viscous (Saffman-Taylor) fingering in Hele-Shaw channels reveal several phenomena that were not observed in previous experiments. At low flow rates, growing fingers undergo width fluctuations that intermittently narrow the finger as they evolve. The magnitude of these fluctuations is proportional to Ca–0.64, where Ca is the capillary number, which is proportional to the finger velocity. This relation holds for all aspect ratios studied up to the onset of tip instabilities. At higher flow rates, finger pinch-off and reconnection events are observed. These events appear to be caused by an interaction between the actively growing finger and suppressed fingers at the back of the channel. Both the fluctuation and pinch-off phenomena are robust but not explained by current theory. ©2003 American Institute of Physic
Complex structure on the smooth dual of GL(n)
Get PDFLet G denote the p-adic group GL(n). With the aid of
Langlands parameters, we equip each Bernstein component Z in the smooth dual of G with the structure of complex
algebraic variety. We prove that the periodic cyclic homology of the corresponding ideal in the Hecke algebra H(G) is isomorphic to the de Rham cohomology of Z. We show how the structure of the variety Z is related to Xi's affirmation of a conjecture of Lusztig for GL(n,C). The smooth dual of G admits a deformation retraction onto the tempered dual of G