There exists a vast literature on convergence rates results for Tikhonov regularized minimizers. We are concerned with the solution of nonlinear ill-posed operator equations. The first convergence rates results for such problems have been developed by Engl, Kunisch and Neubauer in 1989. While these results apply for operator equations formulated in Hilbert spaces, the results of Burger and Osher from 2004, more generally, apply to operators formulated in Banach spaces. Recently, Resmerita et al. presented a modification of the convergence rates result of Burger and Osher which turns out a complete generalization of the rates result of Engl et. al. In all these papers relatively strong regularity assumptions are made. However, it has been observed numerically, that violations of the smoothness assumptions of the operator do not necessarily affect the convergence rate negatively. We take this observation and weaken the smoothness assumptions on the operator and prove a novel convergence rate result. The most significant difference in this result to the previous ones is that the source condition is formulated as a variational inequality and not as an equation as before. As examples we present a phase retrieval problem and a specific inverse option pricing problem, both studied in the literature before. For the inverse finance problem, the new approach allows us to bridge the gap to a singular case, where the operator smoothness degenerates just when the degree of ill-posedness is minimal.

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Dysregulation of the Wnt signalling pathway contributes to developmental abnormalities and carcinogenesis of solid tumours. Here, we examined beta-catenin and adenomatous polyposis coli (APC) by mutational analysis in pituitary adenomas (n=60) and a large series of craniopharyngiomas (n=41). Furthermore, the expression pattern of beta-catenin was immunohistochemically analysed in a cohort of tumours and cysts of the sellar region including pituitary adenomas (n=58), craniopharyngiomas (n=57), arachnoidal cysts (n=8), Rathke's cleft cysts (n=10) and xanthogranulomas (n=6). Whereas APC mutations were not detectable in any tumour entity, beta-catenin mutations were present in 77% of craniopharyngiomas, exclusively of the adamantinomatous subtype. All mutations affected exon 3, which encodes the degradation targeting box of beta-catenin compatible with an accumulation of nuclear beta-catenin protein. In addition, a novel 81-bp deletion of this exonic region was detected in one case. Immunohistochemical analysis confirmed a shift from membrane-bound to nuclear accumulation of beta-catenin in 94% of the adamantinomatous tumours. Aberrant distribution patterns of beta-catenin were never observed in the other tumour entities under study. We conclude that beta-catenin mutations and/or nuclear accumulation serve as diagnostic hallmarks of the adamantinomatous variant, setting it apart from the papillary variant of craniopharyngioma.

Microsurgery remains the treatment of first choice in CD, even though no improvement in remission rates was observed over the years, because complication or remission rates for other treatment options are comparable or worse.

Objective: Medical therapy with dopamine agonists (DA) is the primary treatment of choice in most patients with prolactinomas. 'Classical' surgical indications are intolerance or lack of efficiency of DA therapy. Focusing on a possible shift of recent indications, we retrospectively analyzed our results of surgical treatment in prolactinomas. Patients and methods: Between 1990 and 2005, we have operated on 212 consecutive patients with prolactinomas. Surgical indications were divided into 'classical' indications and 'modern' indications defined as cystic prolactinomas or patients with microprolactinomas who individually decided on a primary surgical treatment. Results: Initial overall remission was accomplished in 53.2% including giant prolactinomas. However, in microadenomas, the remission rate was significantly higher with 91.3%. Overall remission at the latest follow-up was 42.7%, but 72.5% in intrasellar tumors, 80% in cystic prolactinomas, and 84.8% in microprolactinomas. The overall recurrence rate was 18.7%. Relapse of hyperprolactinemia in microprolactinomas was 7.1%. In our series, continually less patients were surgically treated for 'classical' indications. By contrast, the number of patients who individually decided on a primary surgical therapy has increased considerably. Conclusion: Remission rates after surgical treatment of prolactinomas remain excellent, particularly in microadenoma and intrasellar macroadenomas, whereas morbidity of transsphenoidal surgery is low in the hands of experienced pituitary surgeons. Our remission rates not only confirm the already interdisciplinarily accepted surgical indications, but also emphasize the value of primary transsphenoidal surgery as a discussion-worthy alternative to dopaminergic therapy in young patients with microprolactinomas or cystic tumors.

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In this paper we consider the ill-posed nonlinear integral equation x * x = y of autoconvolution defined on the interval [0, I]. We discus conditions for the compacmess, injectivity and weak closedness of the associated integral operator. The general theory of Tikhonov regularization for nonlinear ill-posed problems can be applied, and provides an approach to define different levels and degrees of ill-pasedness in Hilben spaces. For the autoconvolution problem we observe a varying degree of ill-posedness depending on the smoothness of solutions and on the behaviour of solutions and their derivatives for small arguments.* This paper was winen during a stay of B Hofmann at the Free University of Berlin as a fellow of the Alexander von Humboldt Foundation hom October 1992 to March 1993.

In the recent past the authors, with collaborators, have published convergence rate results for regularized solutions of linear ill-posed operator equations by avoiding the usual assumption that the solutions satisfy prescribed source conditions. Instead the degree of violation of such source conditions is expressed by distance functions d(R) depending on a radius R 0 which is an upper bound of the norm of source elements under consideration. If d(R) tends to zero as R → ∞ an appropriate balancing of occurring regularization error terms yields convergence rates results. This approach was called the method of approximate source conditions, originally developed in a Hilbert space setting. The goal of this paper is to formulate chances and limitations of an application of this method to nonlinear ill-posed problems in reflexive Banach spaces and to complement the field of low order convergence rates results in nonlinear regularization theory. In particular, we are going to establish convergence rates for a variant of Tikhonov regularization. To keep structural nonlinearity conditions simple, we update the concept of degree of nonlinearity in Hilbert spaces to a Bregman distance setting in Banach spaces.