123201234
Badly approximable systems of linear forms
A number α is called badly approximable if there is a constant c = c(α)>0 such that 
|q|αq − p| > c(α) 
for any integers p, q ≠ 0. The existence of continuum—many badly approximable numbers follows easily from the theory of continued fractions. In the present paper the notion of a badly approximable number is generalized to badly approximable systems of m linear forms in n variables. Among other results we prove for every given m and n the existence of continuum many badly approximable systems of linear forms.
121229043
Some results in probabilistic geometry
121164029
Über höhere kritische Determinanten von Sternkörpern
120328164
Irregularities of distribution. IV
120952401
The measure of the set of admissible lattices
A0 denotes the lattice of points with integral coordinates. Theorem 4 will be used to prove Theorem 5 which is an improvement of the Minkowski-Hlawka Theorem. We also prove two existence theorems which are in a certain sense converses of the MinkowskiHlawka Theorem (Theorem 6 and Theorem 7). The author is very indebted to the referee who pointed out some errors of the originally submitted paper and made useful suggestions. The originally stated Theorem 1 was wrong.
129296972
Irregularities of distribution. V
120957526
On Stochastic Differential Equations with Reflecting Barriers
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.
121073876
Existence of solutions of certain nonconvex optimal control problems governed by nonlinear integral equations
Optimal control problems with nonlinear equations usually do not have a solution, i.e. an optimal control. Nevertheless, if the cost functional is uniformly concave with respect to the state, the solution may exist. Using the Balder's technique based on a Youngmeasure relaxation, Bauer's external principle and investigation of extreme Young measures; the existence is demonstrated here for optimal control processes described by nonlinear integral equations