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Generalizations of conjectures of Brauer and Olsson
No full textWe investigate suitable generalisations of the conjectures of Brauer and Olsson bounding numbers of irreducible characters of certain degrees in a block by invariants of the defect group
The great circle epidemic model
No full textWe consider a stochastic model for the spread of an epidemic among a population of n individuals that are equally spaced around a circle. Throughout its infectious period, a typical infective, i say, makes global contacts, with individuals chosen independently and uniformly from the whole population, and local contacts, with individuals chosen independently and uniformly according to a contact distribution centred on i. The asymptotic situation in which the local contact distribution converges weakly as n→∞ is analysed. A branching process approximation for the early stages of an epidemic is described and made rigorous as n→∞ by using a coupling argument, yielding a threshold theorem for the model. A central limit theorem is derived for the final outcome of epidemics that take off, by using an embedding representation. The results are specialised to the case of a symmetric, nearest-neighbour local contact distribution
A study of moving mesh applied to a thin flame propagating in a detonator delay element
No full textWe study the application of moving mesh methods to a one-dimensional (time dependent) detonator delay element problem. We consider moving mesh methods based on the equidistribution principle derived by Huang et al. [1]. Adaptive mesh methods have been widely used recently to solve time dependent partial differential equations having large solution gradients. Significant improvements in accuracy and efficiency are achieved by adapting the nodes (mesh points) so that they are concentrated about areas of large solution variations. Each system of equations for the moving mesh methods is solved in conjunction with the detonator problem. In this paper, the system of ordinary differential equations that results (after discretising in space) is solved using the double precision version of the stiff ordinary differential equation solver DASSL. The numerical results clearly demonstrate that the moving mesh methods are capable of tracking the deflagration wave as it travels down the detonator delay element more accurately and more efficiently than a fixed mesh method
Duality and Hermitian Galois Module Structure
No full textSuppose is either the ring of integers of a number field, the ring of integers of a -adic local field, or a field of characteristic . Let be a regular projective scheme which is flat and equidimensional over of relative dimension . Suppose is a finite group acting tamely on . Define to be the Hermitian class group of . Using the duality pairings on the de Rham cohomology groups of the fiber of over , we define a canonical invariant in . When and is either , or , we determine the image of in the adelic Hermitian classgroup by means of -constants. We also show that in this case, the image in of a closely related Hermitian Euler characteristic both determines and is determined by the -constants of the symplectic representations of
MATLAB Toolbox for Classical Matrix Groups
Get PDFWe consider structured matrix groups arising in the context of nondegenerate bilinear or sesquilinear forms on \Rn or \Cn, which remain invariant under similarities by matrices in their associated automorphism group. We develop a \M toolbox for generating random matrices from these automorphism groups with, wherever possible, prescribed condition numbers. The matrix groups considered are the complex orthogonal, real, complex and conjugate symplectic, real perplectic, real and complex pseudo-orthogonal, and pseudo-unitary groups.
We outline all necessary background theory before presenting a self-contained treatment of each group, outlining some applications of the group and deriving an algorithm for their random generation. We first focus our attention on the groups for which a structured SVD or CSD is available, and show that this allows for precise control of the condition number via numerically stable algorithms. We then consider the groups which lack such a decomposition, where we construct matrices via products of generalized \Ga-reflectors. We perform tests which model the behaviour of the condition number in these cases, allowing for its approximate control, and finally we consider the effect of rounding errors on the resultant matrices.
The implementation in \M of these algorithms is hoped to be beneficial for researchers developing structure-preserving algorithms for structured problems
Structured tools for structured matrices
No full textAn extensive and unified collection of structure-preserving
transformations is presented and organized for easy reference.
The structures involved arise in the context of a non-degenerate
bilinear or sesquilinear form on R^n or C^n. A variety of
transformations belonging to the automorphism groups of these
forms, that imitate the action of Givens rotations, Householder
reflectors, and Gauss transformations are constructed. Transformations
for performing structured scaling actions are also described.
The matrix groups considered in this paper are the complex orthogonal,
real, complex and conjugate symplectic, real perplectic, real and
complex pseudo-orthogonal, and pseudo-unitary groups. In addition
to deriving new transformations, this paper collects and unifies
existing structure-preserving tools
Delay Embeddings for Forced Systems. II. Stochastic Forcing
No full textTakens’ Embedding Theorem forms the basis of virtually all approaches
to the analysis of time series generated by nonlinear deterministic dynamical systems.
It typically allows us to reconstruct an unknown dynamical system which gave rise
to a given observed scalar time series simply by constructing a new state space out
of successive values of the time series. This provides the theoretical foundation for
many popular techniques, including those for the measurement of fractal dimensions
and Liapunov exponents, for the prediction of future behaviour, for noise reduction
and signal separation, and most recently for control and targeting. Current versions of
Takens’ Theorem assume that the underlying system is autonomous (and noise-free).
Unfortunately this is not the case for many real systems. In a previous paper, one of us
showed how to extend Takens’ Theorem to deterministically forced systems. Here, we
use similar techniques to prove a number of delay embedding theorems for arbitrarily
and stochastically forced systems. As a special case, we obtain embedding results for
Iterated Functions Systems, and we also briefly consider noisy observations
Local-global principle for the Baum-Connes conjecture with coefficients
Get PDFWe establish the Hasse principle (local-global principle) in the context of the Baum-Connes conjecture with coefficients. We illustrate this principle with the discrete group GL(2,F) where F is any global field
Equivalence of driven and aging fluctuation-dissipation relations in the trap model
No full textWe study the nonequilibrium version of the fluctuation-dissipation (FD) relation in the glass phase of a trap model that is driven into a nonequilibrium steady state by external "shear." This extends our recent study of aging FD relations in the same model, where we found limiting, observable independent FD relations for "neutral" observables that are uncorrelated with the system's average energy. In this work, for such neutral observables, we find the FD relation for a stationary weakly driven system to be the same, to within small corrections, as for an infinitely aged system. We analyze the robustness of this correspondence with respect to non-neutrality of the observable, and with respect to changes in the driving mechanism
Non-normal and stochastic amplification of magnetic energy in the turbulent dynamo: subcritical case
No full textOur attention focuses on the stochastic dynamo equation with non-normal operator that gives an insight into the role of stochastics and non-normality in magnetic field generation. The main point of this Brief Report is a discussion of the generation of a large-scale magnetic field that cannot be explained by traditional linear eigenvalue analysis. The main result is a discovery of nonlinear deterministic instability and growth of finite magnetic field fluctuations in alphabeta dynamo theory. We present a simple stochastic model for the thin-disk axisymmetric alphaOmega dynamo involving three factors: (a) non-normality generated by differential rotation, (b) nonlinearity reflecting how the magnetic field affects the turbulent dynamo coefficients, and (c) stochastic perturbations. We show that even for the subcritical case (all eigenvalues are negative), there are three possible mechanisms for the generation of magnetic field. The first mechanism is a deterministic one that describes an interplay between transient growth and nonlinear saturation of the turbulent alpha effect and diffusivity. It turns out that the trivial state is nonlinearly unstable to small but finite initial perturbations. The second and third are stochastic mechanisms that account for the interaction of non-normal effect generated by differential rotation with random additive and multiplicative fluctuations