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Aus dem Inhalt: 1 Abraham Wald (1902-1950) 2 Einführung der Grundbegriffe. Einige technische bekannte Ergebnisse 2.1 Martingal und Doob-Ungleichung 2.2 Brownsche Bewegung und spezielle Martingale 2.3 Gleichgradige Integrierbarkeit von Prozessen 2.4 Gestopptes Martingal 2.5 Optionaler Stoppsatz von Doob 2.6 Lokales Martingal 2.7 Quadratische Variation 2.8 Die Dichte der ersten einseitigen Überschreitungszeit der Brown- schen Bewegung 2.9 Waldidentitäten für die Überschreitungszeiten der Brownschen Bewegung 3 Erste Waldidentität 3.1 Burkholder, Gundy und Davis Ungleichungen der gestoppten Brown- schen Bewegung 3.2 Erste Waldidentität für die Brownsche Bewegung 3.3 Verfeinerungen der ersten Waldidentität 3.4 Stärkere Verfeinerung der ersten Waldidentität für die Brown- schen Bewegung 3.5 Verfeinerung der ersten Waldidentität für spezielle Stoppzeiten der Brownschen Bewegung 3.6 Beispiele für lokale Martingale für die Verfeinerung der ersten Waldidentität 3.7 Überschreitungszeiten der Brownschen Bewegung für nichtlineare Schranken 4 Zweite Waldidentität 4.1 Zweite Waldidentität für die Brownsche Bewegung 4.2 Anwendungen der ersten und zweitenWaldidentität für die Brown- schen Bewegung 5 Dritte Waldidentität 5.1 Dritte Waldidentität für die Brownsche Bewegung 5.2 Verfeinerung der dritten Waldidentität 5.3 Eine wichtige Voraussetzung für die Verfeinerung der drittenWal- didentität 5.4 Verfeinerung der dritten Waldidentität für spezielle Stoppzeiten der Brownschen Bewegung 6 Waldidentitäten im Mehrdimensionalen 6.1 Erste Waldidentität im Mehrdimensionalen 6.2 Zweite Waldidentität im Mehrdimensionalen 6.3 Dritte Waldidentität im Mehrdimensionalen 7 Appendix
We study resonances for the generator of a diffusion with small noise in R(d) : L = -∈∆ + ∇F * ∇, when the potential F grows slowly at infinity (typically as a square root of the norm). The case when F grows fast is well known, and under suitable conditions one can show that there exists a family of exponentially small eigenvalues, related to the wells of F. We show that, for an F with a slow growth, the spectrum is R+, but we can find a family of resonances whose real parts behave as the eigenvalues of the "quick growth" case, and whose imaginary parts are small.
We consider a class of ergodic Hamilton-Jacobi-Bellman (HJB) equations, related to large time asymptotics of non-smooth multiplicative functional of difusion processes. Under suitable ergodicity assumptions on the underlying difusion, we show existence of these asymptotics, and that they solve the related HJB equation in the viscosity sense.
A multitype Dawson-Watanabe process is conditioned, in subcritical and critical cases, on non-extinction in the remote future. On every nite time interval, its distribution law is absolutely continuous with respect to the law of the unconditioned process. A martingale problem characterization is also given. The explicit form of the Laplace functional of the conditioned process is used to obtain several results on the long time behaviour of the mass of the conditioned and unconditioned processes. The general case is considered first, where the mutation matrix which modelizes the interaction between the types, is irreducible. Several two-type models with decomposable mutation matrices are also analysed.
The accelerated life time model is considered. First, test procedures for testing the parameter of a parametric acceleration function is investigated; this is done under the assumption of parametric and nonparametric baseline distribution. Further, based on nonparametric estimators for regression functions tests are proposed for checking whether a parametric acceleration function is appropriate to model the influence of the covariates. Resampling procedures are discussed for the realization of these methods. Simulations complete the considerations.
