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A continuous version of spherical multiresolution is described, starting from continuous wavelet transform on the sphere. Scale discretization enables us to construct spherical counterparts to Daubechies wavelets and wavelet packets (known from Euclidean theory). Essential tool is the theory of singular integrals on the sphere. It is shown that singular integral operators forming a semigroup of contraction operators of class (Co) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to (pyramidal) algorithms.

The basic theory of spherical singular integrals is recapitulated. Criteria are given for measuring the space-frequency localization of functions on the sphere. The trade off between space localization on the sphere and frequency localization in terms of spherical harmonics is described in form of an uncertainty principle. A continuous version of spherical multiresolution is introduced, starting from continuous wavelet transform corresponding to spherical wavelets with vanishing moments up to a certain order. The wavelet transform is characterized by least-squares properties. Scale discretization enables us to construct spherical counterparts of wavelet packets and scale discrete Daubechies" wavelets. It is shown that singular integral operators forming a semigroup of contraction operators of class (Co) (like Abel-Poisson or Gauß-Weierstraß operators) lead in canonical way to pyramyd algorithms. Fully discretized wavelet transforms are obtained via approximate integration rules on the sphere. Finally applications to (geo-)physical reality are discussed in more detail. A combined method is proposed for approximating the low frequency parts of a physical quantity by spherical harmonics and the high frequency parts by spherical wavelets. The particular significance of this combined concept is motivated for the situation of today" s physical geodesy, viz. the determination of the high frequency parts of the earth" s gravitational potential under explicit knowledge of the lower order part in terms of a spherical harmonic expansion.

Glycine constitutes the major neurotransmitter at inhibitory synapses of lower brain regions.
A rapid removal of glycine from the synaptic cleft and consequent recycling is crucial for
synaptic transmission in systems with high effort on temporal precision. This is mainly
achieved by glycine translocation via two glycine transporters (GlyTs), namely GlyT1 and
GlyT2. At inhibitory synapses, GlyT2 was found to be specifically expressed by neurons,
supplying the presynapse with glycine needed for vesicle filling. In contrast, GlyT1 is attributed
to astrocytes and primarily mediates the termination of synaptic transmission by glycine
removal from the synaptic cleft. Employing patch-clamp recordings from principal neurons of
the lateral superior olive (LSO) in acute brainstem slices of GlyT1b/c knockout (KO) mice and
wildtype (WT) littermates at postnatal day 20, I analyzed how postsynaptic responses are
changed in a GlyT1-depleted environment. During spontaneous vesicle release I found no
change of postsynaptic responses, contradicting my initial hypothesis of prolonged decay
times. Electrical stimulation of fibers of the medial nucleus of the trapezoid body (MNTB),
which are known to form fast, reliable and highly precise synapses with LSO principal neurons,
revealed that GlyT1 is involved in proper synaptic function during sustained, high frequent
synaptic transmission. Stimulation with 50 Hz led to a stronger decay time and latency
prolongation in GlyT1b/c KO, accelerating to 60% longer decay times and 30% longer latencies.
Additionally, a more pronounced frequency-dependent depression and fidelity decrease was
observed during stimulation with 200 Hz in GlyT1b/c KO, resulting in 67% smaller amplitudes
and only 25% of WT fidelity at the end of the challenge. Basic properties like readily releasable
pool, release probability, and quantal size (q) were not altered in GlyT1b/c KO, but
interestingly q decreased during 50 Hz and 100 Hz challenges to about 84%, which was not
observed in WT. I conclude that stronger accumulation of extracellular glycine due to GlyT1
loss leads to prolonged activation of postsynaptic glycine receptors (GlyRs). As a further
consequence, activation of presynaptic GlyRs in the vicinity of the synaptic cleft might be
enhanced, accompanied by a stronger occurrence of shunting inhibition. Furthermore, I
assume a GlyT1-dependent glycine shuttle, which is absent at GlyT1b/c KO synapses. This
could result in a diminished glycine supply to GlyT2 located at more distant sites, causing a
disturbed replenishment during periods with excess release of glycine. Conclusively, my study
reveals a contribution of astrocytes in fast and reliable synaptic transmission at the MNTB-LSO
synapse, which in turn is crucial for proper sound source localization.

