On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Systematics and evolution of the genus Deuterocohnia Mez (Bromeliaceae)

The present study investigates the systematics and evolution of the Neotropical genus Deuterocohnia Mez (Bromeliaceae). It provides a comprehensive taxonomic revision as well as phylogenetic analyses based on chloroplast and nuclear DNA sequences and presents a hypothesis on the evolution of the genus. A broad morphological, anatomical, biogeographical and ecological overview of the genus is given in the first part of the study. For morphological character assessment more than 700 herbarium specimens from 39 herbaria as well as living plant material in the field and in the living collections of botanical gardens were carefully examined. The arid habitats, in which the species of Deuterocohnia grow, are reflected by the morphological and anatomical characters of the species. Important characters for species delimitation were identified, like the length of the inflorescence, the branching order, the density of flowers on partial inflorescences, the relation of the length of the primary bracts to that of the partial inflorescence, the sizes of floral bracts, sepals and petals, flower colour, the presence or absence of a pedicel, the curvature of the stamina and the petals during anthesis. After scrutinizing the nomenclatural history of the taxa belonging to Deuterocohnia – including the 1992 syonymized genus Abromeitiella – 17 species, 4 subspecies and 4 varieties are accepted in the present revision. Taxonomic changes were made in the following cases: (I) New combinations: A. abstrusa (A. Cast.) N. Schütz is re-established – as defined by Castellanos (1931) – and transfered to D. abstrusa; D. brevifolia (Griseb.) M.A. Spencer & L.B. Sm. includes accessions of the former D. lorentziana (Mez) M.A. Spencer & L.B. Sm., which are not assigned to D. abstrusa; D. bracteosa W. Till is synonymized to D. strobilifera Mez; D. meziana Kuntze ex Mez var. carmineo-viridiflora Rauh is classified as a subspecies of D. meziana (ssp. carmineo-viridiflora (Rauh) N. Schütz); D. pedicellata W. Till is classified as a subspecies of D. meziana (ssp. pedicellata (W. Till) N. Schütz); D. scapigera (Rauh & L. Hrom.) M.A. Spencer & L.B. Sm ssp. sanctae-crucis R. Vásquez & Ibisch is classified as a species (D. sanctae-crucis (R. Vásquez & Ibisch) N. Schütz); (II) New taxa: a new subspecies of D. meziana Kuntze ex Mez is established; a new variety of D. scapigera is established; (the new taxa will be validly published elsewhere); (III) New type: an epitype for D. longipetala was chosen. All other species were kept according to Spencer and Smith (1992) or – in the case of more recently described species – according to the protologue. Beside the nomenclatural notes and the detailed descriptions, information on distribution, habitat and ecology, etymology and taxonomic delimitation is provided for the genus and for each of its species. An key was constructed for the identification of currently accepted species, subspecies and varieties. The key is based on easily detectable morphological characters. The former synonymization of the genus Abromeitiella into Deuterocohnia (Spencer and Smith 1992) is re-evalutated in the present study. Morphological as well as molecular investigations revealed Deuterocohnia incl. Abromeitiella as being monophyletic, with some indications that a monophyletic Abromeitiella lineage arose from within Deuterocohnia. Thus the union of both genera is confirmed. The second part of the present thesis describes and discusses the molecular phylogenies and networks. Molecular analyses of three chloroplast intergenic spacers (rpl32-trnL, rps16-trnK, trnS-ycf3) were conducted with a sample set of 119 taxa. This set included 103 Deuterocohnia accessions from all 17 described species of the genus and 16 outgroup taxa from the remainder of Pitcairnioideae s.str. (Dyckia (8 sp.), Encholirium (2 sp.), Fosterella (4 sp.) and Pitcairnia (2 sp.)). With its high sampling density, the present investigation by far represents the most comprehensive molecular study of Deuterocohnia up till now. All data sets were analyzed separately as well as in combination, and various optimality criteria for phylogenetic tree construction were applied (Maximum