We consider the problem of testing whether the density of a mul- tivariate random variable can be expressed by a prespecified copula function and the marginal densities. The proposed test procedure is based on the asymptotic normality of the properly standardized integrated squared distance between a multivariate kernel density estimator and an estimator of its expectation under the hypothesis. The test of independence is a special case of this approach.
We consider an infinite system of hard balls in Rd undergoing Brownian motions and submitted to a pair potential with infinite range and quasi polynomial decay. It is modelized by an infinite-dimensional Stochastic Differential Equation with an infinite-dimensional local time term. Existence and uniqueness of a strong solution is proven for such an equation with deterministic initial condition. We also show that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential.
We consider a system of infinitely many hard balls in Rd undergoing Brownian motions and submitted to a smooth pair potential. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove that the set of all equilibrium measures, solution of a Detailed Balance Equation, coincides with the set of canonical Gibbs measures associated to the hard core potential added to the smooth interaction potential.
The aim of this paper is to demonstrate that nonparametric smoothing methods for estimating functions can be an useful tool in the analysis of life time data. After stating some basic notations we will present a data example. Applying standard parametric methods to these data we will see that this approach fails - basic features of the underlying functions are not reflected by their estimates. Our proposal is to use nonparametric estimation methods. These methods are explained in section 2. Nonparametric approaches are better in the sense that they are more flexible, and misspecifications of the model are avoided. But, parametric models have the advantage that the parameters can be interpreted. So, finally, we will formulate a test procedure to check whether a parametric or a nonparametric model is appropriate.
We give a necessary and sufficient condition for the existence of an increasing coupling of N (N >= 2) synchronous dynamics on S-Zd (PCA). Increasing means the coupling preserves stochastic ordering. We first present our main construction theorem in the case where S is totally ordered; applications to attractive PCAs are given. When S is only partially ordered, we show on two examples that a coupling of more than two synchronous dynamics may not exist. We also prove an extension of our main result for a particular class of partially ordered spaces.
We first introduce some coupling of a finite number of Probabilistic Cellular Automata dynamics (PCA), preserving the stochastic ordering. Using this tool, for a general attractive probabilistic cellular automata on SZd, where S is finite, we prove that a condition (A) is equivalent to the (time-) convergence towards equilibrium of this Markovian parallel dynamics, in the uniform norm, exponentially fast. This condition (A) means the exponential decay of the influence from the boundary for the invariant measures of the system restricted to finite ‘box’-volume. For a class of reversible PCA dynamics on {−1, +1}Zd , with a naturally associated Gibbsian potential ϕ, we prove that a Weak Mixing condition for ϕ implies the validity of the assumption (A); thus the ‘exponential ergodicity’ of the dynamics towards the unique Gibbs measure associated to ϕ holds. On some particular examples of this PCA class, we verify that our assumption (A) is weaker than the Dobrushin-Vasershtein ergodicity condition. For some special PCA, the ‘exponential ergodicity’ holds as soon as there is no phase transition.
The two and k-sample tests of equality of the survival distributions against the alternatives including cross-effects of survival functions, proportional and monotone hazard ratios, are given for the right censored data. The asymptotic power against approaching alternatives is investigated. The tests are applied to the well known chemio and radio therapy data of the Gastrointestinal Tumor Study Group. The P-values for both proposed tests are much smaller then in the case of other known tests. Differently from the test of Stablein and Koutrouvelis the new tests can be applied not only for singly but also to randomly censored data.
We consider a nonparametric survival model with random censoring. To test whether the hazard rate has a parametric form the unknown hazard rate is estimated by a kernel estimator. Based on a limit theorem stating the asymptotic normality of the quadratic distance of this estimator from the smoothed hypothesis an asymptotic ®-test is proposed. Since the test statistic depends on the maximum likelihood estimator for the unknown parameter in the hypothetical model properties of this parameter estimator are investigated. Power considerations complete the approach.
