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We provide geometric inequalities on $R^n$ and on general manifolds with nonnegative Ricci curvature by employing suitable monotone quantities along the flow of capacitary and $p$-capacitary potentials, as well as through related boundary value problems. Among the main achievements, we cite [(i)] a Willmore-type inequality on manifolds with nonnegative Ricci curvature leading in turn to the sharp Isoperimetric Inequality on $3$-manifolds with nonnegative Ricci curvature ; [(ii)] enhanced Kasue/Croke-Kleiner splitting theorems ; [(iii)] a generalised Minkowski-type inequality in $R^n$ holding with no assumptions on the boundary of the domain considered except for smoothness ; [(iv)] a complete discussion of maximal volume solutions to the least area problem with obstacle on Riemannian manifolds and its relation with the variational $p$-capacity.

Detta examensarbete har haft som mål att undersöka tjällyftningen hos en väg i en fältstudie och utvärdera reliabiliteten hos två olika modeller för skattning av tjällyftning. Modellerna som har studerats är Trafikverkets nuvarande, vilken är implementerad i deras vägdimensioneringsverktyg PMS Objekt, samt segregation potential theory. Den studerade vägsträckan är belägen i Kiruna. I vägen har borrkärneprover tagits och vägens obundna lager, inklusive terassmaterial, har blivit analyserade i laboratorium. Dessa tester inkluderar både geoteknisk klassificering och endimensionella frysförsök på terassmaterialet. Materialets tjälfarlighet har klassificerats enligt både Trafikverkets eget och det amerikanska USACEs system. Alla prover, inklusive terrassmaterialet, består huvudsakligen av grus och sand. Enligt båda de använda klassificeringssystemen bedöms alla prover som icke tjälfarliga. Nivåmätningar utförda på vägen visar dock att tjällyftning ändå förekommer i vägen. Denna tjällyftning har inte kunnat modelleras framgångsrikt i PMS Objekt, där ingen tjällyftning alls fås trots att en dimensionerande vinter tillämpas i analysen. Däremot har en god överensstämmelse mellan bakåträknad och empiriskt beräknad tjälfarlighetsklassificering med SP0-värde enligt segregation potential theory kunnat ses. Även när detta beräknas utifrån utförda frysförsök fås en relativt god överensstämmelse. Studiens främsta fynd är att den visar på att även material som bedöms som ej tjällyftande enligt de två systemen för tjälfarlighetsklassificering faktiskt orsakar tjällyftning och att detta omfattar både terrass- och överbyggnadsmaterial. Studien visar att detta tjällyft främst beror på primärt lyft, varför klassificeringarna stämmer ganska väl om endast det sekundära lyftet avses. Studien har även visat på att Trafikverkets nuvarande modell är otillräcklig när det kommer till att modellera tjällyftning för grovkorniga jordar. Slutligen har studien även visat på möjligheten att bedöma jords tjälfarlighet med hjälp av segregation potential theory, även om också denna metod endast kan användas för att modellera sekundärt lyft. / The aim of the master thesis has been to examine the frost heaving of a road in a field study and to evaluate the reliability of two different models for estimation of frost heaving. The studied models are the current one of the Swedish Transport Administration, which is implemented in their road design tool PMS Objekt, and the segregation potential theory. The studied road section is located in Kiruna, in the northern part of Sweden. The road has been cored and the unbound layers of it, including the subgrade, have been analyzed in laboratory. These analyses include both geotechnical characterization and one-dimensional frost heave tests of the subgrade material. The frost susceptibility of the collected material has been classified according to both the Swedish Transport Administrations system and the American USACE’s system. All samples, including the subgrade, consist mainly of gravel and sand. According to both the used classification systems all samples are classified as not frost susceptible. However, the elevation measurements of the road show that frost heaving occurs in it despite this. This heaving has not been possible to model successfully in the program PMS Objekt. In it the frost heave prediction yields no heave at all, despite applying a design winter (worst case) in the analysis. A good conformity has however been observed between back-calculated and empirically evaluated frost susceptibility using segregation potential theory. A quite good conformity has also been observed when the same thing is evaluated from the preformed freeze tests. The most important finding of the study is that even material that is categorized as non-frost susceptible according to the two studied systems actually does heave. This finding includes both subgrade aswell as the unbound material in the superstructure. It has been found that the total frost heave observed in this study originates mainly from primary heaving, meaning the studied systems are quite correct in their frost susceptibility classification if only secondary heaving is intended. The study has also shown that the current model applied in PMS Objekt is poor in predicting the amount of heaving in coarse subgrades. Finally, the study has also shown the possibility to evaluate a soils frost susceptibility using segregation potential theory, although this model aswell is only applicable to secondary heave