Parsimony, Maximum Likelihood, Bayesian inferences and the distance method Neighbour Joining). Congruent topologies were generally obtained with different algorithms and optimality criteria, but individual clades received different degrees of statistical support in some analyses. The rps16-trnK locus was the most informative among the three spacer regions examined. The results of the chloroplast DNA analyses revealed a highly supported paraphyly of Deuterocohnia. Thus, the cpDNA trees divide the genus into two subclades (A and B), of which Deuterocohnia subclade B is sister to the included Dyckia and Encholirium accessions, and both together are sister to Deuterocohnia subclade A. To further examine the relationship between Deuterocohnia and Dyckia/Encholirium at the generic level, two nuclear low copy markers (PRK exon2-5 and PHYC exon1) were analysed with a reduced taxon set. This set included 22 Deuterocohnia accessions (including members of both cpDNA subclades), 2 Dyckia, 2 Encholirium and 2 Fosterella species. Phylogenetic trees were constructed as described above, and for comparison the same reduced taxon set was also analysed at the three cpDNA data loci. In contrast to the cpDNA results, the nuclear DNA data strongly supported the monophyly of Deuterocohnia, which takes a sister position to a clade of Dyckia and Encholirium samples. As morphology as well as nuclear DNA data generated in the present study and in a former AFLP analysis (Horres 2003) all corroborate the monophyly of Deuterocohnia, the apparent paraphyly displayed in cpDNA analyses is interpreted to be the consequence of a chloroplast capture event. This involves the introgression of the chloroplast genome from the common ancestor of the Dyckia/ Encholirium lineage into the ancestor of Deuterocohnia subclade B species. The chloroplast haplotypes are not species-specific in Deuterocohnia. Thus, one haplotype was sometimes shared by several species, where the same species may harbour different haplotypes. The arrangement of haplotypes followed geographical patterns rather than taxonomic boundaries, which may indicate some residual gene flow among populations from different Deuteroccohnia species. Phenotypic species coherence on the background of ongoing gene flow may then be maintained by sets of co-adapted alleles, as was suggested by the porous genome concept (Wu 2001, Palma-Silva et al. 2011). The results of the present study suggest the following scenario for the evolution of Deuterocohnia and its species. Deuterocohnia longipetala may be envisaged as a representative of the ancestral state within the genus. This is supported by (1) the wide distribution of this species; (2) the overlap in distribution area with species of Dyckia; (3) the laxly flowered inflorescences, which are also typical for Dyckia; (4) the yellow petals with a greenish tip, present in most other Deuterocohnia species. The following six extant lineages within Deuterocohnia might have independently been derived from this ancestral state with a few changes each: (I) D. meziana, D. brevispicata and D. seramisiana (Bolivia, lowland to montane areas, mostly reddish-greenish coloured, very laxly to very densely flowered); (II) D. strobilifera (Bolivia, high Andean mountains, yellow flowers, densely flowered); (III) D. glandulosa (Bolivia, montane areas, yellow-greenish flowers, densely flowered); (IV) D. haumanii, D. schreiteri, D. digitata, and D. chrysantha (Argentina, Chile, E Andean mountains and Atacama desert, yellow-greenish flowers, densely flowered); (V) D. recurvipetala (Argentina, foothills of the Andes, recurved yellow flowers, laxly flowered); (VI) D. gableana, D. scapigera, D. sanctae-crucis, D. abstrusa, D. brevifolia, D. lotteae (former Abromeitiella species, Bolivia, Argentina, higher Andean mountains, greenish-yellow flowers, inflorescence usually simple). Originating from the lower montane Andean regions, at least four lineages of the genus (I, II, IV, VI) adapted in part to higher altitudes by developing densely flowered partial inflorescences, shorter flowers and – in at least three lineages (II, IV, VI) – smaller rosettes, whereas species spreading into the lowlands (I, V) developed larger plants, laxly flowered, amply branched inflorescences and in part larger flowers (I).

