Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes

The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.

GROUPS THAT INVOLVE FINITELY MANY PRIMES AND HAVE ALL SUBGROUPS SUBNORMAL II

It is shown that if G is a hypercentral group with all subgroups subnormal, and if the torsion subgroup of G is a pi-group for some finite set pi of primes, then G is nilpotent. In the case where G is not hypercentral there is a section of G that is much like one of the well-known Heineken-Mohamed groups. It is also shown that if G is a residually nilpotent group with all subgroups subnormal whose torsion subgroup satisfies the above condition then G is nilpotent.

Locally targeted cytoprotection with dextran sulfate attenuates experimental porcine myocardial ischaemia/reperfusion injury

AIMS: Intravascular inflammatory events during ischaemia/reperfusion injury following coronary angioplasty alter and denudate the endothelium of its natural anticoagulant heparan sulfate proteoglycan (HSPG) layer, contributing to myocardial tissue damage. We propose that locally targeted cytoprotection of ischaemic myocardium with the glycosaminoglycan analogue dextran sulfate (DXS, MW 5000) may protect damaged tissue from reperfusion injury by functional restoration of HSPG. METHODS AND RESULTS: In a closed chest porcine model of acute myocardial ischaemia/reperfusion injury (60 min ischaemia, 120 min reperfusion), DXS was administered intracoronarily into the area at risk 5 min prior to reperfusion. Despite similar areas at risk in both groups (39+/-8% and 42+/-9% of left ventricular mass), DXS significantly decreased myocardial infarct size from 61+/-12% of the area at risk for vehicle controls to 39+/-14%. Cardioprotection correlated with reduced cardiac enzyme release creatine kinase (CK-MB, troponin-I). DXS abrogated myocardial complement deposition and substantially decreased vascular expression of pro-coagulant tissue factor in ischaemic myocardium. DXS binding, detected using fluorescein-labelled agent, localized to ischaemically damaged blood vessels/myocardium and correlated with reduced vascular staining of HSPG. CONCLUSION: The significant cardioprotection obtained through targeted cytoprotection of ischaemic tissue prior to reperfusion in this model of acute myocardial infarction suggests a possible role for the local modulation of vascular inflammation by glycosaminoglycan analogues as a novel therapy to reduce reperfusion injury.

Physical examination during chemoradiation predicts outcome of locally advanced head and neck cancer. Secondary results of a randomized phase III trial (SAKK 10/94)

OBJECTIVES To analyze the prognostic value of clinical tumor response during chemoradiation for locally advanced head and neck cancer. PATIENTS AND METHODS The locoregional response at 50.4Gy was assessed by physical examination (PE) in patients treated within the randomized trial SAKK 10/94 using hyperfractionated radiotherapy (RT), median total dose 74.4Gy with or without cisplatin 20mg/m(2) chemotherapy on 5 consecutive days during weeks 1 and 5 or 6 of RT. Response was classified as a complete response (CR), complete response with uncertainty (Cru), partial response (PR), stable disease (SD), or progressive disease (PD). The primary endpoint was time to treatment failure (TTF) due to any cause. Secondary endpoints included locoregional-recurrence-free survival (LRRFS), distant metastasis-free survival (DMFS) and overall survival (OS). Univariate and multivariate Cox proportional hazards (PH) models were applied to analyze the associations between survival endpoints and clinical tumor response. RESULTS A total of 136, 131 and 97 patients were evaluable for response at the primary tumor, lymph nodes and both sites combined, respectively. At 50.4Gy 57/136 (42%), 46/131 (35%) and 21/97 (22%) patients had a good response (CR/Cru vs. PR/SD) at the primary tumor, the lymph nodes, and both sites combined, respectively. The median follow-up times were 11.4, 9.6 and 11.4years for the three groups. Good responses were all significantly associated with improved TTF, LRRFS, DMFS and OS in univariate analysis whereas good response at the primary tumor and lymph nodes remained significantly associated with TTF and OS after multivariate Cox PH models. CONCLUSIONS Locoregional response at 50.4Gy was identified as predictor of oncologic outcome. PE during treatment should not be underestimated in clinical practice.

A Finite Model Property for Gödel Modal Logics

A new semantics with the finite model property is provided and used to establish decidability for Gödel modal logics based on (crisp or fuzzy) Kripke frames combined locally with Gödel logic. A similar methodology is also used to establish decidability, and indeed co-NP-completeness for a Gödel S5 logic that coincides with the one-variable fragment of first-order Gödel logic.

