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We describe a natural construction of deformation quantisation on a compact symplectic manifold with boundary. On the algebra of quantum observables a trace functional is defined which as usual annihilates the commutators. This gives rise to an index as the trace of the unity element. We formulate the index theorem as a conjecture and examine it by the classical harmonic oscillator.
We introduce a natural symmetry condition for a pseudodifferential operator on a manifold with cylindrical ends ensuring that the operator admits a doubling across the boundary. For such operators we prove an explicit index formula containing, apart from the Atiyah-Singer integral, a finite number of residues of the logarithmic derivative of the conormal symbol.
For general elliptic pseudodifferential operators on manifolds with singular points, we prove an algebraic index formula. In this formula the symbolic contributions from the interior and from the singular points are explicitly singled out. For two-dimensional manifolds, the interior contribution is reduced to the Atiyah-Singer integral over the cosphere bundle while two additional terms arise. The first of the two is one half of the 'eta' invariant associated to the conormal symbol of the operator at singular points. The second term is also completely determined by the conormal symbol. The example of the Cauchy-Riemann operator on the complex plane shows that all the three terms may be non-zero.
The index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points contains the Atiyah-Singer integral as well as two additional terms. One of the two is the 'eta' invariant defined by the conormal symbol, and the other term is explicitly expressed via the principal and subprincipal symbols of the operator at conical points. In the preceding paper we clarified the meaning of the additional terms for first-order differential operators. The aim of this paper is an explicit description of the contribution of a conical point for higher-order differential operators. We show that changing the origin in the complex plane reduces the entire contribution of the conical point to the shifted 'eta' invariant. In turn this latter is expressed in terms of the monodromy matrix for an ordinary differential equation defined by the conormal symbol.
In the preceding paper we proved an explicit index formula for elliptic pseudodifferential operators on a two-dimensional manifold with conical points. Apart from the Atiyah-Singer integral, it contains two additional terms, one of the two being the 'eta' invariant defined by the conormal symbol. In this paper we clarify the meaning of the additional terms for differential operators.
Using the Riemannian connection on a compact manifold X, we show that the algebra of classical pseudo-differential operators on X generates a canonical deformation quantization on the cotangent manifold T*X. The corresponding Abelian connection is calculated explicitly in terms of the of the exponential mapping. We prove also that the index theorem for elliptic operators may be obtained as a consequence of the index theorem for deformation quantization.
We consider a G-invariant star-product algebra A on a symplectic manifold (M,ω) obtained by a canonical construction of deformation quantization. Under assumptions of the classical Marsden-Weinstein theorem we define a reduction of the algebra A with respect to the G-action. The reduced algebra turns out to be isomorphic to a canonical star-product algebra on the reduced phase space B. In other words, we show that the reduction commutes with the canonical G-invariant deformation quantization. A similar statement in the framework of geometric quantization is known as the Guillemin-Sternberg conjecture (by now completely proved).
We construct a deformation quantization on an infinite-dimensional symplectic space of regular connections on an SU(2)-bundle over a Riemannian surface of genus g ≥ 2. The construction is based on the normal form thoerem representing the space of connections as a fibration over a finite-dimensional moduli space of flat connections whose fibre is a cotangent bundle of the infinite-dimensional gauge group. We study the reduction with respect to the gauge groupe both for classical and quantum cases and show that our quantization commutes with reduction.
We give a construction of an eigenstate for a non-critical level of the Hamiltonian function, and investigate the contribution of Morse critical points to the spectral decomposition. We compare the rigorous result with the series obtained by a perturbation theory. As an example the relation to the spectral asymptotics is discussed.
In this work, the development of a new molecular building block, based on synthetic peptides derived from decorin, is presented. These peptides represent a promising basis for the design of polymer-based biomaterials that mimic the ECM on a molecular level and exploit specific biological recognition for technical applications. Multiple sequence alignments of the internal repeats of decorin that formed the inner and outer surface of the arch-shaped protein were used to develop consensus sequences. These sequences contained conserved sequence motifs that are likely to be related to structural and functional features of the protein. Peptides representative for the consensus sequences were synthesized by microwave-assisted solid phase peptide synthesis and purified by RP-HPLC, with purities higher than 95 mol%. After confirming the desired masses by MALDI-TOF-MS, the primary structure of each peptide was investigated by 1H and 2D NMR, from which a full assignment of the chemical shifts was obtained. The characterization of the peptides conformation in solution was performed by CD spectroscopy, which demonstrated that using TFE, the peptides from the outer surface of decorin show a high propensity to fold into helical structures as observed in the original protein. To the contrary, the peptides from the inner surface did not show propensity to form stable secondary structure. The investigation of the binding capability of the peptides to Collagen I was performed by surface plasmon resonance analyses, from which all but one of the peptides representing the inner surface of decorin showed binding affinity to collagen with values of dissociation constant between 2•10-7 M and 2.3•10-4 M. On the other hand, the