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On the length of finite groups and of fixed points
Get PDFThe generalized Fitting height of a finite group is
the least number such that , where the is the generalized Fitting series: and is the inverse image of . It is proved that if admits a soluble group of automorphisms of coprime order, then is bounded in terms of , where is the fixed-point subgroup, and the number of prime factors of counting multiplicities. The result follows from the special case when is of prime order, where it is proved that .
The nonsoluble length of a finite group is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if is a group of automorphisms of of coprime order, then is bounded in terms of and the number of prime factors of counting multiplicities
On the length of finite factorized groups
Get PDFThe nonsoluble length~ of a finite group~ is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group~ is the least number such that , where is the generalized Fitting subgroup, and is the inverse image of . It is proved that if a finite group is factorized by two subgroups of coprime orders, then the nonsoluble length of~ is bounded in terms of the generalized Fitting heights of~ and~. It is also proved that if, say, is soluble of derived length~, then the generalized Fitting height of~ is bounded in terms of~ and the generalized Fitting height of~
Classification of Boundary Equilibrium Bifurcations in planar Filippov systems
Get PDFIf a family of piecewise smooth systems depending on a real parameter is defined on two different regions of the plane separated by a switching surface then a boundary equilibrium bifurcation occurs if a stationary point of one of the systems intersects the switching surface at a critical value of the parameter. We derive the leading order terms of a normal form for boundary equilibrium bifurcations of planar systems. This makes it straightforward to derive a complete classification of the bifurcations that can occur. We are thus able to confirm classic results of Filippov\cite{Fil} using different and more transparent methods, and explain why the `missing' cases of Hogan \emph{et al}\cite{Hog} are the only cases omitted in more recent work
BioPreDyn-bench: a suite of benchmark problems for dynamic modelling in systems biology
Get PDFBackground
Dynamic modelling is one of the cornerstones of systems biology. Many research efforts are currently being invested in the development and exploitation of large-scale kinetic models. The associated problems of parameter estimation (model calibration) and optimal experimental design are particularly challenging. The community has already developed many methods and software packages which aim to facilitate these tasks. However, there is a lack of suitable benchmark problems which allow a fair and systematic evaluation and comparison of these contributions.
Results
Here we present BioPreDyn-bench, a set of challenging parameter estimation problems which aspire to serve as reference test cases in this area. This set comprises six problems including medium and large-scale kinetic models of the bacterium E. coli, baker�s yeast S. cerevisiae, the vinegar fly D. melanogaster, Chinese Hamster Ovary cells, and a generic signal transduction network. The level of description includes metabolism, transcription, signal transduction, and development. For each problem we provide (i) a basic description and formulation, (ii) implementations ready-to-run in several formats, (iii) computational results obtained with specific solvers, (iv) a basic analysis and interpretation.
Conclusions
This suite of benchmark problems can be readily used to evaluate and compare parameter estimation methods. Further, it can also be used to build test problems for sensitivity and identifiability analysis, model reduction and optimal experimental design methods. The suite, including codes and documentation, can be freely downloaded from the BioPreDyn-bench website, https://sites.google.com/site/biopredynbenchmarks
Chebyshev-Fiedler pencils
Get PDFFiedler pencils are a family of strong linearizations for polynomials expressed in the monomial basis, that include the classical Frobenius companion pencils as special cases.
We generalize the definition of a Fiedler pencil from monomials to a larger class of orthogonal polynomial bases.
In particular, we derive comrade-Fiedler pencils for two bases that are extremely important in practical applications: the Chebyshev polynomials of the first and second kind.
The new approach allows one to construct linearizations having limited bandwidth: a Chebyshev analogue of the pentadiagonal Fiedler pencils in the monomial basis. Moreover, our theory allows for linearizations of square matrix polynomials expressed in the Chebyshev basis (and in other bases), regardless of whether the matrix polynomial is regular or singular, and for recovery formulae for eigenvectors, and minimal indices and bases
Hermitian flag manifolds and orbits of the Euclidean group
No full textWe study the adjoint and coadjoint representations of a class of Lie group including the Euclidean group. Despite the fact that these representations are not in general isomorphic, we show that there is a geometrically defined bijection between the sets of adjoint and coadjoint orbits of such groups. In addition, we show that the corresponding orbits, although different, are homotopy equivalent. We also provide a geometric description of the adjoint and coadjoint orbits of the Euclidean and orthogonal groups as a special class of flag manifold which we call a Hermitian flag manifold. These manifolds consist of flags endowed with complex structures equipped to the quotient spaces that define the flag
Completions of the Goldschmidt G3-amalgam and Alternating Groups
Get PDFHere we show that the alternating group of degree n is a completion of the Goldschmidt G3-amalgam if and only if n is in the set {1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 16, 17, 19, 23
Polynomial Eigenvalue Problems: Theory, Computation, and Structure
Get PDFMatrix polynomial eigenproblems arise in many application areas, both directly and as approximations for more general nonlinear eigenproblems.
One of the most common strategies for solving a polynomial eigenproblem is via a linearization, which replaces the matrix polynomial
by a matrix pencil with the same spectrum,
and then computes with the pencil.
Many matrix polynomials arising from applications have additional
algebraic structure, leading to symmetries in the spectrum
that are important for any computational method to respect.
Thus it is useful to employ a structured linearization
for a matrix polynomial with structure.
This essay surveys the progress over the last decade
in our understanding of linearizations and their construction,
both with and without structure,
and the impact this has had on numerical practice
The effects of fewer projection tomographic scans on the image reconstruction
Get PDFReconstructing a 2D slice or a 3D volume from a set of insufficient tomographic data is a difficult problem, and it is often tackled with analytical reconstruction algorithms. However, these types of methods fall short on delivering a quality image due to the severe artefacts introduced by the insufficiency of the data. The presented work shows the effects of taking tomographic scans with fewer radiographs on the quality of the reconstructed images. The aim here is to show the advantages of using iterative reconstruction methods over analytical methods, which are demonstrated by a quantitative comparison
Estimating the Largest Elements of a Matrix
Get PDFWe derive an algorithm for estimating the largest p >= 1 values a ij or |a ij | for an
m x n matrix A, along with their locations in the matrix. The matrix is accessed using only matrix-vector or matrix-matrix products. For p = 1 the algorithm estimates the norm A M := max i,j |a ij |
or max i,j a ij . The algorithm is based on a power method for mixed subordinate matrix norms and
iterates on n x t matrices, where t > p is a parameter. For p = t = 1 we show that the algorithm
is essentially equivalent to rook pivoting in Gaussian elimination; we also obtain a bound for the
expected number of matrix-vector products for random matrices and give a class of counter-examples.
Our numerical experiments show that for p = 1 the algorithm usually converges in just two iterations,
requiring the equivalent of 4t matrix-vector products, and for t = 2 the algorithm already provides
excellent estimates that are usually within a factor 2 of the largest element and frequently exact.
For p > 1 we incorporate deflation to improve the performance of the algorithm. Experiments on
real-life datasets show that the algorithm is highly effective in practice