Finite element analysis in asymptomatic, symptomatic, and ruptured abdominal aortic aneurysms: in search of new rupture risk predictors.

OBJECTIVES To compare biomechanical rupture risk parameters of asymptomatic, symptomatic and ruptured abdominal aortic aneurysms (AAA) using finite element analysis (FEA). STUDY DESIGN Retrospective biomechanical single center analysis of asymptomatic, symptomatic, and ruptured AAAs. Comparison of biomechanical parameters from FEA. MATERIALS AND METHODS From 2011 to 2013 computed tomography angiography (CTA) data from 30 asymptomatic, 15 symptomatic, and 15 ruptured AAAs were collected consecutively. FEA was performed according to the successive steps of AAA vessel reconstruction, segmentation and finite element computation. Biomechanical parameters Peak Wall Rupture Risk Index (PWRI), Peak Wall Stress (PWS), and Rupture Risk Equivalent Diameter (RRED) were compared among the three subgroups. RESULTS PWRI differentiated between asymptomatic and symptomatic AAAs (p < .0004) better than PWS (p < .1453). PWRI-dependent RRED was higher in the symptomatic subgroup compared with the asymptomatic subgroup (p < .0004). Maximum AAA external diameters were comparable between the two groups (p < .1355). Ruptured AAAs showed the highest values for external diameter, total intraluminal thrombus volume, PWS, RRED, and PWRI compared with asymptomatic and symptomatic AAAs. In contrast with symptomatic and ruptured AAAs, none of the asymptomatic patients had a PWRI value >1.0. This threshold value might identify patients at imminent risk of rupture. CONCLUSIONS From different FEA derived parameters, PWRI distinguishes most precisely between asymptomatic and symptomatic AAAs. If elevated, this value may represent a negative prognostic factor for asymptomatic AAAs.

Consensus and differences in primary radiotherapy for localized and locally advanced prostate cancer in Switzerland: A survey on patterns of practice

INTRODUCTION External beam radiotherapy (EBRT), with or without androgen deprivation therapy (ADT), is an established treatment option for nonmetastatic prostate cancer. Despite high-level evidence from several randomized trials, risk group stratification and treatment recommendations vary due to contradictory or inconclusive data, particularly with regard to EBRT dose prescription and ADT duration. Our aim was to investigate current patterns of practice in primary EBRT for prostate cancer in Switzerland. MATERIALS AND METHODS Treatment recommendations on EBRT and ADT for localized and locally advanced prostate cancer were collected from 23 Swiss radiation oncology centers. Written recommendations were converted into center-specific decision trees, and analyzed for consensus and differences using a dedicated software tool. Additionally, specific radiotherapy planning and delivery techniques from the participating centers were assessed. RESULTS The most commonly prescribed radiation dose was 78 Gy (range 70-80 Gy) across all risk groups. ADT was recommended for intermediate-risk patients for 6 months in over 80 % of the centers, and for high-risk patients for 2 or 3 years in over 90 % of centers. For recommendations on combined EBRT and ADT treatment, consensus levels did not exceed 39 % in any clinical scenario. Arc-based intensity-modulated radiotherapy (IMRT) is implemented for routine prostate cancer radiotherapy by 96 % of the centers. CONCLUSION Among Swiss radiation oncology centers, considerable ranges of radiotherapy dose and ADT duration are routinely offered for localized and locally advanced prostate cancer. In the vast majority of cases, doses and durations are within the range of those described in current evidence-based guidelines.

Locally rare species influence grassland ecosystem multifunctionality

Species diversity promotes the delivery of multiple ecosystem functions (multifunctionality). However, the relative functional importance of rare and common species in driving the biodiversity–multifunctionality relationship remains unknown. We studied the relationship between the diversity of rare and common species (according to their local abundances and across nine different trophic groups), and multifunctionality indices derived from 14 ecosystem functions on 150 grasslands across a land-use intensity (LUI) gradient. The diversity of above- and below-ground rare species had opposite effects, with rare above-ground species being associated with high levels of multifunctionality, probably because their effects on different functions did not trade off against each other. Conversely, common species were only related to average, not high, levels of multifunctionality, and their functional effects declined with LUI. Apart from the community-level effects of diversity, we found significant positive associations between the abundance of individual species and multifunctionality in 6% of the species tested. Species-specific functional effects were best predicted by their response to LUI: species that declined in abundance with land use intensification were those associated with higher levels of multifunctionality. Our results highlight the importance of rare species for ecosystem multifunctionality and help guiding future conservation priorities.

