Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes $\Gamma$. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe $H_{\lambda_\Gamma^+}$ modulo forcing in $\Gamma$, for some cardinal $\lambda_\Gamma$ naturally associated to $\Gamma$. These axioms naturally extend projective absoluteness for arbitrary set-forcing--in this situation $\lambda_\Gamma=\omega$--to classes $\Gamma$ with $\lambda_\Gamma>\omega$. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms, but can be forced under mild large cardinal assumptions on $V$. We also show the existence of many classes $\Gamma$ with $\lambda_\Gamma=\omega_1$, and giving rise to pairwise incompatible theories for $H_{\omega_2}$.Comment: arXiv admin note: substantial text overlap with arXiv:1805.0873

Incompatible bounded category forcing axioms

We introduce bounded category forcing axioms for well-behaved classes Γ. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe Hλ+Γ modulo forcing in Γ, for some cardinal λΓ naturally associated to Γ. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation λΓ=ω — to classes Γ with λΓ>ω. Unlike projective absoluteness, these higher bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on V. We also show the existence of many classes Γ with λΓ=ω1 giving rise to pairwise incompatible theories for Hω2

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

The relationship between educational television and mathematics capability in Tanzania

Previous studies have frequently demonstrated that educational television viewing can have a positive effect on learning in low-income country contexts when shows are delivered in controlled settings. However, the consequence of day-to-day viewing in such contexts has scarcely been considered. Additionally, no recent published research has provided any information on the costs of educational television. The lack of research in these areas is striking. Examining educational television viewing in monitored settings provides limited information on the influence of routine television consumption. Further, the broad reach of numerous educational television programmes should provide low per-viewer costs and, resultantly, strong cost-effectiveness findings. This PhD study therefore examined (1) the association between educational television exposure and mathematics capability and (2) the cost effectiveness of educational television interventions. To achieve this, research was carried out that centred on Ubongo Kids – a popular Tanzanian-produced show delivering mathematics-focused content. Quantitative investigation into the association between educational television exposure and mathematics capability used nationally representative data, collected by Uwezo Tanzania. A household fixed-effects model showed that exposure to educational television was significantly associated with mathematics capability among children aged 7-16, when controlling for age, sex, school enrolment and Kiswahili attainment. Findings from this model were used in cost-effectiveness calculations, alongside cost data and an estimate of the number of Ubongo Kids viewers. Results compared favourably against those for other interventions, with calculations regarding Ubongo Kids’ ongoing activities suggesting it to have been more cost effective than any other intervention previously investigated using the same cost-effectiveness approach. These findings indicate that in low-income contexts: educational television programmes can aid learning; and, that directing a greater proportion of available educational resources towards educational television interventions may benefit educational outcomes

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2