Ausbreitung einer Infektion durch ein System Brownscher Bewegungen

In dieser Arbeit werden wir ein Modell untersuchen, welches die Ausbreitung einer Infektion beschreibt. Bei diesem Modell werden zunächst Partikel gemäß eines Poissonschen Punktprozesses auf der reellen Achse verteilt. Bis zu einem gewissen Punkt auf der reellen Achse sind alle Partikel von einer Infektion befallen. Während sich nicht infizierte Partikel nicht bewegen, folgen die infizierten Partikel den Pfaden von voneinander unabhängigen Brownschen Bewegungen und verbreitet die Infektion dabei an den Orten, welche sie betreten. Wenn sie dabei auf ein nicht infiziertes Partikel treffen, ist dieses von diesem Moment an auch infiziert und beginnt ebenfalls, dem Pfad einer Brownschen Bewegung zu folgen und die Infektion auszubreiten. Auf diese Art verschiebt sich nun der am weitesten rechts liegende Ort R_t, an dem die Infektion bereits verbreitet wurde. Wir werden mit Hilfe des subadditiven Ergodensatzes zeigen, dass sich dieser Ort mit linearer Geschwindigkeit fortbewegt. Ferner werden wir eine obere und eine untere Schranke für die Ausbreitungsgeschwindkeit angeben. Danach werden wir zeigen, dass der Prozess Regenerationszeiten hat, nämlich solche zufällige Zeiten, zu denen er eine Art Neustart unter speziellen Startbedingungen durchführt. Wir werden diese für eine weitere Charakterisierung der Ausbreitungsgeschwingkeit nutzen. Ferner erhalten wir durch die Regenerationszeiten auch einen Zentralen Grenzwertsatz für R_t und können zeigen, dass die Verteilung der infizierten Partikel aus Sicht des am weitesten rechts liegenden infizierten Ortes gegen eine invariante Verteilung konvergiert.

Uptake mechanism, intracellular trafficking and endo-lysosomal pH monitoring of polystyrene nanoparticles

What is the intracellular fate of nanoparticles (NPs) taken up by the cells? This question has been investigated for polystyrene NPs of different sizes with a set of molecular biological and biophysical techniques.rnTwo sets of fluorescent NPs, cationic and non-ionic, were synthesized with three different polymerization techniques. Non-ionic particles (132 – 846 nm) were synthesized with dispersion polymerization in an ethanol/water solution. Cationic NPs with 120 nm were synthesized by miniemulsion polymerization Particles with 208, 267 and 603 nm were produced by seeding the 120 nm particle obtained by miniemulsion polymerization with drop-wise added monomer and polymerization of such. The colloidal characterization of all particles showed a comparable amount of the surface groups. In addition, particles were characterized with regard to their size, morphology, solid content, amount of incorporated fluorescent dye and zeta potential. The fluorescent intensities of all particles were measured by fluorescence spectroscopy for calibration in further cellular experiments. rnThe uptake of the NPs to HeLa cells after 1 – 24 h revealed a much higher uptake of cationic NPs in comparison to non-ionic NPs. If the same amount of NPs with different sizes is introduced to the cell, a different amount of particles is present in the cell medium, which complicates a comparison of the uptake. The same conclusion is valid for the particles’ overall surface area. Therefore, HeLa cells were incubated with the same concentration, amount and surface area of NPs. It was found that with the same concentration always the same polymer amount is taking up by cells. However, the amount of particles taken up decreases for the biggest. A correlation to the surface area could not be found. We conclude that particles are endocytosed by an excavator-shovel like mechanism, which does not distinguish between different sizes, but is only dependent on the volume that is taken up. For the decreased amount of large particles, an overload of this mechanism was assumed, which leads to a decrease in the uptake. rnThe participation of specific endocytotic processes has been determined by the use of pharmacological inhibitors, immunocytological staining and immunofluorescence. The uptake of NPs into the endo-lysosomal machinery is dominated by a caveolin-mediated endocytosis. Other pathways, which include macropinocytosis and a dynamin-dependent mechanism but exclude clathrin mediated endocytosis, also occur as competing processes. All particles can be found to some extent in early endosomes, but only bigger particles were proven to localize in late endosomes. No particles were found in lysosomes; at least not in lysosomes that are labeled with Lamp1 and cathepsin D. However, based on the character of the performed experiment, a localization of particles in lysosomes cannot be excluded.rnDuring their ripening process, vesicles undergo a gradual acidification from early over late endosomes to lysosomes. It is hypothesized that NPs in endo-lysosomal compartments experience the same change in pH value. To probe the environmental pH of NPs after endocytosis, the pH-sensitive dye SNARF-4F was grafted onto amino functionalized polystyrene NPs. The pH value is a ratio function of the two emission wavelengths of the protonated and deprotonated form of the dye and is hence independent of concentration changes. The particles were synthesized by the aforementioned miniemulsion polymerization with the addition of the amino functionalized copolymer AEMH. The immobilization of SNARF-4F was performed by an EDC-coupling reaction. The amount of physically adsorbed dye in comparison to covalently bonded dye was 15% as determined by precipitation of the NPs in methanol, which is a very good solvent for SNARF-4F. To determine influences of cellular proteins on the fluorescence properties, a intracellular calibration fit was established with platereader measurements and cLSM imaging by the cell-penetrable SNARF-4F AM ester. Ionophores equilibrated the extracellular and intracellular pH.rnSNARF-4F NPs were taken up well by HeLa cells and showed no toxic effects. The pH environment of SNARF-4F NPs has been qualitatively imaged as a movie over a time period up to 1 h in pseudo-colors by a self-written automated batch program. Quantification revealed an acidification process until pH value of 4.5 over 24 h, which is much slower than the transport of nutrients to lysosomes. NPs are present in early endosomes after min. 1 h, in late endosomes at approx. 8 h and end up in vesicles with a pH value typical for lysosomes after > 24 h. We therefore assume that NPs bear a unique endocytotic mechanism, at least with regards to the kinetic involvedrn