Finite elastic deformations of transversely isotropic circular cylindrical tubes

We consider the finite radially symmetric deformation of a circular cylindrical tube of a homogeneous transversely isotropic elastic material subject to axial stretch, radial deformation and torsion, supported by axial load, internal pressure and end moment. Two different directions of transverse isotropy are considered: the radial direction and an arbitrary direction in planes normal locally to the radial direction, the only directions for which the considered deformation is admissible in general. In the absence of body forces, formulas are obtained for the internal pressure, and the resultant axial load and torsional moment on the ends of the tube in respect of a general strain-energy function. For a specific material model of transversely isotropic elasticity, and material and geometrical parameters, numerical results are used to illustrate the dependence of the pressure, (reduced) axial load and moment on the radial stretch and a measure of the torsional deformation for a fixed value of the axial stretch.

Quantum groups with invariant integrals

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

The structure of Selmer groups

The purpose of this article is to describe certain results and conjectures concerning the structure of Galois cohomology groups and Selmer groups, especially for abelian varieties. These results are analogues of a classical theorem of Iwasawa. We formulate a very general version of the Weak Leopoldt Conjecture. One consequence of this conjecture is the nonexistence of proper Λ-submodules of finite index in a certain Galois cohomology group. Under certain hypotheses, one can prove the nonexistence of proper Λ-submodules of finite index in Selmer groups. An example shows that some hypotheses are needed.

The adsorption of water in finite carbon pores

Grand canonical Monte Carlo simulations were applied to the adsorption of SPCE model water in finite graphitic pores with different configurations of carbonyl functional groups on only one surface and several pore sizes. It was found that almost all finite pores studied exhibit capillary condensation behaviour preceded by adsorption around the functional groups. Desorption showed the reverse transitions from a filled to a near empty pore resulting in a clear hysteresis loop in all pores except for some of the configurations of the 1.0nm pore. Carbonyl configurations had a strong effect on the filling pressure of all pores except, in some cases, in 1.0nm pores. A decrease in carbonyl neighbour density would result in a higher filling pressure. The emptying pressure was negligibly affected by the configuration of functional groups. Both the filling and emptying pressures increased with increasing pore size but the effect on the emptying pressure was much less. At pressures lower than the pore filling pressure, the adsorption of water was shown to have an extremely strong dependence on the neighbour density with adsorption changing from Type IV to Type III to linear as the neighbour density decreased. The isosteric heat was also calculated for these configurations to reveal its strong dependence on the neighbour density. These results were compared with literature experimental results for water and carbon black and found to qualitatively agree.

Response-based segmentation using Finite Mixture Partial Least Squares: theoretical foundations and an application to American Customer Satisfaction Index Data

When applying multivariate analysis techniques in information systems and social science disciplines, such as management information systems (MIS) and marketing, the assumption that the empirical data originate from a single homogeneous population is often unrealistic. When applying a causal modeling approach, such as partial least squares (PLS) path modeling, segmentation is a key issue in coping with the problem of heterogeneity in estimated cause-and-effect relationships. This chapter presents a new PLS path modeling approach which classifies units on the basis of the heterogeneity of the estimates in the inner model. If unobserved heterogeneity significantly affects the estimated path model relationships on the aggregate data level, the methodology will allow homogenous groups of observations to be created that exhibit distinctive path model estimates. The approach will, thus, provide differentiated analytical outcomes that permit more precise interpretations of each segment formed. An application on a large data set in an example of the American customer satisfaction index (ACSI) substantiates the methodology’s effectiveness in evaluating PLS path modeling results.

Polynomial Automorphisms Over Finite Fields

It is shown that the invertible polynomial maps over a finite field Fq , if looked at as bijections Fn,q −→ Fn,q , give all possible bijections in the case q = 2, or q = p^r where p > 2. In the case q = 2^r where r > 1 it is shown that the tame subgroup of the invertible polynomial maps gives only the even bijections, i.e. only half the bijections. As a consequence it is shown that a set S ⊂ Fn,q can be a zero set of a coordinate if and only if #S = q^(n−1).