Homogeneous model theory of metric structures

This thesis studies homogeneous classes of complete metric spaces. Over the past few decades model theory has been extended to cover a variety of nonelementary frameworks. Shelah introduced the abstact elementary classes (AEC) in the 1980s as a common framework for the study of nonelementary classes. Another direction of extension has been the development of model theory for metric structures. This thesis takes a step in the direction of combining these two by introducing an AEC-like setting for studying metric structures. To find balance between generality and the possibility to develop stability theoretic tools, we work in a homogeneous context, thus extending the usual compact approach. The homogeneous context enables the application of stability theoretic tools developed in discrete homogeneous model theory. Using these we prove categoricity transfer theorems for homogeneous metric structures with respect to isometric isomorphisms. We also show how generalized isomorphisms can be added to the class, giving a model theoretic approach to, e.g., Banach space isomorphisms or operator approximations. The novelty is the built-in treatment of these generalized isomorphisms making, e.g., stability up to perturbation the natural stability notion. With respect to these generalized isomorphisms we develop a notion of independence. It behaves well already for structures which are omega-stable up to perturbation and coincides with the one from classical homogeneous model theory over saturated enough models. We also introduce a notion of isolation and prove dominance for it.

Total hip arthroplasty in young patients : with special references to patients under 55 years of age and to patients with developmental dysplasia of the hip

There is an ongoing controversy as to which methods in total hip arthroplasty (THA) could provide young patients with best long-term results. THA is an especially demanding operation in patients with severely dysplastic hips. The optimal surgical treatment for these patients also remains controversial. The aim of this study was to evaluate the long-term survival of THA in young patients (<55 years at the time of the primary operation) on a nation-wide level, and to analyze the long-term clinical and radio-graphical outcome of uncemented THA in patients with severely dysplastic joints. Survival of 4661 primary THAs performed for primary osteoarthritis (OA), 2557 primary THAs per-formed for rheumatoid arthritis (RA), and modern uncemented THA designs performed for primary OA in young patients, were analysed from the Finnish Arthroplasty Register. A total of 68 THAs were per-formed in 56 consecutive patients with high congenital hip dislocation between 1989-1994, and 68 THAs were performed in 59 consecutive patients with severely dysplastic hips and a previous Schanz osteotomy of the femur between 1988-1995 at the Orton Orthopaedic Hospital, Helsinki, Finland. These patients underwent a detailed physical and radiographical evaluation at a mean of 12.3 years and 13.0 years postoperatively, respectively. The risk of stem revision due to aseptic loosening in young patients with primary OA was higher for cemented stems than for proximally porous-coated or HA-coated uncemented stems implanted over the 1991-2001 period. There was no difference in the risk of revision between all-poly cemented-cups and press-fit porous-coated uncemented cups implanted during the same period, when the end point was defined as any revision (including exchange of liner). All uncemented stem designs studied in young patients with primary OA had >90% survival rates at 10 years. The Biomet Bi-Metric stem had a 95% (95% CI 93-97) survival rate even at 15 years. When the end point was defined as any revision, 10 year survival rates of all uncemented cup designs except the Harris-Galante II decreased to <80%. In young patients with RA, the risk of stem revision due to aseptic loosening was higher with cemented stems than with proximally porous-coated uncemented stems. In contrast, the risk of cup revision was higher for all uncemented cup concepts than for all-poly cemented cups with any type of cup revision as the end point. The Harris hip score increased significantly (p<0.001) both in patients with high con-genital hip dislocation and in patients with severely dysplastic hips and a previous Schanz osteotomy, treated with uncemented THA. There was a negative Trendelenburg sign in 92% and in 88% of hips, respectively. There were 12 (18%) and 15 (22%) perioperative complications. The rate of survival for the CDH femoral components, with revision due to aseptic loosening as the end point, was 98% (95% CI 97-100) at 10 years in patients with high hip dislocation and 92% (95% CI, 86-99) at 14 years in patients with a previous Schanz osteotomy. The rate of survival for press-fit, porous-coated acetabular components, with revision due to aseptic loosening as the end point, was 95% (95% CI 89-100) at 10 years in patients with high hip dislocation, and 98% (95% CI 89-100) in patients with a previous Schanz osteotomy. When revision of the cup for any reason was defined as the end point, 10 year sur-vival rates declined to 88% (95% CI 81-95) and to 69% (95% CI, 56-82), respectively. For young patients with primary OA, uncemented proximally circumferentially porous- and HA-coated stems are the implants of choice. However, survival rates of modern uncemented cups are no better than that of all-poly cemented cups. Uncemented proximally circumferentially porous-coated stems and cemented all-poly cups are currently the implants of choice for young patients with RA. Uncemented THA, with placement of the cup at the level of the true acetabulum, distal advancement of the greater trochanter and femoral shortening osteotomy provided patients with high congenital hip dislocation good long-term outcomes. Most of the patients with severely dysplastic hips and a previous Schanz osteotomy can be successfully treated with the same method. However, the subtrochanteric segmental shortening with angular correction gives better leg length correction for the patients with a previous low-seated unilateral Schanz osteotomy.

Global, Local and Vector-Valued Tb Theorems on Non-Homogeneous Metric Spaces

Various Tb theorems play a key role in the modern harmonic analysis. They provide characterizations for the boundedness of Calderón-Zygmund type singular integral operators. The general philosophy is that to conclude the boundedness of an operator T on some function space, one needs only to test it on some suitable function b. The main object of this dissertation is to prove very general Tb theorems. The dissertation consists of four research articles and an introductory part. The framework is general with respect to the domain (a metric space), the measure (an upper doubling measure) and the range (a UMD Banach space). Moreover, the used testing conditions are weak. In the first article a (global) Tb theorem on non-homogeneous metric spaces is proved. One of the main technical components is the construction of a randomization procedure for the metric dyadic cubes. The difficulty lies in the fact that metric spaces do not, in general, have a translation group. Also, the measures considered are more general than in the existing literature. This generality is genuinely important for some applications, including the result of Volberg and Wick concerning the characterization of measures for which the analytic Besov-Sobolev space embeds continuously into the space of square integrable functions. In the second article a vector-valued extension of the main result of the first article is considered. This theorem is a new contribution to the vector-valued literature, since previously such general domains and measures were not allowed. The third article deals with local Tb theorems both in the homogeneous and non-homogeneous situations. A modified version of the general non-homogeneous proof technique of Nazarov, Treil and Volberg is extended to cover the case of upper doubling measures. This technique is also used in the homogeneous setting to prove local Tb theorems with weak testing conditions introduced by Auscher, Hofmann, Muscalu, Tao and Thiele. This gives a completely new and direct proof of such results utilizing the full force of non-homogeneous analysis. The final article has to do with sharp weighted theory for maximal truncations of Calderón-Zygmund operators. This includes a reduction to certain Sawyer-type testing conditions, which are in the spirit of Tb theorems and thus of the dissertation. The article extends the sharp bounds previously known only for untruncated operators, and also proves sharp weak type results, which are new even for untruncated operators. New techniques are introduced to overcome the difficulties introduced by the non-linearity of maximal truncations.