Effect Of Confined Impinging Jet Mixing Processing Parameters On Poly (Lactic Acid) Nanoparticle Morphology

Biodegradable nanoparticles are at the forefront of drug delivery research as they provide numerous advantages over traditional drug delivery methods. An important factor affecting the ability of nanoparticles to circulate within the blood stream and interact with cells is their morphology. In this study a novel processing method, confined impinging jet mixing, was used to form poly (lactic acid) nanoparticles through a solvent-diffusion process with Pluronic F-127 being used as a stabilizing agent. This study focused on the effects of Reynolds number (flow rate), surfactant presence in mixing, and polymer concentration on the morphology of poly (lactic acid) nanoparticles. In addition to looking at the parameters affecting poly (lactic acid) morphology, this study attempted to improve nanoparticle isolation and purification methods to increase nanoparticle yield and ensure specific morphologies were not being excluded during isolation and purification. The isolation and purification methods used in this study were centrifugation and a stir cell. This study successfully produced particles having pyramidal and cubic morphologies. Despite successful production of these morphologies the yield of non-spherical particles was very low, additionally great variability existed between redundant trails. Surfactant was determined to be very important for the stabilization of nanoparticles in solution but appears to be unnecessary for the formation of nanoparticles. Isolation and purification methods that produce a high yield of surfactant free particles have still not been perfected and additional testing will be necessary for improvement.¿

The Value of Adding Optics to Ecosystem Models: A Case Study

Many ecosystem models have been developed to study the ocean's biogeochemical properties, but most of these models use simple formulations to describe light penetration and spectral quality. Here, an optical model is coupled with a previously published ecosystem model that explicitly represents two phytoplankton (picoplankton and diatoms) and two zooplankton functional groups, as well as multiple nutrients and detritus. Surface ocean color fields and subsurface light fields are calculated by coupling the ecosystem model with an optical model that relates biogeochemical standing stocks with inherent optical properties (absorption, scattering); this provides input to a commercially available radiative transfer model (Ecolight). We apply this bio-optical model to the equatorial Pacific upwelling region, and find the model to be capable of reproducing many measured optical properties and key biogeochemical processes in this region. Our model results suggest that non-algal particles largely contribute to the total scattering or attenuation (> 50% at 660 nm) but have a much smaller contribution to particulate absorption (< 20% at 440 nm), while picoplankton dominate the total phytoplankton absorption (> 95% at 440 nm). These results are consistent with the field observations. In order to achieve such good agreement between data and model results, however, key model parameters, for which no field data are available, have to be constrained. Sensitivity analysis of the model results to optical parameters reveals a significant role played by colored dissolved organic matter through its influence on the quantity and quality of the ambient light. Coupling explicit optics to an ecosystem model provides advantages in generating: (1) a more accurate subsurface light-field, which is important for light sensitive biogeochemical processes such as photosynthesis and photo-oxidation, (2) additional constraints on model parameters that help to reduce uncertainties in ecosystem model simulations, and (3) model output which is comparable to basic remotely-sensed properties. In addition, the coupling of biogeochemical models and optics paves the road for future assimilation of ocean color and in-situ measured optical properties into the models.

