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Grazing-sliding bifurcations, the border collision normal form, and the curse of dimensionality for nonsmooth bifurcation theory
Get PDFIn this paper we show that the border collision normal form of continuous but non-differentiable discrete time maps
is affected by a curse of dimensionality: it is impossible to reduce the study
of the general case to low dimensions, since in every dimension the bifurcation produces fundamentally different attractors (contrary to the case of smooth systems). In particular we show that the -dimensional border collision normal form can have invariant sets of dimension for
integer from to . We also show that the border collision normal form is related to grazing-sliding bifurcations of
switching dynamical systems. This implies that the dynamics of these
two apparently distinct bifurcations (one for discrete time dynamics, the other for continuous time dynamics) are closely related
and hence that a similar curse of dimensionality holds for this bifurcation
Finite and Infinite Elementary Divisors of Matrix Polynomials: A Global Approach
Get PDFThere is general agreement on the definition of the finite elementary divisors of a matrix polynomial Q(\l)\in\F[\l]^{m\times n}, where \F an arbitrary field.
Regarding the elementary divisors at infinity, or infinite elementary divisors, such an agreement has not been so unanimous.
We define the infinite elementary divisors of Q(\l) to be the elementary divisors of \l^\ell Q(\l^{-1}) at 0, where is the degree of Q(\l).
We show that this is the most natural definition if one applies the usual geometric technique of using homogeneous coordinates to deal with the point at infinity.
We call our approach global because the homogeneous invariant factors of Q(\l) are defined for all points of the projective line and to distinguish it from
another possible approach that, using local rings, leads to the same conclusions
Triangularizing Quadratic Matrix Polynomials
Get PDFWe show that any regular quadratic matrix polynomial can be reduced to an upper
triangular quadratic matrix polynomial over the complex numbers
preserving the finite and infinite elementary divisors.
We characterize the real quadratic matrix polynomials that are
triangularizable over the real numbers and show that those that are
not triangularizable over the real numbers are quasi-triangularizable
with diagonal blocks of sizes and .
We also derive complex and real Schur-like theorems for linearizations of
quadratic matrix polynomials with nonsingular leading coefficients.
In particular, we show that for any monic linearization \l I+A of an
quadratic matrix polynomial,
there exists a nonsingular matrix defined in terms of
orthonormal vectors that transforms to a companion
linearization of a (quasi)-triangular quadratic matrix polynomial.
This provides the foundation for designing numerical algorithms for the
reduction of quadratic matrix polynomials to upper (quasi)-triangular form
Efficient sparse matrix multiple-vector multiplication using a bitmapped format
Get PDFThe problem of obtaining high computational throughput from sparse matrix multiple--vector multiplication routines is considered.
Current sparse matrix formats and algorithms have high bandwidth requirements and poor reuse of cache and register loaded entries, which restrict their performance.
We propose the mapped blocked row format: a bitmapped sparse matrix format that stores entries as blocks without a fill overhead, thereby offering blocking without additional storage and bandwidth overheads.
An efficient algorithm decodes bitmaps using de Bruijn sequences and minimizes the number of conditionals evaluated.
Performance is compared with that of popular formats, including vendor implementations of sparse BLAS.
