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The module structure of a group action on a polynomial ring: A finiteness theorem
No full textConsider a group acting on a polynomial ring over a finite field. We study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries. For example, any indecomposable module that appears as a summand must also appear in low degree, and so the number of isomorphism types of such summands is finite. There are also applications to invariant theory, giving a priori bounds on the degrees of the generators
Spaces of polytopes and cobordism of quasitoric manifolds
No full textOur aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with non-singular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors' approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details
Locally polynomially bounded structures
Get PDFWe prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally definable in a fixed o-minimal and polynomially bounded reduct. As an application we show that in certain o-minimal structures definable functions are piecewise implicitly defined over the basic functions in the language
D-optimal minimum support mixture designs in blocks
Get PDFThis paper is concerned with the statistical properties of experimental designs where the factor levels cannot be set precisely. When the errors in setting the factor levels cannot be measured, design robustness is explored. However, when the actual design could be measured at the end of the investigation, its optimality is of interest. D-optimality could be assessed in different ways. Several measures are compared. Evaluating them is difficult even in simple cases. Therefore, in general, simulations are used to obtain their values. It is shown that if D-optimality is measured by the expected value of the determinant of the information matrix of the experimental design, as has been suggested in the past, on average the designs appear to improve with the variance of the error in setting the factor levels. However, we argue that the criterion of D-optimality should be based on the inverse of the information matrix. In this case it is shown that the experiment could be better or worse than the planned one. It is also recognized that setting the factor levels with error could lead to an increased risk of losing observations, which on its own could reduce considerably the optimality of the experimental designs. Advice on choosing the design region in such a way that such a risk is controlled to an acceptable level is given
Optimum Experimental Designs, With SAS
Get PDFExperiments on patients, processes or plants all have random error, making statistical methods essential for their efficient design and analysis. This book presents the theory and methods of optimum experimental design, making them available through the use of SAS programs. Little previous statistical knowledge is assumed. The first part of the book stresses the importance of models in the analysis of data and introduces least squares fitting and simple optimum experimental designs. The second part presents a more detailed discussion of the general theory and of a wide variety of experiments. The book stresses the use of SAS to provide hands-on solutions for the construction of designs in both standard and non-standard situations. The mathematical theory of the designs is developed in parallel with their construction in SAS, so providing motivation for the development of the subject. Many chapters cover self-contained topics drawn from science, engineering and pharmaceutical investigations, such as response surface designs, blocking of experiments, designs for mixture experiments and for nonlinear and generalized linear models. Understanding is aided by the provision of "SAS tasks" after most chapters as well as by more traditional exercises and a fully supported website. The authors are leading experts in key fields and this book is ideal for statisticians and scientists in academia, research and the process and pharmaceutical industries
Pricing financial claims contingent upon an underlying asset monitored at discrete times
Get PDFExotic option contracts typically specify a contingency upon an underlying asset price monitored at a discrete set of times. Yet, techniques used to price such options routinely assume continuous monitoring leading to often substantial price discrepancies. A brief review of relevant option-pricing methods is presented. The pricing problem is transformed into one of Wiener–Hopf type using a z-transform in time and a Fourier transform in the logarithm of asset prices. The Wiener–Hopf technique is used to obtain probabilistic identities for the related random walks killed by an absorbing boundary. An accurate and efficient approximation is obtained using Padé approximants and an approximate inverse z-transform based on the trapezoidal rule. For simplicity, European barrier options in a Gaussian Black–Scholes framework are used to exemplify the technique (for which exact analytic expressions are obtained). Extensions to different option contracts and options driven by other Lévy processes are discussed
The equivariant cohomology of weighted projective spaces
Get PDFWe describe the integral equivariant cohomology of a weighted projective space in terms of piecewise polynomials, as well as by generators and relations. Unlike the ordinary integral cohomology, this ring distinguishes among weighted projective spaces. We also prove a Chern class formula for weighted
projective bundles
Long-term persistence of solar active longitudes and its implications for the solar dynamo theory
No full textWe present an overview of the observational results related to the existence of long-lived sunspot active longitudes. These are affected
by the solar differential rotation. The existence of such migrating active longitudes imposes an important constraint on the dynamo theory.