The dependence between survival times and covariates is described e.g. by proportional hazard models. We consider partly parametric Cox models and discuss here the estimation of interesting parameters. We represent the ma- ximum likelihood approach and extend the results of Huang (1999) from linear to nonlinear parameters. Then we investigate the least squares esti- mation and formulate conditions for the a.s. boundedness and consistency of these estimators.
We give a survey on procedures for testing functions which are based on quadratic deviation measures. The following problems are considered: Testing whether a density function lies in a parametric class of functions, whether continuous random variables are independent; testing cell probabilities and independence in sparse data sets; testing the parametric fit of a regression homoscedasticity in a regression model and testing the hazard rate in survival models with censoring and with and without covariates.
In diesem Beitrag wird der Zusammenhang zwischen Algebrodifferentialgleichungen (ADGL) und Vektorfeldern auf Mannigfaltigkeiten untersucht. Dazu wird zunächst der Begriff der regulären ADGL eingeführt, wobei unter eirter regulären ADGL eine ADGL verstanden wird, deren Lösungsmenge identisch mit der Lösungsmenge eines Vektorfeldes ist. Ausgehend von bekannten Aussagen über die Lösungsmenge eines Vektorfeldes werden analoge Aussagen für die Lösungsmenge einer regulären ADGL abgeleitet. Es wird eine Reduktionsmethode angegeben, die auf ein Kriterium für die Begularität einer ADGL und auf die Definition des Index einer nichtlinearen ADGL führt. Außerdem wird gezeigt, daß beliebige Vektorfelder durch reguläre ADGL so realisiert werden können, daß die Lösungsmenge des Vektorfeldes mit der der realisierenden ADGL identisch ist. Abschließend werden die für autonome ADGL gewonnenen Aussagen auf den Fall der nichtautonomen ADGL übertragen.
A theoretical famework for the investigation of the qualitative behavior of differential-algebraic equations (DAEs) near an equilibrium point is established. The key notion of our approach is the notion of regularity. A DAE is called regular locally around an equilibrium point if there is a unique vector field such that the solutions of the DAE and the vector field are in one-to-one correspondence in a neighborhood of this equili Drium point. Sufficient conditions for the regularity of an equilibrium point are stated. This in turn allows us to translate several local results, as formulated for vector fields, to DAEs that are regular locally around a g: ven equilibrium point (e.g. Local Stable and Unstable Manifold Theorem, Hopf theorem). It is important that ihese theorems are stated in terms of the given problem and not in terms of the corresponding vector field.
An existence and uniqueness theory is developed for general nonlinear and nonautonomous differential-algebraic equations (DAEs) by exploiting their underlying differential-geometric structure. A DAE is called regular if there is a unique nonautonomous vector field such that the solutions of the DAE and the solutions of the vector field are in one-to-one correspondence. Sufficient conditions for regularity of a DAE are derived in terms of constrained manifolds. Based on this differential-geometric characterization, existence and uniqueness results are stated for regular DAEs. Furthermore, our not ons are compared with techniques frequently used in the literature such as index and solvability. The results are illustrated in detail by means of a simple circuit example.
The subject of this paper is the relation of differential-algebraic equations (DAEs) to vector fields on manifolds. For that reason, we introduce the notion of a regular DAE as a DAE to which a vector field uniquely corresponds. Furthermore, a technique is described which yields a family of manifolds for a given DAE. This socalled family of constraint manifolds allows in turn the formulation of sufficient conditions for the regularity of a DAE. and the definition of the index of a regular DAE. We also state a method for the reduction of higher-index DAEs to lowsr-index ones that can be solved without introducing additional constants of integration. Finally, the notion of realizability of a given vector field by a regular DAE is introduced, and it is shown that any vector field can be realized by a regular DAE. Throughout this paper the problem of path-tracing is discussed as an illustration of the mathematical phenomena.
Technical and physical systems, especially electronic circuits, are frequently modeled as a system of differential and nonlinear implicit equations. In the literature such systems of equations are called differentialalgebraic equations (DAEs). It turns out that the numerical and analytical properties of a DAE depend on an integer called the index of the problem. For example, the well-known BDF method of Gear can be applied, in general, to a DAE only if the index does not exceed one. In this paper we give a geometric interpretation of higherindex DAEs and indicate problems arising in connection with such DAEs by means of several examples.