Our sparse matrix multiple-vector multiplication algorithm achieves high throughput on all platforms and is implemented using platform neutral optimizations
Predicting the pressure-volume curve of an elastic microsphere composite
Get PDFThe effective macroscopic response of nonlinear elastomeric inhomogeneous materials is of great interest in many applications including nonlinear composite materials and soft biological tissues. The interest of the present work is associated with a microsphere composite material, which is modelled as a matrix-inclusion composite. The matrix phase is a homogeneous isotropic
nonlinear rubber-like material and the inclusion phase is more complex, consisting of a distribution of sizes of stiff thin spherical shells filled with gas. Experimentally, such materials have been shown to undergo complex deformation under cyclic loading. Here, we consider microspheres embedded in an unbounded host material and assume that a hydrostatic pressure is applied in the "far-field". Taking into account a variety of effects including buckling of the spherical shells, large deformation of the host phase and evolving microstructure, we derive a model predicting the pressure-relative volume change load curves. Nonlinear constitutive behaviour of the matrix medium is accounted for by employing neo-Hookean and Mooney-Rivlin incompressible models. Moreover a nearly-incompressible solution is derived via asymptotic analysis
for a spherical cavity embedded in un unbounded isotropic homogeneous hyperelastic medium loaded hydrostatically. The load-curve predictions reveal a strong dependence on the microstructure of the composite, including distribution of microspheres, the stiffness of the shells, and on the initial volume fraction of the inclusions, whereas there is only a modest dependence on the characteristic properties of the nonlinear elastic model used for the rubber host
Fitting height of a finite group with a Frobenius group of automorphisms
Get PDFSuppose that a
finite group admits a Frobenius
group of automorphisms with kernel and complement such that acts without
non-trivial fixed points (that is, such that ). It is proved that the Fitting height of is
equal to the Fitting height of the fixed-point subgroup and the Fitting series
of coincides with the intersections of with the Fitting series of .
As a corollary, it is also proved that for any set of primes
the -length of is equal to the -length of
Geometric structure and the local Langlands conjecture
Get PDFWe prove that a strengthened form of the local Langlands conjecture is valid throughout the principal series of any connected split reductive -adic group. The method of proof is to establish the presence of a very simple geometric structure, in both the smooth dual and the Langlands parameters. We prove that this geometric structure is present, in the same way, for the general linear group, including all of its inner forms. With these results as evidence, we give a detailed formulation of a general geometric structure conjecture
Critical path statistics of max-plus linear systems with Gaussian noise
No full textThe critical paths of a max-plus linear systems with noise are random variables. In this paper we introduce the edge criticalities which measure how often the critical paths traverse each edge in the precedence graph. We also present the parallel path approximation, a novel method for approximating these new statistics as well as the previously studied max-plus exponent. We show that for low amplitude noise the critical paths spend most of their time traversing the deterministic maximally weighted cycle and that as the noise amplitude is increased the critical paths become more random and their distribution over the edges in the precedence graph approaches a highly uniform measure of maximal entropy
A spatially adaptive iterative method for a class of nonlinear operator eigenproblems
Get PDFWe present a new algorithm for the iterative solution of nonlinear operator eigenvalue problems arising from partial differential equations. This algorithm combines automatic spatial resolution of linear operators with the infinite Arnoldi method for nonlinear matrix eigenproblems proposed in [E. JARLEBRING, W. MICHIELS, AND K. MEERBERGEN, A linear eigenvalue algorithm for the nonlinear eigenvalue problem, Numer. Math., 122 (2012)]. The iterates in this infinite Arnoldi method are functions, and each iteration requires the solution of an inhomogeneous differential equation. This formulation is independent of the spatial representation of the functions, which allows us to employ a dynamic representation with an accuracy of about the level of machine precision at each iteration, similar to what is done in the Chebfun system [Z. BATTLES AND L. N. TREFETHEN, An extension of MATLAB to continuous functions and operators, SIAM J. Sci. Comput., 25 (2004)] with its chebop functionality [T. A. DRISCOLL, F. BORNEMANN, AND L. N. TREFETHEN, The chebop system for automatic solution of differential equations, BIT, 48 (2008)], although our function representation is entirely based on coefficients instead of function values. Our approach also allows for nonlinearities in the boundary conditions of the PDE. The algorithm is illustrated with several examples, e.g., the study of eigenvalues of a vibrating string with delayed boundary feedback control
The hyperbolic Schur decomposition
No full textWe propose a hyperbolic counterpart of the Schur decomposition, with the emphasis on the preservation of structures related to some given hyperbolic scalar product. We give results regarding the existence of such a decomposition and research the properties of its block triangular factor for various structured matrices