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A Schur--Pad\'e Algorithm for Fractional Powers of a Matrix
No full textA new algorithm is developed for computing arbitrary real powers of a matrix . The algorithm starts with a Schur decomposition, takes square roots of the triangular factor , evaluates an Pad\'e approximant of at , and squares the result times. The parameters and are chosen to minimize the cost subject to achieving double precision accuracy in the evaluation of the Pad\'e approximant, making use of a result that bounds the error in the matrix Pad\'e approximant by the error in the scalar Pad\'e approximant with argument the norm of the matrix. The Pad\'e approximant is evaluated from the continued fraction representation in bottom-up fashion, which is shown to be numerically stable. In the squaring phase the diagonal and first superdiagonal are computed from explicit formulae for , yielding increased accuracy. Since the basic algorithm is designed for , a criterion for reducing an arbitrary real to this range is developed, making use of bounds for the condition number of the problem. How best to compute for a negative integer is also investigated. In numerical experiments the new algorithm is found to be superior in accuracy and stability to several alternatives, including the use of an eigendecomposition and approaches based on the formula
A logical characterization of coherence for imprecise probabilities
No full textWhilst supported by compelling arguments, the consensus on representing uncertainty by means of (subjective) probability is not unanimous. A substantial part of the relevant criticisms point to its alleged inadequacy for representing ignorance as opposed to uncertainty. The purpose of this paper is to show how a strong justification for taking belief as probability, namely the Dutch Book argument, can be extended naturally so as to provide a logical characterization of coherence for imprecise probability, a framework which is widely believed to accommodate some fundamental features of reasoning under ignorance. The appropriate logic for our purposes is an algebraizable logic whose equivalent algebraic semantics is a variety of MV-algebras with
an additional internal unary operation representing upper probability (these algebras will be called UMV-algebras)
Palindromic Companion Forms for Matrix Polynomials of Odd Degree
No full textThe standard way to solve polynomial eigenvalue problems P(\la)x=0
is to convert the matrix polynomial P(\la)
into a matrix pencil
that preserves its spectral information --
a process known as linearization.
When P(\la) is palindromic,
the eigenvalues, elementary divisors,
and minimal indices of P(\la)
have certain symmetries that can be lost
when using the classical first and second
Frobenius companion linearizations
for numerical computations,
since these linearizations
do not preserve the palindromic structure.
Recently new families of pencils have been introduced
with the goal of finding linearizations that retain whatever structure
the original P(\la) might possess,
with particular attention to the preservation
of palindromic structure.
However, no general construction of palindromic linearizations valid for all palindromic polynomials
has as yet been achieved.
In this paper we present a family of linearizations
for odd degree polynomials P(\la)
which are palindromic whenever P(\la) is,
and which are valid for all palindromic polynomials
of odd degree.
We illustrate our construction with several examples.
In addition, we establish a simple way to recover the minimal indices of the polynomial
from those of the linearizations in the new family
On the decidability of the real field with a generic power function
No full textWe show that the theory of the real eld with a generic real power
function is decidable, relative to an oracle for the rational cut of the
exponent of the power function. We show the existence of generic
computable real numbers, hence providing an example of a decidable
o-minimal proper expansion of the real eld by an analytic function
Solving Stochastic Collocation Systems with Algebraic Multigrid
No full textStochastic collocation methods facilitate the numerical solution of partial differential equations (PDEs) with random data and give rise to long sequences of similar discrete linear systems. When elliptic PDEs with random diffusion coefficients are discretized with standard finite element methods in the physical domain, the resulting collocation systems are symmetric and positive definite. Such systems can be solved iteratively with the conjugate gradient (cg) method and algebraic multigrid
(amg) is a highly robust preconditioner. When mixed finite
element methods are applied, amg is also a key tool for
solving the resulting sequence of saddle point systems via the preconditioned minimal residual (minres) method. In both
cases, the stochastic collocation systems are trivial to solve when considered individually. The challenge lies in exploiting the systems' similarities to recycle information and minimize the cost of solving the entire sequence.