This thesis is concerned with the issue of extinction of populations composed of different types of individuals, and their behavior before extinction and in case of a very late extinction. We approach this question firstly from a strictly probabilistic viewpoint, and secondly from the standpoint of risk analysis related to the extinction of a particular model of population dynamics. In this context we propose several statistical tools. The population size is modeled by a branching process, which is either a continuous-time multitype Bienaymé-Galton-Watson process (BGWc), or its continuous-state counterpart, the multitype Feller diffusion process. We are interested in different kinds of conditioning on non-extinction, and in the associated equilibrium states. These ways of conditioning have been widely studied in the monotype case. However the literature on multitype processes is much less extensive, and there is no systematic work establishing connections between the results for BGWc processes and those for Feller diffusion processes. In the first part of this thesis, we investigate the behavior of the population before its extinction by conditioning the associated branching process X_t on non-extinction (X_t≠0), or more generally on non-extinction in a near future 0≤θ<∞ (X_{t+θ}≠0), and by letting t tend to infinity. We prove the result, new in the multitype framework and for θ>0, that this limit exists and is non-degenerate. This reflects a stationary behavior for the dynamics of the population conditioned on non-extinction, and provides a generalization of the so-called Yaglom limit, corresponding to the case θ=0. In a second step we study the behavior of the population in case of a very late extinction, obtained as the limit when θ tends to infinity of the process conditioned by X_{t+θ}≠0. The resulting conditioned process is a known object in the monotype case (sometimes referred to as Q-process), and has also been studied when X_t is a multitype Feller diffusion process. We investigate the not yet considered case where X_t is a multitype BGWc process and prove the existence of the associated Q-process. In addition, we examine its properties, including the asymptotic ones, and propose several interpretations of the process. Finally, we are interested in interchanging the limits in t and θ, as well as in the not yet studied commutativity of these limits with respect to the high-density-type relationship between BGWc processes and Feller processes. We prove an original and exhaustive list of all possible exchanges of limit (long-time limit in t, increasing delay of extinction θ, diffusion limit). The second part of this work is devoted to the risk analysis related both to the extinction of a population and to its very late extinction. We consider a branching population model (arising notably in the epidemiological context) for which a parameter related to the first moments of the offspring distribution is unknown. We build several estimators adapted to different stages of evolution of the population (phase growth, decay phase, and decay phase when extinction is expected very late), and prove moreover their asymptotic properties (consistency, normality). In particular, we build a least squares estimator adapted to the Q-process, allowing a prediction of the population development in the case of a very late extinction. This would correspond to the best or to the worst-case scenario, depending on whether the population is threatened or invasive. These tools enable us to study the extinction phase of the Bovine Spongiform Encephalopathy epidemic in Great Britain, for which we estimate the infection parameter corresponding to a possible source of horizontal infection persisting after the removal in 1988 of the major route of infection (meat and bone meal). This allows us to predict the evolution of the spread of the disease, including the year of extinction, the number of future cases and the number of infected animals. In particular, we produce a very fine analysis of the evolution of the epidemic in the unlikely event of a very late extinction.
On a method for solution of the ordinary differential equations connected with Huygens' equations
(2010)
We establish elements of a new approach to ellipticity and parametrices within operator algebras on manifolds with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaes. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The "full" calculus involves two separate theories, one near the tip of the corner and another one at the conical exit to infinity. However, concerning the conical exit to infinity, we establish here a new concrete calculus of edge-degenerate operators which can be iterated to higher singularities.
