Doctor of Philosophy / Department of Mathematics / Ivan Blank / In the Fermi Lectures on the obstacle problem in 1998, Caffarelli gave a proof of the mean value theorem which extends to general divergence form uniformly elliptic operators. In the general setting, the result shows that for any such operator L and at any point [chi]₀ in the domain, there exists a nested family of sets { D[subscript]r([chi]₀) } where the average over any of those sets is related to the value of the function at [chi]₀. Although it is known that the { D[subscript]r([chi]₀) } are nested and are comparable to balls in the sense that there exists c, C depending only on L such that B[subscript]cr([chi]₀) ⊂ D[subscript]r([chi]₀) ⊂ B[subscript]Cr([chi]₀) for all r > 0 and [chi]₀ in the domain, otherwise their geometric and topological properties are largely unknown. In this work we begin the study of these topics and we prove a few results about the geometry of these sets and give a couple of applications of the theorems.

"There is no more fundamental theorem in calculus than the mean-value theorem. Much of the theory of calculus depends, either directly or indirectly, on this theorem. As a consequence of its importance, the theorem has been investigated by a number of mathematicians with the result that various modifications and extensions of the basic theorem have been made"--Introduction. / "May 1956." / Typescript. / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Includes bibliographical references (leaf 17).

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This study covers means, mean value theorems of the differential calculus, and mean value theorems of integral calculus.

A variety of new mean value theorems are presented along with interesting proofs and generalizations of the standard theorems. Three proofs are given for the ordinary Mean Value Theorem for derivatives, the third of which is interesting in that it is independent of of Rolle's Theorem. The Second Mean Value Theorem for derivatives is generalized, with the use of determinants, to three functions and also generalized in terms of nth order derivatives. Observing that under certain conditions the tangent line to the curve of a differentiable function passes through the initial point, we find a new type of mean value theorem for derivatives. This theorem is extended to two functions and later in the paper an integral analog is given together with integral mean value theorems. Many new mean value theorems are presented in their respective settings including theorems for the total variation of a function, the arc length of the graph of a function, and for vector-valued functions. A mean value theorem in the complex plane is given in which the difference quotient is equal to a linear combination of the values of the derivative. Using a regular derivative, the ordinary Mean Value Theorem for derivatives is extended into Rn, n>1.

Please read the PDF file of my thesis.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 121-123).

<p>The emission legislations and the new On Board Diagnostics (OBD) legislations are becoming more strict and making the demands on control and fault detection higher. One way to</p><p>control and diagnose the engine is to use a control/diagnose strategy based on physical models and therefore better models are necessary. Also, to be competitive and meet the markets demand of higher power, longer durability and better fuel economy, the models needs to be improved continuously. In this thesis a mean value model of the intake system that predicts the charge air temperature has been developed. Three models of different complexity for the intercooler heat-exchanger have been investigated and validated with various results. The suggested intercooler heat-exchanger model is implemented in the mean value model of the intake system and the whole model is validated on three different data sets. The model predicts the intake manifold temperature with a maximum absolute error of 10.12 K.</p>

The mean value engine model (MVEM) is a mathematical model derived from basic physical principles such as conservation of mass and energy equations. Although the MVEM is based on some simplified assumptions and time averaged combustion engine parameters, it models the engine with a reasonable approximation and gives a satisfactory amount of information about the physics of the fluid energy passing through an engine system. MVEM can predict an engine’s main external variables such as crankshaft speed and manifold pressure, and important internal variables, such as volumetric and thermal efficiencies. Usually, the differential equations used in MVEM will predict fuel film flow, manifold pressure, and crankshaft speed. Because of its simplicity and short simulation time, the MVEM is widely used for engine control development. A mean value engine based on mathematical and parametric equations has recently been developed in the new MapleSim software. The model consists of three main components: the throttle body, the manifold, and the engine. The new MVEM uses combinations of causal and acausal components along with lookup tables and parametric equations. Adjusting the parameters allows the model to be used for new engines of interest. The model is forward-looking and so benefits from both Maple’s powerful mathematical tool and Modelica’s modern equation-based language. A set of throttle angle and mass flow data is used to find the throttle angle function, and to validate the throttle mass flow rates obtained from the model and the experiment.

published_or_final_version / Mathematics / Master / Master of Philosophy