History of the Infinitely Small and the Infinitely Large in Calculus

The infinitely small and the infinitely large are essential in calculus. They have appeared throughout its history in various guises: infinitesimals, indivisibles, differentials, evanescent quantities, moments, infinitely large and infinitely small magnitudes, infinite sums, power series, limits, and hyperreal numbers. And they have been fundamental at both the technical and conceptual levels – as underlying tools of the subject and as its foundational underpinnings. We will consider examples of these aspects of the infinitely small and large as they unfolded in the history of calculus from the 17^th through the 20^th centuries. We will also present `didactic observations' at relevant places in the historical account. This revised version was published online in July 2006 with corrections to the Cover Date.     

Flavor preferences conditioned with starch in rats

Rats were trained to associate artificial cherry or grape flavors with 1% starch suspensions. Conditioning was assessed by offering the rats a choice of the cherry versus grape flavors without starch. Conditioned preferences were moderately strong and persistent; 3 days of conditioning produced a preference that did not fully extinguish within 18 days. Food deprivation substantially increased intake of 1% starch suspension, However, the degree of preference conditioned was not influenced by the availability of food during the conditioning period; the rats that had been food deprived during training acquired as strong a preference as did those fed freely during training. The degree of preference obtained was similar in the rats given reinforcing and nonreinforcing fluids simultaneously or sequentially. Starch conditioned a slightly stronger preference than did the same concentration of glucose, even though these substances contain the same amount of calories. A statistically significant, but weak, preference was conditioned by 0.5%, but not by 0.25%, starch. It is proposed that the flavor of starch, independent of its calories, is reinforcing to rats.     

Discriminative control of interspecific killing in hungry and satiated rats

Hungry and satiated killer Long-Evans rats were exposed to two species of prey, mice and frogs. Experiment 1 demonstrated that the rats learned to discriminate between prey when attacks upon one of the prey were punished by electric shocks and attacks upon the alternative were not punished. Thus, killing of the “dangerous” prey was suppressed, while killing of the alternative, “safe” prey continued. However, in Experiment 2—in which the consequences of killing the prey differed in that one was allowed to be eaten following a kill but the other was not—no evidence of discriminative attack resulted. Hungry and satiated rats did not differ in their responsiveness toward prey as objects of attack, but hungry killers were more responsive to prey as food, thus demonstrating a dissociation of killing and eating control.     

Carbohydrate-induced stimulation of saccharin intake: Yoked controls

Dilute, intragastric infusions of carbohydrate stimulate saccharin intake. To determine whether this effect is due to an associative process, a yoked control was employed. One group of rats was trained in the conventional fashion: they received carbohydrate (maltodextrin) infusions whenever they drank saccharin. Whenever these rats were infused with carbohydrate, yoked rats were infused with the same amount of carbohydrate. In an experiment in which saccharin was available continuously, carbohydrate infusions stimulated saccharin intake in conventionally trained rats but not in yoked rats. In a subsequent experiment, in which rats were offered saccharin for only 2.5 h/day, carbohydrate infusions stimulated intake in conventionally trained and in yoked rats. However, when these rats were switched to continuous saccharin availability, saccharin intake declined in yoked rats but remained elevated in conventionally trained rats. Thus, carbohydrate infusions stimulate saccharin intake via an associative process.     

Characteristics impacting on session rating of perceived exertion training load in Australian footballers

The relationship between external training load and session rating of perceived exertion (s-RPE) training load and the impact that playing experience, playing position and 2-km time-trial performance had on s-RPE training load were explored. From 39 Australian Football players, 6.9 ± 4.6 training sessions were analysed, resulting in 270 samples. Microtechnology devices provided external training load (distance, average speed, high-speed running distance, player load (PL) and player load_slow (PL_slow)). The external training load measures had moderate to very large associations (r, 95% CI) with s-RPE training load, average speed (0.45, 0.35–0.54), high-speed running distance (0.51, 0.42–0.59), PL_slow (0.80, 0.75–0.84), PL (0.86, 0.83–0.89) and distance (0.88, 0.85–0.90). Differences were described using effect sizes (d ±95% CL). When controlling for external training load, the 4- to 5-year players had higher s-RPE training load than the 0- to 1- (0.44 ± 0.33) and 2- to 3-year players (0.51 ± 0.30), ruckmen had moderately higher s-RPE training load than midfielders (0.82 ± 0.58), and there was a 0.2% increase in s-RPE training load per 1 s increase in time-trial (95% CI: 0.07–0.34). Experience, position and time-trial performance impacted the relationship between external training load and s-RPE training load. This suggests that a given external training load may result in different internal responses between athletes, potentially leaving individuals at risk of overtraining or failing to elicit positive adaptation. It is therefore vital that coaches and trainers give consideration to these mediators of s-RPE training load.     

Four social psychological lenses for developmental psychology

How can the advances of social and developmental psychology be integrated? This conceptual paper proposes to examine four basic theoretical models of social situations through which learning and development have been observed in the post-piagetian tradition: the psychosocial triangle, the frame, models of transfer and transitions, and models integrating the notion of cultural instrument. We show different dynamics highlighted by series of studies using these models. We finally discuss the notions of development and of the person emerging through such social developmental approaches.     

The effects of football match congestion in an international tournament on hip adductor squeeze strength and pain in elite youth players

The purpose of this study was to investigate the effect of a congested international tournament match schedule on adductor strength and pain in elite youth football players. Twenty-two male players (age: 15.53 ± 0.48 years, height: 174.87 ± 7.59 cm, weight: 67.45 ± 7.40 kg) were included. The 5-second adductor squeeze strength was captured daily using a hand-held dynamometer during a 7-game international tournament. Pain during the squeeze test was recorded using numerical pain ratings (0–10) to quantify groin pain. Sessional rate of perceived exertion (sRPE) was collected during the tournament. Adductor strength changed significantly during the tournament in relation to time (F(14,294.94) = 1.89, p = 0.027) and cumulative sRPE (F(1,314) = 5.59, p = 0.019). Cumulative sRPE displayed a negative relationship with strength (B = ?0.008, SE = 0.0032, 95%CI = ?0.014,-0.002). The results indicate that for every 100 match sRPE arbitrary units the squeeze peak force reduced by 0.8N. Sixteen (72.7%) players demonstrated clinically meaningful strength reductions (>15%) during the tournament. Match congestion impacts on hip adductor squeeze strength in male youth football players. A negative relationship between match sRPE and adductor strength exists. Player monitoring involving the 5-second adductor squeeze test can be captured effectively and is suitable to include as part of secondary injury prevention during or immediately after a congested tournament.