Computing with Classical Real Numbers

There are two incompatible Coq libraries that have a theory of the real numbers; the Coq standard library gives an axiomatic treatment of classical real numbers, while the CoRN library from Nijmegen defines constructively valid real numbers. Unfortunately, this means results about one structure cannot easily be used in the other structure. We present a way interfacing these two libraries by showing that their real number structures are isomorphic assuming the classical axioms already present in the standard library reals. This allows us to use O'Connor's decision procedure for solving ground inequalities present in CoRN to solve inequalities about the reals from the Coq standard library, and it allows theorems from the Coq standard library to apply to problem about the CoRN reals

A Computer Verified Theory of Compact Sets

Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and displaying images with a computer. In this paper, I build upon existing work about complete metric spaces to define compact sets as the completion of the space of finite sets under the Hausdorff metric. This definition allowed me to quickly develop a computer verified theory of compact sets. I applied this theory to compute provably correct plots of uniformly continuous functions.Comment: This paper is to be part of the proceedings of the Symbolic Computation in Software Science Austrian-Japanese Workshop (SCSS 2008

Classical Mathematics for a Constructive World

Interactive theorem provers based on dependent type theory have the flexibility to support both constructive and classical reasoning. Constructive reasoning is supported natively by dependent type theory and classical reasoning is typically supported by adding additional non-constructive axioms. However, there is another perspective that views constructive logic as an extension of classical logic. This paper will illustrate how classical reasoning can be supported in a practical manner inside dependent type theory without additional axioms. We will see several examples of how classical results can be applied to constructive mathematics. Finally, we will see how to extend this perspective from logic to mathematics by representing classical function spaces using a weak value monad.Comment: v2: Final copy for publicatio

A Representation Theorem for Second-Order Functionals

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of second-order functionals which are polymorphic over a class of functors. Types polymorphic over a class of functors are easily representable in languages such as Haskell, but are difficult to analyse and reason about. The concrete representation provided by the theorem is easier to analyse, but it might not be as convenient to implement. Therefore, depending on the task at hand, the change of representation may prove valuable in one direction or the other. We showcase the usefulness of the representation theorem with a range of examples. Concretely, we show how the representation theorem can be used to show that traversable functors are finitary containers, how parameterised coalgebras relate to very well-behaved lenses, and how algebraic effects might be implemented in a functional language

WEDGED FOOTWEAR PERTURBATIONS AFFECT LOWER EXTREMITY COORDINATION DYNAMICS

The purpose of this study was to investigate the coordinative changes that occur with a footwear perturbation consisting of a neutral shoe and varus and valgus wedged shoes. This type of footwear is often prescribed for clinical use. Lower extremity kinematics were collected as six male subjects ran overground at 3.6 m⋅s-1±5%. A modified vector coding technique assessed coordination between rearfoot motion and leg rotation. It was determined that there were clinically relevant differences between the footwear during the middle and late stance period. The differences were most evident between the varus and valgus conditions. However, the varus condition was closer in coordination structure to the neutral condition. The difference in coordination during the wedged conditions indicated that the valgus wedge perturbation may have implications in producing soft tissue injury

The Muslim Sports Tourist and the Olympic Games Movement

This paper explores the view that the Olympic Games Movement and the modern day Olympic Games while seeking to be a secular event is regarded by many participants and spectators as an event where they are able to demonstrate expressions of their religious faith. The first Olympic Games were staged in ancient Greece and they comprised a series of sporting events staged as a way of honouring the Greek God; Zeus, considered by the Romans to be a pagan god. The origins and mythology surrounding the early Olympic Games had a non-secular basis and led to an almost intrinsic belief that to attend the Olympic Games was a way of undertake a sporting pilgrimage to affirm one’s faith and to give praise to one’s God. The aim of this paper is to examine the relevant beliefs and values associated with one of the World’s fastest growing religions Islam, which has over 1.5 billion worshippers worldwide, and the challenges these beliefs and values may present to the Olympic Games Movement and future host cities. The context for this examination is that the Olympic Games Movement is seeking to engage and encourage greater numbers of Muslim athletes and Muslim sport tourists to become part of this global mega event and movement. Within this paper’s conceptual findings the influences that Islamic beliefs and values may have upon the modern day Olympic Games movement and future Olympic Games will be assessed. It will also be asserted that this influence is set to become more profound as the Olympic Games Movement seeks to attract greater numbers of Muslim athletes, especially Muslim females, and spectators both in person and via the global reach of modern day media. Additional growth factors examined relate to the staging of major and mega sporting events by emerging nations as part of their tourism destination development strategies. It is today being strongly suggested that prospects of a wealthy nation and city within the Muslim world hosting and staging a future Olympic Game are fast coming one step closer towards becoming a reality. Consequently the possible implications associated with this future occurrence requires acknowledgement and understanding

Instantaneous effects of mindfulness meditation on tennis return performance in elite junior athletes completing an implicitly sequenced serve return task

Single-session meditation augmentation of sport-specific skill performance was tested with elite junior tennis athletes. Athletes completed one of two styles of mindfulness meditation (focused-attention or open-monitoring) or a control listening condition prior to performing an implicitly sequenced tennis serve return task involving the goal of hitting a target area placed on the service court. Unbeknownst to athletes, six distinct serves followed a repeating second-order conditional sequence for two task blocks before the sequence was altered in a third transfer block. Task performance was operationalized as serve return outcome and analyzed using beta regression modeling. Models analyzed group by block differences in the proportion of returned serves (i.e., non-aces), returns placed in the service court, and target hits. Contrary to previous laboratory findings, results did not support meditation-related augmentation of performance and/or sequence learning. In fact, compared to control, meditation may have impaired performance improvements and acquisition of serve sequence information. It is possible that the effects of single-session meditation seen in laboratory research may not extend to more complex motor tasks, at least in highly-trained adolescents completing a well-learned skill. Further research is required to elucidate the participant, task, and meditation-related characteristics that might promote single-session meditation performance enhancement

Chain transfer to solvent in the radical polymerization of structurally diverse acrylamide monomers using straight-chain and branched alcohols as solvents

Chain transfer to solvent in conventional radical polymerizations of N-tert-butylacrylamide (TBAM) and N-(2-morpholin-4-ylethyl) acrylamide (MEA) in a range of alcohol solvents is investigated. Mayo analysis of polymerization of TBAM in linear alcohols (C-3-C-9) resulted in an approximately linear increase in chain transfer to solvent constant (C-tr,(S)) with the number of methylene (CH2) units in the solvent. The branched alcohol 3-methyl-3-pentanol gave the smallest C-tr,C-S (using Mayo analysis), and thus allowed attainment of higher molecular weights (MWs) in the nitroxide-mediated polymerizations (NMP) of TBAM. Overall, the data show that MEA is more prone to chain transfer to solvent than TBAM (higher C-tr,C-S), and further analysis of the conventional radical polymerization of MEA in 3-methyl-3-pentanol indicate chain transfer to monomer may also be occurring. The first controlled/ living polymerizations of MEA are detailed with chain transfer having a greater impact on maximum achievable MWs in NMP in comparison to TBAM