text:
Most animals, including reptiles, lower body temperature(Tb) under hypoxic conditions. Numerous physiological and behavioural traits significant to the regulation of Tb are altered by hypoxia in ways that suggest an orchestrated adjustment of Tb at a new and lower regulated level. We examined this matter in bearded dragons, Pogona vitticeps, a species of reptile that naturally exhibits open mouth gaping at high temperatures, presumably in order to promote evaporation and thus prevent or avoid further increases in Tb. The threshold for the onset of gaping (assessed as the temperature at which lizards spent 50% of their time gaping) was reduced from 36.9°C in normoxia to 35.5°C at 10% and 34.3°C at 6%O2. The overall magnitude or degree of gaping, measured qualitatively, was more pronounced at lower temperatures in hypoxia. Females consistently had lower gaping threshold temperatures than did males, and this difference was retained throughout exposure to hypoxia. In addition to gaping,evaporative water loss from the cloaca may also play a significant role in temperature regulation, since the ambient temperature at which cloacal discharge occurred was also reduced significantly in hypoxia. The results reported herein strongly support the view that hypoxia reduces temperature set-point in lizards and that such changes are coordinated by specific behavioural thermoeffectors that modulate evaporative water loss and thus facilitate a high potential for controlling or modifying Tb.

The bearded dragon, Pogona vitticeps (Ahl 1926), is an agamid lizard that naturally inhabits the inland regions of Australia. In the wild,it prefers temperatures of approximately 33°C(Melville and Schulte, 2001)and will adopt behavioural and physiological strategies to achieve this preferred body temperature. Lizards achieve these kinds of preferred temperatures in the wild through shuttling between warm and cold environments(Brattstrom, 1971), modifying heart rates during heat and cooling(Seebacher and Franklin,2001), altering peripheral blood flow(Grigg and Seebacher, 1999),changing cloacal evaporative water loss(DeNardo et al., 2004) and via ventilatory mechanisms, manifesting as gaping or panting responses (Heatwole et al.,1973). Numerous lizard species open the mouth widely when heated. The temperature at which this gaping occurs in some lizard species is often well above the preferred temperature, acting as a last-ditch attempt to survive nearly lethal temperatures (Veron and Heatwole, 1970; Webb et al., 1972). Many lizards, however, begin panting at temperatures at or very close to their preferred body temperature(Heatwole et al., 1973),suggesting that ventilatory heat loss through evaporation is an important route for fine-tuning body temperature (Tb) regulation in these species. Australian bearded dragons are lizards that employ the latter strategy, spending much of their time basking and gaping, rather than shuttling back and forth between warm and cool environments.

Numerous physical factors have been shown to alter the temperature at which lizards gape or pant. Panting thresholds have been shown to be altered by dehydration (Parmenter and Heatwole,1975), circadian rhythms (Chong et al., 1973) and the rate or source of heating(Heatwole et al., 1973). The direction of these changes reflects the constraints of other physiological processes on thermoregulation; to conserve water, lizards do not gape until higher temperatures, and, since daily metabolic demand decreases at night in many reptiles (Rismiller and Heldmaier,1987), a concomitant decline in the threshold for gaping occurs as the thermoregulatory set-point (Tset) is reduced along with the reduced metabolic requirements(Rismiller and Heldmaier,1982).

It is well established that low oxygen causes many animals, including reptiles, to lower Tb(Wood and Gonzales, 1996). Changes in preferred temperatures are often taken as proof of a change in temperature set-point in the brain, since preferred temperature selection is a behavioural and thus, ultimately, a neurophysiological phenomenon. Numerous lizards (e.g. genera Iguana, Dipsosaurus and Anolis) select temperatures in hypoxia approximately 10°C lower than normoxic preferred temperatures (Hicks and Wood,1985; Petersen et al.,2003). If the Tset is decreased in hypoxia,then the thresholds for any autonomic or behavioural response that results in a decline or an attempt to decrease Tb should be decreased. Since many lizards exhibit panting or gaping responses as ambient temperature rises (Crawford and Barber,1974; Crawford and Gatz,1974; Crawford et al.,1977; Heatwole et al.,1973; Pough and McFarland,1976), this is taken to imply that reptiles exhibit some degree of control of Tb, albeit manifesting as a behavioural response.

With respect to behavioural thresholds for thermoregulation, Dupréet al. (1986) showed that the evaporative cooling threshold is diminished in hypoxic lizards; however, no attempt to assess the magnitude or persistence of the gaping response was made. Lizards that gape or pant may do so intermittently. It would be fruitful to assess the entire gaping response across a wide range of temperatures in order to determine whether the response is sustained, is proportional to temperature and the severity of hypoxia and is dramatic enough to have an effect on Tb regulation.

Studying the control of Tb in lizards under hypoxic conditions allows for the exploration of the existence of thermal threshold responses and thus makes inferences regarding the presence of thermoregulatory set-points. The objectives of this study, however, were to examine whether the gaping behaviour was proportionately related to ambient temperature and whether the threshold temperature at which gaping occurred was proportionately lowered in hypoxia. Understanding these two questions will shed light on the mechanism of thermoregulation in reptiles and other vertebrates and will provide evidence for whether hypoxia elicits a reduction in a hitherto seldom-studied behavioural response in reptiles. For the purpose of this study, we examined the gaping response (i.e. simple mouth opening) rather than panting (i.e. an altered breathing pattern). It is plausible that bearded dragons actually adopt panting, where the breathing frequency rises and tidal volume declines; however, assessing tidal volume accurately is difficult to do in lizards that open their mouths. Since simply opening the mouth may achieve a similar result of eliciting evaporative water loss across the mucosa within the mouth and throat, it is not necessarily required that panting accompanies gaping behaviour.

