Refine
Has Fulltext
- yes (53) (remove)
Year of publication
- 2001 (53) (remove)
Document Type
- Preprint (29)
- Doctoral Thesis (9)
- Article (7)
- Working Paper (3)
- Habilitation Thesis (2)
- Postprint (2)
- Monograph/Edited Volume (1)
Language
- English (53) (remove)
Keywords
- chaos (2)
- synchronization (2)
- 1799-1804 (1)
- Affixabfolge (1)
- Akkretionsscheiben (1)
- Autokorrelation (1)
- Berlin (1)
- Boosting (1)
- Boundary value problems (1)
- Brewsterwinkel-Mikroskopie (1)
- Calculus of conormal symbols (1)
- Chaos (1)
- Chaostheorie (1)
- Chaotische Sattel (1)
- Cholesterin (1)
- Daphnia (1)
- Dissoziation (1)
- Distributed Morphology (1)
- Dynamische Systeme (1)
- Effekt (1)
- Elektrolyte (1)
- European Alps (1)
- Feld (1)
- Fettsäure (1)
- Fettsäuren (1)
- Flexion (1)
- GIXD (1)
- Hidden Markov Model (HMM) (1)
- Humboldt Digital Library (1)
- Hungarian-German bilinguals (1)
- Hydrodynamik (1)
- Interaktion (1)
- Isothermen (1)
- Jean-Baptiste Boussingault (1)
- Klassifikation mit großem Margin (1)
- Klimaveränderung (1)
- Kosmopolit (1)
- Lake Constance (1)
- Langmuir Monoschicht (1)
- Large Margin Classification (1)
- Leistungsspektrum (1)
- Linienspannung (1)
- Mathematical Optimization (1)
- Mathematische Optimierung (1)
- Merapi (1)
- Monoschicht (1)
- Morphologie (1)
- Multistability (1)
- Multistabilität (1)
- Netzwerke des Wissens (1)
- Nichtlineare Dynamik (1)
- Nichtlineare Systeme (1)
- Nichtlineares dynamisches System / Harmonische Analyse / Fraktal (1)
- Oberflächenpotential (1)
- Optimality Theory (1)
- Optimalitätstheorie (1)
- P hasensynchronisierung (1)
- PUFA (1)
- Phase (1)
- Power Monitoring (1)
- Rauschen (1)
- Regression (1)
- Regularisierung (1)
- Regularization (1)
- Seismic Array Methods (1)
- Simulation (1)
- Sternwinde (1)
- Stochastische Prozesse (1)
- Strahlungstransport (1)
- Stromverbrauchüberwachung (1)
- Strukturbildung (1)
- Support Vectors (1)
- Support-Vector Lernen (1)
- Synchronisation (1)
- Synchronisierung (1)
- System (1)
- Theorie (1)
- Thermodynamische Eigenschaft (1)
- Time Series Analysis (1)
- Toeplitz operators (1)
- Venezuela (1)
- Wasseroberfläche (1)
- Weltbürger (1)
- accretion disks (1)
- affix order (1)
- alga (1)
- anisotropic spaces (1)
- autocorrelation (1)
- boundary value problem (1)
- chaotic (1)
- chaotic saddle (1)
- chaotisch (1)
- charged systems (1)
- classification (1)
- cognates (1)
- complex (1)
- complex systems (1)
- conormal asymptotic expansions (1)
- consumer (1)
- coupled (1)
- covering (1)
- discrete saymptotic types (1)
- dynamical systems (1)
- electric double layer (1)
- electrolytes (1)
- elektrische Doppelschicht (1)
- elliptic operator (1)
- elliptic operators (1)
- fatty acid (1)
- field (1)
- finiteness theorem (1)
- food quality (1)
- foreign language acquisition (1)
- gekoppelt (1)
- geladene Systeme (1)
- good-inner function (1)
- grazer (1)
- herbivore (1)
- hydrodynamics (1)
- index (1)
- index theory (1)
- inflection (1)
- interaction (1)
- interference (1)
- komplex (1)
- komplexe Systeme (1)
- language transfer (1)
- manifold with singularities (1)
- meanfield (1)
- modn-index (1)
- monitoring (1)
- morphology (1)
- noise (1)
- nonlinear dynamics (1)
- nonlinear systems (1)
- nonlocal problem (1)
- o (1)
- operators on manifolds with singularities (1)
- pattern recognition (1)
- phase (1)
- phase synchronization (1)
- polydisc (1)
- power spectrum (1)
- pseudo-differential boundary value problems (1)
- pseudodifferential operators (1)
- radiative transfer (1)
- relative index (1)
- relative η-invariant (1)
- seston (1)
- simulation (1)
- stellar winds (1)
- stochastic processes (1)
- surgery (1)
- system (1)
- theory (1)
- transition (1)
- volcanic seismology (1)
- weighted Sobolev spaces with discrete saymptotics (1)
- Übergang (1)
Institute
- Institut für Mathematik (29)
- Institut für Physik und Astronomie (6)
- Extern (3)
- Institut für Romanistik (3)
- Wirtschaftswissenschaften (3)
- Department Linguistik (2)
- Department Psychologie (1)
- Historisches Institut (1)
- Institut für Anglistik und Amerikanistik (1)
- Institut für Biochemie und Biologie (1)
Research on monolayers of amphiphilic lipids on aqueous solution is of basic importance in surface science. Due to the applicability of a variety of surface sensitive techniques, floating insoluble monolayers are very suitable model systems for the study of order, structure formation and material transport in two dimensions or the interactions of molecules at the interface with ions or molecules in the bulk (headword 'molecular recognition'). From the behavior of monolayers conclusions can be drawn on the properties of lipid layers on solid substrates or in biological membranes. This work deals with specific and fundamental interactions in monolayers both on the molecular and on the microscopic scale and with their relation to the lattice structure, morphology and thermodynamic behavior