46 research outputs found
Algorithms for Mappings and Symmetries of Differential Equations
Differential Equations are used to mathematically express the laws of physics and models in biology, finance, and many other fields. Examining the solutions of related differential equation systems helps to gain insights into the phenomena described by the differential equations. However, finding exact solutions of differential equations can be extremely difficult and is often impossible. A common approach to addressing this problem is to analyze solutions of differential equations by using their symmetries. In this thesis, we develop algorithms based on analyzing infinitesimal symmetry features of differential equations to determine the existence of invertible mappings of less tractable systems of differential equations (e.g., nonlinear) into more tractable systems of differential equations (e.g., linear). We also characterize features of the map if it exists. An algorithm is provided to determine if there exists a mapping of a non-constant coefficient linear differential equation to one with constant coefficients. These algorithms are implemented in the computer algebra language Maple, in the form of the MapDETools package. Our methods work directly at the level of systems of equations for infinitesimal symmetries. The key idea is to apply a finite number of differentiations and eliminations to the infinitesimal symmetry systems to yield them in the involutive form, where the properties of Lie symmetry algebra can be explored readily without solving the systems. We also generalize such differential-elimination algorithms to a more frequently applicable case involving approximate real coefficients. This contribution builds on a proposal by Reid et al. of applying Numerical Algebraic Geometry tools to find a general method for characterizing solution components of a system of differential equations containing approximate coefficients in the framework of the Jet geometry. Our numeric-symbolic algorithm exploits the fundamental features of the Jet geometry of differential equations such as differential Hilbert functions. Our novel approach establishes that the components of a differential equation can be represented by certain points called critical points
Geometric Analysis of Nonlinear Partial Differential Equations
This book contains a collection of twelve papers that reflect the state of the art of nonlinear differential equations in modern geometrical theory. It comprises miscellaneous topics of the local and nonlocal geometry of differential equations and the applications of the corresponding methods in hydrodynamics, symplectic geometry, optimal investment theory, etc. The contents will be useful for all the readers whose professional interests are related to nonlinear PDEs and differential geometry, both in theoretical and applied aspects
Applied stochastic eigen-analysis
Submitted in partial fulfillment of the requirements for the degree of
Doctor of Philosophy at the Massachusetts Institute of Technology and the
Woods Hole Oceanographic Institution February 2007The first part of the dissertation investigates the application of the theory of large
random matrices to high-dimensional inference problems when the samples are drawn
from a multivariate normal distribution. A longstanding problem in sensor array processing
is addressed by designing an estimator for the number of signals in white noise
that dramatically outperforms that proposed by Wax and Kailath. This methodology is
extended to develop new parametric techniques for testing and estimation. Unlike techniques
found in the literature, these exhibit robustness to high-dimensionality, sample
size constraints and eigenvector misspecification.
By interpreting the eigenvalues of the sample covariance matrix as an interacting
particle system, the existence of a phase transition phenomenon in the largest (“signal”)
eigenvalue is derived using heuristic arguments. This exposes a fundamental limit on
the identifiability of low-level signals due to sample size constraints when using the
sample eigenvalues alone.
The analysis is extended to address a problem in sensor array processing, posed by
Baggeroer and Cox, on the distribution of the outputs of the Capon-MVDR beamformer
when the sample covariance matrix is diagonally loaded.
The second part of the dissertation investigates the limiting distribution of the
eigenvalues and eigenvectors of a broader class of random matrices. A powerful method
is proposed that expands the reach of the theory beyond the special cases of matrices
with Gaussian entries; this simultaneously establishes a framework for computational
(non-commutative) “free probability” theory.
The class of “algebraic” random matrices is defined and the generators of this class
are specified. Algebraicity of a random matrix sequence is shown to act as a certificate
of the computability of the limiting eigenvalue distribution and, for a subclass, the limiting
conditional “eigenvector distribution.” The limiting moments of algebraic random
matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion
so that they may often be efficiently enumerated in closed form. The method is applied
to predict the deterioration in the quality of the sample eigenvectors of large algebraic
empirical covariance matrices due to sample size constraints.I am grateful to the National Science Foundation for supporting this work via grant
DMS-0411962 and the Office of Naval Research Graduate Traineeship awar
Computational aspects of singularity theory.
In this thesis we develop computational methods suitable for performing the symbolic
calculations common to local singularity theory. For classification theory
we employ the unipotent determinacy techniques of Bruce, du Plessis, Wall and
complete transversal theorems of Bruce, du Plessis. The latter results are, as yet,
unpublished and we spend some time reviewing these results, extending them to
filtrations of the module m,,,.E (n, p) other than the standard filtration by degree.
Weighted filtrations and filtrations induced by the action of a nilpotent Lie algebra
are considered. A computer package called Transversal is developed. This
is written in the mathematical language Maple and performs calculations such
as those mentioned above and those central to unfolding theory. We discuss the
package in detail and give examples of calculations performed in this thesis.
Several classifications are obtained. The first is an extensive classification
of map-germs (R2,0) -p (R4,0) under A-equivalence. We also consider the
classification of function-germs (CP, O) -f (C, 0) under R(D)-equivalence: the
restriction of R-equivalence to source coordinate changes which preserve a discriminant
variety, D. We consider the cases where D is the discriminant of the
A2 and A3 singularities, extending the results of Arnol'd. Several other simple
singularities are discussed briefly; in particular, we consider the cases where D is
the discriminant of the A4, D4, D5, D6, and Ek singularities.
The geometry of the singularities (R2,0) -f (R4,0) is investigated by calculating
the adjacencies and several geometrical invariants. For the given source
and target dimensions, the invariants associated to the double point schemes and
L-codimension of the germs are particularly significant.
Finally we give an application of computer graphics to singularity theory. A
program is written (in C) which calculates and draws the family of profiles of a
surface rotating about a fixed axis in R3, the resulting envelope of profiles, and
several other geometrical features. The program was used in recent research by
Rycroft. We review some of the results and conclude with computer produced
images which demonstrate certain transitions of the singularities on the envelope
Residue distributions, iterated residues, and the spherical automorphic spectrum
Let be a split reductive group over a number field . We consider the
computation of the inner product of two -spherical pseudo Eisenstein series
of supported in by means of residues, following a
classical approach initiated by Langlands. We show that only the singularities
of the intertwining operators due to the poles of the completed Dedekind zeta
function contribute to the spectrum, while the singularities caused
by the zeroes of do not contribute to any of the iterated residues
which arise as a result of the necessary contour shifts.
In the companion paper [DMHO] we use this result to explicitly determine the
spectral measure of by a comparison of the iterated
residues with the residue distributions of [HO1].Comment: 56 pages, contains a list of symbols in the en
Computer Science for Continuous Data:Survey, Vision, Theory, and Practice of a Computer Analysis System
Building on George Boole's work, Logic provides a rigorous foundation for the powerful tools in Computer Science that underlie nowadays ubiquitous processing of discrete data, such as strings or graphs. Concerning continuous data, already Alan Turing had applied "his" machines to formalize and study the processing of real numbers: an aspect of his oeuvre that we transform from theory to practice.The present essay surveys the state of the art and envisions the future of Computer Science for continuous data: natively, beyond brute-force discretization, based on and guided by and extending classical discrete Computer Science, as bridge between Pure and Applied Mathematics