Evaluating partial root-zone irrigation and mulching in okra (Abelmoschus esculentus L.) under a sub-humid tropical climate

The field experiments were conducted to compare the alternate partial root-zone irrigation (APRI) with and without black plastic mulch (BPM) with full root-zone irrigation (FRI) in furrow-irrigated okra (Abelmoschus esculentus L. Moench) at Bhubaneswar, India. APRI means that one of the two neighbouring furrows was alternately irrigated during consecutive watering. FRI was the conventional method where every furrow was irrigated during each watering. The used irrigation levels were 25% available soil moisture depletion (ASMD), 50% ASMD, and 75% ASMD. The plant growth and yield parameters were observed to be significantly (p < 0.05) higher with frequent irrigation (at 25% ASMD) under all irrigation strategies. However, APRI + BPM produced the maximum plant growth and yield using 22% and 56% less water over APRI without BPM and FRI, respectively. The highest pod yield (10025 kg ha^-1) was produced under APRI at 25% ASMD + BPM, which was statistically at par with the pod yield under APRI at 50% ASMD + BPM. Irrigation water use efficiency (IWUE), which indicates the pod yield per unit quantity of irrigation water, was estimated to be highest (12.3 kg m^-3) under APRI at 50% ASMD + BPM, followed by APRI at 25% ASMD + BPM. Moreover, the treatment APRI at 50% ASMD + BPM was found economically superior to other treatments, generating more net return (US $ 952 ha^-1) with higher benefit–cost ratio (1.70).

A Compilation Strategy for Numerical Programs Based on Partial Evaluation

This work demonstrates how partial evaluation can be put to practical use in the domain of high-performance numerical computation. I have developed a technique for performing partial evaluation by using placeholders to propagate intermediate results. For an important class of numerical programs, a compiler based on this technique improves performance by an order of magnitude over conventional compilation techniques. I show that by eliminating inherently sequential data-structure references, partial evaluation exposes the low-level parallelism inherent in a computation. I have implemented several parallel scheduling and analysis programs that study the tradeoffs involved in the design of an architecture that can effectively utilize this parallelism. I present these results using the 9- body gravitational attraction problem as an example.

Exploiting the Parallelism Exposed by Partial Evaluation

We describe the key role played by partial evaluation in the Supercomputing Toolkit, a parallel computing system for scientific applications that effectively exploits the vast amount of parallelism exposed by partial evaluation. The Supercomputing Toolkit parallel processor and its associated partial evaluation-based compiler have been used extensively by scientists at MIT, and have made possible recent results in astrophysics showing that the motion of the planets in our solar system is chaotically unstable.

Application of the european customer satisfaction index to postal services. Structural equation models versus partial least squars

Customer satisfaction and retention are key issues for organizations in today’s competitive market place. As such, much research and revenue has been invested in developing accurate ways of assessing consumer satisfaction at both the macro (national) and micro (organizational) level, facilitating comparisons in performance both within and between industries. Since the instigation of the national customer satisfaction indices (CSI), partial least squares (PLS) has been used to estimate the CSI models in preference to structural equation models (SEM) because they do not rely on strict assumptions about the data. However, this choice was based upon some misconceptions about the use of SEM’s and does not take into consideration more recent advances in SEM, including estimation methods that are robust to non-normality and missing data. In this paper, both SEM and PLS approaches were compared by evaluating perceptions of the Isle of Man Post Office Products and Customer service using a CSI format. The new robust SEM procedures were found to be advantageous over PLS. Product quality was found to be the only driver of customer satisfaction, while image and satisfaction were the only predictors of loyalty, thus arguing for the specificity of postal services

Partial disjoint path for multi-layer protection in GMPLS networks

In this paper, different recovery methods applied at different network layers and time scales are used in order to enhance the network reliability. Each layer deploys its own fault management methods. However, current recovery methods are applied to only a specific layer. New protection schemes, based on the proposed partial disjoint path algorithm, are defined in order to avoid protection duplications in a multi-layer scenario. The new protection schemes also encompass shared segment backup computation and shared risk link group identification. A complete set of experiments proves the efficiency of the proposed methods in relation with previous ones, in terms of resources used to protect the network, the failure recovery time and the request rejection ratio

Partial Diff pdf

Exam questions and solutions in PDF

Partial Diff

Exam questions and solutions in LaTex

Applications of Partial Differentiation

Exercises and solutions for an applications of partial differentiation course. Diagrams for the questions are all together in the support.zip file, as .eps files

Partial Differentiation

Exercises and solutions about partial differentiation.

INFO2009 Group Website Resource - The Digital Divide

INFO2009 Group website as main resource for the group coursework. Covers the topic of the Digital Divide.