Bounded variation and the strength of Helly's selection theorem

We analyze the strength of Helly's selection theorem HST, which is the most important compactness theorem on the space of functions of bounded variation. For this we utilize a new representation of this space intermediate between $L_1$ and the Sobolev space W1,1, compatible with the, so called, weak* topology. We obtain that HST is instance-wise equivalent to the Bolzano-Weierstra\ss\ principle over RCA0. With this HST is equivalent to ACA0 over RCA0. A similar classification is obtained in the Weihrauch lattice

On the strength of weak compactness

We study the logical and computational strength of weak compactness in the separable Hilbert space \ell_2. Let weak-BW be the statement the every bounded sequence in \ell_2 has a weak cluster point. It is known that weak-BW is equivalent to ACA_0 over RCA_0 and thus that it is equivalent to (nested uses of) the usual Bolzano-Weierstra{\ss} principle BW. We show that weak-BW is instance-wise equivalent to the \Pi^0_2-CA. This means that for each \Pi^0_2 sentence A(n) there is a sequence (x_i) in \ell_2, such that one can define the comprehension functions for A(n) recursively in a cluster point of (x_i). As consequence we obtain that the Turing degrees d > 0" are exactly those degrees that contain a weak cluster point of any computable, bounded sequence in \ell_2. Since a cluster point of any sequence in the unit interval [0,1] can be computed in a degree low over 0', this show also that instances of weak-BW are strictly stronger than instances of BW. We also comment on the strength of weak-BW in the context of abstract Hilbert spaces in the sense of Kohlenbach and show that his construction of a solution for the functional interpretation of weak compactness is optimal

Non-principal ultrafilters, program extraction and higher order reverse mathematics

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. Let U be the statement that a non-principal ultrafilter exists and let ACA_0^{\omega} be the higher order extension of ACA_0. We show that ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that ACA_0^{\omega}+\U is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U a realizing term in G\"odel's system T can be extracted. This means that one can extract a term t, such that A(f,t(f))

Non-commutative tachyon action and D-brane geometry

We analyse open string correlators in non-constant background fields, including the metric $g$, the antisymmetric $B$-field, and the gauge field $A$. Working with a derivative expansion for the background fields, but exact in their constant parts, we obtain a tachyonic on-shell condition for the inserted functions and extract the kinetic term for the tachyon action. The 3-point correlator yields a non-commutative tachyon potential. We also find a remarkable feature of the differential structure on the D-brane: Although the boundary metric $G$ plays an essential role in the action, the natural connection on the D-brane is the same as in closed string theory, i.e. it is compatible with the bulk metric and has torsion $H=dB$. This means, in particular, that the parallel transport on the brane is independent of the gauge field $A$.Comment: 12 pages, no figure