MIMS EPrints
Not a member yet
2151 research outputs found
Sort by
Stochastic Evolution Equations Driven by Compensated Poisson Measures: Existence, Uniqueness and Large Deviation Estimates
Get PDFExistence and uniqueness results are established for
solutions of stochastic evolution equations driven by
Poisson point processes. Large deviation estimates are
obtaines for the case of additive Poisson noise.
Examples are provided
A new bound for the smallest x with \pi(x) > \li(x)
No full textWe reduce the dominant term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays & Hudson. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [1.39792101 \times 10^316, 1.39847603 \times 10^316] for which \pi(x) > \li(x). This interval is strictly a sub-interval of the interval in Bays & Hudson [1], and is narrower by a factor of about 10
Computing the Condition Number of Tridiagonal and Diagonal-Plus-Semiseparable Matrices in Linear Time
No full textFor an tridiagonal matrix we exploit the structure of
its QR factorization to devise two new algorithms for computing the 1-norm
condition number in operations. The algorithms avoid
underflow and overflow, and are simpler than existing algorithms since
tests are not required for degenerate cases. An error analysis of the
first algorithm is given, while the second algorithm is shown to be
competitive in speed with existing algorithms. We then turn our
attention to an diagonal-plus-semiseparable matrix, ,
for which several algorithms have recently been developed to solve
in operations. We again exploit the QR factorization of
the matrix to present an algorithm that computes the 1-norm condition
number in operations
Structured Condition Numbers and Backward Errors in Scalar Product Spaces
No full textWe investigate the effect of structure-preserving perturbations on the solution
to a linear system, matrix inversion, and distance to singularity.
Particular attention is paid to linear and nonlinear
structures that form Lie algebras,
Jordan algebras and automorphism groups of a scalar product.
These include complex symmetric, pseudo-symmetric, persymmetric, skew-symmetric, Hamiltonian,
unitary, complex orthogonal and symplectic matrices.
We show that under reasonable assumptions on the scalar product,
there is little or no difference between structured and unstructured condition
numbers and distance to singularity for matrices in Lie and Jordan algebras.
Hence, for these classes of matrices, the usual unstructured perturbation
analysis is sufficient.
We show this is not true in general for structures in automorphism groups.
Bounds and computable expressions for the structured condition numbers
for a linear system and matrix inversion are derived for these nonlinear
structures.
Structured backward errors for the approximate solution of linear systems
are also considered.
Conditions are given for the structured backward error to be finite.
We prove that for Lie and Jordan algebras, whenever
the structured backward error is finite, it is within a small factor of or equal
to the unstructured one.
The same conclusion holds for orthogonal and unitary structures but cannot
easily be extended to other matrix groups.
This work extends and unifies earlier analyses
Embedding Nonlinear Dynamical Systems: A Guide to Takens' Theorem
Get PDFThe embedding theorem forms a bridge between the theory of nonlinear dynamical systems and the analysis of experimental time series. This memorandum describes the theorem and gives a detailed account of its proof. The necessary differential topology is briefly reviewed, and then a proof of the theorem is presented; this proof follows broadly the argument of Takens, although it differs in some details. Some extensions to the theorem, which facilitate its use in applications, are described. The memo concludes with a brief discussion of what the theorem implies about time series, viewed as the raw material for signal processing algorithms
Grading infantile cataracts
No full textPurpose: To introduce and describe two methods of grading the severity of infantile cataracts, and
thereby propose a useful clinical guide for early surgical intervention.
Methods: Thirty-three subjects, aged 1 week to 8 years, participated in the study. Twenty-two were
evaluated soon after birth (1 week), and 11 in childhood (3–8 years). All had isolated infantile
cataracts, of which 16 were bilateral and 17 unilateral. Nine cataract types were examined; nuclear
(n = 9), lamellar (n = 9), posterior lenticonus (n = 4), persistent hyperplastic primary vitreous
(n = 4), posterior polar (n = 3) and single cases of total, cortical, sutural and anterior polar. Grading
the infantile cataracts was performed subjectively based on the cataract morphology, density and
position using an 11-point (0–10) ordinal scale. Objective measures of the cataracts were performed
by scanning and then digitising photo-slit lamp images to provide cataract intensity profiles. Subjects
without cataracts acted as controls.
Results: Subjective gradings of 0 and 10 were assigned to the clear, cataract-free lens and the total
cataract, respectively. Fixed grades of 1 (anterior polar, sutural) and 6 (posterior polar) were
assigned to the three remaining cataracts with static morphologies. The five cataracts which were all
progressive were given grading ranges, reflecting the initial and likely final morphological states.
Objective measures were found to be valuable in indicating the exact position and relative density of
the cataract, as well as accurately defining boundaries.
Conclusions: The magnitude and severity of infantile cataracts can be usefully characterised by an
11-point ordinal subjective grading scale. Although subjective grading alone is satisfactory, it can be
greatly assisted by objective measures, particularly in the documentation of cataract progression.
Cataracts assigned grades 1–4 were considered minor obstructions to vision and therefore not
candidates for early surgery. Cataracts graded 5 and above were considered major visual defects,
and ideally should be removed early in life
Symmetric Linearizations for Matrix Polynomials
Get PDFA standard way of treating the polynomial eigenvalue problem
P(\l)x = 0 is to convert it
into an equivalent matrix pencil---a process known as linearization.
Two vector spaces of pencils
\Ell_1(P) and \Ell_2(P), and their intersection \DL(P),
have recently been defined and studied by
Mackey, Mackey, Mehl, and Mehrmann.
The aim of our work is to gain new insight into these spaces
and the extent to which their constituent pencils inherit
structure from \@.
For arbitrary polynomials we show that every pencil in \DL(P)
is block symmetric
and we obtain a convenient basis for \DL(P) built from block Hankel matrices.
This basis is then exploited to
prove that
the first pencils in a sequence constructed by
Lancaster in the 1960s generate \DL(P).
When is symmetric, we show that
the symmetric pencils in \Ell_1(P) comprise
\DL(P),
while for Hermitian the Hermitian pencils in \Ell_1(P)
form a proper subset of \DL(P) that we explicitly characterize.
Almost all pencils in
each of these subsets are shown to be linearizations.
In addition to obtaining new results, this work provides a self-contained
treatment of some of the key properties of \DL(P)
together with some new, more concise proofs
Equalisers of frames in constructive set theory
No full textIn a recent note Erik Palmgren has shown that the category of set-presented formal topologies has coequalisers in a sufficiently strong version of Martin-Lof's Type Theory. Here we get a version of Palmgren's result in a sufficiently strong version of constructive set theory. We prefer to work with the category of set-presented class frames, a category that is equivalent to the opposite of the category of set-presented formal topologies