Sublimation of water ice mixed with silicates and tholins: Evolution of surface texture and reflectance spectra, with implications for comets

The surfaces of many objects in the Solar System comprise substantial quantities of water ice sometimes mixed with minerals and/or organic molecules. The sublimation of the ice changes the structural and optical properties of these objects. We present laboratory data on the evolution of the structure and the visible and near-infrared spectral reflectance of icy surface analogues of cometary ices, made of water ice, complex organic matter (tholins) and silicates, as they undergo sublimation under low temperature (<-70°C) and pressure (10-⁵mbar) conditions inside the SCITEAS simulation chamber. As the water ice sublimated, we observed in situ the formation of a porous sublimation lag deposit, or sublimation mantle, at the top of the ice. This mantle is a network of filaments made of the non-volatile particles. Organics or phyllosilicates grains, able to interact via stronger inter-particulate forces than olivine grains, can form a foam-like structure having internal cohesiveness, holding olivine grains together. As this mantle builds-up, the band depths of the sub-surface water ice are attenuated until complete extinction under only few millimeters of mantle. Optically thick sublimation mantles are mainly featureless in the near infrared. The absorption bands of the minerals present in the mantle are weak, or even totally absent if minerals are mixed with organics which largely dominate the VIS–NIR reflectance spectrum. During sublimation, ejections of large fragments of mantle, triggered by the gas flow, expose ice particles to the surface. The contrast of brightness between mantled and ice-exposed areas depends on the wavelength range and the dust/ice ratio considered. We describe how the chemical nature of the non-volatiles, the size of their particles, the way they are mixed with the ice and the dust/ice mass ratio influence the texture, activity and spectro-photometric properties of the sublimation mantles. These data provide useful references for interpreting remote-sensing observations of comets and also icy satellites or trans-neptunian objects.

Paleointensity record of Site 320-U1336

We present a high-resolution magnetostratigraphy and relative paleointensity (RPI) record derived from the upper 85 meters of IODP Site U1336, an equatorial Pacific early to middle Miocene succession recovered during Expedition 320/321. The magnetostratigraphy is well resolved with reversals typically located to within a few centimeters resulting in a well-constrained age model. The lowest normal polarity interval, from 85 to 74.87 meters, is interpreted as the upper part of Chron C6n (18.614-19.599 Ma). Another 33 magnetozones occur from 74.87 to 0.85 m, which are interpret to represent the continuous sequence of chrons from Chron C5Er (18.431-18.614 Ma) up to the top of Chron C5An.1n (12.014 Ma). We identify three new possible subchrons within Chron C5Cn.1n, Chron 5Bn.1r, and C5ABn. Sedimentation rates vary from about 7 to 15 m/Myr with a mean of about 10 m/Myr. We observe rapid, apparent changes in the sedimentation rate at geomagnetic reversals between ~16 and 19 Ma that indicate a calibration error in geomagnetic polarity timescale (ATNTS2004). The remanence is carried mainly by non-interacting particles of fine-grained magnetite, which have FORC distributions characteristic of biogenic magnetite. Given the relative homogeneity of the remanence carriers throughout the 85-m-thick succession and the quality with which the remanence is recorded, we have constructed a relative paleointensity (RPI) record that provides new insights into middle Miocene geomagnetic field behavior. The RPI record indicates a gradual decline in field strength between 18.5 Ma and 14.5 Ma, and indicates no discernible link between RPI and either chron duration or polarity state.

