Elliptic Reconstruction and a Posteriori Error Estimates for Fully Discrete Semilinear Parabolic Optimal Control Problems

This article studies a posteriori error analysis of fully discrete finite element approximations for semilinear parabolic optimal control problems. Based on elliptic reconstruction approach introduced earlier by Makridakis and Nochetto [25], a residual based a posteriori error estimators for the state, co-state and control variables are derived. The space discretization of the state and co-state variables is done by using the piecewise linear and continuous finite elements, whereas the piecewise constant functions are employed for the control variable. The temporal discretization is based on the backward Euler method. We derive a posteriori error estimates for the state, co-state and control variables in the $L^\infty(0,T;L^2(\Omega))$-norm. Finally, a numerical experiment is performed to illustrate the performance of the derived estimators.     

Morse Index of Multiple Blow-up Solutions to the Two-Dimensional Sinh-Poisson Equation

In this paper we consider the Dirichlet problemwhere $\rho$ is a small parameter and $\Omega$ is a $C^2$ bounded domain in $\mathbb{R}^2$. In [1], the author proves the existence of a $m$-point blow-up solution $u_\rho$ jointly with its asymptotic behaviour. We compute the Morse index of $u_\rho$ in terms of the Morse index of the associated Hamilton function of this problem. In addition, we give an asymptotic estimate for the first $4m$ eigenvalues and eigenfunctions.     

Exponential stability of discrete-time systems with slowly time-varying delays

Slowly time-varying delays are seldom, but do need to be, considered in the context of discrete-time systems. This paper addresses the exponential stability issue of discrete-time systems with slowly time-varying delays. The basic idea is to transform, by utilizing the switching transformation approach, the original system with slowly time-varying delays into an equivalent switched system with special switching signal. Different types of delays correspond to different types of switching signals, and the stability issue of the original system is converted into that of a switched system. It is the first time that the method of switched homogeneous polynomial Lyapunov function is applied to general delayed systems. Some sufficient exponential stability conditions for the original system are proposed in several situations. It is numerically shown that the conservativeness of the proposed conditions reduces as the degree of the switched homogeneous polynomial Lyapunov function increases.     

Uniqueness and clustering properties of Gibbs states for classical and quantum unbounded spin systems

We consider quantum unbounded spin systems (lattice boson systems) in -dimensional lattice space Z. Under appropriate conditions on the interactions we prove that in a region of high temperatures the Gibbs state is unique, is translationally invariant, and has clustering properties. The main methods we use are the Wiener integral representation, the cluster expansions for zero boundary conditions and for general Gibbs state, and explicitly -dependent probability estimates. For one-dimensional systems we show the uniqueness of Gibbs states for any value of temperature by using the method of perturbed states. We also consider classical unbounded spin systems. We derive necessary estimates so that all of the results for the quantum systems hold for the classical systems by straightforward applications of the methods used in the quantum case.     

Exponential distributions on semigroups

Three fundamental characterizations of the standard exponential distribution on [0, ) are the remaining life, memoryless and constant failure properties. Analogs of these properties are studied for distributions on a class of semigroups in which the semigroup operation replaces addition, a compatible partial order replaces the ordinary order, and a left-invariant measure replaces Lebesgue measure. Partial characterizations of exponential distributions on such semigroups are obtained and the semigroup formulation provides new characterizations of certain aging properties studied in reliability-increasing failure rate, new better than used, and increasing failure rate average.     

Dynamic structure factor in a random diffusion model

Let {X _t:0} denote random walk in the random waiting time model, i.e., simple random walk with jump ratew ^–1(X _t), where {w(x):x^d} is an i.i.d. random field. We show that (under some mild conditions) theintermediate scattering function F(q,t)=E ₀ (q^d) is completely monotonic int (E ₀ denotes double expectation w.r.t. walk and field). We also show that thedynamic structure factor S(q, w)=2 ₀ cos(t)F(q, t) exists for 0 and is strictly positive. Ind=1, 2 it diverges as 1/||^1/2, resp. –ln(||), in the limit 0; ind3 its limit value is strictly larger than expected from hydrodynamics. This and further results support the conclusion that the hydrodynamic region is limited to smallq and small such that ||D |q|², whereD is the diffusion constant.     