Due to the increasing demand of renewable energy production facilities, modeling geothermal reservoirs is a central issue in today's engineering practice. After over 40 years of study, many models have been proposed and applied to hundreds of sites worldwide. Nevertheless, with increasing computational capabilities new efficient methods are becoming available. The aim of this paper is to present recent progress on seismic processing as well as fluid and thermal flow simulations for porous and fractured subsurface systems. The commonly used methods in industrial energy exploration and production such as forward modeling, seismic migration, and inversion methods together with continuum and discrete flow models for reservoir monitoring and management are reviewed. Furthermore, for two specific features numerical examples are presented. Finally, future fields of studies are described.

The inverse problem of recovering the Earth's density distribution from data of the first or second derivative of the gravitational potential at satellite orbit height is discussed for a ball-shaped Earth. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbitheight is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG (satellite gravity gradiometry) satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust. Moreover, the noise sensitivity of the regularization technique is analyzed numerically.

The inverse problem of recovering the Earth's density distribution from satellite data of the first or second derivative of the gravitational potential at orbit height is discussed. This problem is exponentially ill-posed. In this paper a multiscale regularization technique using scaling functions and wavelets constructed for the corresponding integro-differential equations is introduced and its numerical applications are discussed. In the numerical part the second radial derivative of the gravitational potential at 200 km orbit height is calculated on a point grid out of the NASA/GSFC/NIMA Earth Geopotential Model (EGM96). Those simulated derived data out of SGG satellite measurements are taken for convolutions with the introduced scaling functions yielding a multiresolution analysis of harmonic density variations in the Earth's crust.

We show that the intersection local times \(\mu_p\) on the intersection of \(p\) independent planar Brownian paths have an average density of order three with respect to the gauge function \(r^2\pi\cdot (log(1/r)/\pi)^p\), more precisely, almost surely, \[ \lim\limits_{\varepsilon\downarrow 0} \frac{1}{log |log\ \varepsilon|} \int_\varepsilon^{1/e} \frac{\mu_p(B(x,r))}{r^2\pi\cdot (log(1/r)/\pi)^p} \frac{dr}{r\ log (1/r)} = 2^p \mbox{ at $\mu_p$-almost every $x$.} \] We also show that the lacunarity distributions of \(\mu_p\), at \(\mu_p\)-almost every point, is given as the distribution of the product of \(p\) independent gamma(2)-distributed random variables. The main tools of the proof are a Palm distribution associated with the intersection local time and an approximation theorem of Le Gall.

Integral equations on the half of line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by apositie number called finite-section parameter. In this paper we consider the finite-section approximation for first kind intgral equations which are typically ill-posed and call for regularization. For some classes of such equations corresponding to inverse problems from optics and astronomy we indicate the finite-section parameters that allows to apply standard regularization techniques. Two discretization schemes for the finite-section equations ar also proposed and their efficiency is studied.

Power-ordered sets are not always lattices. In the case of distributive lattices we give a description by disjoint of chains. Finite power-ordered sets have a polarity. We introduct the leveled lattices and show examples with trivial tolerance. Finally we give a list of Hasse diagrams of power-ordered sets.

Algebraic Systems Theory
(2004)

Control systems are usually described by differential equations, but their properties of interest are most naturally expressed in terms of the system trajectories, i.e., the set of all solutions to the equations. This is the central idea behind the so-called "behavioral approach" to systems and control theory. On the other hand, the manipulation of linear systems of differential equations can be formalized using algebra, more precisely, module theory and homological methods ("algebraic analysis"). The relationship between modules and systems is very rich, in fact, it is a categorical duality in many cases of practical interest. This leads to algebraic characterizations of structural systems properties such as autonomy, controllability, and observability. The aim of these lecture notes is to investigate this module-system correspondence. Particular emphasis is put on the application areas of one-dimensional rational systems (linear ODE with rational coefficients), and multi-dimensional constant systems (linear PDE with constant coefficients).