### Animals

Fourteen (eight male and six female) inland bearded dragons, P. vitticeps, were used in this study. All were captive-bred individuals ranging in age from approximately 6 to 12 months, as well as one adult lizard(>5 years of age). The lizards ranged in mass from approximately 50 g to 220 g, with the adult weighing 350 g. The animals were fed a daily diet consisting of crickets and mealworms and were provided with chopped vegetables ad libitum. Tb in the home cages (using a non-contact infrared thermometer) and body masses were monitored once a week at 11.00 h. Lizards had access to a 100 W light bulb, which could allow their Tbs to rise as high as 39°C or as low as 28°C in the housing environment. Animal experimentation was approved by the Brock University Animal Care Committee (ACUC Protocol #031001).

### Experimental set-up

Throughout experimentation, an individual lizard was housed in a 24×24×40 cm clear acrylic box, which was blacked out on three sides to prevent distractions and to minimise reflections. Once inside the box, the lizard sat elevated 3.5 cm from the bottom on a perforated, clear,acrylic platform, which allowed gas to be pumped into the box from a tube below the platform and helped to collect urine and faeces. The box was located within an environmental chamber (Thermo Forma; Marietta, OH, USA) to enable changing temperatures between 30 and 40°C. A non-radiant heat source was chosen in order to simplify the exposure regime. Since lizards reacted to human presence, a small surveillance camera was affixed to the environmental chamber inside wall and aimed towards the acrylic box to allow for observation of the undisturbed lizard's behaviour on a video monitor outside the chamber. An infrared (IR) thermal imaging camera (Mikron 7515; Oakland, NJ, USA) was positioned on top of the box, looking down onto the lizard, to obtain body surface temperature data. The IR imager was hooked up to a computer outside the chamber to obtain computerised images. In order to allow for varying oxygen levels inside the box, the IR camera rested on a set of bellows',sealed to the box with weather stripping, making the box relatively air-tight. A small tube was inserted beneath the bellows into the box to allow for gas sampling during hypoxic conditions to verify oxygen levels of 21, 10 and 6%.

### Data collection

The animals were placed in the chamber to start experimentation in the morning, preferably before feeding. Since it has been shown that lizards have different gaping thresholds between night and day(Chong et al., 1973), we elected to perform the gaping measurements during the same time of the day,between 10.00 h and 16.00 h. Lizards were given time to acclimate to the new temperature, as well as for the box to reach hypoxic levels if necessary. This usually required 20–30 min.

### Assessing gaping behaviour

The lizard was observed for 15 min at each temperature of interest (30, 32,34, 36, 38, 40°C), and the time spent gaping was recorded together with the degree or type of opening of the lizard's mouth(Fig. 1). Type I represented a barely open mouth, Type II was a typical gape and Type III was a wide open mouth, usually accompanied by a head-back posture, with the tongue partially protruding and a puffing out of the throat. The observation periods were started once the lizard's dorsal surface temperature was within 0.5°C of the ambient temperature of interest. The only exceptions were the observation periods at 40°C (in normoxia and 10% oxygen) and 38°C (at 6% oxygen),when the animal was observed before it had reached ambient temperature. This was due to the long time required for the animal to reach the highest temperatures. The animals were observed sooner in order to prevent heat damage, since at this temperature all animals were already gaping maximally.

Fig. 1.

The three types of gaping categorised in this study. Type I was ascribed to situations when the mouth was visibly, but barely, open (i.e. there was no obvious sealing of the upper jaw with the lower jaw), which is readily distinguished from a normal closed mouth. Type II was ascribed to situations when the mouth was obviously open by more than a few millimetres. Type III was easily distinguished from other types as the lizard's tongue was easily visible and the throat was obviously distended due to the open mouth.

Fig. 1.

The three types of gaping categorised in this study. Type I was ascribed to situations when the mouth was visibly, but barely, open (i.e. there was no obvious sealing of the upper jaw with the lower jaw), which is readily distinguished from a normal closed mouth. Type II was ascribed to situations when the mouth was obviously open by more than a few millimetres. Type III was easily distinguished from other types as the lizard's tongue was easily visible and the throat was obviously distended due to the open mouth.

### Assessing animal surface temperatures

The camera software, MikroSpec RT Version 2.1394 (Oakland, NJ, USA), was set to capture an image of the thermal data every minute, up to a maximum of 500 frames. Image capturing was started upon the beginning of heating to 32°C (or 30°C at 6% oxygen). Images from the time during the 15 min observation period were analyzed, and the lizard surface temperatures recorded every minute. The mean temperature of a circular region of interest was recorded from the lizard's back, head (between the eyes), tip of the nose,tongue tip (if visible when gaping) and eye.

### Experimental design and test conditions

The animals were each tested under normoxic and two hypoxic conditions (10 and 6% O2), which was accomplished by mixing air with nitrogen to achieve the desired oxygen level. In all cases, gases entered the box at 5 l min–1. Since no previous data were known on the effects of hypoxia in P. vitticeps, initial observations were made of the animals at oxygen levels between 5 and 10%. An oxygen level of 6% was chosen as appropriate for this experiment, since this placed a significant enough stress on the animal while not seriously risking damage to the animal after the 6 h ofexposure required (i.e. lizards appeared distressed at levels below 6% O2 whereas at 6% O2 or above they remained calm throughout the procedures). Throughout the trials, the oxygen level was kept to within ±0.2% of the desired level. One animal was tested per day at each oxygen level. Two to three weeks passed before the performance of another experiment on the same individual at a new level of oxygen. At each level of oxygen, lizards were tested at 4–5 different ambient temperatures between 30 and 40°C (32, 34, 36, 38 and 40 for 21 and 10% O2and 30, 32, 34, 36 and 38°C for 6% O2). Lizards were exposed to these temperatures in a step-wise fashion, with the 2°C increments lasting either 1 h each or as long as it took to achieve skin surface temperature equilibration with ambient temperature. It usually took approximately 1 h for the lizard's body surface temperature to come into equilibrium with the environment. The highest temperature (40°C) was not used in the 6%O2 group out of concern for the lizard's survival. Previous studies have shown that lizards held at high temperatures under hypoxic conditions will die (Hicks and Wood,1985).