This thesis considers on the one hand the construction of point processes via conditional intensities, motivated by the partial Integration of the Campbell measure of a point process. Under certain assumptions on the intensity the existence of such a point process is shown. A fundamental example turns out to be the Pólya sum process, whose conditional intensity is a generalisation of the Pólya urn dynamics. A Cox process representation for that point process is shown. A further process considered is a Poisson process of Gaussian loops, which represents a noninteracting particle system derived from the discussion of indistinguishable particles. Both processes are used to define particle systems locally, for which thermodynamic limits are determined.
We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ0, σ1) where σ0 is the standard (degenerate) homogeneous principal symbol, and σ1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dimX, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.
The Cauchy problem of the vacuum Einstein's equations aims to find a semimetric g(αβ) of a spacetime with vanishing Ricci curvature Rα,β and prescribed initial data. Under the harmonic gauge condition, the equations Rα,β = 0 are transferred into a system of quasi-linear wave equations which are called the reduced Einstein equations. The initial data for Einstein's equations are a proper Riemannian metric h(αβ) and a second fundamental form K(αβ). A necessary condition for the reduced Einstein equation to satisfy the vacuum equations is that the initial data satisfy Einstein constraint equations. Hence the data (h(αβ),K(αβ)) cannot serve as initial data for the reduced Einstein equations. Previous results in the case of asymptotically flat spacetimes provide a solution to the constraint equations in one type of Sobolev spaces, while initial data for the evolution equations belong to a different type of Sobolev spaces. The goal of our work is to resolve this incompatibility and to show that under the harmonic gauge the vacuum Einstein equations are well-posed in one type of Sobolev spaces.
We give a survey on the calculus of (pseudo-differential) boundary value problems with the transmision property at the boundary, and ellipticity in the Shapiro-Lopatinskij sense. Apart from the original results of the work of Boutet de Monvel we present an approach based on the ideas of the edge calculus. In a final section we introduce symbols with the anti-transmission property.
We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27 - 31, 2008, at the MSRI, University of Berkeley.
We prove a local in time existence and uniqueness theorem of classical solutions of the coupled Einstein{Euler system, and therefore establish the well posedness of this system. We use the condition that the energy density might vanish or tends to zero at infinity and that the pressure is a certain function of the energy density, conditions which are used to describe simplified stellar models. In order to achieve our goals we are enforced, by the complexity of the problem, to deal with these equations in a new type of weighted Sobolev spaces of fractional order. Beside their construction, we develop tools for PDEs and techniques for hyperbolic and elliptic equations in these spaces. The well posedness is obtained in these spaces.
We consider a mixed problem for a degenerate differentialoperator equation of higher order. We establish some embedding theorems in weighted Sobolev spaces and show existence and uniqueness of the generalized solution of this problem. We also give a description of the spectrum for the corresponding operator.
The ellipticity of boundary value problems on a smooth manifold with boundary relies on a two-component principal symbolic structure (σψ; σ∂), consisting of interior and boundary symbols. In the case of a smooth edge on manifolds with boundary we have a third symbolic component, namely the edge symbol σ∧, referring to extra conditions on the edge, analogously as boundary conditions. Apart from such conditions in integral form' there may exist singular trace conditions, investigated in [6] on closed' manifolds with edge. Here we concentrate on the phenomena in combination with boundary conditions and edge problem.
We establish elements of a new approch to ellipticity and parametrices within operator algebras on a manifold with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaces. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The “full” calculus is voluminous; so we content ourselves here with some typical aspects such as symbols in terms of order reducing families, classes of relevant examples, and operators near the conical exit to infinity.
In this article we construct the fundamental solutions for the wave equation arising in the de Sitter model of the universe. We use the fundamental solutions to represent solutions of the Cauchy problem and to prove the Lp − Lq-decay estimates for the solutions of the equation with and without a source term.
