Cycle-Level Products in Equivariant Cohomology of Toric Varieties

In this paper, we define an action of the group of equivariant Cartier divisors on a toric variety X on the equivariant cycle groups of X, arising naturally from a choice of complement map on the underlying lattice. If X is nonsingular, this gives a lifting of the multiplication in equivariant cohomology to the level of equivariant cycles. As a consequence, one naturally obtains an equivariant cycle representative of the equivariant Todd class of any toric variety. These results extend to equivariant cohomology the results of Thomas and Pommersheim. In the case of a complement map arising from an inner product, we show that the equivariant cycle Todd class obtained from our construction is identical to the result of the inductive, combinatorial construction of Berline-Vergne. In the case of arbitrary complement maps, we show that our Todd class formula yields the local Euler-Maclarurin formula introduced in Garoufalidis-Pommersheim.Comment: 15 pages, to be published in Michigan Mathematical Journal; LaTe

A K_0K\_0-avoiding dimension group with an order-unit of index two

We prove that there exists a dimension group $G$ whose positive cone is not isomorphic to the dimension monoid Dim$L$ of any lattice $L$. The dimension group $G$ has an order-unit, and can be taken of any cardinality greater than or equal to $\aleph\_2$. As to determining the positive cones of dimension groups in the range of the Dim functor, the $\aleph\_2$ bound is optimal. This solves negatively the problem, raised by the author in 1998, whether any conical refinement monoid is isomorphic to the dimension monoid of some lattice. Since $G$ has an order-unit of index two, this also solves negatively a problem raised in 1994 by K.R. Goodearl about representability, with respect to $K\_0$, of dimension groups with order-unit of index 2 by unit-regular rings.Comment: To appear in Journal of Algebr

Effects of Multirate Systems on the Statistical Properties of Random Signals

In multirate digital signal processing, we often encounter time-varying linear systems such as decimators, interpolators, and modulators. In many applications, these building blocks are interconnected with linear filters to form more complicated systems. It is often necessary to understand the way in which the statistical behavior of a signal changes as it passes through such systems. While some issues in this context have an obvious answer, the analysis becomes more involved with complicated interconnections. For example, consider this question: if we pass a cyclostationary signal with period K through a fractional sampling rate-changing device (implemented with an interpolator, a nonideal low-pass filter and a decimator), what can we say about the statistical properties of the output? How does the behavior change if the filter is replaced by an ideal low-pass filter? In this paper, we answer questions of this nature. As an application, we consider a new adaptive filtering structure, which is well suited for the identification of band-limited channels. This structure exploits the band-limited nature of the channel, and embeds the adaptive filter into a multirate system. The advantages are that the adaptive filter has a smaller length, and the adaptation as well as the filtering are performed at a lower rate. Using the theory developed in this paper, we show that a matrix adaptive filter (dimension determined by the decimator and interpolator) gives better performance in terms of lower error energy at convergence than a traditional adaptive filter. Even though matrix adaptive filters are, in general, computationally more expensive, they offer a performance bound that can be used as a yardstick to judge more practical "scalar multirate adaptation" schemes

Concepts for on-board satellite image registration, volume 1

The NASA-NEEDS program goals present a requirement for on-board signal processing to achieve user-compatible, information-adaptive data acquisition. One very specific area of interest is the preprocessing required to register imaging sensor data which have been distorted by anomalies in subsatellite-point position and/or attitude control. The concepts and considerations involved in using state-of-the-art positioning systems such as the Global Positioning System (GPS) in concert with state-of-the-art attitude stabilization and/or determination systems to provide the required registration accuracy are discussed with emphasis on assessing the accuracy to which a given image picture element can be located and identified, determining those algorithms required to augment the registration procedure and evaluating the technology impact on performing these procedures on-board the satellite