### Data analysis

All values reported are means ± s.e.m., unless otherwise stated. The percentage time spent gaping at the different ambient temperatures was initially analysed to determine a temperature at which 50% of the animal's time was spent gaping. This was done by fitting individual Hill equations to each animal using:
$\ P_{\mathrm{gape}}=100{\times}\{(T_{\mathrm{a}})^{N}{/}[(T_{\mathrm{a}})^{N}+(ET_{50})^{N}]\},$
(1)
where Pgape refers to the percentage of time spent gaping, Ta is the ambient temperature, ET50 is the effective temperature at which 50% of the animal's time was spent gaping,and the exponent, N, is Hill's constant (a larger Nindicates a steeper sigmoidal relationship between percentage gaping time and Ta). The parameters that minimised the root mean square error were determined using an iterative procedure facilitated by Microsoft™ Excel's solver tool.

Data (either time spent gaping or surface temperatures) were analysed using repeated-measures two-way ANOVA, with oxygen and ambient temperature as the two treatments. On occasions where normality was not met, log transformations were performed and the test repeated. In all cases, residuals from the individual ANOVAs were examined to verify a normal distribution. If residuals were non-normal, a non-parametric repeated-measures ANOVA of ranks was performed. We were able to examine the effects of gender once we had calculated the ET50s and N. On that occasion, we used two-way repeated-measures ANOVA to test for significant ET50 and N, with oxygen level and sex as treatments. In cases where ANOVAs yielded significant effects, post-hoc multiple comparisons were performed using the Holm-Sidak method. All statistics were considered significant at P<0.05.

### General observations in hypoxia

Lizards initially responded to the test chambers by turning around and searching their environments, settling down within 30 min, moving only occasionally thereafter. At the lower levels of oxygen, lizards would often close their eyes, appearing to fall asleep, though we did not make attempts to quantify this. Most lizards defecated or urinated in the test chamber during the course of observations (Fig. 2), even though they had been fasted for 16 h prior to experimentation. This discharge led to a dramatic and often prolonged decline in tail surface temperature (Fig. 2). Lizards (10/14=71.4% of the individuals) kept under normoxic conditions first discharged faeces and urine (cloacal discharge) at 36.1±0.6°C. At 10% O2, the cloacal discharge temperature(TCD; back surface temperature at which cloacal discharge first occurs) fell significantly (P=0.02) to 34.4±0.8°C in 71.4% of lizards, and at 6% O2 the TCD was also significantly lower (P=0.02) than normoxia at 33.2±0.4°C(Table 1) in 78.6% of individuals. There was no significant difference in the TCD between 10 and 6% O2. The lizards that did not exhibit cloacal discharge (21.4–28.6% of individuals) were not factored into the calculations for TCD. Of those lizards that exhibited cloacal discharge during exposure to temperatures between 30 and 40°C, 1.8±0.2 discharges occurred at 21% O2,1.5±0.3 discharges occurred at 10% O2, and 3.4±0.4 discharges occurred at 6% O2. The latter value was significantly higher than the values at 21% O2 (P<0.05; Table 1).

Fig. 2.

Infrared thermal images of lizards at different ambient temperatures showing Type II gaping (A), Type III gaping with the inside of the mouth clearly visible (B) and cloacal discharge in two different lizards (C, in hypoxia; D, in normoxia). In all cases, regional temperature differences can be observed across the body surfaces. Note the different temperature keys to the right of each image.

Fig. 2.

Infrared thermal images of lizards at different ambient temperatures showing Type II gaping (A), Type III gaping with the inside of the mouth clearly visible (B) and cloacal discharge in two different lizards (C, in hypoxia; D, in normoxia). In all cases, regional temperature differences can be observed across the body surfaces. Note the different temperature keys to the right of each image.

Table 1.

Estimates of ET50, Hill constant(N) and cloacal discharge temperature(TCD) from lizards breathing 21, 10 and 6%O2

ET50NTCD (°C)No. of cloacal discharges
21% O2 36.9±0.2 69.8±11.0 36.1±0.6 1.8±0.2 (71.4)
Male 37.2±0.3
Female 36.6±0.4
10% O2 35.5±0.4* 78.4±10.8 34.4±0.8* 1.5±0.3 (71.4)
Male 35.8±0.5
Female 35.0±0.6
6% O2 34.3±0.4* 113.4±15.6* 33.2±0.4* 3.4±0.4 (78.6)*
Male 34.8±0.4
Female 33.6±0.6
ET50NTCD (°C)No. of cloacal discharges
21% O2 36.9±0.2 69.8±11.0 36.1±0.6 1.8±0.2 (71.4)
Male 37.2±0.3
Female 36.6±0.4
10% O2 35.5±0.4* 78.4±10.8 34.4±0.8* 1.5±0.3 (71.4)
Male 35.8±0.5
Female 35.0±0.6
6% O2 34.3±0.4* 113.4±15.6* 33.2±0.4* 3.4±0.4 (78.6)*
Male 34.8±0.4
Female 33.6±0.6

Values are means ± s.e.m. Mean male and female values are shown for those parameters in which there was a significant difference between sexes using repeated-measures ANOVA, with oxygen level and sex as factors.

