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Girsanov and Feynman-Kac type transformations for symmetric Markov processes
No full textStudied in this paper is the transformation of an arbitrary symmetric Markov process X by multiplicative functionals which are the exponential of continuous additive functionals of X having zero quadratic variations. We characterize the transformed semigroups by their associated quadratic forms. This is done by first identifying the symmetric Markov process under Girsanov transform, which may be of independent interest, and then applying Feynman–Kac transform to the Girsanov transformed process. Stochastic analysis for discontinuous martingales is used in our approach
Torus Actions and their Applications in Topology and Combinatorics
No full textHere, the study of torus actions on topological spaces is presented as a bridge connecting combinatorial and convex geometry with commutative and homological algebra, algebraic geometry, and topology. This established link helps in understanding the geometry and topology of a space with torus action by studying the combinatorics of the space of orbits. Conversely, subtle properties of a combinatorial object can be realized by interpreting it as the orbit structure for a proper manifold or as a complex acted on by a torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement, etc., while the combinatorial objects include simplicial and cubical complexes, polytopes, and arrangements. This approach also provides a natural topological interpretation in terms of torus actions of many constructions from commutative and homological algebra used in combinatorics. The exposition centers around the theory of moment-angle complexes, providing an effective way to study invariants of triangulations by methods of equivariant topology. The book includes many new and well-known open problems and would be suitable as a textbook. It will be useful for specialists both in topology and in combinatorics and will help to establish even tighter connections between the subjects involved
Computing the Nearest Correlation Matrix---A Problem from Finance
Get PDFGiven a symmetric matrix what is the nearest correlation matrix,
that is, the nearest symmetric positive semidefinite matrix with unit diagonal?
This problem arises in the finance industry,
where the correlations are between stocks.
For distance measured in two weighted Frobenius norms
we characterize the solution using convex analysis.
We show how the modified alternating projections method
can be used to compute the solution for the more commonly used of the
weighted Frobenius norms.
In the finance application the original matrix has many zero or negative
eigenvalues;
we show that for a certain class of weights
the nearest correlation matrix has
correspondingly many zero eigenvalues and that this fact
can be exploited in the computation
The Hooley-Huxley contour method for problems in number fields II: factorization and divisibility
Get PDFOn transonic viscous-inviscid interaction
Get PDFThe paper is concerned with the interaction between the boundary layer on a smooth body surface and the outer inviscid compressible flow in the vicinity of a sonic point. First, a family of local self-similar solutions of the Kármán–Guderley equation describing the inviscid flow behaviour immediately outside the interaction region is analysed; one of them was found to be suitable for describing the boundary-layer separation. In this solution the pressure has a singularity at the sonic point with the pressure gradient on the body surface being inversely proportional to the cubic root dpw/dx [similar] ([minus sign]x)[minus sign]1/3 of the distance ([minus sign]x) from the sonic point. This pressure gradient causes the boundary layer to interact with the inviscid part of the flow. It is interesting that the skin friction in the boundary layer upstream of the interaction region shows a characteristic logarithmic decay which determines an unusual behaviour of the flow inside the interaction region. This region has a conventional triple-deck structure. To study the interactive flow one has to solve simultaneously the Prandtl boundary-layer equations in the lower deck which occupies a thin viscous sublayer near the body surface and the Kármán–Guderley equations for the upper deck situated in the inviscid flow outside the boundary layer. In this paper a numerical solution of the interaction problem is constructed for the case when the separation region is entirely contained within the viscous sublayer and the inviscid part of the flow remains marginally supersonic. The solution proves to be non-unique, revealing a hysteresis character of the flow in the interaction region
Nonlinear thoughts about linear signal processing
Get PDFRecent work on modelling digital channels using iterated function systems suggests
a general approach to the theory of signal processing in digital communications
which uses so-called delay methods developed for deterministic nonlinear
timeseries analysis. Here we make the connection between this work and the
more conventional approach to digital communications by casting linear channel
models as iterated function systems and showing how the use of delay methods
gives a nice connection with the theory of observability in the control of linear
systems
Suppression of puberty with long-acting goserelin (zoladex-LA): effect on gonadotrophin response to GnRH in the first treatment cycle
No full textbackground and objectivesDepot GnRH analogues are widely used in the treatment of precocious puberty, or suppression of relatively early puberty where growth or psychosocial well-being may be compromised. One example is Zoladex (Z goserelin 3·6 mg), which can be given every 4 weeks. This injection frequency may not always achieve adequate suppression of pubertal signs. A long-acting form, Zoladex-LA 10·8 mg, has now been introduced with a potential duration of action of 12 weeks. In order to assess the efficacy of Zoladex-LA in gonadotrophin suppression we have measured LH and FSH responses to GnRH at diagnosis and 8 and 12 weeks after injection in a group of children treated with Zoladex-LA for central precocious or early puberty.
