Polynomial time ultrapowers and the consistency of circuit lower bounds

A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory ∀PV of all polynomial time functions. Generalizing a theorem of Hirschfeld (Israel J Math 20(2):111–126, 1975), we show that every countable model of ∀PV is isomorphic to an existentially closed substructure of a polynomial time ultrapower. Moreover, one can take a substructure of a special form, namely a limit polynomial time ultrapower in the classical sense of Keisler (in: Bergelson, V., Blass, A., Di Nasso, M., Jin, R. (eds.) Ultrafilters across mathematics, contemporary mathematics vol 530, pp 163–179. AMS, New York, 1963). Using a polynomial time ultrapower over a nonstandard Herbrand saturated model of ∀PV we show that ∀PV is consistent with a formal statement of a polynomial size circuit lower bound for a polynomial time computable function. This improves upon a recent result of Krajíček and Oliveira (Logical methods in computer science 13 (1:4), 2017).Peer ReviewedPostprint (author's final draft

Effect of measurement conditions on barkhausen noise parameters

The Barkhausen noise was measured in plastically deformed low-carbon steel at various measurement conditions. Strip samples magnetized by a single yoke were used for investigation. The measurement results for different magnetizing frequencies, waveforms of the magnetizing field and cut-off frequencies of the processing filter were compared and the differences, mainly concerning the sensitivity of the Barkhausen noise on the plastic deformation, will be discussed

Cohen forcing and its properties

This bachelor thesis studies properties of Cohen Forcing and its relation to the unprovability of Continuum Hypothesis and Generalised Continuum Hypothesis. The thesis is divided into four parts. In the first part the technique of forcing based on partial orders is introduced. The second part introduces a notion of Cohen forcing, shows properties of cardinal arithmetic sufficient to preservation of cardinals by Cohen forcing and focuses mainly on generic sets added by concrete variations of Cohen Forcing. Finally some of the properties of Cohen reals are shown in this part. The third part reconstruct a proof of unprovability of Continuum Hypothesis and shows a use of Cohen Forcing in relation to the statements about the Generalised Continuum Hypothesis. The last part discusses briefly a non-minimality of generic filters on Cohen forcing and introduce a notion of Sacks forcing in order to show an existence of forcing notion whose generic filters are minimal. Keywords Cohen forcing, CH, GCH, Cohen reals

Influence of substrate on magneto-elastic sensor transfer characteristics

Calculation of the dynamic elektromagnetic field in a two-coil strain sensor with amorphous ferromagnetic ribbon core is presented. Open-core set-up and material properties are considered of a model, the reluctivity of which depends on the mechanical stress being measured

Polynomial time ultrapowers and the consistency of circuit lower bounds

A polynomial time ultrapower is a structure given by the set of polynomial time computable functions modulo some ultrafilter. They model the universal theory ∀PV of all polynomial time functions. Generalizing a theorem of Hirschfeld (Israel J Math 20(2):111–126, 1975), we show that every countable model of ∀PV is isomorphic to an existentially closed substructure of a polynomial time ultrapower. Moreover, one can take a substructure of a special form, namely a limit polynomial time ultrapower in the classical sense of Keisler (in: Bergelson, V., Blass, A., Di Nasso, M., Jin, R. (eds.) Ultrafilters across mathematics, contemporary mathematics vol 530, pp 163–179. AMS, New York, 1963). Using a polynomial time ultrapower over a nonstandard Herbrand saturated model of ∀PV we show that ∀PV is consistent with a formal statement of a polynomial size circuit lower bound for a polynomial time computable function. This improves upon a recent result of Krajíček and Oliveira (Logical methods in computer science 13 (1:4), 2017).Peer Reviewe