*

Significant difference (P<0.05) from 21% O2values.

Numbers in parentheses refer to the percentage of lizards exhibiting cloacal discharge.

### Gaping times in normoxia and hypoxia

Overall gaping time was significantly affected by inspired oxygen level(P<0.001) and by temperature (P<0.001; Fig. 3). In all cases, the total gaping time increased sigmoidally at the higher ambient temperatures;however, there was no significant interaction between oxygen and temperature(P=0.112). The mean ET50 values for 21%, 10% and 6% O2 were 36.9±0.2°C, 35.5±0.4°C, and 34.3±0.4°C, respectively (Table 1), and the effect of hypoxia (10 and 6% O2) was found to be significantly lower than normoxia. There was also a significant trend for the Hill constant, N, to be higher in hypoxia, although this effect was only significant at 6% O2 (P=0.05). Nwas 69.8±11.0 at 21% O2, 78.4±10.8 at 10%O2 and 113.4±15.6 at 6% O2(Table 1). The appropriateness of ET50 as an overall estimate of the threshold temperature is demonstrated by the good fit (r2=0.873, P<0.001) of the regression of ET50versus the total time spent gaping between 30 and 38°C (i.e. the experimental duration).

Fig. 3.

Mean times (± s.e.m.) spent gaping in lizards at three different levels of oxygen – 21% (filled circles),10% (open circles) and 6% O2 (filled triangles) – during 15 min periods of observations of lizards that were in thermal equilibrium with different environmental temperatures (30–40°C). The broken lines represent Hill equations (see equation 1) fitted through the mean data. The vertical dotted lines represent the mean time at which lizards spent 50% of their time gaping (ET50) at each level of oxygen inspired,as calculated from each lizard. The inset graph refers to the ET50 values for male (filled circles) and female lizards(open circles) at 21, 10 and 6% O2.

Fig. 3.

Mean times (± s.e.m.) spent gaping in lizards at three different levels of oxygen – 21% (filled circles),10% (open circles) and 6% O2 (filled triangles) – during 15 min periods of observations of lizards that were in thermal equilibrium with different environmental temperatures (30–40°C). The broken lines represent Hill equations (see equation 1) fitted through the mean data. The vertical dotted lines represent the mean time at which lizards spent 50% of their time gaping (ET50) at each level of oxygen inspired,as calculated from each lizard. The inset graph refers to the ET50 values for male (filled circles) and female lizards(open circles) at 21, 10 and 6% O2.

### Effect of sex on thermal preferences andET50

Using a two-way repeated-measures ANOVA to test ET50with oxygen and sex as the treatments, there was no significant interaction(P=0.49), although sex and oxygen level each had a significant effect(P=0.046 and P<0.001, respectively). Male lizards exhibited significantly higher ET50 values than females at all levels of oxygen tested (Table 1). Interestingly, this sex difference was also borne out in the mean preferred temperatures of lizards in their home cages during the 3-month period of experimentation; just as in the ET50 estimates,there was a significant effect of sex on home cage preferred temperature(P=0.02). Males exhibited a slightly, though significantly, higher Tb of 35.2±0.17 versus34.2±0.34°C in females.

### Gaping type in normoxia and hypoxia

There was a significant effect of temperature on the percentage time spent in Type I gaping (P=0.03), but no significant effect of oxygen(P=0.31) nor a significant interaction effect (P=0.18). At all three levels of oxygen tested, the percentage time spent in Type I gaping gradually increased with increasing temperature before falling back down to 0%at the highest temperatures (Fig. 4).

Fig. 4.

Mean times (± s.e.m.) spent engaged in (A) Type I, (B) Type II and (C) Type III gaping at 21, 10 and 6%O2. Data for 40°C are shown for comparison, although not included in statistical analysis, since data points were not available at 6%O2. refers to a significant difference between 10 and 21% O2, and * refers to a significant difference between 6 and 21% O2 with post-hoc tests.

Fig. 4.

Mean times (± s.e.m.) spent engaged in (A) Type I, (B) Type II and (C) Type III gaping at 21, 10 and 6%O2. Data for 40°C are shown for comparison, although not included in statistical analysis, since data points were not available at 6%O2. refers to a significant difference between 10 and 21% O2, and * refers to a significant difference between 6 and 21% O2 with post-hoc tests.

Type II gaping was significantly affected by inspired oxygen(P=0.02) and ambient temperature (P<0.001). Furthermore,there was a significant interaction between oxygen and temperature(P<0.001). Type II gaping predominated at lower temperatures at 6%O2. At all three levels of oxygen, Type II gaping initially increased with higher temperatures before decreasing at the highest temperatures (Fig. 4).

There was a significant effect of oxygen and temperature on the percentage time spent in Type III gaping (P<0.001 for oxygen and P<0.001 for temperature) and a significant interaction between oxygen and temperature (P<0.001). Qualitatively, Type III gaping was initiated earlier (i.e. at lower temperatures) in hypoxia than in normoxia(Fig. 4). Indeed, at temperatures above 36°C, significantly more time (>50%) was spent engaged in Type III gaping at 6% O2 than at 21% O2.

