We analyze approximation rates by deep ReLU networks of a class multi-variate solutions Kolmogorov equations which arise in option pricing. Key technical devices are architectures capable efficiently approximating tensor products. Combining this with results concerning the well behaved (i.e. fulfilling some smoothness properties) univariate functions, provides insights into functions structures...

This paper is concerned with two improved variants of the Hutch++ algorithm for estimating trace a square matrix, implicitly given through matrix-vector products. combines randomized low-rank approximation in first phase stochastic estimation second phase. In turn, only requires $O\left(\varepsilon^{-1}\right)$ products to approximate within relative error $\varepsilon$ high probability, provid...

The main purpose of this paper is to consider the t-best co-approximation and t-best simultaneous co-approximation in fuzzy normed spaces. We develop the theory of t-best co-approximation and t-best simultaneous co-approximation in quotient spaces. This new concept is employed us to improve various characterisations of t-co-proximinal and t-co-Chebyshev sets.

The main purpose of this paper is to consider the t-best co-approximation and t-best simultaneous co-approximation in fuzzy normed spaces. We develop the theory of t-best co-approximation and t-best simultaneous co-approximation in quotient spaces. This new concept is employed us to improve various characterisations of t-co-proximinal and t-co-Chebyshev sets.

We investigate the problem of best simultaneous Diophantine approximation under a constraint on the denominator, as proposed by Jurkat. New lower estimates for optimal approximation constants are given in terms of critical determinants of suitable star bodies. Tools are results on simultaneous Diophantine approximation of rationals by rationals with smaller denominator. Finally, the approximati...

We study nonlinear m-term approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f ∈ H and any dictionary D an expansion into a series f = ∞ X j=1 cj(f)φj(f), φj(f) ∈ D, j = 1, 2, . . . with the Parseval property: ‖f‖2 = j |cj(f)|. Following the paper of A. Lutoborsk...

Motivated by the problem to approximate all feasible schedules by one schedule in a given scheduling environment, we introduce in this paper the concepts of strong simultaneous approximation ratio (SAR) and weak simultaneous approximation ratio (WAR). Then we study the two parameters under various scheduling environments, such as, nonpreemptive, preemptive or fractional scheduling on identical,...