Rademacher and Gaussian complexities are successfully used in learning theory for measuring the capacity of the class of functions to be learned. One of the most important properties for these complexities is their Lipschitz property: a composition of a class of functions with a fixed Lipschitz function may increase its complexity by at most twice the Lipschitz constant. The proof of this prope...

In this paper we consider the computability of the solution of the initialvalue problem for differential inclusions with semicontinuous right-hand side. We present algorithms for the computation of the solution using the “ten thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of an upper-semicontin...

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipsch...

Suppose we are given a black-box evaluator (an oracle that returns the function value at a given point) for a Lipschitz function with a known Lipschitz constant. We consider queries that can be answered about the function by using a finite number of black-box evaluations. Specifically, we study the problems of approximating a Lipschitz function, approximately integrating a Lipschitz function, a...

Let $mathfrak{L}$ be the Virasoro-like algebra and $mathfrak{g}$ itsderived algebra, respectively. We investigate the structure of the Lie triplederivation algebra of $mathfrak{L}$ and $mathfrak{g}$. We provethat they are both isomorphic to $mathfrak{L}$, which provides twoexamples of invariance under triple derivation.

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit $\mathcal{C}^{1,1}$-parametrization and they solve the geodesic equation in sense of Filippov this parametrization. Our proof shows are either everywhere lightlike or timelike. Furthermore, demonstrates an $\alpha$-H\"older $\mathcal{C}^{1,\frac{\alpha}{4}}$-parametrization.

in this paper we investigate some hereditary properties of amenability modulo an ideal of banach algebras. we show that if (e) is a bounded approximate identity modulo i of a banach algebra a and x is a neo-unital modulo i, then (e) is a bounded approximate identity for x. moreover we show that amenability modulo an ideal of a banach algebra a can be only considered by the neo-unital modulo...