Studying Algebraic Structures Using Prover9 and Mace4

In this chapter we present a case study, drawn from our research work, on the application of a fully automated theorem prover together with an automatic counter-example generator in the investigation of a class of algebraic structures. We will see that these tools, when combined with human insight and traditional algebraic methods, help us to explore the problem space quickly and effectively. The counter-example generator rapidly rules out many false conjectures, while the theorem prover is often much more efficient than a human being at verifying algebraic identities. The specific tools in our case study are Prover9 and Mace4; the algebraic structures are generalisations of Heyting algebras known as hoops. We will see how this approach helped us to discover new theorems and to find new or improved proofs of known results. We also make some suggestions for how one might deploy these tools to supplement a more conventional approach to teaching algebra.Comment: 21 pages, to appear as Chapter 5 in "Proof Technology in Mathematics Research and Teaching", Mathematics Education in the Digital Era 14, edited by G. Hanna et al. (eds.), published by Springe

Physical activity and genitourinary cancer survivorship

In this chapter we discuss common toxicities arising from genitourinary cancer treatments, in particular the adverse effects from androgen deprivation therapy (ADT) for prostate cancer given its well established detrimental effects on physical, physiological, and psychological function, and existing physical activity research in the postdiagnosis period including studies focusing on supportive care and some limited data on disease outcomes. Overall, consistent positive outcomes have been reported across studies showing that exercise is beneficial to reduce a number of treatment-related toxicities and improve symptoms. Additional studies are needed in genitourinary cancers other than prostate to establish specific physical activity requirements and implementation strategies