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.

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

Correcting Measurement Error Bias in Interaction Models with Small Samples

Several methods have been suggested to estimate non-linear models with interaction terms in the presence of measurement error. Structural equation models eliminate measurement error bias, but require large samples. Ordinary least squares regression on summated scales, regression on factor scores and partial least squares are appropriate for small samples but do not correct measurement error bias. Two stage least squares regression does correct measurement error bias but the results strongly depend on the instrumental variable choice. This article discusses the old disattenuated regression method as an alternative for correcting measurement error in small samples. The method is extended to the case of interaction terms and is illustrated on a model that examines the interaction effect of innovation and style of use of budgets on business performance. Alternative reliability estimates that can be used to disattenuate the estimates are discussed. A comparison is made with the alternative methods. Methods that do not correct for measurement error bias perform very similarly and considerably worse than disattenuated regression

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.

Efectos de la ventilación mecánica con heliox en niños y adolescentes con patología bronquial obstructiva

Evaluar si el Heliox reduce la resistencia en la vía aérea en niños y adolescentes con patología bronquial obstructiva que requieren ventilación mecánica. Materiales y Métodos: Estudio prospectivo observacional descriptivo en niños y adolescentes con patología bronquial obstructiva y ventilación mecánica con Fi02 ≤ 0,5. Medición de variables: resistencia, presión pico, presión media de la vía aérea, presión meseta, volumen corriente, autoPEEP, distensibilidad, PetCO2, ventilación de espacio muerto antes de inicio de heliox y a los 30 minutos, 2, 4, 6, 12, 18 y 24 horas y diariamente hasta suspenderlo por extubación o FiO2 > 0,5. Resultados: Resultados parciales, incluyó 9 pacientes encontrando descenso significativo de resistencia espiratoria a los 30 minutos (51,2 vs 32,3; p=0,0008 ), 2 horas ( 51,2 vs 33,4; p=0,0019) y 4 horas (51,2 vs 30,7; p=0,0012) así como de la resistencia inspiratoria a la hora 2 (48,6 vs 36,2; p = 0,013) y hora 4 (48,6 vs 30 ; p=0,004). Se observó tendencia al descenso de la PetCO2 que no fue significativa (52,3 vs 34,3: p=0,06). No se evidenció cambios en las variables; autoPEEP, presión pico, presión media de la vía aérea, distensibilidad, ventilación de espacio muerto, presión meseta y volumen corriente antes y después del inicio del Heliox. Conclusión: La ventilación mecánica con Heliox en niños con patología bronquial obstructiva parece ser que reduce de manera significativa la resistencia de la vía aérea, con tendencia al descenso de la PetC02. Se necesitan estudios prospectivos al menos observacionales analíticos que corroboren estos hallazgos.

A prediction model for establishing the disability degree in adults with spinal cord injury: results based on WHO-DAS II perspective

Objective: To establish a prediction model of the degree of disability in adults with Spinal CordInjury (SCI ) based on the use of the WHO-DAS II . Methods: The disability degree was correlatedwith three variable groups: clinical, sociodemographic and those related with rehabilitation services.A model of multiple linear regression was built to predict disability. 45 people with sci exhibitingdiverse etiology, neurological level and completeness participated. Patients were older than 18 andthey had more than a six-month post-injury. The WHO-DAS II and the ASIA impairment scale(AIS ) were used. Results: Variables that evidenced a significant relationship with disability were thefollowing: occupational situation, type of affiliation to the public health care system, injury evolutiontime, neurological level, partial preservation zone, ais motor and sensory scores and number ofclinical complications during the last year. Complications significantly associated to disability werejoint pain, urinary infections, intestinal problems and autonomic disreflexia. None of the variablesrelated to rehabilitation services showed significant association with disability. The disability degreeexhibited significant differences in favor of the groups that received the following services: assistivedevices supply and vocational, job or educational counseling. Conclusions: The best predictiondisability model in adults with sci with more than six months post-injury was built with variablesof injury evolution time, AIS sensory score and injury-related unemployment.

Partial decentralization as a safeguard against favoritism

In this paper I investigate the optimal level of decentralization of tasks for the provision of a local public good. I enrich the well-known trade-off between internalization of spillovers (that favors centralization) and accountability (that favors decentralization) by considering that public goods are produced through multiple tasks. This adds an additional institutional setting, partial decentralization, to the classical choice between full decentralization and full centralization. The main results are that partial decentralization is optimal when both the variance of exogenous shocks to electorate’s utility is large and the electorate expects high performance from politicians. I also show that the optimal institutional setting depends on the degree of substitutability / complementarity between tasks. In particular, I show that a large degree of substitutability between tasks makes favoritism more likely, which increases the desirability of partial decentralization as a safeguard against favoritism.

Admissible statistics of educational test scores. 'Estadisticas admisibles de puntuaciones en test educativos'.

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