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Toric Topology of Stasheff Polytopes
Get PDFThe Stasheff polytopes , , first arose in his paper
``Homotopy associativity of -spaces'' (1963) as the space of
homotopy parameters for maps determining associativity conditions
for a product ; they were defined
via binary bracketings of the formal monomials .
Stasheff polytopes are in the limelight of several research areas,
recently especially in connection with physical applications of
operad theory.
We will describe geometry and combinatorics of Stasheff polytopes
using the several different ways to introduce these polytopes and
the methods and results of toric topology.
We will show that the two-parameter generating function
that enumerates the number of -dimensional faces of the -th
Stasheff polytope satisfies the famous Burgers--Hopf equation
.
We will discuss the applications of this result including an
interpretation of Dehn--Sommerville relations in the term of Cauchy
problem, and Cayley formula in the term of conservation laws
On group actions on free Lie algebras
Get PDFWe first study the module structure of the free Lie algebra in characteristic zero under the action of the general linear group. Here we give a new, purely combinatorial, proof of Klyachko's celebrated theorem on Lie representations using the Kra\'{s}kiewicz-Weyman theorem.
We then give a new factorisation of the Dynkin-Specht-Wever
idempotent and use this to prove that is a -module direct summand of , for an arbitrary group, a field of characteristic and a -module. For finite-dimensional modules , this follows immediately from the Decomposition Theorem of Bryant and Schocker. We consider a small example of this theorem, namely the sixth Lie power over a field of
characteristic . Here we show explicitly that decomposes into a direct sum of the modules and , where denotes the symmetric square of . We give a description, up to isomorphism, of the modules occurring in the Decomposition Theorem.
Finally, we apply our knowledge of Lie powers to a group theoretic problem. We show that the torsion subgroup of the quotient is bounded as follows, for or , where is an arbitrary prime, :
\begin{eqnarray*}
2t_{2p^m} = 0 \;\;\;\mbox{provided } G=F/R \mbox{ has no 2-torsion and no p-torsion,}\\
3t_{3p^m} = 0 \;\;\;\mbox{provided } G=F/R \mbox{ has no 3-torsion
and no p-torsion.}
\end{eqnarray*}
Thus, we have that is torsion-free, provided that has no elements of order or
Detecting and Solving Hyperbolic Quadratic Eigenvalue Problems
Get PDFHyperbolic quadratic matrix polynomials are an important class of
Hermitian matrix polynomials
with real eigenvalues, among which the overdamped quadratics are those
with nonpositive eigenvalues.
Neither the definition of overdamped nor any of the standard
characterizations provides an efficient way to test if a given
has this property.
We show that a quadratically convergent matrix iteration based on
cyclic reduction, previously studied by Guo and Lancaster,
provides necessary and sufficient conditions for to be overdamped.
For weakly overdamped the iteration is shown to be generically linearly
convergent with constant at worst 1/2,
which implies that
the convergence of the iteration is reasonably fast in almost all
cases of practical interest.
We show that the matrix iteration can be implemented in such a way
that when overdamping is detected a scalar is
provided that lies in the gap between the largest and
smallest eigenvalues of the
quadratic eigenvalue problem (QEP) .
Once such a is known, the QEP can be solved by linearizing to a
definite pencil that can be reduced using already available
Cholesky factorizations to a standard Hermitian eigenproblem.