We apply full tensor and sparse grid stochastic collocation schemes to a model stochastic elliptic problem and discretize in physical space using standard piecewise linear finite elements and Raviart-Thomas mixed finite elements. We propose efficient solvers for the resulting sequences of linear systems, that are more robust than other solution strategies in the literature. In particular, we show that it is feasible to use finely-tuned amg
preconditioning for each system if key set-up information is reused. Crucially, the preconditioners are robust with respect to variations in the discretization and statistical parameters for both stochastically linear and nonlinear diffusion coefficients
Mosaic vaccines elicit CD8+ T lymphocyte responses that confer enhanced immune coverage of diverse HIV strains in monkeys
No full textAn effective HIV vaccine must elicit immune responses that recognize genetically diverse viruses. It must generate CD8+ T lymphocytes that control HIV replication and CD4+ T lymphocytes that provide help for the generation and maintenance of both cellular and humoral immune responses against the virus. Creating immunogens that can elicit cellular immune responses against the genetically varied circulating isolates of HIV presents a key challenge for creating an HIV vaccine. Polyvalent mosaic immunogens derived by in silico recombination of natural strains of HIV are designed to induce cellular immune responses that recognize genetically diverse circulating virus isolates8. Here we immunized rhesus monkeys by plasmid DNA prime and recombinant vaccinia virus boost with vaccine constructs expressing either consensus or polyvalent mosaic proteins. As compared to consensus immunogens, the mosaic immunogens elicited CD8+ T lymphocyte responses to more epitopes of each viral protein than did the consensus immunogens and to more variant sequences of CD8+ T lymphocyte epitopes. This increased breadth and depth of epitope recognition may contribute both to protection against infection by genetically diverse viruses and to the control of variant viruses that emerge as they mutate away from recognition by cytotoxic T lymphocytes
Finite Elements and Anisotropic EIT reconstruction
Get PDFWe study the indeterminacy of the inverse problem for the finite element approximation to the anisotropic inverse problem of EIT
Episodic, transient systemic acidosis delays evolution of the malignant phenotype: Possible mechanism for cancer prevention by increased physical activity
Get PDFBackground:
The transition from premalignant to invasive tumour growth is a prolonged multistep process governed by phenotypic adaptation to changing microenvironmental selection pressures. Cancer prevention strategies are required to interrupt or delay somatic evolution of the malignant invasive phenotype. Empirical studies have consistently demonstrated that increased physical activity is highly effective in reducing the risk of breast cancer but the mechanism is unknown.
Results:
Here we propose the hypothesis that exercise-induced transient systemic acidosis will alter the in situ tumour microenvironment and delay tumour adaptation to regional hypoxia and acidosis in the later stages of carcinogenesis. We test this hypothesis using a hybrid cellular automaton approach. This model has been previously applied to somatic evolution on epithelial surfaces and demonstrated three phases of somatic evolution, with cancer cells escaping in turn from the constraints of limited space, nutrient supply and waste removal. In this paper we extend the model to test our hypothesis that transient systemic acidosis is sufficient to arrest, or at least delay, transition from in situ to invasive cancer.
Conclusions:
Model simulations demonstrate that repeated episodes of transient systemic acidosis will interrupt critical evolutionary steps in the later stages of carcinogenesis resulting in substantial delay in the evolution to the invasive phenotype. Our results suggest transient systemic acidosis may mediate the observed reduction in cancer risk associated with increased physical activity
Large particle segregation, transport and accumulation in granular free-surface flows
No full textParticle size segregation can have a significant feedback on the motion of many hazardous geophysical mass flows such as debris flows, dense pyroclastic flows and snow avalanches. This paper develops a new depth-averaged theory for segregation that can easily be incorporated into the existing depth-averaged structure of typical models of geophysical mass flows. The theory is derived by depth-averaging the segregation-remixing equation for a bi-disperse mixture of large and small particles and assuming that (i) the avalanche is always inversely graded and (ii) there is a linear downslope velocity profile through the avalanche depth. Remarkably, the resulting ‘large particle transport equation’ is very closely related to the segregation equation from which it is derived. Large particles are preferentially transported towards the avalanche front and then accumulate there. This is important, because when this is combined with mobility feedback effects, the larger less mobile particles at the front can be continuously shouldered aside to spontaneously form lateral levees that channelize the flow and enhance run-out. The theory provides a general framework that will enable segregation-mobility feedback effects to be studied in detail for the first time. While the large particle transport equation has a very simple representation of the particle size distribution, it does a surprisingly good job of capturing solutions to the full theory once the grains have segregated into inversely graded layers. In particular, we show that provided the inversely graded interface does not break it has precisely the same solution as the full theory. When the interface does break, a concentration shock forms instead of a breaking size segregation wave, but the net transport of large particles towards the flow front is exactly the same. The theory can also model more complex effects in small-scale stratification experiments, where particles may either be brought to rest by basal deposition or by the upslope propagation of a granular bore. In the former case the resulting deposit is normally graded, while in the latter case it is inversely graded. These completely opposite gradings in the deposit arise from a parent flow that is inversely graded, which raises many questions about how to interpret geological deposits