of monolayers at the air-water interface. As model system especially monolayers of long chain fatty acids are used, since there the molecular interactions can be gradually adjusted by varying the degree of dissociation by means of the suphase pH value. For manipulating the molecular interactions besides the subphase composition also temperature and monolayer composition are systematically varied. The change in the monolayer properties as a function of an external parameter is analyzed by means of isotherm and surface potential measurements, Brewster-angle microscopy, X-ray diffraction at grazing incidence and polarization modulated infrared reflection absorption spectroscopy. For this a quantitative measure for the molecular interactions and for the chain conformational order is derived from the X-ray data. The most interesting results of this work are the elucidation of the origin of regular polygonal and dendritic domain shapes, the various effects of cholesterol on molecular packing and lattice order of long chain amphiphiles, as well as the detection of an abrupt change in the head group bonding interactions, the chain conformational order and the phase transition pressure between tilted phases in fatty acid monolayers near pH 9. For the interpretation of the latter point a model of the head group bonding structure in fatty acid monolayers as a function of the pH value is developed.
Merapi volcano is one of the most active and dangerous volcanoes of the earth. Located in central part of Java island (Indonesia), even a moderate eruption of Merapi poses a high risk to the highly populated area. Due to the close relationship between the volcanic unrest and the occurrence of seismic events at Mt. Merapi, the monitoring of Merapi's seismicity plays an important role for recognizing major changes in the volcanic activity. An automatic seismic event detection and classification system, which is capable to characterize the actual seismic activity in near real-time, is an important tool which allows the scientists in charge to take immediate decisions during a volcanic crisis. In order to accomplish the task of detecting and classifying volcano-seismic signals automatically in the continuous data streams, a pattern recognition approach has been used. It is based on the method of hidden Markov models (HMM), a technique, which has proven to provide high recognition rates at high confidence levels in classification tasks of similar complexity (e.g. speech recognition). Any pattern recognition system relies on the appropriate representation of the input data in order to allow a reasonable class-decision by means of a mathematical test function. Based on the experiences from seismological observatory practice, a parametrization scheme of the seismic waveform data is derived using robust seismological analysis techniques. The wavefield parameters are summarized into a real-valued feature vector per time step. The time series of this feature vector build the basis for the HMM-based classification system. In order to make use of discrete hidden Markov (DHMM) techniques, the feature vectors are further processed by applying a de-correlating and prewhitening transformation and additional vector quantization. The seismic wavefield is finally represented as a discrete symbol sequence with a finite alphabet. This sequence is subject to a maximum likelihood test against the discrete hidden Markov models, learned from a representative set of training sequences for each seismic event type of interest. A time period from July, 1st to July, 5th, 1998 of rapidly increasing seismic activity prior to the eruptive cycle between July, 10th and July, 19th, 1998 at Merapi volcano is selected for evaluating the performance of this classification approach. Three distinct types of seismic events according to the established classification scheme of the Volcanological Survey of Indonesia (VSI) have been observed during this time period. Shallow volcano-tectonic events VTB (h < 2.5 km), very shallow dome-growth related seismic events MP (h < 1 km) and seismic signals connected to rockfall activity originating from the active lava dome, termed Guguran. The special configuration of the digital seismic station network at Merapi volcano, a combination of small-aperture array deployments surrounding Merapi's summit region, allows the use of array methods to parametrize the continuously recorded seismic wavefield. The individual signal parameters are analyzed to determine their relevance for the discrimination of seismic event classes. For each of the three observed event types a set of DHMMs has been trained using a selected set of seismic events with varying signal to noise ratios and signal durations. Additionally, two sets of discrete hidden Markov models have been derived for the seismic noise, incorporating the fact, that the wavefield properties of the ambient vibrations