By incorporating an initial preprocessing stage that shifts a
hyperbolic so that it is overdamped,
we obtain an efficient algorithm that identifies and solves a hyperbolic or
overdamped QEP maintaining symmetry throughout and guaranteeing real
computed eigenvalues
Midplane-symmetry breaking in the flow between two counter-rotating disks
Get PDFThis paper considers the axisymmetric steady flow driven by exact counter rotation of two co-axial disks of finite radius. At the edges of the rotating disks one of three conditions is (typically) imposed: (i) zero velocity, corresponding to a stationary, impermeable, cylindrical shroud (ii) zero normal velocity and zero tangential fluid traction, corresponding to a (confined) free surface and (iii) an edge constraint that is consistent with a similarity solution of von Kármán form. The similarity solution is valid in an infinite geometry and possesses a pitchfork bifurcation that breaks the midplane symmetry at a critical Reynolds number. In this paper, similar bifurcations of the global (finite-domain) flow are sought and comparisons are made between the resulting bifurcation structure and that found for the similarity solution. The aim is to assess the validity of the nonlinear similarity solutions in finite domains and to explore the sensitivity of the solution structure to edge conditions that are implicitly neglected when assuming a self-similar flow. It is found that, whilst the symmetric similarity solution can be quantitatively useful for a range of boundary conditions, the bifurcated structure of the finite-domain flow is rather different for each boundary condition and bears little resemblance to the self-similar flow
A note on the geometry of linear Hamiltonian systems of signature 0 in R4
Get PDFIt is shown that a linear Hamiltonian system on R4 is elliptic or hyperbolic according to the number of Lagrangian planes in the null-cone H^−1(0), or equivalently the number of invariant Lagrangian planes. Some extension to higher dimensions is described
A brief historical perspective of the Wiener-Hopf technique
Get PDFIt is a little over 75 years since two of the most important mathematicians of the 20th century collaborated on finding the exact solution of a particular equation with semi-infinite convolution type integral operator. The elegance and analytical sophistication of the method, now called the Wiener–Hopf technique, impress all who use it. Its applicability to almost all branches of engineering, mathematical physics and applied mathematics is borne out by the many thousands of papers published on the subject since its conception. The Wiener–Hopf technique remains an extremely important tool for modern scientists, and the areas of application continue to broaden. This special issue of the Journal of Engineering Mathematics is dedicated to the work of Wiener and Hopf, and includes a number of articles which demonstrate the relevance of the technique to a representative range of model problems
Total Variation Regularization in Electrical Impedance Tomography
No full textThis paper presents an evaluation of the use of Primal Dual Methods for efficiently regularizing the electric impedance tomography (EIT) problem with the Total Variation (TV) functional. The Total Variation functional is assuming an important role in the regularization of inverse problems thanks to its ability to preserve dis-
continuities in reconstructed profiles. This property is desirable in many fields of application of EIT imaging, such as the medical and the industrial, where inter-organ boundaries, in the first case, and inter-phase boundaries, in the latter case, present step changes in electrical
properties which are difficult to be reconstructed with traditional regularization methods, as they tend to smooth the reconstructed image. Though desirable, the TV functional leads to the formulation of the inverse problem as a minimization of a non-differentiable function whichcannot be efficiently solved with traditional optimization techniques such as the Newton Method. In this paper we demonstrate the use of Primal Dual - Interior Point Methods (PD-IPM) as a framework for TV regularized inversion
Stability and Convergence of the Method of Fundamental Solutions for Helmholtz problems on analytic domains
Get PDFThe Method of Fundamental Solutions (MFS) is a popular tool to solve Laplace
and Helmholtz boundary value problems. Its main drawback is that it often leads
to ill-conditioned systems of equations. In this paper we investigate for the
interior Helmholtz problem
on analytic domains how the singularities (charge points) of the MFS basis
functions have to be chosen such that approximate solutions can be represented
by the MFS basis in a numerically stable way. For Helmholtz problems on the
unit disc we give a full analysis
which includes the high frequency (short wavelength) limit.
For more difficult and nonconvex
domains such as crescents we demonstrate how the
right choice of charge points is connected to how far
into the complex plane the solution of the boundary value problem can be
analytically continued, which in turn depends on both domain shape
and boundary data.
Using this we develop a recipe for locating charge points which
allows us to reach error norms of typically on a wide variety
of analytic domains.
At high frequencies of order only 3 points per wavelength are
needed, which compares very favorably to boundary integral methods
Prediction of regulatory networks: identification of transcription factor-target relationship from gene ontology information and gene expression data
Get PDFDefining regulatory networks, linking transcription factors (TFs) to their targets, is a central problem in post-genomic biology. Here we apply an approach based on Nearest Neighbour (NN) Algorithm to predict the targets of a transcription factor by combining gene ontology (GO) and gene expression data. In particular, we used NN algorithm to predict the regulatory targets for 36 transcription factors in the Saccharomyces cerevisiae (Qian J. et al., 2003, Bioinformatics. 19(15):1917-26) based on the gene ontology and microarray expression data from various physiological conditions. We trained and tested our NN algorithm on a data set which contains a number of both positive and negative examples. The overall success rate by the jackknife test for the dataset was 97%, and that for the regulatory targets(positive) was 58%, suggesting that such a hybrid approach particularly by incorporating the knowledge of gene ontology) may become a useful high-throughput tool in the area of regulatory networks modelling
Infinite dimensional second order differential equations via
Get PDFThe vector bundle structure obtained on the second order
(acceleration) tangent bundle T^2M of a smooth manifold M by
means of a linear connection on the base provides an alternative
way for the study of second order differential equations on
manifolds of finite and infinite dimension. Second order vector
fields and their integral curves provide a new way of solving a
wide class of second order differential equations on Frechet
manifolds and may be used also to describe geodesic curves on a
Riemannian manifold. The new technique proposed is illustrated by
concrete examples within the framework of Banach and Frechet
spaces as well as on Lie groups