### Effect of body mass on gaping times

There was no significant effect of body mass on the ET50 estimates at 21, 10 or 6% O2, despite the large range of body masses examined (P=0.18, 0.29 and 0.45 and r2=0.14, 0.09 and 0.05, respectively, determined through linear regressions). Furthermore, body mass had no significant effect on N estimates for 21 and 10% O2 (P=0.26 and 0.76 and r2=0.10 and 0.008, respectively); however, Nestimates were significantly and negatively correlated with body mass in the lizards at 6% O2 (P=0.0005 and r2=0.66), demonstrating that small lizards exhibited a more rapid transition to continuous gaping as temperature increased.

Fig. 5.

Mean surface temperatures (± s.e.m.) exposed to (A) 21, (B) 10 and (C) 6%O2 during changes in ambient temperature ranging from 30 to 40°C. Shown are surface temperatures of the head, body, nose, eye and tongue (when visible). The dotted lines represent the line of equality for surface temperature and ambient temperature.

Fig. 5.

Mean surface temperatures (± s.e.m.) exposed to (A) 21, (B) 10 and (C) 6%O2 during changes in ambient temperature ranging from 30 to 40°C. Shown are surface temperatures of the head, body, nose, eye and tongue (when visible). The dotted lines represent the line of equality for surface temperature and ambient temperature.

### Effect of ambient temperature and oxygen on surface temperatures

Ambient temperature significantly affected all surface temperatures (body,head, nose, eye; P<0.001 for all values), except for tongue surface temperature, where there was a nearly significant difference(P=0.062). Oxygen level had no significant effect on any of the surface temperatures (except tongue) nor was there a significant interaction between oxygen and temperature for either body, head, nose or eye temperature(P=0.466, 0.10, 0.011 and 0.261, respectively; Fig. 5).

The only meaningful significant effects were on the tongue temperature. To simplify comparisons and account for some inter-individual responses, we also examined the body surface–tongue surface temperature difference. Oxygen level and ambient temperature had significant effects (P=0.011 and P<0.001, respectively) on the body surface–tongue surface temperatures, although there was not a significant interaction between oxygen and ambient temperature (P=0.92; Fig. 6). Overall, the body–tongue difference was greater at 6% O2 than at 21%O2, an effect that was most apparent at the lower ambient temperatures.

### Heating times in normoxia and hypoxia

Although not specifically controlled for, the times required to heat body surface temperature from 30 to 38°C over the course of the entire experiment were significantly affected by inspired oxygen (P=0.003). It took 225.6±12.7 min in normoxia to warm body surface temperature to 38°C, 249.4±9.2 min at 10% O2, and 283.1±9.7 min at 6% O2 (the latter value being significantly higher than the normoxic value with post-hoc comparisons).

Fig. 6.

Body surface temperature minus tongue temperature at 21, 10 and 6%O2 between ambient temperatures of 32–38°C. There was a significant effect of oxygen and ambient temperature on this difference(two-way repeated-measures ANOVA), suggesting that hypoxic conditions led to a higher value at lower ambient temperatures. The dotted line represents an extrapolation beyond 38°C in the 6% O2 group.

Fig. 6.

Body surface temperature minus tongue temperature at 21, 10 and 6%O2 between ambient temperatures of 32–38°C. There was a significant effect of oxygen and ambient temperature on this difference(two-way repeated-measures ANOVA), suggesting that hypoxic conditions led to a higher value at lower ambient temperatures. The dotted line represents an extrapolation beyond 38°C in the 6% O2 group.

The results of this study support the notion that the Tb set-point for gaping is reduced in proportion to the severity of hypoxia and that evaporative cooling in the bearded dragon seems critical to normal Tb regulation. That an apparently simple behavioural response such as gaping can be so elegantly controlled, in both magnitude and duration, speaks volumes for the importance of thermoregulation in this species. In terms of behavioural strategies for thermoregulation, an argument could be made for the importance of this response since the costs of gaping are relatively inexpensive, provided a lizard is hydrated. In general, animal behaviour is believed to exhibit optimality, such that the costs and benefits of a given behaviour balance out. Provided adequate moisture is available, gaping or panting can be an effective strategy that does not require the movement into or out of certain thermal environments. Increased movement in the wild may lead to increased visibility and thus increased risk of predation, so if subtle behavioural adjustments can modulate Tb adequately, then the less expensive behavioural strategies should prevail. Whether the altered gaping response to hypoxia would occur in those lizard species that only utilise gaping and panting as a last resort remains to be seen.

Fig. 7.

Various threshold parameters in bearded dragons as they pertain to the appropriateness of thermoregulation. (A) The plot of ET50versus cumulative time spent gaping during experimental procedures demonstrates that ET50 estimates provided a good fit across all three levels of O2. (B) Cloacal discharge threshold(TCD) in normoxia versus mean cage temperature(see Materials and methods) showed no significant relationships. (C) A significant relationship between ET50 in normoxia and mean cage temperature was not apparent, although a significant intercept did occur,suggesting that the ET50 tends to occur at temperatures higher than preferred cage temperatures. (D) There was a significant relationship between ET10 (the temperature at which the lizards were gaping for 10% of the time) and mean cage temperature, although no significant intercept, suggesting an isometric relationship between the two variables.

Fig. 7.

Various threshold parameters in bearded dragons as they pertain to the appropriateness of thermoregulation. (A) The plot of ET50versus cumulative time spent gaping during experimental procedures demonstrates that ET50 estimates provided a good fit across all three levels of O2. (B) Cloacal discharge threshold(TCD) in normoxia versus mean cage temperature(see Materials and methods) showed no significant relationships. (C) A significant relationship between ET50 in normoxia and mean cage temperature was not apparent, although a significant intercept did occur,suggesting that the ET50 tends to occur at temperatures higher than preferred cage temperatures. (D) There was a significant relationship between ET10 (the temperature at which the lizards were gaping for 10% of the time) and mean cage temperature, although no significant intercept, suggesting an isometric relationship between the two variables.