Vertical distribution of suspended aggregates at time series station DYNAPROC

Day/night variations in the size distribution of the particulate matter >0.15 mm (PM) were studied in May 1995 during the DYNAPROC time-series cruise in the northwestern Mediterranean Sea. Data on vertical distributions of PM (>0.15 mm) and zooplankton were collected with the Underwater Video Profiler (UVP). The comparisons of the UVP data with plankton net data and POC data from water bottles indicated that more than 97% of the particles detected by the UVP were non-living particles (0.15 mm) and that the PM contributed 4-34% of the total dry weight measured on GF/F filters. Comparison of seven pairs of day and night vertical profiles performed during the cruise showed that in the upper 800 m, the mean size and the volume of particles was higher at night than during the day. During the night, the integrated volume of the PM increased on average by 32±20%. This increase corresponded to a shift of smaller size classes (<0.5 mm) towards the larger ones (>0.5 mm). During the day, the pattern was reversed, and the quantity of PM >0.5 mm decreased. During the study period, the standing stock of PM (60-800 m) decreased from 7.5 to less than 2 g m?2 but the diel variations persisted, except for two short periods in the superficial layer following a wind event. The cyclic feeding activity induced by the diel vertical migration of zooplankton could be the best candidate to explain the observed diel fluctuations in the size classes of PM in the water column. However, our results also suggest that in the upper layer additional driving forces such as the increase of the level of turbulence after a wind event or the modification of the zoo- and phytoplankton community can influence the PM temporal evolution.

Asymptotic Description of Flutter Amplitude Saturation by Nonlinear Friction Forces

The computation of the non-linear vibration dynamics of an aerodynamically unstable bladed-disk is a formidable numerical task, even for the simplified case of aerodynamic forces assumed to be linear. The nonlinear friction forces effectively couple dif- ferent travelling waves modes and, in order to properly elucidate the dynamics of the system, large time simulations are typically required to reach a final, saturated state. Despite of all the above complications, the output of the system (in the friction microslip regime) is basically a superposition of the linear aeroelastic un- stable travelling waves, which exhibit a slow time modulation that is much longer than the elastic oscillation period. This slow time modulation is due to both, the small aerodynamic effects and the small nonlinear friction forces, and it is crucial to deter- mine the final amplitude of the flutter vibration. In this presenta- tion we apply asymptotic techniques to obtain a new simplified model that captures the slow time dynamics of the amplitudes of the travelling waves. The resulting asymptotic model is very re- duced and extremely cheap to simulate, and it has the advantage that it gives precise information about the characteristics of the nonlinear friction models that actually play a role in the satura- tion of the vibration amplitude.

Asymptotic Description of Flutter Amplitude Saturation by Nonlinear Friction Forces

The computation of the non-linear vibration dynamics of an aerodynamically unstable bladed-disk is a formidable numerical task, even for the simplified case of aerodynamic forces assumed to be linear. The nonlinear friction forces effectively couple dif- ferent travelling waves modes and, in order to properly elucidate the dynamics of the system, large time simulations are typically required to reach a final, saturated state. Despite of all the above complications, the output of the system (in the friction microslip regime) is basically a superposition of the linear aeroelastic un- stable travelling waves, which exhibit a slow time modulation that is much longer than the elastic oscillation period. This slow time modulation is due to both, the small aerodynamic effects and the small nonlinear friction forces, and it is crucial to deter- mine the final amplitude of the flutter vibration. In this presenta- tion we apply asymptotic techniques to obtain a new simplified model that captures the slow time dynamics of the amplitudes of the travelling waves. The resulting asymptotic model is very re- duced and extremely cheap to simulate, and it has the advantage that it gives precise information about the characteristics of the nonlinear friction models that actually play a role in the satura- tion of the vibration amplitude.

Numerical simulation of magnetite segregation in a dense medium cyclone

Numerical simulations of turbulent driven flow in a dense medium cyclone with magnetite medium have been conducted using Fluent. The predicted air core shape and diameter were found to be close to the experimental results measured by gamma ray tomography. It is possible that the Large eddy simulation (LES) turbulence model with Mixture multi-phase model can be used to predict the air/slurry interface accurately although the LES may need a finer grid. Multi-phase simulations (air/water/medium) are showing appropriate medium segregation effects but are over-predicting the level of segregation compared to that measured by gamma-ray tomography in particular with over prediction of medium concentrations near the wall. Further, investigated the accurate prediction of axial segregation of magnetite using the LES turbulence model together with the multi-phase mixture model and viscosity corrections according to the feed particle loading factor. Addition of lift forces and viscosity correction improved the predictions especially near the wall. Predicted density profiles are very close to gamma ray tomography data showing a clear density drop near the wall. The effect of size distribution of the magnetite has been fully studied. It is interesting to note that the ultra-fine magnetite sizes (i.e. 2 and 7 mu m) are distributed uniformly throughout the cyclone. As the size of magnetite increases, more segregation of magnetite occurs close to the wall. The cut-density (d(50)) of the magnetite segregation is 32 gm, which is expected with superfine magnetite feed size distribution. At higher feed densities the agreement between the [Dungilson, 1999; Wood, J.C., 1990. A performance model for coal-washing dense medium cyclones, Ph.D. Thesis, JKMRC, University of Queensland] correlations and the CFD are reasonably good, but the overflow density is lower than the model predictions. It is believed that the excessive underflow volumetric flow rates are responsible for under prediction of the overflow density. (c) 2006 Elsevier Ltd. All rights reserved.

