Smooth approximations by continuous choice-functions

We explore the existence of rational-valued approximation processes by continuous functions of two variables, such that the output continuously depends of the imposed error-bound. To this sake we prove that the theory of densely ordered sets with generic predicates is ℵ0- categorical. A model of the theory and a particular continuous choice-function are constructed. This function transfers to all other models by the respective isomorphisms. If some common-sense conditions are fulfilled, the processes are computable. As a byproduct, other functions with surprising properties can be constructed.

On some smooth approximations on thick periodic multi-structures

In this paper we study the approximation properties of measurable and square-integrable functions. In particular we show that any $L^2$-bounded function can be approximated in $L^2$-norm by smooth functions defined on a highly oscillating boundary of thick multi-structures in ${\mathbb{R}}^n$. We derive the norm estimates for the approximating functions and study their asymptotic behaviour.

Sigmoid functions for the smooth approximation to the absolute value function

We present smooth approximations to the absolute value function |x| using sigmoid functions. In particular, x erf(x/μ) is proved to be a better smooth approximation for |x| than x tanh(x/μ) and \sqrt {{x^2} + \mu } with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for the error function.

Smooth approximations and their applications to homotopy types

Let $M, N$ the be smooth manifolds, $\mathcal{C}^{r}(M,N)$ the space of ${C}^{r}$ maps endowed with the corresponding weak Whitney topology, and $\mathcal{B} \subset \mathcal{C}^{r}(M,N)$ an open subset.It is proved that for $0<r<s\leq\infty$ the inclusion $\mathcal{B} \cap \mathcal{C}^{s}(M,N) \subset \mathcal{B}$ is a weak homotopy equivalence.It is also established a parametrized variant of such a result.In particular, it is shown that for a compact manifold $M$, the inclusion of the space of $\mathcal{C}^{s}$ isotopies $\eta:[0,1]\times M \to M$ fixed near $\{0,1\}\times M$ into the space of loops $\Omega(\mathcal{D}^{r}(M), \mathrm{id}_{M})$ of the group of $\mathcal{C}^{r}$ diffeomorphisms of $M$ at $\mathrm{id}_{M}$ is a weak homotopy equivalence.

Sigmoid functions for the smooth approximation to $ \vert{x}\vert $

We present smooth approximations to $ \vert{x} \vert $ using sigmoid functions. In particular $ x\,erf(x/\mu) $ is proved to be better smooth approximation for $ \vert{x} \vert $ than $ x\,tanh(x/\mu) $ and $ \sqrt{x^2 + \mu} $ with respect to accuracy. To accomplish our goal we also provide sharp hyperbolic bounds for error function.