differ considerably during working hours and night time. A total recognition accuracy of 67% is obtained. The mean false alarm (FA) rate can be given by 41 FA/class/day. However, variations in the recognition capabilities for the individual seismic event classes are significant. Shallow volcano-tectonic signals (VTB) show very distinct wavefield properties and (at least in the selected time period) a stable time pattern of wavefield attributes. The DHMM-based classification performs therefore best for VTB-type events, with almost 89% recognition accuracy and 2 FA/day. Seismic signals of the MP- and Guguran-classes are more difficult to detect and classify. Around 64% of MP-events and 74% of Guguran signals are recognized correctly. The average false alarm rate for MP-events is 87 FA/day, whereas for Guguran signals 33 FA/day are obtained. However, the majority of missed events and false alarms for both MP and Guguran events are due to confusion errors between these two event classes in the recognition process. The confusion of MP and Guguran events is interpreted as being a consequence of the selected parametrization approach for the continuous seismic data streams. The observed patterns of the analyzed wavefield attributes for MP and Guguran events show a significant amount of similarity, thus providing not sufficient discriminative information for the numerical classification. The similarity of wavefield parameters obtained for seismic events of MP and Guguran type reflect the commonly observed dominance of path effects on the seismic wave propagation in volcanic environments. The recognition rates obtained for the five-day period of increasing seismicity show, that the presented DHMM-based automatic classification system is a promising approach for the difficult task of classifying volcano-seismic signals. Compared to standard signal detection algorithms, the most significant advantage of the discussed technique is, that the entire seismogram is detected and classified in a single step.
In this work we consider statistical learning problems. A learning machine aims to extract information from a set of training examples such that it is able to predict the associated label on unseen examples. We consider the case where the resulting classification or regression rule is a combination of simple rules - also called base hypotheses. The so-called boosting algorithms iteratively find a weighted linear combination of base hypotheses that predict well on unseen data. We address the following issues: o The statistical learning theory framework for analyzing boosting methods. We study learning theoretic guarantees on the prediction performance on unseen examples. Recently, large margin classification techniques emerged as a practical result of the theory of generalization, in particular Boosting and Support Vector Machines. A large margin implies a good generalization performance. Hence, we analyze how large the margins in boosting are and find an improved algorithm that is able to generate the maximum margin solution. o How can boosting methods be related to mathematical optimization techniques? To analyze the properties of the resulting classification or regression rule, it is of high importance to understand whether and under which conditions boosting converges. We show that boosting can be used to solve large scale constrained optimization problems, whose solutions are well characterizable. To show this, we relate boosting methods to methods known from mathematical optimization, and derive convergence guarantees for a quite general family of boosting algorithms. o How to make Boosting noise robust? One of the problems of current boosting techniques is that they are sensitive to noise in the training sample. In order to make boosting robust, we transfer the soft margin idea from support vector learning to boosting. We develop theoretically motivated regularized algorithms that exhibit a high noise robustness. o How to adapt boosting to regression problems? Boosting methods are originally designed for classification problems. To extend the boosting idea to regression problems, we use the previous convergence results and relations to semi-infinite programming to design boosting-like algorithms for regression problems. We show that these leveraging algorithms have desirable theoretical and practical properties. o Can boosting techniques be useful in practice? The presented theoretical results are guided by simulation results either to illustrate properties of the proposed algorithms or to show that they work well in practice. We report on successful applications in a non-intrusive power monitoring system, chaotic time series analysis and a drug discovery process. --- Anmerkung: Der Autor ist Träger des von der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Potsdam vergebenen Michelson-Preises für die beste Promotion des Jahres 2001/2002.
