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

Mathematics in Medical Diagnostics - 2022 Proceedings of the 4th International Conference on Trauma Surgery Technology

The 4th event of the Giessen International Conference Series on Trauma Surgery Technology took place on April, the 23rd 2022 in Warsaw, Poland. It aims to bring together practical application research, with a focus on medical imaging, and the TDA experts from Warsaw. This publication contains details of our presentations and discussions

Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(\alpha_n, \beta_n)}$ is studied, assuming that $$\lim_{n\to\infty} \frac{\alpha_n}{n}=A, \qquad \lim_{n\to\infty} \frac{\beta _n}{n}=B,$$ with $A$ and $B$ satisfying $A > -1$, $B>-1$, $A+B < -1$. The asymptotic analysis is based on the non-Hermitian orthogonality of these polynomials, and uses the Deift/Zhou steepest descent analysis for matrix Riemann-Hilbert problems. As a corollary, asymptotic zero behavior is derived. We show that in a generic case the zeros distribute on the set of critical trajectories $\Gamma$ of a certain quadratic differential according to the equilibrium measure on $\Gamma$ in an external field. However, when either $\alpha_n$, $\beta_n$ or $\alpha_n+\beta_n$ are geometrically close to $\Z$, part of the zeros accumulate along a different trajectory of the same quadratic differential.Comment: 31 pages, 12 figures. Some references added. To appear in Journal D'Analyse Mathematiqu

Motor expertise modulates the unconscious processing of human body postures

Little is known about the cognitive background of unconscious visuomotor control of complex sports movements. Therefore, we investigated the extent to which novices and skilled high-jump athletes are able to identify visually presented body postures of the high jump unconsciously. We also asked whether or not the manner of processing differs (qualitatively or quantitatively) between these groups as a function of their motor expertise. A priming experiment with not consciously perceivable stimuli was designed to determine whether subliminal priming of movement phases (same vs. different movement phases) or temporal order (i.e. natural vs. reversed movement order) affects target processing. Participants had to decide which phase of the high jump (approach vs. flight phase) a target photograph was taken from. We found a main effect of temporal order for skilled athletes, that is, faster reaction times for prime-target pairs that reflected the natural movement order as opposed to the reversed movement order. This result indicates that temporal-order information pertaining to the domain of expertise plays a critical role in athletes’ perceptual capacities. For novices, data analyses revealed an interaction between temporal order and movement phases. That is, only the reversed movement order of flight-approach pictures increased processing time. Taken together, the results suggest that the structure of cognitive movement representation modulates unconscious processing of movement pictures and points to a functional role of motor representations in visual perception

The role of motor simulation in action perception: a neuropsychological case study

Research on embodied cognition stresses that bodily and motor processes constrain how we perceive others. Regarding action perception the most prominent hypothesis is that observed actions are matched to the observer’s own motor representations. Previous findings demonstrate that the motor laws that constrain one’s performance also constrain one’s perception of others’ actions. The present neuropsychological case study asked whether neurological impairments affect a person’s performance and action perception in the same way. The results showed that patient DS, who suffers from a frontal brain lesion, not only ignored target size when performing movements but also when asked to judge whether others can perform the same movements. In other words DS showed the same violation of Fitts’s law when performing and observing actions. These results further support the assumption of close perception action links and the assumption that these links recruit predictive mechanisms residing in the motor system

The Self in Social Interactions: Sensory Attenuation of Auditory Action Effects Is Stronger in Interactions with Others

Weiss C, Herwig A, Schuetz-Bosbach S. The Self in Social Interactions: Sensory Attenuation of Auditory Action Effects Is Stronger in Interactions with Others. PLoS ONE. 2011;6(7): e22723.The experience of oneself as an agent not only results from interactions with the inanimate environment, but often takes place in a social context. Interactions with other people have been suggested to play a key role in the construal of self-agency. Here, we investigated the influence of social interactions on sensory attenuation of action effects as a marker of pre-reflective self-agency. To this end, we compared the attenuation of the perceived loudness intensity of auditory action effects generated either by oneself or another person in either an individual, non-interactive or interactive action context. In line with previous research, the perceived loudness of self-generated sounds was attenuated compared to sounds generated by another person. Most importantly, this effect was strongly modulated by social interactions between self and other. Sensory attenuation of self-and other-generated sounds was increased in interactive as compared to the respective individual action contexts. This is the first experimental evidence suggesting that pre-reflective self-agency can extend to and is shaped by interactions between individuals

The impact of the Great Exhibition of 1851 on the development of technical education during the second half of the nineteenth century

This paper examines the contribution made by the mechanics’ institute movement in Britain just prior to, and following, the opening of the Great Exhibition of 1851 in London. It argues that far from making little contribution to education, as often portrayed by historians, the movement was ideally positioned to respond to the findings of the Exhibition, which were that foreign goods on display were often more advanced than those produced in Britain. The paper highlights, through a regional study, how well suited mechanics’ institutes were in organising their own exhibitions, providing the idea of this first international exhibition. Subsequently, many offered nationally recognised technical subject examinations through relevant education as well as informing government commissions, prior to the passing of the Technical Instruction Acts in 1889 and the Local Taxation Act of 1890. These acts effectively put mechanics’ institutes into state ownership as the first step in developing further education for all in Britai

The prediction of visual stimuli influences auditory loudness discrimination

The brain combines information from different senses to improve performance on perceptual tasks. For instance, auditory processing is enhanced by the mere fact that a visual input is processed simultaneously. However, the sensory processing of one modality is itself subject to diverse influences. Namely, perceptual processing depends on the degree to which a stimulus is predicted. The present study investigated the extent to which the influence of one processing pathway on another pathway depends on whether or not the stimulation in this pathway is predicted. We used an action–effect paradigm to vary the match between incoming and predicted visual stimulation. Participants triggered a bimodal stimulus composed of a Gabor and a tone. The Gabor was either congruent or incongruent compared to an action–effect association that participants learned in an acquisition phase.We tested the influence of action–effect congruency on the loudness perception of the tone. We observed that an incongruent–task-irrelevant Gabor stimulus increases participant’s sensitivity to loudness discrimination. An identical result was obtained for a second condition in which the visual stimulus was predicted by a cue instead of an action. Our results suggest that prediction error is a driving factor of the crossmodal interplay between vision and audition

Keeping an eye on the violinist: motor experts show superior timing consistency in a visual perception task

Common coding theory states that perception and action may reciprocally induce each other. Consequently, motor expertise should map onto perceptual consistency in specific tasks such as predicting the exact timing of a musical entry. To test this hypothesis, ten string musicians (motor experts), ten non-string musicians (visual experts), and ten non-musicians were asked to watch progressively occluded video recordings of a first violinist indicating entries to fellow members of a string quartet. Participants synchronised with the perceived timing of the musical entries. Results revealed significant effects of motor expertise on perception. Compared to visual experts and non-musicians, string players not only responded more accurately, but also with less timing variability. These findings provide evidence that motor experts’ consistency in movement execution—a key characteristic of expert motor performance—is mirrored in lower variability in perceptual judgements, indicating close links between action competence and perception