methodsForty-nine children (40 girls) with clinical evidence of central precocious puberty (CPP) or early puberty (EP) were started on Zoladex-LA, either de novo (n = 29) or on changing from Zoladex. Ages at diagnosis ranged from 1·7 to 10·6 years (median 7·8 years). Twenty-three had a structural cause with abnormality on magnetic resonance/computerized tomography (MR/CT) head scan, nine had a syndrome or nonspecific brain injury, and in 17 the cause was idiopathic.
resultsAt diagnosis, in the de novo group, median peak LH was 13·6 IU/l and median peak FSH was 12·0 IU/l. By 12 weeks gonadotrophins were suppressed to 0·9 and 0·8 IU/l, respectively. In the previously treated group, median peak LH at diagnosis was 12·8 IU/l and median peak FSH was 15·0 IU/l with suppression to 0·8 and 1·1 IU/l, respectively, at 12 weeks. In the latter group peak FSH was higher than peak LH at both 8 and 12 weeks (P < 0·05) and there was a significant rise in peak LH (P < 0·05) and FSH (P = 0·01) between 8 and 12 weeks. There was no correlation between age at diagnosis and peak LH or FSH at 8 or 12 weeks. Nevertheless, individual patients in both groups showed evidence of incomplete gonadotrophin suppression at 12 weeks.
conclusionZoladex-LA induces a significant reduction in gonadotrophins over 12 weeks. However, there are individuals, particularly those previously on Zoladex, in whom gonadotrophin suppression is waning by 12 weeks. As found with Zoladex, some children with precocious puberty treated with Zoladex-LA may require increased injection frequency, although correlation with clinical evidence of suppression needs to be studied further
Characterisation of congenital nystagmus waveforms in terms of periodic orbits
No full textBecause the oscillatory eye movements of congenital nystagmus vary from cycle to cycle, there is no clear relationship between the waveform produced and the underlying abnormality of the ocular motor system. We consider the durations of successive cycles of nystagmus which could be (1) completely determined by the lengths of the previous cycles, (2) completely independent of the lengths of the previous cycles or (3) a mixture of the two. The behaviour of a deterministic system can be characterised in terms of a collection of (unstable) oscillations, referred to as periodic orbits, which make up the system. By using a recently developed technique for identifying periodic orbits in noisy data, we find evidence for periodic orbits in nystagmus waveforms, eliminating the possibility that each cycle is independent of the previous cycles. The technique also enables us to identify the waveforms which correspond to the deterministic behaviour of the ocular motor system. These waveforms pose a challenge to our understanding of the ocular motor system because none of the current extensions to models of the normal behaviour of the ocular motor system can explain the range of identified waveforms
Limit at zero of the brownian first-passage density
No full textLet (Bt)t S 0) be a standard Brownian motion started at zero, let g : Â_+ M Â be an upper function for B satisfying g(0)=0, and let
be the first-passage time of B over g. Assume that g is C1 on d0,X¢, increasing (locally at zero), and concave (locally at zero). Then the following identities hold for the density function f of F:
in the sense that if the second and third limit exist so does the first one and the equalities are valid (here is the standard normal density). These limits can take any value in [0,X]. The method of proof relies upon the strong Markov property of B and makes use of real analysis