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For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question.
The paper considers quantitati
We prove that, for every theory $T$ which is given by an ${mathcal L}_{omega_1,omega}$ sentence, $T$ has less than $2^{aleph_0}$ many countable models if and only if we have that, for every $Xin 2^omega$ on a cone of Turing degrees, every $X$-hyperar
We study the computational content of the Radon-Nokodym theorem from measure theory in the framework of the representation approach to computable analysis. We define computable measurable spaces and canonical representations of the measures and the i
The TTE computability notion in effective metric spaces is usually defined by using Cauchy representations. Under some weak assumptions, we characterize this notion in a way which avoids using the representations.
Let $mathfrak{n}$ be a nonempty, proper, convex subset of $mathbb{C}$. The $mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $mathfrak{n}$ and are maximal with this property. Typical examples of these are the max