peptides representative for the outer surface of decorin did not show any significant interaction to collagen. This information was then used to develop experimental demonstration for the binding capabilities of the peptides from the inner surface of decorin to collagen even when used in more complicated situations close to possible appications. With this purpose, the peptide (LRELHLNNN) which showed the highest binding affinity to collagen (2•10-7 M) was functionalized with an N-terminal triple bond in order to obtain a peptide dimer via copper(I)-catalyzed cycloaddition reaction with 4,4'-diazidostilbene-2,2'-disulfonic acid. Rheological measurements showed that the presence of the peptide dimer was able to enhance the elastic modulus (G') of a collagen gel from ~ 600 Pa (collagen alone) to ~ 2700 Pa (collagen and peptide dimer). Moreover, it was shown that the mechanical properties of a collagen gel can be tailored by using different molar ratios of peptide dimer respect to collagen. The same peptide, functionalized with the triple bond, was used to obtain a peptide-dye conjugate by coupling it with N-(5'-azidopentanoyl)-5-aminofluorescein. An aqueous solution (5 vol% methanol) of the peptide dye conjugate was injected into a collagen and a hyaluronic acid (HA) gel and images of fluorescence detection showed that the diffusion of the peptide was slower in the collagen gel compared to the HA gel. The third experimental demonstration was gained using the peptide (LSELRLHNN) which showed the lower binding affinity (2.3•10-4 M) to collagen. This peptide was grafted to hyaluronic acid via EDC-chemistry, with a degree of functionalization of 7 ± 2 mol% as calculated by 1H-NMR. The grafting was further confirmed by FTIR and TGA measurements, which showed that the onset of decomposition for the HA-g-peptide decreased by 10 °C compared to the native HA. Rheological measurements showed that the elastic modulus of a system based on collagen and HA-g-peptide increased by almost two order of magnitude (G' = 200 Pa) compared to a system based on collagen and HA (G' = 0.9 Pa). Overall, this study showed that the synthetic peptides, which were identified from decorin, can be applied as potential building blocks for biomimetic materials that function via biological recognition.
One of the most significant current discussions in Astrophysics relates to the origin of high-energy cosmic rays. According to our current knowledge, the abundance distribution of the elements in cosmic rays at their point of origin indicates, within plausible error limits, that they were initially formed by nuclear processes in the interiors of stars. It is also believed that their energy distribution up to 1018 eV has Galactic origins. But even though the knowledge about potential sources of cosmic rays is quite poor above „ 1015 eV, that is the “knee” of the cosmic-ray spectrum, up to the knee there seems to be a wide consensus that supernova remnants are the most likely candidates. Evidence of this comes from observations of non-thermal X-ray radiation, requiring synchrotron electrons with energies up to 1014 eV, exactly in the remnant of supernovae. To date, however, there is not conclusive evidence that they produce nuclei, the dominant component of cosmic rays, in addition to electrons. In light of this dearth of evidence, γ-ray observations from supernova remnants can offer the most promising direct way to confirm whether or not these astrophysical objects are indeed the main source of cosmic-ray nuclei below the knee. Recent observations with space- and ground-based observatories have established shell-type supernova remnants as GeV-to- TeV γ-ray sources. The interpretation of these observations is however complicated by the different radiation processes, leptonic and hadronic, that can produce similar fluxes in this energy band rendering ambiguous the nature of the emission itself. The aim of this work is to develop a deeper understanding of these radiation processes from a particular shell-type supernova remnant, namely RX J1713.7–3946, using observations of the LAT instrument onboard the Fermi Gamma-Ray Space Telescope. Furthermore, to obtain accurate spectra and morphology maps of the emission associated with this supernova remnant, an improved model of the diffuse Galactic γ-ray emission background is developed. The analyses of RX J1713.7–3946 carried out with this improved background show that the hard Fermi-LAT spectrum cannot be ascribed to the hadronic emission, leading thus to the conclusion that the leptonic scenario is instead the most natural picture for the high-energy γ-ray emission of RX J1713.7–3946. The leptonic scenario however does not rule out the possibility that cosmic-ray nuclei are accelerated in this supernova remnant, but it suggests that the ambient density may not be high enough to produce a significant hadronic γ-ray emission. Further investigations involving other supernova remnants using the improved back- ground developed in this work could allow compelling population studies, and hence prove or disprove the origin of Galactic cosmic-ray nuclei in these astrophysical objects. A break- through regarding the identification of the radiation mechanisms could be lastly achieved with a new generation of instruments such as CTA.
Ground-based gamma-ray astronomy has had a major breakthrough with the impressive results obtained using systems of imaging atmospheric Cherenkov telescopes. Ground-based gamma-ray astronomy has a huge potential in astrophysics, particle physics and cosmology. CTA is an international initiative to build the next generation instrument, with a factor of 5-10 improvement in sensitivity in the 100 GeV-10 TeV range and the extension to energies well below 100 GeV and above 100 TeV. CTA will consist of two arrays (one in the north, one in the south) for full sky coverage and will be operated as open observatory. The design of CTA is based on currently available technology. This document reports on the status and presents the major design concepts of CTA.
The aim of this paper is to bring together two areas which are of great importance for the study of overdetermined boundary value problems. The first area is homological algebra which is the main tool in constructing the formal theory of overdetermined problems. And the second area is the global calculus of pseudodifferential operators which allows one to develop explicit analysis.