Nonlinear multistable systems under the influence of noise exhibit a plethora of interesting dynamical properties. A medium noise level causes hopping between the metastable states. This attractorhopping process is characterized through laminar motion in the vicinity of the attractors and erratic motion taking place on chaotic saddles, which are embedded in the fractal basin boundary. This leads to noise-induced chaos. The investigation of the dissipative standard map showed the phenomenon of preference of attractors through the noise. It means, that some attractors get a larger probability of occurrence than in the noisefree system. For a certain noise level this prefernce achieves a maximum. Other attractors are occur less often. For sufficiently high noise they are completely extinguished. The complexity of the hopping process is examined for a model of two coupled logistic maps employing symbolic dynamics. With the variation of a parameter the topological entropy, which is used together with the Shannon entropy as a measure of complexity, rises sharply at a certain value. This increase is explained by a novel saddle merging bifurcation, which is mediated by a snapback repellor. Scaling laws of the average time spend on one of the formerly disconnected parts and of the fractal dimension of the connected saddle describe this bifurcation in more detail. If a chaotic saddle is embedded in the open neighborhood of the basin of attraction of a metastable state, the required escape energy is lowered. This enhancement of noise-induced escape is demonstrated for the Ikeda map, which models a laser system with time-delayed feedback. The result is gained using the theory of quasipotentials. This effect, as well as the two scaling laws for the saddle merging bifurcation, are of experimental relevance.
One of the rules-of-thumb of colloid and surface physics is that most surfaces are charged when in contact with a solvent, usually water. This is the case, for instance, in charge-stabilized colloidal suspensions, where the surface of the colloidal particles are charged (usually with a charge of hundreds to thousands of e, the elementary charge), monolayers of ionic surfactants sitting at an air-water interface (where the water-loving head groups become charged by releasing counterions), or bilayers containing charged phospholipids (as cell membranes). In this work, we look at some model-systems that, although being a simplified version of reality, are expected to capture some of the physical properties of real charged systems (colloids and electrolytes). We initially study the simple double layer, composed by a charged wall in the presence of its counterions. The charges at the wall are smeared out and the dielectric constant is the same everywhere. The Poisson-Boltzmann (PB) approach gives asymptotically exact counterion density profiles around charged objects in the weak-coupling limit of systems with low-valent counterions, surfaces with low charge density and high temperature (or small Bjerrum length). Using Monte Carlo simulations, we obtain the profiles around the charged wall and compare it with both Poisson-Boltzmann (in the low coupling limit) and the novel strong coupling (SC) theory in the opposite limit of high couplings. In the latter limit, the simulations show that the SC leads in fact to asymptotically correct density profiles. We also compare the Monte Carlo data with previously calculated corrections to the Poisson-Boltzmann theory. We also discuss in detail the methods used to perform the computer simulations. After studying the simple double layer in detail, we introduce a dielectric jump at the charged wall and investigate its effect on the counterion density distribution. As we will show, the Poisson-Boltzmann description of the double layer remains a good approximation at low coupling values, while the strong coupling theory is shown to lead to the correct density profiles close to the wall (and at all couplings). For very large couplings, only systems where the difference between the dielectric constants of the wall and of the solvent is small are shown to be well described by SC. Another experimentally relevant modification to the simple double layer is to make the charges at the plane discrete. The counterions are still assumed to be point-like, but we constraint the distance of approach between ions in the plane and counterions to a minimum distance D. The ratio between D and the distance between neighboring ions in the plane is, as we will see, one of the important quantities in determining the influence of the discrete nature of the charges at the wall over the density profiles. Another parameter that plays an important role, as in the previous case, is the coupling as we will demonstrate, systems with higher coupling are more subject to discretization effects than systems with low coupling parameter. After studying the isolated double layer, we look at the interaction between two double layers. The system is composed by two equally charged walls at distance d, with the counterions confined between them. The charge at the walls is smeared out and the dielectric constant is the same everywhere. Using Monte-Carlo simulations we obtain the inter-plate pressure in the global parameter space, and the pressure is shown to be negative (attraction) at certain conditions. The simulations also show that the equilibrium plate separation (where the pressure changes from attractive to repulsive) exhibits a novel unbinding transition. We compare the Monte Carlo results with the strong-coupling theory, which is shown to describe well the bound states of systems with moderate and high couplings. The regime where the two walls are very close to each other is also shown to be well described by the SC theory. Finally, Using a field-theoretic approach, we derive the exact low-density ("virial") expansion of a binary mixture of positively and negatively charged hard spheres (two-component hard-core plasma, TCPHC). The free energy obtained is valid for systems where the diameters d_+ and d_- and the charge valences q_+ and q_- of positive and negative ions are unconstrained, i.e., the same expression can be used to treat dilute salt solutions (where typically d_+ ~ d_- and q_+ ~ q_-) as well as colloidal suspensions (where the difference in size and valence between macroions and counterions can be very large). We also discuss some applications of our results.