Jobs, Unemployment, and Macroeconomic Transmission

Measuring Job Openings: Evidence from Swedish Plant Level Data. In modern macroeconomic models “job openings'' are a key component. Thus, when taking these models to the data we need an empirical counterpart to the theoretical concept of job openings. To achieve this, the literature relies on job vacancies measured either in survey or register data. Insofar as this concept captures the concept of job openings well we should see a tight relationship between vacancies and subsequent hires on the micro level. To investigate this, I analyze a new data set of Swedish hires and job vacancies on the plant level covering the period 2001-2012. I find that vacancies contain little power in predicting hires over and above (i) whether the number of vacancies is positive and (ii) plant size. Building on this, I propose an alternative measure of job openings in the economy. This measure (i) better predicts hiring at the plant level and (ii) provides a better fitting aggregate matching function vis-à-vis the traditional vacancy measure. Firm Level Evidence from Two Vacancy Measures. Using firm level survey and register data for both Sweden and Denmark we show systematic mis-measurement in both vacancy measures. While the register-based measure on the aggregate constitutes a quarter of the survey-based measure, the latter is not a super-set of the former. To obtain the full set of unique vacancies in these two databases, the number of survey vacancies should be multiplied by approximately 1.2. Importantly, this adjustment factor varies over time and across firm characteristics. Our findings have implications for both the search-matching literature and policy analysis based on vacancy measures: observed changes in vacancies can be an outcome of changes in mis-measurement, and are not necessarily changes in the actual number of vacancies. Swedish Unemployment Dynamics. We study the contribution of different labor market flows to business cycle variations in unemployment in the context of a dual labor market. To this end, we develop a decomposition method that allows for a distinction between permanent and temporary employment. We also allow for slow convergence to steady state which is characteristic of European labor markets. We apply the method to a new Swedish data set covering the period 1987-2012 and show that the relative contributions of inflows and outflows to/from unemployment are roughly 60/30. The remaining 10\% are due to flows not involving unemployment. Even though temporary contracts only cover 9-11\% of the working age population, variations in flows involving temporary contracts account for 44\% of the variation in unemployment. We also show that the importance of flows involving temporary contracts is likely to be understated if one does not account for non-steady state dynamics. The New Keynesian Transmission Mechanism: A Heterogeneous-Agent Perspective. We argue that a 2-agent version of the standard New Keynesian model---where a ``worker'' receives only labor income and a “capitalist'' only profit income---offers insights about how income inequality affects the monetary transmission mechanism. Under rigid prices, monetary policy affects the distribution of consumption, but it has no effect on output as workers choose not to change their hours worked in response to wage movements. In the corresponding representative-agent model, in contrast, hours do rise after a monetary policy loosening due to a wealth effect on labor supply: profits fall, thus reducing the representative worker's income. If wages are rigid too, however, the monetary transmission mechanism is active and resembles that in the corresponding representative-agent model. Here, workers are not on their labor supply curve and hence respond passively to demand, and profits are procyclical.

