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An Improved Schur--Padé Algorithm for Fractional Powers of a Matrix and their Fréchet Derivatives
Get PDFThe Schur--Padé algorithm [N. J. Higham and L. Lin, A Schur--Padé algorithm for fractional powers of a matrix, SIAM J. Matrix Anal. Appl., 32(3):1056--1078, 2011] computes arbitrary real powers of a matrix using the building blocks of Schur decomposition, matrix square roots, and Padé approximants. We improve the algorithm by basing the underlying error analysis on the quantities , for several small , instead of . We extend the algorithm so that it computes along with one or more Fréchet derivatives, with reuse of information when more than one Fréchet derivative is required, as is the case in condition number estimation. We also derive a version of the extended algorithm that works entirely in real arithmetic when the data is real. Our numerical experiments show the new algorithms to be superior in accuracy to, and often faster than, the original Schur--Padé algorithm for computing matrix powers and more accurate than several alternative methods for computing the Fréchet derivative. They also show that reliable estimates of the condition number of are obtained by combining the algorithms with a matrix norm estimator
Grothendieck Rings of Theories of Modules
Get PDFThe model-theoretic Grothendieck ring of a first order structure, as defined by Krajic\v{e}k and Scanlon, captures some combinatorial properties of the definable subsets of finite powers of the structure. In this paper we compute the Grothendieck ring, , of a right -module , where is any unital ring. As a corollary we prove a conjecture of Prest that is non-trivial, whenever is non-zero. The main proof uses various techniques from the homology theory of simplicial complexes
Maximising the Size of Non-Redundant Protein Datasets Using Graph Theory
Get PDFAnalysis of protein data sets often requires prior removal of redundancy, so that data is not biased by having multiple copies of similar proteins. This is usually achieved by pairwise comparison of sequences, followed by purging so that no two pairs have similarities above a chosen threshold. From a starting set, such as the PDB or a genome, one should remove as few sequences as possible, to give the largest possible non-redundant set for subsequent analysis. Protein redundancy can be represented as a graph, with proteins as nodes connected by undirected edges, if they have a pairwise similarity above the chosen threshold. The problem is then equivalent to finding the maximum independent set (MIS), where as few nodes are removed as possible to remove all edges. We tested seven MIS algorithms, three of which are new. We applied the methods to the PDB, subsets of the PDB, various genomes and the BHOLSIB benchmark datasets. For PDB subsets of up to 1000 proteins, we could compare to the exact MIS, found by the Cliquer algorithm.
The best algorithm was the new method, Leaf. This works by adding clique members that have no edges to nodes outside the clique to the MIS, starting with the smallest cliques. For PDB subsets of up to 1000 members, it usually finds the MIS and is fast enough to apply to data sets of tens of thousands of proteins. It gives sets that are around 10% larger than the commonly used PISCES algorithm, that are of identical quality. We therefore suggest that Leaf should be the method of choice for generating non-redundant protein data sets, though it is ineffective on dense graphs, such as the BHOLSIB benchmarks. The Leaf algorithm and sets from genomes and the PDB are available at: http://www.bioinf.manchester.ac.uk/leaf/
Dimensionality reduction for classification of stochastic fibre radiographs
No full textDimensionality reduction helps to identify small numbers of essential features
of stochastic fibre networks for classification of image pixel density
datasets from experimental radiographic measurements of commercial samples and simulations.
Typical commercial macro-fibre networks use finite length fibres suspended in a fluid from which
they are continuously deposited onto a moving bed to make a continuous web;
the fibres can cluster to differing degrees,
primarily depending on the fluid turbulence, fibre dimensions and flexibility.
Here we use information geometry of trivariate Gaussian spatial distributions of pixel density among
first and second neighbours to reveal features related to sizes and density of fibre clusters
Geometric structure in smooth dual and local Langlands conjecture
No full textThis expository note is based on the Takagi lectures given by the second-named author in November, 2012.
Topics in the lectures:
#1. Review of the LL (Local Langlands) conjecture.
#2. Statement of the ABPS(Aubert-Baum-Plymen-Solleveld) conjecture. #3. Brief indication of the proof that for any connected split reductive p-adic group G both ABPS and LL are valid throughout the principal series of G
On the local Langlands correspondence for non-tempered representations
No full textLet G be a reductive p-adic group. We study how a local Langlands
correspondence for irreducible tempered G-representations can be extended to a local Langlands correspondence for all irreducible smooth representations of G. We prove that, under a natural condition involving compatibility with unramified twists, this is possible in a canonical way.
To this end we introduce analytic R-groups associated to non-tempered essentially square-integrable representations of Levi subgroups of G. We establish the basic properties of these new R-groups, which generalize Knapp�Stein R-groups
Nonsoluble and non--soluble length of finite groups
No full textEvery finite group has a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. We define the nonsoluble length as the number of nonsoluble factors in a shortest series of this kind. Upper bounds for appear in the study of various problems on finite, residually finite, and profinite groups.
We prove that is bounded in terms of the maximum -length of soluble subgroups of , and that
is bounded by the maximum Fitting height of soluble subgroups. For an odd prime , the non--soluble length is introduced, and it is proved that does not exceed
the maximum -length of -soluble subgroups. We conjecture that for a given prime and a given proper group variety the non--soluble length of finite groups whose Sylow -subgroups belong to is bounded.
In this paper we prove this conjecture for any variety
that is a product of several soluble varieties and varieties of finite exponent.
As an application of the results obtained, an error is corrected in the proof of the main result of the second author's paper ``Multilinear commutators in residually finite groups'', \emph{Israel J. Math.} \textbf{189} (2012), 207--224
Words and pronilpotent subgroups in profinite groups
No full textLet be a multilinear commutator word, that is, a commutator of weight in different group variables. It is proved that if is a profinite group in which all pronilpotent subgroups generated by -values are periodic, then the verbal subgroup is locally finite
Near-optimal perfectly matched layers for indefinite Helmholtz problems
No full textA new construction of an absorbing boundary condition for indefinite Helmholtz problems on unbounded domains is presented. This construction is based on a near-best uniform rational interpolant of the inverse square root function on the union of a negative and positive real interval, designed with the help of a classical result by Zolotarev. Using Krein's interpretation of a Stieltjes continued fraction, this interpolant can be converted into a three-term finite difference discretization of a perfectly matched layer (PML) which converges exponentially fast in the number of grid points. The convergence rate is asymptotically optimal for both propagative and evanescent wave modes. Several numerical experiments and illustrations are included
Adaptive Finite Element Method Assisted by Stochastic Simulation of Chemical Systems
Get PDFStochastic models of chemical systems are often analyzed by solving the corresponding Fokker--Planck equation, which is a drift-diffusion partial differential equation for the probability distribution function. Efficient numerical solution of the Fokker--Planck equation requires adaptive mesh refinements. In this paper, we present a mesh refinement approach which makes use of a stochastic simulation of the underlying chemical system. By observing the stochastic trajectory for a relatively short amount of time, the areas of the state space with nonnegligible probability density are identified. By refining the finite element mesh in these areas, and coarsening elsewhere, a suitable mesh is constructed and used for the computation of the stationary probability density. Numerical examples demonstrate that the presented method is competitive with existing a posteriori methods