Line driven winds are accelerated by the momentum transfer from photons to a plasma, by absorption and scattering in numerous spectral lines. Line driving is most efficient for ultraviolet radiation, and at plasma temperatures from 10^4 K to 10^5 K. Astronomical objects which show line driven winds include stars of spectral type O, B, and A, Wolf-Rayet stars, and accretion disks over a wide range of scales, from disks in young stellar objects and cataclysmic variables to quasar disks. It is not yet possible to solve the full wind problem numerically, and treat the combined hydrodynamics, radiative transfer, and statistical equilibrium of these flows. The emphasis in the present writing is on wind hydrodynamics, with severe simplifications in the other two areas. I consider three topics in some detail, for reasons of personal involvement. 1. Wind instability, as caused by Doppler de-shadowing of gas parcels. The instability causes the wind gas to be compressed into dense shells enclosed by strong shocks. Fast clouds occur in the space between shells, and collide with the latter. This leads to X-ray flashes which may explain the observed X-ray emission from hot stars. 2. Wind runaway, as caused by a new type of radiative waves. The runaway may explain why observed line driven winds adopt fast, critical solutions instead of shallow (or breeze) solutions. Under certain conditions the wind settles on overloaded solutions, which show a broad deceleration region and kinks in their velocity law. 3. Magnetized winds, as launched from accretion disks around stars or in active galactic nuclei. Line driving is assisted by centrifugal forces along co-rotating poloidal magnetic field lines, and by Lorentz forces due to toroidal field gradients. A vortex sheet starting at the inner disk rim can lead to highly enhanced mass loss rates.
One of the classical ways to describe the dynamics of nonlinear systems is to analyze theur Fourier spectra. For periodic and quasiperiodic processes the Fourier spectrum consists purely of discrete delta-functions. On the contrary, the spectrum of a chaotic motion is marked by the presence of the continuous component. In this work, we describe the peculiar, neither regular nor completely chaotic state with so called singular-continuous power spectrum. Our investigations concern various cases from most different fields, where one meets the singular continuous (fractal) spectra. The examples include both the physical processes which can be reduced to iterated discrete mappings or even symbolic sequences, and the processes whose description is based on the ordinary or partial differential equations.
The dissertation examines aspects of the interlingual lexical processes of word recognition and word retrieval in Hungarian-German bilinguals learning English as a foreign language, with particular respect to the role of cognates. The purpose of the study is to describe the process of lexical activaton in a polyglot system and to model the mental lexicons and the ways entries in the lexicons are connected and activated (e.g. activation through direct word association or through concept mediation). Three dependent variables are studied in quantitative and qualitative analysis of empirical data taken from experiments: rate of accurate responses, response latencies and phonological interference. The results of the experiments are interpreted in the framework of a multiple language network model.