### Applicability of gaping thresholds to reptilian thermoregulation

It is possible that the wide range in body masses (50–350 g) used in this study led to some variability in the data. We found, however, no significant trends with body mass on the ET50 estimates,which is consistent with Heatwole et al.(1973), who found no effect of size on the panting threshold in a related species, Amphibolurus muricatus. Another possible concern is that we were only measuring surface temperatures of the lizards rather than core Tb. Since core Tb (which should be the regulated variable)will always lag behind peripheral temperatures during changes in ambient temperature, the percentage time estimates for gaping might actually be underestimates since core temperature could still have been rising during our assessments of gaping behaviour. Thus, the ET50 values may be slight overestimates (since ET50s are negatively correlated with total time spent gaping; see Fig. 7), although this effect should be consistent across all oxygen levels. The use of ambient temperature as the reference point rather than a particular surface temperature may have circumvented this problem. Furthermore, it could be argued that lizards utilise peripheral thermal information in a feed-forward fashion to predict changes in core Tb. If that were the case, the use of peripheral temperatures in our experiments might be more relevant in the assessment of gaping threshold.

During heating, the periphery would be the first to warm up and,considering the fact that ET50s are always higher than preferred Tb values in normoxia(Fig. 7C,D), this suggests that the majority of gaping occurs at or above the normal Tset,acting as an important contributor to Tb regulation when ambient temperatures exceed Tset. The fact that, in normoxia, a regression of normoxic ET10versusmean Tb in home cage yielded a significant regression suggests that the gaping thresholds determined here are meaningful values with respect to the individual lizard's normal preferred temperatures and that gaping is not simply a randomly displayed behavioural response.

One final concern was that at the highest temperatures of 38–40°C, lizard surfaces did not come fully into equilibrium with ambient temperature within the experimental time period. This suggests the efficacy with which bearded dragons can defend a lower Tbthrough increased evaporative cooling, since at 40°C lizards spent 100% of their time gaping. Thus, although the surface temperatures (and thus also Tb) were not in true equilibrium with ambient temperature,this would not have affected the ET50 estimates, since the maximum gaping efforts were already reached (i.e. values greater than 100%would not have been possible and thus the Hill equations would not have been affected).

### Hypoxia reduces ET50 through a proportional regulator

The main hypothesis tested in this study, that hypoxia would lower the gaping threshold in a proportional fashion, was supported by the data. The central hypothalamic thermostat is thought to operate as a proportional controller (Mrosovsky, 1990). In other words, the further the regulated variable deviates from a set-point,the larger will be the corrective response. In this case, as lizards are warmed up, they gape for progressively longer periods of time. This regulator responds, as well, to magnitude; lizards in hypoxia had a greater tendency to exhibit the more pronounced Type II and Type III gaping at lower ambient temperatures than in normoxia (Fig. 4).

One interesting result from this study is that the three levels of O2 yielded a proportionate reduction in the gaping threshold. Previous work examining the preferred Tb of hypoxic lizards seemed to indicate that a critical level of hypoxia lower than 10%O2 was required before a proportionate drop in Tb would occur (Hicks and Wood, 1985). There are no data on the preferred temperatures of P. vitticeps in hypoxia; however, it is possible that the gaping threshold responds slightly differently to low oxygen than set-point driven behaviour that requires activity or movement (i.e. preferred temperature). Strict or precise' thermoregulatory behaviour has been thought only to occur in reptiles when inhabiting low-cost' environments with low risk of predation(Huey and Slatkin, 1976). If the costs for thermoregulating (e.g. locomotory costs or risk of predation) are too high, then precise thermoregulation will not occur. In this context, gaping or panting can be viewed as a low-cost strategy for thermoregulation, inasmuch as its instantaneous costs are low. Thus, it might be expected that the threshold for gaping could more easily be sensitive and responsive to factors that alter set-point than a locomotory means of regulating Tb could.

### Regional differences in surface temperatures

During the present experiments, lizards were gradually warmed up from a temperature of 30°C (their surface temperature in the early morning at the beginning of experiments) to a final temperature of 38 or 40°C (depending on O2 level; see Materials and methods). The overall heating times between temperatures of 30–38°C were progressively longer in the hypoxic trials. This was probably due to their propensity to gape and exhibit cloacal discharge at lower thresholds, and hence the augmented evaporative cooling allowed for greater attempts to defend a lower core Tb at the higher ambient temperatures. Whether other thermoeffectors operate in a similar fashion remains to be shown. Despite the slightly different heating times, most surface temperatures were not affected by hypoxia, except for the tongue surface temperatures, where the changes that occurred strongly suggest that some internal temperatures (i.e. brain) may be differentially controlled in hypoxia.

Previously, Pough and McFarland(1976) showed that brain temperatures of lizards housed at temperatures greater than 40°C exhibit a substantial difference from the body, a response that was not observed in dead lizards held at similar temperatures. In extreme cases, lizard brain temperature can be up to 6°C lower than Tb(Crawford and Barber, 1974; Warburg, 1965). It is tempting to ascribe a physiological role (i.e. preferential blood flow that favours brain cooling at high ambient temperature) for this response, although one has yet to be shown. Since we were only examining surface temperatures, we cannot comment on brain temperature or its regulation. Previous work by Webb et al.(1972) showed significant body–head temperature differences in a wide variety of lizards. We did not see very large differences in the present study, although this could be because surface temperatures are not as informative as internal temperatures. Interestingly, a dragon lizard in the Webb et al.(1972) study exhibited panting at high temperatures (usually greater than 40°C), compared with P. vitticeps in the present study. This could have been related to the rapidly induced thermal changes, which is quite opposite to the present study. Our study was looking at steady-state changes, where lizards had time to equilibrate with their environmental temperatures, and, as such, we notice lizards gaping at much lower temperatures.

