Digitisation, representation, and formalisation - Digital libraries of mathematics

One of the main tasks of the mathematical knowledge management community must surely be to enhance access to mathematics on digital systems. In this paper we present a spectrum of approaches to solving the various problems inherent in this task, arguing that a variety of approaches is both necessary and useful. The main ideas presented are about the differences between digitised mathematics, digitally represented mathematics and formalised mathematics. Each has its part to play in managing mathematical information in a connected world. Digitised material is that which is embodied in a computer file, accessible and displayable locally or globally. Represented material is digital material in which there is some structure (usually syntactic in nature) which maps to the mathematics contained in the digitised information. Formalised material is that in which both the syntax and semantics of the represented material, is automatically accessible. Given the range of mathematical information to which access is desired, and the limited resources available for managing that information, we must ensure that these resources are applied to digitise, form representations of or formalise, existing and new mathematical information in such a way as to extract the most benefit from the least expenditure of resources. We also analyse some of the various social and legal issues which surround the practical tasks.

Can wavelets improve the representation of forecast error covariances in variational data assimilation?

Two wavelet-based control variable transform schemes are described and are used to model some important features of forecast error statistics for use in variational data assimilation. The first is a conventional wavelet scheme and the other is an approximation of it. Their ability to capture the position and scale-dependent aspects of covariance structures is tested in a two-dimensional latitude-height context. This is done by comparing the covariance structures implied by the wavelet schemes with those found from the explicit forecast error covariance matrix, and with a non-wavelet- based covariance scheme used currently in an operational assimilation scheme. Qualitatively, the wavelet-based schemes show potential at modeling forecast error statistics well without giving preference to either position or scale-dependent aspects. The degree of spectral representation can be controlled by changing the number of spectral bands in the schemes, and the least number of bands that achieves adequate results is found for the model domain used. Evidence is found of a trade-off between the localization of features in positional and spectral spaces when the number of bands is changed. By examining implied covariance diagnostics, the wavelet-based schemes are found, on the whole, to give results that are closer to diagnostics found from the explicit matrix than from the nonwavelet scheme. Even though the nature of the covariances has the right qualities in spectral space, variances are found to be too low at some wavenumbers and vertical correlation length scales are found to be too long at most scales. The wavelet schemes are found to be good at resolving variations in position and scale-dependent horizontal length scales, although the length scales reproduced are usually too short. The second of the wavelet-based schemes is often found to be better than the first in some important respects, but, unlike the first, it has no exact inverse transform.

The representation of conditional expectations of non-Gaussian noise

For Wiener spaces conditional expectations and $L^{2}$-martingales w.r.t. the natural filtration have a natural representation in terms of chaos expansion. In this note an extension to larger classes of processes is discussed. In particular, it is pointed out that orthogonality of the chaos expansion is not required.

View-based approaches to spatial representation in human vision

In an immersive virtual environment, observers fail to notice the expansion of a room around them and consequently make gross errors when comparing the size of objects. This result is difficult to explain if the visual system continuously generates a 3-D model of the scene based on known baseline information from interocular separation or proprioception as the observer walks. An alternative is that observers use view-based methods to guide their actions and to represent the spatial layout of the scene. In this case, they may have an expectation of the images they will receive but be insensitive to the rate at which images arrive as they walk. We describe the way in which the eye movement strategy of animals simplifies motion processing if their goal is to move towards a desired image and discuss dorsal and ventral stream processing of moving images in that context. Although many questions about view-based approaches to scene representation remain unanswered, the solutions are likely to be highly relevant to understanding biological 3-D vision.