Experiments to assess the transport of fibrous insulation debris

Loss of coolant accidents (LOCA) in the primary cooling circuit of a nuclear reactor may result in damage to insulation materials that are located near to the leak. The insulation materials released may compromise the operation of the emergency core cooling system (ECCS). Insulation material in the form of mineral wool fibre agglomerates (MWFA) maybe transported to the containment sump strainers mounted at the inlet of the emergency cooling pumps, where the insulation fibres may block or penetrate the strainers. In addition to the impact of MWFA on the pressure drop across the strainers, corrosion products formed over time may also accumulate in the fibre cakes on the strainers, which can lead to a significant increase in the strainer pressure drop and result in cavitation in the ECCS. Thus, knowledge of transport characteristics of the damaged insulation materials in various scenarios is required to help plan for the long-term operability of nuclear reactors, which undergo LOCA. An experimental and theoretical study performed by the Helmholtz-Zentrum Dresden-Rossendorf and the Hochschule Zittau/Görlitz1 is investigating the phenomena that maybe observed in the containment vessel during a LOCA. The study entails the generation of fibre agglomerates, the determination of their transport properties in single and multi-effect experiments and the long-term effect that corrosion of the containment internals by the coolant has on the strainer pressure drop. The focus of this presentation is on the experiments performed that characterize the horizontal transport of MWFA, whereas the corresponding CFD simulations are described in an accompanying contribution (see abstract of Cartland Glover et al.). The experiments were performed a racetrack type channel that provided a near uniform horizontal flow. The channel is 0.1 wide by 1.2 m high with a straight length of 5 m and two bends of 0.5 m. The measurement techniques include particle imaging (both wide-angle and macro lens), concurrent particle image velocimetry, ultravelocimetry, laser detection sensors to sense the presence of absence of MWFA and pertinent measurements of the MWFA concentration and quiescent settling characteristics. The transport of the MWFA was observed at velocities of 0.1 and 0.25 m s-1 to verify numerical model behaviour in and just beyond expected velocities in the containment sump of a nuclear reactor.

Numerical models used for the modelling of the transport of fibrous insulation debris

Mineral wool insulation material applied to the primary cooling circuit of a nuclear reactor maybe damaged in the course of a loss of coolant accident (LOCA). The insulation material released by the leak may compromise the operation of the emergency core cooling system (ECCS), as it maybe transported together with the coolant in the form of mineral wool fiber agglomerates (MWFA) suspensions to the containment sump strainers, which are mounted at the inlet of the ECCS to keep any debris away from the emergency cooling pumps. In the further course of the LOCA, the MWFA may block or penetrate the strainers. In addition to the impact of MWFA on the pressure drop across the strainers, corrosion products formed over time may also accumulate in the fiber cakes on the strainers, which can lead to a significant increase in the strainer pressure drop and result in cavitation in the ECCS. Therefore, it is essential to understand the transport characteristics of the insulation materials in order to determine the long-term operability of nuclear reactors, which undergo LOCA. An experimental and theoretical study performed by the Helmholtz-Zentrum Dresden-Rossendorf and the Hochschule Zittau/Görlitz1 is investigating the phenomena that maybe observed in the containment vessel during a primary circuit coolant leak. The study entails the generation of fiber agglomerates, the determination of their transport properties in single and multi-effect experiments and the long-term effects that particles formed due to corrosion of metallic containment internals by the coolant medium have on the strainer pressure drop. The focus of this presentation is on the numerical models that are used to predict the transport of MWFA by CFD simulations. A number of pseudo-continuous dispersed phases of spherical wetted agglomerates can represent the MWFA. The size, density, the relative viscosity of the fluid-fiber agglomerate mixture and the turbulent dispersion all affect how the fiber agglomerates are transported. In the cases described here, the size is kept constant while the density is modified. This definition affects both the terminal velocity and volume fraction of the dispersed phases. Only one of the single effect experimental scenarios is described here that are used in validation of the numerical models. The scenario examines the suspension and horizontal transport of the fiber agglomerates in a racetrack type channel. The corresponding experiments will be described in an accompanying presentation (see abstract of Seeliger et al.).

Wave kinetics of random fibre lasers

Traditional wave kinetics describes the slow evolution of systems with many degrees of freedom to equilibrium via numerous weak non-linear interactions and fails for very important class of dissipative (active) optical systems with cyclic gain and losses, such as lasers with non-linear intracavity dynamics. Here we introduce a conceptually new class of cyclic wave systems, characterized by non-uniform double-scale dynamics with strong periodic changes of the energy spectrum and slow evolution from cycle to cycle to a statistically steady state. Taking a practically important example—random fibre laser—we show that a model describing such a system is close to integrable non-linear Schrödinger equation and needs a new formalism of wave kinetics, developed here. We derive a non-linear kinetic theory of the laser spectrum, generalizing the seminal linear model of Schawlow and Townes. Experimental results agree with our theory. The work has implications for describing kinetics of cyclical systems beyond photonics.

Matrix Power S-box Analysis

* Work supported by the Lithuanian State Science and Studies Foundation.