121309269
Stochastic Integration for Some Rough Non‐adapted Processes
For the Wiener process W on the unit interval and 0 ⩽ θ ⩽ 1/4 consider the non-adapted process ut =sgn(Wt−θ)sgn(W1−WT+θ), θ⩽t⩽1−θ. By giving explicit descriptions of their expansions in terms of generalized Hermite polynomials we show that they possess Skorohod integral processes. In the more interesting case θ = 0, a component due to the interaction of forwards and backwards Wiener process seems to dominate the non-interaction part, whereas for θ > 0 interaction drops geometrically with respect to non-interaction. We translate these purely analytic properties into more familiar terms of samples of stochastic processes. Firstly, we construct simple approximations of the integral process which are of the type of Riemann sums. We then show that their quadratic variations are the same as for the Wiener process on [θ, 1 − θ]. Finally, using the fact that W remains a semimartingale if its filtration is enlarged by W1, we compare the resulting Ito integral process to the Skorohod integral process of u in case θ = 0. We show that they differ by a local time. Consequently the Skorohod integral process of u is a semimartingale.
120957526
On Stochastic Differential Equations with Reflecting Barriers
121164029
Über höhere kritische Determinanten von Sternkörpern
121422615
Flächenapproximation beim Jacobialgorithmus
119634267
On Thue's equation
and more precisely for the number of primitive solutions to (1.1), that is, solutions in coprime integers x, y. The first important results on the number of solutions of Thue's equation were obtained by C.L. Siegel, in the case r = 3 and in the binomial case F ( x , y ) = a x r b y r. In view of his results and comments, we should attribute to Siegel the problem to decide whether the number of primitive solutions to (1.1), for irreducible F of degree r > 3, could be bounded by a function depending only on r and h, but otherwise independent of F. In 1983, Siegel's question was answered in the affirmative by J.-H. Evertse [2]. As a special case of his results (he also treats equations in number fields), Evertse obtains the bound
122911860
Quantentheorie der Felder als Distributionstheorie
ZusammenfassungDie Distributionstheorie vonL. Schwartz wird zu einer mathematisch konsistenten Formulierung der Quantenfeldtheorie verwendet. Die Einführung von Distributionen impliziter Funktionen gestattet es, viele Formeln mit geringfügigen Änderungen in den strengen Formalismus zu übernehmen. Die Annahme, daß es eine raumzeitliche Feldoperatordistribution gibt, führt auf einen vonWigiitman angegebenen Hilbertraum. Er ist durch eine Folge von Distributionen, die Vakuumerwartungswerte der Theorie, bestimmt. Diese werden für die freien Felder (Schrödinger-Gordon-Feld, Maxwellfeld, Diracfeld) aus einfachen Annahmen hergeleitet. Beim Maxwellfeld ergibt sich zwanglos, daß die Metrik nur in dem durch die Lorentzbedingung eingeschränkten Hilbertraum definit ist. Die Definition einer Dichte-Distribution, welche den üblichen Doppelpunktprodukten entspricht, erweist sich auch in dieser Fassung der Quantenfeldtheorie als möglich.RiassuntoViene applicata la teoria delle distribuzioni diL. Schwartz nella formulazione della teoria quantistica dei campi onde risulti matematicamente consistente. L’introduzione di funzioni di distribuzione permette di ottenere moite formule nel formalismo rigoroso e con cambiamenti semplicissimi rispetto alla formulazione corrente. Facendo l’ipotesi dell’esistenza di un operatore di campo spazio-temporale avente l’aspetto distribuzionale, si arriva alla formulazione di uno spazio Hilbertiano introdotto dalWightman: taie spazio è metricamente definite da una suecessione di distribuzioni le quali altro non sono che i valori medi del vuoto risultanti dalla teoria. Tali valori vengono ottenuti con semplici ipotesi per i campi di Schrödinger-Gordon, di Maxwell, di Dirac. A proposito del campo elettromagnetico si osaerva che la metrica è definita solo in un sottospazio dell’intero spazio di Hilbert definito dalla condizione di Lorentz. Infine si rende possibile la definizione di densità di corrente nella struttura della teoria, densità che corrisponde all’abituale.