We used body–tongue temperature differences to estimate the degree of evaporative cooling. The difference between body surface temperature and tongue temperature increases at higher ambient temperatures, suggesting a greater degree of evaporative cooling at higher temperatures. It is also apparent that the continuous gaping that occurred in hypoxia, combined with a presumably higher ventilatory rate (i.e. an hypoxic ventilatory response), led to a higher degree of evaporative cooling in the hypoxic lizards across all the test temperatures. This is not surprising given that preservation of brain function would be more critically challenged under hypoxic conditions. Small changes in brain temperatures have been shown to produce large changes in brain damage in either hypothermic or febrile animals(Herrmann et al., 2003; Kataoka and Yanase, 1998; Katz et al., 2004; Trescher et al., 1997); higher temperatures exacerbate excitotoxic damage, and lower temperatures lead to less damage.

### Qualitative aspects of the gaping response

It is possible that using a non-radiant means for changing Tb is not the most appropriate for lizards that tend to bask in the sun. Indeed, Heatwole (1973) showed that thresholds for gaping were less variable in radiantly heated individuals than in non-radiantly heated individuals, even though gaping thresholds were not affected by the source of heat. Variability, however, did not appear to be a problem in our study due to the paired, repeated nature of the experimental design and the fact that we assessed percentage time spent gaping to estimate the ET50 threshold (Fig. 2). Previous studies have used the first attempt at gaping as the gaping threshold (Chong et al.,1973; Dupré et al.,1986; Heatwole et al.,1973; Parmenter and Heatwole,1975). It is possible that assessing a simple value such as the initial gape has much more variability inherent in the estimate, particularly if the response is truly sigmoidal, as we have demonstrated. At the lower temperatures, lizards spend so little time gaping that there is little effect of increasing temperature on the time spent gaping. Furthermore, the magnitude of gaping is much less at lower temperatures(Fig. 4), begging the question of how much evaporative cooling occurs when lizards are using Type I gaping. Using an estimate such as the ET50 incorporates the entire response over a range of temperatures and allows for an objective estimate of the animal's overall gaping response to increasing temperature.

There was, however, a significant negative correlation between Nand body size at 6% O2, suggesting that the smaller lizards exhibited a more typically on–off approach (i.e. a steep sigmoidal relationship between gaping time and ambient temperature) to gaping in severe hypoxia rather than a graded transition into increasingly more time spent gaping. This might reflect the fact that, as the smaller-sized lizards were undergoing more rapid changes in Tb, they responded with a more dramatic increase in time spent gaping as their Tbwas raised above Tset and evaporative cooling mechanisms were required.

### Effect of sex on gaping threshold

Few attempts have been made to examine the role of sex on normal thermoregulatory behaviours in reptiles outside of the breeding season (see Lailvaux et al., 2003). Even with pregnancy, there are inconsistent results regarding whether males and females exhibit consistently different preferred Tb. In one previous study, females had higher mean 24 h Tb than did males (Sievert and Hutchison,1989), although the reverse or lack of difference occurs just as often, with field data often contradicting laboratory thermal gradient experiments (Lailvaux et al.,2003). Interestingly, we have shown that sex had a significant effect on the gaping threshold (Table 1).Females consistently initiated gaping at a lower temperature than males, even during hypoxia. Previously, Heatwole et al.(1973) had shown that female Jacky dragons had a tendency toward lower (although not significant) gaping thresholds, a result consistent with the present study.

To the best of our knowledge, this is the first example of a sex difference related to a physiological response to hypoxia in a reptile. If a lowered set-point in hypoxia is truly adaptive, then it could be argued that all lizards, regardless of sex, should lower Tb and gaping threshold to an equal extent or as far as possible. The fact that the difference between males and females is retained in hypoxia suggests that sex differences in Tb regulation are of overriding importance. A corollary of the above is that females will presumably have greater tolerance to hypoxia if their lower gaping thresholds also translate into a lower overall Tb regulation. Wood and Stabenau(1998) showed that female rats exhibit a lower Tb in hypoxia, which helped translate into a greater tolerance to hypoxia. Survival times in hypoxic female rats were also significantly longer than in male rats. Whether the same sensitivities occur in reptiles is unknown.

### Concluding remarks

It is apparent from the present study that bearded dragons make use of a subtle behavioural response to effect changes in Tb. As an ectotherm, they may not be able to artificially augment (for exceptions, see Tattersall et al., 2004) or decrease their overall Tb that far from ambient temperature; however, they can use behavioural responses to serve as brakes on thermal changes. The fact that gaping is proportionately controlled in both duration and magnitude with respect to both ambient temperature and oxygen strongly supports the importance of precise thermoregulatory control in as much as other physiological and behavioural constraints will allow. It remains to be shown exactly how effective gaping is at controlling Tb for prolonged periods, how often this behaviour occurs in the wild and whether it is routinely used in response to other stressors in addition to hypoxia. Further information on the altered gaping or panting thresholds in those species of lizards that do not routinely use panting as alow-cost' component to thermoregulation could be a fruitful avenue of future research.

We would like to thank Bill Mears for providing us with the bearded dragons, and acknowledge the excellent animal care provided by Dayle Belme,Nicole Handley, and Sara Stewart from Brock University. This project was funded by an NSERC operating grant to Glenn Tattersall and by an NSERC undergraduate research assistantship (USRA) to Rebecca Gerlach.