Devolution and the political representation of business interests in the UK

The devolution of political power in Scotland, Wales and Northern Ireland and the developing regional agenda in England are widely read as a significant reconfiguration of the institutions and scales of economic governance. The process is furthest developed in Scotland while Wales and Northern Ireland, in their own distinct ways, provide intermediate cases. Devolution is least developed in England where regional political identities are generally weak and the historical legacy of regional institutions is limited. Within the overall context of devolution government policy has continued to emphasize partnership forms of. governance. Accordingly, the political representation of business interests has a particular salience in the new arrangements. This paper reports on findings from a study designed to examine the relationship between devolution and changes in the political representation of business interests in the territories and regions of the UK. It highlights a number of changes in the nature and extent of business representation. While some of these are significant the evidence suggests that they fail to mark a fundamental shift in the institutional foundation for sub-national business interest representation in the UK. Indeed the political geography of business representation remains dominated by an overarching centralism that is likely to provide a significant check on the further devolution of political power and democratic authority. (c) 2004 Elsevier Ltd. All rights reserved.

Patterns of pollen and seed dispersal in a small, fragmented population of the wind-pollinated tree Araucaria angustifolia in southern Brazil

Paternity analysis based on eight microsatellite loci was used to investigate pollen and seed dispersal patterns of the dioecious wind- pollinated tree, Araucaria angustifolia. The study sites were a 5.4 ha isolated forest fragment and a small tree group situated 1.7 km away, located in Paran alpha State, Brazil. In the forest fragment, 121 males, 99 females, 66 seedlings and 92 juveniles were mapped and genotyped, together with 210 seeds. In the tree group, nine male and two female adults were mapped and genotyped, together with 20 seeds. Paternity analysis within the forest fragment indicated that at least 4% of the seeds, 3% of the seedlings and 7% of the juveniles were fertilized by pollen from trees in the adjacent group, and 6% of the seeds were fertilized by pollen from trees outside these stands. The average pollination distance within the forest fragment was 83 m; when the tree group was included the pollination distance was 2006m. The average number of effective pollen donors was estimated as 12.6. Mother- trees within the fragment could be assigned to all seedlings and juveniles, suggesting an absence of seed immigration. The distance of seedlings and juveniles from their assigned mother- trees ranged from 0.35 to 291m ( with an average of 83m). Significant spatial genetic structure among adult trees, seedlings, and juveniles was detected up to 50m, indicating seed dispersal over a short distance. The effective pollination neighborhood ranged from 0.4 to 3.3 ha. The results suggest that seed dispersal is restricted but that there is longdistance pollen dispersal between the forest fragment and the tree group; thus, the two stands of trees are not isolated.

A reconstructed Siberian High index since A. D. 1599 from Eurasian and North American tree rings

The long-term variability of the Siberian High, the dominant Northern Hemisphere anticyclone during winter, is largely unknown. To investigate how this feature varied prior to the instrumental record, we present a reconstruction of a Dec-Feb Siberian High (SH) index based on Eurasian and North American tree rings. Spanning 1599-1980, it provides information on SH variability over the past four centuries. A decline in the instrumental SH index since the late 1970s, related to Eurasian warming, is the most striking feature over the past four hundred years. It is associated with a highly significant (p < 0.0001) step change in 1989. Significant similar to 3-4 yr spectral peaks in the reconstruction fall within the range of variability of the East Asian winter monsoon (which has also declined recently) and lend further support to proposed relationships between these largescale features of the climate system.

Wiggle-match dating of tree-ring sequences

Given the non-monotonic form of the radiocarbon calibration curve, the precision of single C-14 dates on the calendar timescale will always be limited. One way around this limitation is through comparison of time-series, which should exhibit the same irregular patterning as the calibration curve. This approach can be employed most directly in the case of wood samples with many years growth present (but not able to be dated by dendrochronology), where the tree-ring series of unknown date can be compared against the similarly constructed C-14 calibration curve built from known-age wood. This process of curve-fitting has come to be called "wiggle-matching." In this paper, we look at the requirements for getting good precision by this method: sequence length, sampling frequency, and measurement precision. We also look at 3 case studies: one a piece of wood which has been independently dendrochronologically dated, and two others of unknown age relating to archaeological activity at Silchester, UK (Roman) and Miletos, Anatolia (relating to the volcanic eruption at Thera).

