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Dynamic homogenization in periodic fibre reinforced media. Quasi-static limit for SH waves.
No full textThe effective response of a periodic fibre reinforced material to SH wave propagation is studied using the method of asymptotic homogenization, complex variable theory and multipole expansions. The quasi-static limit of the effective properties is calculated when the wavelength is much longer than the defining lengthscale of the microstructure. The method developed allows the determination of the elastic properties in the most general (monoclinic) fibre reinforced media and the resulting expressions for the effective moduli are concise. The method is therefore both more general and provides neater closed form solutions than extant methods. Results are shown to be excellent even for very high volume fractions of fibres
Multiple speeds of flame edge propagation for Lewis numbers above one
No full textEdges of diffusion flames in a counterflow burner are examined numerically for Lewis greater than unity. When the speed of propagation is plotted against Damköhler for a range of Lewis a fold bifurcation is observed. It is shown that there exist stable positively and negatively propagating edges for some Damköhler and Lewis number pairs. It is further shown that changed local conditions can lead to a transition from positive (advancing into the unburnt gasses) to negative (receding) propagation
Computational approahes to parameter estimation and model selection in immunology
No full textOne of the significant challenges in biomathematics (and other areas of science) is to formulate meaningful mathematical models. Our problem is to decide on a parametrized model which is, in some sense, most likely to represent the information in a set of observed data. In this paper, we illustrate the computational implementation of an information-theoretic approach (associated with a maximum likelihood treatment) to modelling in immunology.
The approach is illustrated by modelling LCMV infection using a family of models based on systems of ordinary differential and delay differential equations. The models (which use parameters that have a scientific interpretation) are chosen to fit data arising from experimental studies of virus-cytotoxic T lymphocyte kinetics; the parametrized models that result are arranged in a hierarchy by the computation of Akaike indices. The practical illustration is used to convey more general insight. Because the mathematical equations that comprise the models are solved numerically, the accuracy in the computation has a bearing on the outcome, and we address this and other practical details in our discussion
Onset of flow-induced fingering in bushfires
Get PDFOn a relatively large length-scale, the fire-fronts of wind-driven bushfires are sometimes seen to develop into curved shapes, suggesting that a linear fire-front becomes unstable. A mechanism for this instability can be identified if the hot plume of the fire is considered to partially block the air-flow from below, while stratification of the atmosphere resists upward displacement. Downwind of the fire this causes a speeding up of the component of the average horizontal flow in the direction of the fire-front. The perturbation in the horizontal wind that results from a perturbed shape of the fire-front shows an increase in the flow of air into the fire at more advanced parts of the front, normally resulting in an increased burning rate which would therefore increase the size of the perturbation
Lyapunov exponents for linear delay equations in arbitrary phase spaces
Get PDFA linear integral equation with infinite delay is considered where the admissible function space of initial conditions is as usually only described axiomatically. Merely using this axiomatic description, the long time behavior of the solutions is determined by calculating the Lyapunov exponents. The calculation is based on a representation of the solution in the second dual space of and on a connection between the asymptotic behavior of the solutions of the integral equation under consideration and its adjoint equation subject to the spectral decomposition of the space of initial functions. We apply the result to an example of a stochastic differential equation with infinite delay
Triangulated categories and the Ziegler spectrum
Get PDFThe relationship between the Ziegler spectrum of (the category of modules over) a ring and the Ziegler spectrum of its derived category is investigated. Over von Neumann regular rings and hereditary rings the spectrum of the derived category is a disjoint union of copies of the spectrum of the ring but in general there are further indecomposable pure-injective objects of the derived category
A change-of-variable formula with local time on curves
No full textLet be a continuous semimartingale and let be a continuous function of bounded variation. Setting suppose that a continuous function is given such that F is C1,2 on and F is on . Then the following change-of-variable formula holds: \eqalign{ F(t,X_t) = F(0,X_0)+\int_0^{t} {1 \over 2} (F_t(s, X_s+) + F_t(s,X_s-)) ds\cr + \int_0^t {1 \over 2} (F_x(s,X_s+) + F_x(s,X_s-))dX_s\cr + {1 \over 2} \int_0^t F_{xx} (s,X_s)I (X_s \neq b(s)) d \langle X, X \rangle_s\cr + {1 \over 2} \int_0^t (F_x(s,X_s+)-F_x(s,X_s-)) I(X_s = b(s)) d\ell_{s}^{b} (X),\cr} where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping
Computed eigenmodes of planar regions
Get PDFRecently developed numerical methods make possible the high-accuracy computation of eigenmodes of the Laplacian for a variety of "drums" in two dimensions. A number of computed examples are presented together with a discussion of their implications concerning bound and continuum states, isospectrality, symmetry and degeneracy, eigenvalue avoidance, resonance, localization, eigenvalue optimization, perturbation of eigenvalues and eigenvectors, and other matters
The Scaling and Squaring Method for the Matrix Exponential Revisited
Get PDFThe scaling and squaring method is the most widely used method for computing the matrix exponential, not least because it is the method implemented in MATLAB's {\tt expm} function. The method scales the matrix by a power of 2 to reduce the norm to order 1, computes a Padé approximant to the matrix exponential, and then repeatedly squares to undo the effect of the scaling. We give a new backward error analysis of the method (in exact arithmetic) that employs sharp bounds for the truncation errors and leads to an implementation of essentially optimal efficiency. We also give new rounding error analysis that shows the computed Padé approximant of the scaled matrix to be highly accurate. For IEEE double precision arithmetic the best choice of degree of Padé approximant turns out to be 13, rather than the 6 or 8 used by previous authors. Our implementation of the scaling and squaring method always requires at least two fewer matrix multiplications than {\tt expm} when the matrix norm exceeds 1, which can amount to a 37% saving in the number of multiplications, and it is typically more accurate, owing to the fewer required squarings. We also investigate a different scaling and squaring algorithm proposed by Najfeld and Havel that employs a Padé approximation to the function . This method is found to be essentially a variation of the standard one with weaker supporting error analysis
Algorithms for the Matrix p'th Root
No full textNew theoretical results are presented about the principal matrix pth root. In particular, we
show that the pth root is related to the matrix sign function and to the Wiener–Hopf factorization,
and that it can be expressed as an integral over the unit circle. These results are used
in the design and analysis of several new algorithms for the numerical computation of the pth
root. We also analyze the convergence and numerical stability properties of Newton’s method
for the inverse pth root. Preliminary computational experiments are presented to compare the
methods