Die vorliegende Arbeit stellt eine kritische Übersicht über den Forschungsstand zu multiplen Wh-Konstruktionen im Slavischen dar. Das Ziel ist es, die Unklarheit der Datenlage und die Widersprüchlichkeit der auf solchen "unklaren" Daten basierten Theorien aufzuzeigen. Inhalt: Historischer Hintergrund (Wachowicz 1974) Einige ältere Ansätze Höhepunkt: die folgenschwere Arbeit von Rudin (1988) Probleme: - Das Problem der Zuverlässlichkeit von Daten - Das Problem der Relevanz von Daten "Harte" Fakten: - Strikte Superioritätseffekte im Bulgarischen - Obligatorische Wh-Anhebung im Slavischen Neuere Ansätze: - "Qualitative" Ansätze - "Quantitative" Ansätze - Alternative Ansätze
This study examines how the size of trade unions relative to the la- bor force impacts on the desirability of different organizational forms of self-financing unemployment insurance (UI) for workers, firms, and with reference to an efficiency criterion. For this purpose, we respectively nu- merically compare the outcome of a model with a uniform payroll tax to a model where workers pay taxes according to their systematic risk of unemployment. Our results highlight the importance of the bargaining structure for the assessment of a particular UI scheme. Most importantly, it depends on the size of the unions whether efficiency favors a uniform or a differentiated UI scheme.
We examine the effects of regionalising the budget of unemployment insurance (UI) on wages, employment, and on UI parameters, which, for their part, determine the agents’ preferences concerning such a reform. A numerical example shows that, under reasonable assumptions, the intuition that the reform would enhance efficiency and improve the economic situation of agents from the low- unemployment region to the disadvantage of agents from the high- unemployment region is not valid in general.
Our analysis is concerned with the impact of a regionalisation of unemployment insurance (UI) on workers’ preferences, on firms’ profits, and on effciency. The existence and the extent of UI are endogenously derived by maximising an objective function of the state. Three different types of regionalisation are considered which differ with respect to the area the UI objective function is related to, and with respect to the policy variable used to maximise it. It comes to light that workers are always in favour of central UI, while it depends on the type of regionalisation whether or not firms are better off with regional or with central UI. The same somewhat surprising result applies for efficiency.
If Humboldt had a laptop
(2001)
The difficult publication history and expensive editions of Alexander von Humboldt’s volumes on the expedition to the Americas have resulted in incomplete library holdings which has limited scholarly access and sometimes caused unbalanced scholarship. A plan for a Humboldt Digital Library examines the structures and features of this representational system in print and proposes models for converting these materials to electronic form. Several issues posed by Humboldt’s works include: establishing authoritative standard editions in several languages, creating high-resolution access to the many visual innovations in the works, and using software to restore the grand concept that all of the separate disciplines of study can be seen as interrelated parts of the whole. Using techniques of geographic visualization, a prototype is planned which will connect this historical body of knowledge with modern scientific databases.
In the middle of the 19th century the question whether expanding civilization and industrialization had an effect on climate was discussed intensely worldwide. It was feared that increasing deforestation would lead to continuous decrease in rainfall. This first scientific discussion about climate change as the result of human intervention was strongly influenced by the research Alexander von Humboldt and Jean-Baptiste Boussingault had undertaken when they investigated the falling water levels of Lake Valencia in Venezuela. This essay aims to clarify the question whether Alexander von Humboldt can be counted among the leading figures of modern environmentalism on account of this research as is being claimed by Richard H. Grove in his influential book Green Imperialism. Colonial Expansion, Tropical Island Edens and Origins of Environmentalism, 1600-1860 (1995).
Function spaces with asymptotics is a usual tool in the analysis on manifolds with singularities. The asymptotics are singular ingredients of the kernels of pseudodifferential operators in the calculus. They correspond to potentials supported by the singularities of the manifold, and in this form asymptotics can be treated already on smooth configurations. This paper is aimed at describing refined asymptotics in the Dirichlet problem in a ball. The beauty of explicit formulas highlights the structure of asymptotic expansions in the calculi on singular varieties.
A function has vanishing mean oscillation (VMO) on R up(n) if its mean oscillation - the local average of its pointwise deviation from its mean value - both is uniformly bounded over all cubes within R up(n) and converges to zero with the volume of the cube. The more restrictive class of functions with vanishing lower oscillation (VLO) arises when the mean value is replaced by the minimum value in this definition. It is shown here that each VMO function is the difference of two functions in VLO.
Contents: 1 Introduction 2 Statement of the problem and definitions 3 The main results 4 Proof of theorem 2 4.1 Reduction of problem (2) to functional - integral equations 4.2 The uniqueness of a solution of equation (2) 4.3 The existence of a solution of equation (2) 5 Proof of theorem 1 6 Proof of theorem 3 7 First boundary problem for hyperbolic differential equations 7.1 Statement of the problem 7.2 The formulation of the result and a sketch of the proof