Competition in tree row agroforestry systems. 3. Soil water distribution and dynamics

The purpose of this study was to test the hypothesis that soil water content would vary spatially with distance from a tree row and that the effect would differ according to tree species. A field study was conducted on a kaolinitic Oxisol in the sub-humid highlands of western Kenya to compare soil water distribution and dynamics in a maize monoculture with that under maize (Zea mays L.) intercropped with a 3-year-old tree row of Grevillea robusta A. Cunn. Ex R. Br. (grevillea) and hedgerow of Senna spectabilis DC. (senna). Soil water content was measured at weekly intervals during one cropping season using a neutron probe. Measurements were made from 20 cm to a depth of 225 cm at distances of 75, 150, 300 and 525 cm from the tree rows. The amount of water stored was greater under the sole maize crop than the agroforestry systems, especially the grevillea-maize system. Stored soil water in the grevillea-maize system increased with increasing distance from the tree row but in the senna-maize system, it decreased between 75 and 300 cm from the hedgerow. Soil water content increased least and more slowly early in the season in the grevillea-maize system, and drying was also evident as the frequency of rain declined. Soil water content at the end of the cropping season was similar to that at the start of the season in the grevillea-maize system, but about 50 and 80 mm greater in the senna-maize and sole maize systems, respectively. The seasonal water balance showed there was 140 mm, of drainage from the sole maize system. A similar amount was lost from the agroforestry systems (about 160 mm in the grevillea-maize system and 145 mm in the senna-maize system) through drainage or tree uptake. The possible benefits of reduced soil evaporation and crop transpiration close to a tree row were not evident in the grevillea-maize system, but appeared to greatly compensate for water uptake losses in the senna-maize system. Grevillea, managed as a tree row, reduced stored soil water to a greater extent than senna, managed as a hedgerow.

The social relations of business representation and devolved governance in the United Kingdom

The reform of regional governance in the United Kingdom has been, in part, premised on the notion that regions provide new territories of action in which cooperative networks between business communities and state-agencies can be established. Promoting business interests is seen as one mechanism for enhancing the economic competitiveness and performance of 'laggard' regions. Yet, within this context of change, business agendas and capacities are often assumed to exist 'out there, as a resource waiting to be tapped by state institutions. There is little recognition that business organisations' involvement in networks of governance owes much to historical patterns and practices of business representation, to the types of activities that exist within the business sector, and to interpretations of their own role and position within wider policymaking and implementation networks. This paper, drawing on a study of business agendas in post-devolution Scotland, demonstrates that in practice business agendas are highly complex. Their formation in any particular place depends on the actions of reflexive agents, whose perspectives and capacities are shaped by the social, economic, and political contexts within which they are operating. As such, any understanding of business agendas needs to identify the social relations of business as a whole, rather than assuming away such complexities.

Devolution and the political representation of business interests in the UK

The devolution of political power in Scotland, Wales and Northern Ireland and the developing regional agenda in England are widely read as a significant reconfiguration of the institutions and scales of economic governance. The process is furthest developed in Scotland while Wales and Northern Ireland, in their own distinct ways, provide intermediate cases. Devolution is least developed in England where regional political identities are generally weak and the historical legacy of regional institutions is limited. Within the overall context of devolution government policy has continued to emphasize partnership forms of. governance. Accordingly, the political representation of business interests has a particular salience in the new arrangements. This paper reports on findings from a study designed to examine the relationship between devolution and changes in the political representation of business interests in the territories and regions of the UK. It highlights a number of changes in the nature and extent of business representation. While some of these are significant the evidence suggests that they fail to mark a fundamental shift in the institutional foundation for sub-national business interest representation in the UK. Indeed the political geography of business representation remains dominated by an overarching centralism that is likely to provide a significant check on the further devolution of political power and democratic authority. (c) 2004 Elsevier Ltd. All rights reserved.