Perturbed polyhedra and the construction of local Euler-Maclaurin formulas

A polyhedron P is a subset of a rational vector space V bounded by hyperplanes. If we fix a lattice in V , then we may consider the exponential integral and sum, two meromorphic functions on the dual vector space which serve to generalize the notion of volume of and number of lattice points contained in P, respectively. In 2007, Berline and Vergne constructed an Euler-Maclaurin formula that relates the exponential sum of a given polyhedron to the exponential integral of each face. This formula was "local", meaning that the coefficients in this formula had certain properties independent of the given polyhedron. In this dissertation, the author finds a new construction for this formula which is very different from that of Berline and Vergne. We may 'perturb' any polyhedron by tranlsating its bounding hyperplanes. The author defines a ring of differential operators R(P) on the exponential volume of the perturbed polyhedron. This definition is inspired by methods in the theory of toric varieties, although no knowledge of toric varieties is necessary to understand the construction or the resulting Euler-Maclaurin formula. Each polyhedron corresponds to a toric variety, and there is a dictionary between combinatorial properties of the polyhedron and algebro-geometric properties of this variety. In particular, the equivariant cohomology ring and the group of equivariant algebraic cycles on the corresponding toric variety are equal to a quotient ring and subgroup of R(P), respectively. Given an inner product (or, more generally, a complement map) on V , there is a canonical section of the equivariant cohomology ring into the group of algebraic cycles. One can use the image under this section of a particular differential operator called the Todd class to define the Euler-Maclaurin formula. The author shows that this formula satisfies the same properties which characterize the Berline-Vergne formula

Interpolation for de-Dopplerisation

'De-Dopplerisation' is one aspect of a problem frequently encountered in experimental acoustics: deducing an emitted source signal from received data. It is necessary when source and receiver are in relative motion, and requires interpolation of the measured signal. This introduces error. In acoustics, typical current practice is to employ linear interpolation and reduce error by over-sampling. In other applications, more advanced approaches with better performance have been developed. Associated with this work is a large body of theoretical analysis, much of which is highly specialised. Nonetheless, a simple and compact performance metric is available: the Fourier transform of the `kernel' function underlying the interpolation method. Furthermore, in the acoustics context, it is a more appropriate indicator than other, more abstract, candidates. On this basis, interpolators from three families previously identified as promising --- piecewise-polynomial, windowed-sinc, and B-spline-based --- are compared. The results show that significant improvements over linear interpolation can straightforwardly be obtained. The recommended approach is B-spline-based interpolation, which performs best irrespective of accuracy specification. Its only drawback is a pre-filtering requirement, which represents an additional implementation cost compared to other methods. If this cost is unacceptable, and aliasing errors (on re-sampling) up to approximately 1% can be tolerated, a family of piecewise-cubic interpolators provides the best alternative

Interpolating point spread function anisotropy

Planned wide-field weak lensing surveys are expected to reduce the statistical errors on the shear field to unprecedented levels. In contrast, systematic errors like those induced by the convolution with the point spread function (PSF) will not benefit from that scaling effect and will require very accurate modeling and correction. While numerous methods have been devised to carry out the PSF correction itself, modeling of the PSF shape and its spatial variations across the instrument field of view has, so far, attracted much less attention. This step is nevertheless crucial because the PSF is only known at star positions while the correction has to be performed at any position on the sky. A reliable interpolation scheme is therefore mandatory and a popular approach has been to use low-order bivariate polynomials. In the present paper, we evaluate four other classical spatial interpolation methods based on splines (B-splines), inverse distance weighting (IDW), radial basis functions (RBF) and ordinary Kriging (OK). These methods are tested on the Star-challenge part of the GRavitational lEnsing Accuracy Testing 2010 (GREAT10) simulated data and are compared with the classical polynomial fitting (Polyfit). We also test all our interpolation methods independently of the way the PSF is modeled, by interpolating the GREAT10 star fields themselves (i.e., the PSF parameters are known exactly at star positions). We find in that case RBF to be the clear winner, closely followed by the other local methods, IDW and OK. The global methods, Polyfit and B-splines, are largely behind, especially in fields with (ground-based) turbulent PSFs. In fields with non-turbulent PSFs, all interpolators reach a variance on PSF systematics $\sigma_{sys}^2$ better than the $1\times10^{-7}$ upper bound expected by future space-based surveys, with the local interpolators performing better than the global ones