We review different approaches to model non-axisymmetry in solar dynamo models and find that, in principle, plausible mechanisms
exist to reproduce the observed non-axisymmetry. The most favorable interpretation is suggested by the ‘stroboscopic effect’, where a
quasi-rigidly rotating non-axisymmetric mean-field can produce seemingly migrating active longitudes in sunspots. Other scenarios
are less favorable but cannot yet be excluded
A model to investigate the feasibility of FDG as a surrogate marker of hypoxia
Get PDFFmiso-PET is a non-invasive modality used for the assessment of tumour hypoxia, and increasingly for planning radiotherapy. However, the availability and contrast properties of Fmiso are not ideal. Recent efforts to compare FDG binding with that of Fmiso, in order to ascertain FDG's potential as a marker of hypoxia, have met with mixed results. The potential reasons for correlated and disparate binding patterns between the two tracers have been postulated, but not formally outlined as yet. We present a model of a key component of the image formation process - tracer pharmacokinetics. This involves a series of coupled PDEs, describing the interplay between concentrations of oxygen, glucose, HIF, Fmiso and FDG. We use this model to assess the general feasibility of FDG as a surrogate marker of hypoxia and find that its utility is dependent on activity of oncogenic pathways
O-Minimal Structures
Get PDFThe notion of an o-minimal expansion of the ordered field of real numbers was
invented by L van den Dries [vdD1] as a framework for investigating the model theory
of the real exponential function exp : R ! R : x ! ex, and thereby settle an old
problem of Tarski. More on this later, but for the moment it is best motivated as being
a candidate for Grothendieck’s idea of “tame topology” as expounded in his Esquisse
d’un Programme [Gr]. It seems to me that such a candidate should satisfy (at least)
the following criteria.
(A) It should be a framework that is flexible enough to carry out many geometrical
and topological constructions on real functions and on subsets of real euclidean spaces.
(B) But at the same time it should have built in restrictions so that we are a priori
guaranteed that pathological phenomena can never arise. In particular, there should
be a meaningful notion of dimension for all sets under consideration and any that can
be constructed from these by use of the operations allowed under (A).
(C) One must be able to prove finiteness theorems that are uniform over fibred collections.
None of the standard restrictions on functions that arise in elementary real analysis
satisfy both (A) and (B). For example, there exists a continuous function G : (0, 1) !
(0, 1)2 which is surjective, thereby destroying any hope of a dimension theory for a
framework that admits all continuous functions. Restricting to the smooth (i.e. C1)
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environment fares no better. For every closed subset of any euclidean space, in particular
the subset graph(G) of R3, is the set of zeros of some smooth function. So by the
use of a few simple constructions that we would certainly wish to allow under (A), we
soon arrive at dimension-destroying phenomena. The same is even true (though this is
harder to prove) if we start from just those smooth functions that are everywhere real
analytic (i.e. equal the sum of their Taylor series on a neighbourhood of every point),
although, as we shall see, this class of functions is locally well-behaved and as such can
serve as a model for the three criteria above.
Rather than enumerate analytic conditions on sets and functions sufficient to guarantee
the criteria (A), (B) and (C) however, we shall give one succinct axiom, the
o-minimality axiom, which implies them. Of course, this is a rather open-ended (and
currently flourishing) project because of the large number of questions that one can ask
under (C). One must also provide concrete examples of collections of sets and functions
that satisfy the axiom and this too is an active area of research. In this talk I shall
survey both aspects of the theory.
Our formulation of the o-minimality axiom makes use of definability theory from
mathematical logic. We begin with a collection F of real valued functions of real
variables (not necessarily all of the same number of arguments). We consider the
ordered field structure on R augmented by the functions in F. This gives us a first-order
structure (or model ) RF := hR;+, ·,−,<,Fi, and we denote the corresponding firstorder
logical language by L(F). We then call the structure RF o-minimal if whenever
(x) is an L(F)-formula (with parameters) then the subset of R defined by (x) is a
finite union of open intervals and points (i.e. it is the union of finitely many connected
sets). I shall elucidate what is meant by an L(F)-formula and by the subset of R (and,
more generally, of Rn) defined by such a formula in the next two sections. However, I
should emphasize at this stage that such a formula not only defines a subset , denoted
(RF), of Rn, but also a subset (R) of Rn where R is any ordered ring augmented
by a collection of functions, F say, such that F and F are in correspondence via a
bijection that preserves the number of places (arity) of the functions. One can, and
should, define the notion o-minimality for such structures hR;Fi and it was at (rather
more than) this level of generality that the true foundations of the subject were laid by
Pillay and Steinhorn in [P-S], shortly after van den Dries’ work on the real field. Indeed,
it turned out that the solution to Tarski’s problem on the real exponential function (the
case F = {exp} in the above notation) relied heavily on the Pillay-Steinhorn theory of
o-minimality for structures based on ordered fields other than the reals. This having
been said, I shall concentrate in this lecture on the real case, alluding only occasionally
to the more general situation, and leave the reader to adapt the definitions and theorems
to the setting of o-minimal expansions of arbitrary ordered fields