The development of a three dimensional B-Spline based method, which is suitable for the steady-state potential flow analysis of free surface piercing bodies in hydrodynamics, is presented. The method requires the B-Spline or Non Uniform Rational B-Spline (NURBS) representation of the body as an input. In order to solve for the unknown potential, the source surface, both for the body as well as the free surface, is represented by NURBS surfaces. The method does not require the body surface to be discritized into flat panels. Therefore, instead of a mere panel approximation, the exact body geometry is utilized for the computation. The technique does not use a free surface Green's function, which already satisfies the linear free surface boundary conditions, but uses a separate source patch for the free surface. By eliminating the use of a free surface Green's function, the method can be extended to considering non-linear free surface conditions, thus providing the possibility for wave resistance calculations. The method is first applied to the double body flow problem around a sphere and a Wigley hull. Some comparisons are made with exact solutions to validate the accuracy of the method. Results of linear free surface conditions are then presented.

Thesis. 1978. Ph.D.--Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Vita. / Bibliography: leaves 266-276. / Ph.D.

This paper deals with approximating an upper bound for the number of equilibrium points of a potential field produced by point charges in the plane. This is a simplified form of a problem posed by Maxwell [4], who considered spatial configurations of the point charges. Using algebraic techniques, we will give an upper bound for planar charges that is sharper than the bound given in [6] for most general configurations of charges. Then we will study an example of a configuration of charges that has exactly the number of equilibrium points that Maxwell's conjecture predicts, and we will look into the nature of the extremal points in this case. We will conclude with a solution to the twin problem for the logarithmic potential, followed by a discussion of the conditions necessary for a degenerate case in the plane.

In this dissertation we investigate Lagrange interpolation. Our first result will deal with a hierarchy of interpolation schemes. Specifically, we will show that given a triangular array of points in a regular compact set K, such that the corresponding Lebesgue constants are subexponential, one always has the uniform convergence of Ln(f) to f for all functions analytic on K. We will then show that uniform convergence of Ln(f) to f for all analytic functions f is equivalent to the fact that the probability measures γn = 1/n Σn j=1 δzn,j , which are associated with our triangular array, converge weak star to the equilibrium distribution for K. Motivated by our hierarchy, we will then come to our main result, namely that the Lebesgue constants associated with Leja sequences on fairly general compact sets are subexponential. More generally, considering Newton interpolation on a sequence of points, we will show that the weak star convergence of their corresponding probability measures to the equilibrium distribution, together with a certain distancing rule, implies that their corresponding Lebesgue constants are sub-exponential.

"November, 2001" / Includes bibliographical references (leaves 229-246) / xxiv, 254 leaves : ill. (col.) ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 2002

In this thesis, we examined some Dirichlet type problems of the form: \begin{eqnarray*} \triangle u & = & 0\ {\rm in\ } \mathbb{R}^n \\ u & = & f\ {\rm on\ } \psi = 0, \end{eqnarray*} and we were particularly interested in finding entire solutions when entire data was prescribed. This is an extension of the work of D. Siegel, M. Mouratidis, and M. Chamberland, who were interested in finding polynomial solutions when polynomial data was prescribed. In the cases where they found that polynomial solutions always existed for any polynomial data, we tried to show that entire solutions always existed given any entire data. For half space problems we were successful, but when we compared this to the heat equation, we found that we needed to impose restrictions on the type of data allowed. For problems where data is prescribed on a pair of intersecting lines in the plane, we found a surprising dependence between the existence of an entire solution and the number theoretic properties of the angle between the lines. We were able to show that for numbers $\alpha$ with $\omega_1$ finite according to Mahler's classification of transcendental numbers, there will always be an entire solution given entire data for the angle $2\alpha\pi$ between the lines. We were also able to construct an uncountable, dense set of angles of measure 0, much in the spirit of Liouville's number, for which there will not always be an entire solution for all entire data. Finally, we investigated a problem where data is given on the boundary of an infinite strip in the plane. We were unable to settle this problem, but we were able to reduce it to other {\it a priori} more tractable problems.