We study the asymptotic regularity of solutions of Hartree-Fock equations for Coulomb systems. In order to deal with singular Coulomb potentials, Fock operators are discussed within the calculus of pseudo-differential operators on conical manifolds. First, the non-self-consistent-field case is considered which means that the functions that enter into the nonlinear terms are not the eigenfunctions of the Fock operator itself. We introduce asymptotic regularity conditions on the functions that build up the Fock operator which guarantee ellipticity for the local part of the Fock operator on the open stretched cone R+ × S². This proves existence of a parametrix with a corresponding smoothing remainder from which it follows, via a bootstrap argument, that the eigenfunctions of the Fock operator again satisfy asymptotic regularity conditions. Using a fixed-point approach based on Cances and Le Bris analysis of the level-shifting algorithm, we show via another bootstrap argument, that the corresponding self-consistent-field solutions of the Hartree-Fock equation have the same type of asymptotic regularity.
In this paper, we discuss the global existence of solutions for Chemotaxis models with saturation growth. If the coe±cients of the equations are all positive smooth T-periodic functions, then the problem has a positive T-periodic solution, and meanwhile we discuss here the stability problems for the T-periodic solutions.
Formal Poincaré lemma
(2007)
We show how the multiple application of the formal Cauchy-Kovalevskaya theorem leads to the main result of the formal theory of overdetermined systems of partial differential equations. Namely, any sufficiently regular system Au = f with smooth coefficients on an open set U ⊂ Rn admits a solution in smooth sections of a bundle of formal power series, provided that f satisfies a compatibility condition in U.
Contents: Part I: Symplectic Geometry Chapter 1: Symplectic Spaces and Lagrangian Planes Chapter 2: The Symplectic Group Chapter 3: Multi-Oriented Symplectic Geometry Chapter 4: Intersection Indices in Lag(n) and Sp(n) Part II: Heisenberg Group, Weyl Calculus, and Metaplectic Representation Chapter 5: Lagrangian Manifolds and Quantization Chapter 6: Heisenberg Group and Weyl Operators Chapter 7: The Metaplectic Group Part III: Quantum Mechanics in Phase Space Chapter 8: The Uncertainty Principle Chapter 9: The Density Operator Chapter 10: A Phase Space Weyl Calculus
Differential and pseudo-differential operators on a manifold with (regular) geometric singularities can be studied within a calculus, inspired by the concept of classical pseudo-differential operators on a C1 manifold. In the singular case the operators form an algebra with a principal symbolic hierarchy σ = (σj)0≤j≤k, with k being the order of the singularity and σk operator-valued for k ≥ 1. The symbols determine ellipticity and the nature of parametrices. It is typical in this theory that, similarly as in boundary value problems (which are special edge problems, where the edge is just the boundary), there are trace, potential and Green operators, associated with the various strata of the configuration. The operators, obtained from the symbols by various quantisations, act in weighted distribution spaces with multiple weights. We outline some essential elements of this calculus, give examples and also comment on new challenges and interesting problems of the recent development.
We develop an approach to the problem of optimal recovery of continuous linear functionals in Banach spaces through information on a finite number of given functionals. The results obtained are applied to the problem of the best analytic continuation from a finite set in the complex space Cn, n ≥ 1, for classes of entire functions of exponential type which belong to the space Lp, 1 < p < 1, on the real subspace of Cn. These latter are known as Wiener classes.
On a manifold with edge we construct a specific class of (edgedegenerate) elliptic differential operators. The ellipticity refers to the principal symbolic structure σ = (σψ, σ^) of the edge calculus consisting of the interior and edge symbol, denoted by σψ and σ^, respectively. For our choice of weights the ellipticity will not require additional edge conditions of trace or potential type, and the operators will induce isomorphisms between the respective edge spaces.
We prove the existence of a class of local in time solutions, including static solutions, of the Einstein-Euler system. This result is the relativistic generalisation of a similar result for the Euler-Poisson system obtained by Gamblin [8]. As in his case the initial data of the density do not have compact support but fall off at infinity in an appropriate manner. An essential tool in our approach is the construction and use of weighted Sobolev spaces of fractional order. Moreover, these new spaces allow us to improve the regularity conditions for the solutions of evolution equations. The details of this construction, the properties of these spaces and results on elliptic and hyperbolic equations will be presented in a forthcoming article.
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.