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Periodic orbits in some classes of Hamiltonian systems with symmetry
Get PDFWe study the existence of families of periodic orbits near a symmetric equilibrium
point in different classes of Hamiltonian systems with symmetry. We center our attention
to special types of symmetry less-studied in the literature, such as systems with
(semi-)invariant Hamiltonian and reversible equivariant Hamiltonian systems, when
the linearisation has two pairs of purely imaginary eigenvalues.
In each case, we provide normal forms for the symmetries, the linear structure
map and the linearisation. Moreover, the existence of symmetric and non-symmetric
periodic orbits is proved. Another result we found is the classification of Hamiltonian
systems with dihedral symmetry, of order eight, with all different possible combinations
of time-reversing and symplectic-reversing actions.
The method used in finding periodic orbits is the Liapunov-Schmidt reduction.
The symmetry plays a vital role in determining the set of (semi-)invariants, in order
to write the reduced problem and then to distinguish the solutions according to their
symmetry type
Analysis of optimal liquidation in limit order books
Get PDFIn this paper we study the optimal trading strategy of a passive trader who is trading in the limit order book. Using a combined approach of accurate numerical methods and asymptotical analysis we examine the problem using different stochastic processes to model the asset price, as well as introducing a proportional resilience for the limit order book.
This results in more complex equations to solve than when examined under the case of standard Brownian motion, allowing us to perform interesting analytical (asymptotic) analysis which adds insight into the solution space.
Under Geometric Brownian Motion, we reduce the resulting four-dimensional Hamilton-Jacobi-Bellman partial differential equation (PDE) to a novel three-dimensional non-linear PDE, as well as rescaling the variables to reduce the number of input parameters by two. We use numerical methods to solve the PDE before asymptotically examining it in several limits, with each approach informing and confirming the other. We find the transition from a time-varying solution to a perpetual-type solution results in the development of singular behaviour, and this transition is examined in some detail. Finally we emphasise the adaptability of our proposed methodologies by implementing the same methods on a mean-reverting process for the asset price.
Throughout the paper we also analyse the resulting trading strategies from a financial perspective. The trading strategies we develop are asset-price dependent, which to our knowledge is a unique concept in the passive optimal trading literature, and is arguably more realistic
Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective
Get PDFThese notes provide a self-contained account of the combinatorial theory of intrinsic volumes for polyhedral cones. Streamlined derivations of the General Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In addition, a connection between the characteristic polynomial of a hyperplane arrangement and the intrinsic volumes of the regions of the arrangement, due to Klivans and Swartz, is generalized and some applications presented
Essential partial differential equations
Get PDFThis volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these; the qualitative behaviour of solutions being established using classical concepts: maximum principles and energy methods.
Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the approximation of multidimensional advection–diffusion problems.
The underlying theory is illustrated by numerous examples and there are around 300 exercises, designed to promote and test understanding. They are starred according to level of difficulty. Solutions to odd-numbered exercises are available to all readers while even-numbered solutions are available to authorised instructors
Conductivity perturbations in EIT
Get PDFDifference EIT reconstructs a signal generated
by conductivity contrasting regions. We explore the effect that, for large contrasts, conductive regions produce a
greater signal than non-conductive ones, and this difference
is determined by the region shap
Matching Exponential-Based and Resolvent-Based Centrality Measures
Get PDFThe relative importance of nodes in a network can be quantified via functions of the adjacency matrix. Two popular choices of function are the exponential, which is parameter-free, and the resolvent function, which yields the Katz centrality measure. Katz centrality can be the more computationally efficient, especially for large directed networks, and has the benefit of generalizing naturally to time-dependent network sequences, but it depends on a parameter. We give a prescription for selecting the Katz parameter based on the objective of matching the centralities of the exponential counterpart. For our new choice of parameter the resolvent can be very ill conditioned, but we argue that the centralities computed in floating point arithmetic can nevertheless reliably be used for ranking. Experiments on five real networks show that the new choice of Katz parameter leads to rankings of nodes that match those from the exponential centralities well in practice
An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations
Get PDFStochastic Galerkin finite element approximation of PDEs with random inputs leads to linear systems of equations with coefficient matrices that have a characteristic Kronecker product structure. By reformulating the systems as multi-term linear matrix equations, we develop an efficient solution algorithm which generalizes ideas from rational Krylov subspace approximation. The new approach determines a low-rank approximation to the solution matrix by performing a projection onto a low-dimensional space and provides an efficient solution strategy whose convergence rate is independent of the spatial approximation. Moreover, it requires far less memory than the standard preconditioned conjugate gradient method applied to the Kronecker formulation of the linear systems
Polynomial Zigzag Matrices, Dual Minimal Bases, and the Realization of Completely Singular Polynomials
Get PDFMinimal bases of rational vector spaces are a well-known
and important tool in systems theory.
If minimal bases for two subspaces of rational -space
are displayed as the rows of polynomial matrices
and , respectively,
then and are said to be
dual minimal bases
if the subspaces have complementary dimension,
i.e., ,
and .
In other words, each provides a minimal basis
for the nullspace of the other.
It has long been known that for any dual minimal bases and ,
the row degree sums of and are the same.
In this paper we show that this is the only constraint on the row degrees,
thus characterizing the possible row degrees of dual minimal bases.
The proof is constructive, making extensive use
of a new class of sparse, structured polynomial matrices
that we have baptized zigzag matrices.
Another application of these polynomial zigzag matrices
is the constructive solution of the following inverse problem for minimal indices --
given a list of left and right minimal indices and a desired degree ,
does there exist a completely singular matrix polynomial
(i.e., a matrix polynomial with no elementary divisors whatsoever)
of degree
having exactly the prescribed minimal indices?
We show that such a matrix polynomial exists
if and only if divides the sum of the minimal indices.
The constructed realization is simple,
and explicitly displays the desired minimal indices
in a fashion analogous
to the classical Kronecker canonical form
of singular pencils