A priori error estimates for numerical methods for scalar conservation laws. Part I: The general approach

In this paper, we construct a general theory of a priori error estimates for scalar conservation laws by suitably modifying the original Kuznetsov approximation theory. As a first application of this general technique, we show that error estimates for conservation laws can be obtained without having to use explicitly any regularity properties of the approximate solution. Thus, we obtain optimal error estimates for the Engquist-Osher scheme without using the fact (i) that the solution is uniformly bounded, (ii) that the scheme is total variation diminishing, and (iii) that the discrete semigroup associated with the scheme has the -contraction property, which guarantees an upper bound for the modulus of continuity in time of the approximate solution.

     

The stability of the exponential equation

We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an -approximately exponential function to the set of all exponential functions tends to zero as tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers .

     

Superstability estimates for anharmonic systems

We consider an anharmonic crystal described by variablesS _x ,x ^d ,S _x , with one-body interaction ¦S _x ¦ and nearest neighbor (n.n.) two body interaction ¦S _x –S _y ¦. We prove that, for ^d bounded, , where is the correlation function for the free boundary condition Gibbs state in ,>0 and are suitable constants independent of and . This generalizes previous results obtained in the case.Research partially supported by Consiglio Nazionale delle Ricerche.     

Strukturuntersuchungen an naßersponnenen Polyacrylnitrilfasern nach der Gefriertrocknung

Zusammenfassung An unterschiedlich verstreckten und hydrothermisch nachbehandelten, danach gefriergetrockneten Polyacrylnitril-Gelfasern (PAN) wurden Messungen der Röntgenkleinwinkelstreuung (RKWS), Röntgenweitwinkelstreuung (RWWS) und Tieftemperatur-Gassorption (TTGA) durchgeführt. Die RKWS-Messungen zeigen, daß sowohl in den isotropen als auch in den orientierten Proben eine exponentialstatistische Verteilung der Form und Größe der Inhomogenitäten (Hohlräume, Cluster) vorliegt, welche durch die Kohärenzlängel _c bestimmt wird. Eine erhöhte Röntgenabsorption wurde bei den verstreckten Proben festgestellt. Die Analyse der aus der Kohärenzlänge und den Volumsanteilen der Phasen berechneten charakteristischen RK WS-Parameter (z.B. innere Oberfläche, Durchschußlänge) ergibt, daß jede Behandlung der Gelfasern eine Zu- oder Abnahme der etwa gleich großen Hohlräume verursacht (nucleotic change). Das relative Ausmaß dieser Veränderungen wird besprochen.Der Vergleich mit den Ergebnissen der TTGA an den gleichen Fasern zeigt, daß die Tendenzen bei der spezifischen Oberfläche und dem Porenvolumen (Zu- bzw. Abnahme) bei beiden Methoden annähernd gleich sind. Die Absolutwerte des Hohlraumvolumens unterscheiden sich aber z.T. um den Faktor 40. Dies führt zur Schlußfolgerung, daß mit der RK WS Faserbereiche als Hohlräume erfaßt werden, die bei der TTGA nicht angezeigt werden, weil sie für das Meßgas unzugänglich sind.

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Structural studies on wet-spun freeze-dried polyacrylonitrile fibers

Summary Polyacrylonitrile fibers (PAN) were drawn to different extents, hydrothermally aftertreated, and at last freeze-dried. The products were studied by small angle X-ray scattering (SAXS), wide angle X-ray scattering (WAXS), and low temperature gassorption (LTGS). The SAXS data show that the shape and the size of the inhomogeneities (voids, clusters) have an exponential statistical distribution determined by the correlation lengthl _c, both in the isotropic fibers and in the fiber-oriented ones. It has also been observed that stretching always causes a higher X-ray absorption. An analysis of the characteristic SAXS parameters (e.g., internal surface, mean chord length), which were calculated from the correlation length and the volume fractions of the phases, evidences that in all samples an augmentation or diminution of the number of the voids with nearly the same shape and size takes place (nucleotic change). Its relative number has been pointed out.Comparison with the results of the LTGS investigations at the same fibers shows that the tendencies of the specific surface and porosity (increase, decrease) are approximately the same with both methods. However, the absolute values of void volume differ in some cases by the factor 40. This leads to the conclusion that the SAXS method records domains of these fibers as voids which are not registered by LTGS (voids not accessible for the gas).

Herrn em.o.Univ. Prof. Dr.Josef Schurz zum 70. Geburtstag gewidmet