Brattstrom, B. H. (
1971
). Social and thermoregulatory behavior of the bearded dragon, Amphibolurus barbatus.
Copeia
3
,
484
-497.
Chong, G., Heatwole, H. and Firth, B. T.(
1973
). Panting thresholds of lizards-II. Diel variation in panting threshold of Amphibolurus muricatus.
Comp. Biochem. Physiol.
46
,
827
-829.
Crawford, E. C. and Barber, B. J. (
1974
). Effects of core, skin, and brain temperature on panting in lizard Sauromalus obesus.
Am. J. Physiol.
226
,
569
-573.
Crawford, E. C. and Gatz, R. N. (
1974
). Respiratory alkalosis in a panting lizard (Sauromalus obesus).
Experientia
30
,
638
-639.
Crawford, E. C., Palomeque, J. and Barber, B. J.(
1977
). Physiological basis for head-body temperature differences in a panting lizard.
Comp. Biochem. Physiol. A
56
,
161
-163.
DeNardo, D. F., Zubal, T. E. and Hoffman, T. C. M.(
2004
). Cloacal evaporative cooling: a previously undescribed means of increasing evaporative water loss at higher temperatures in a desert ectotherm, the Gila monster Heloderma suspectum.
J. Exp. Biol.
207
,
945
-953.
Dupré, R. K., Hicks, J. W. and Wood, S. C.(
1986
). The effect of hypoxia on evaporative cooling thresholds of lizards.
J. Therm. Biol.
11
,
223
-227.
Grigg, G. C. and Seebacher, F. (
1999
). Field test of a paradigm: hysteresis of heart rate in thermoregulation by a free-ranging lizard (Pogona barbata).
Proc. R. Soc. London B Biol. Sci.
266
,
1291
-1297.
Heatwole, H., Firth, B. T. and Webb, G. J. W.(
1973
). Panting thresholds of lizards-I. Some methodological and internal influences on panting threshold of an Agamid, Amphibolurus muricatus.
Comp. Biochem. Physiol.
46
,
799
-826.
Herrmann, O., Tarabin, V., Suzuki, S., Attigah, N., Coserea, I.,Schneider, A., Vogel, J., Prinz, S., Schwab, S., Monyer, H. et al.(
2003
). Regulation of body temperature and neuroprotection by endogenous interleukin-6 in cerebral ischemia.
J. Cereb. Blood Flow Metab.
23
,
406
-415.
Hicks, J. W. and Wood, S. C. (
1985
). Temperature regulation in lizards: effects of hypoxia.
Am. J. Physiol.
248
,
R595
-R600.
Kataoka, K. and Yanase, H. (
1998
). Mild hypothermia – a revived countermeasure against ischemic neuronal damages.
Neurosci. Res.
32
,
103
-117.
Katz, L. M., Young, A. S., Frank, J. E., Wang, Y. F. and Park,K. (
2004
). Regulated hypothermia reduces brain oxidative stress after hypoxic-ischemia.
Brain Res.
1017
,
85
-91.
Lailvaux, S. P., Alexander, G. J. and Whiting, M. J.(
2003
). Sex-based differences and similarities in locomotor performance, thermal preferences, and escape behaviour in the lizard Platysaurus intermedius wilhelmi.
Physiol. Biochem. Zool.
76
,
511
-521.
Melville, J. and Schulte, J. A. (
2001
). Correlates of active body temperatures and microhabitat occupation in nine species of central Australian agamid lizards.
Austral Ecol.
26
,
660
-669.
Mrosovsky, N. (
1990
).
Rheostasis:The Physiology of Change
. New York: Oxford University Press.
Parmenter, C. J. and Heatwole, H. (
1975
). Panting thresholds of lizards.4. Effect of dehydration on panting threshold of Amphibolurus barbatus and Amphibolurus muricatus.
J. Exp. Zool.
191
,
327
-332.
Petersen, A. M., Gleeson, T. T. and Scholnick, D. A.(
2003
). The effect of oxygen and adenosine on lizard thermoregulation.
Physiol. Biochem. Zool.
76
,
339
-347.
Pough, F. H. and McFarland, W. N. (
1976
). Physical basis for head–body temperature differences in reptiles.
Comp. Biochem. Physiol. A
53
,
301
-303.
Rismiller, P. D. and Heldmaier, G. (
1982
). The effect of photoperiod on temperature selection in the European green lizard, Lacerta viridis.
Oecologia
53
,
222
-226.
Rismiller, P. D. and Heldmaier, G. (
1987
). Seasonal changes in daily metabolic patterns of Lacerta viridis.
J. Comp. Physiol.
161
,
482
-488.
Seebacher, F. and Franklin, C. E. (
2001
). Control of heart rate during thermoregulation in the heliothermic lizard Pogona barbata: importance of cholinergic and adrenergic mechanisms.
J. Exp. Biol.
204
,
4361
-4366.
Sievert, L. M. and Hutchison, V. H. (
1989
). Influences of season, time of day, light and sex on the thermoregulatory behaviour of Crotphytus collaris.
J. Therm. Biol.
14
,
159
-165.
Tattersall, G. J., Milsom, W. K., Abe, A. S., Brito, S. P. and Andrade, D. V. (
2004
). The thermogenesis of digestion in rattlesnakes.
J. Exp. Biol.
207
,
579
-585.
Trescher, W. H., Ishiwa, S. and Johnston, M. V.(
1997
). Brief post-hypoxic-ischemic hypothermia markedly delays neonatal brain injury.
Brain Dev.
19
,
326
-338.
Veron, J. and Heatwole, H. (
1970
). Temperature relations of the water skink, Sphenomorphys quoyi.
J. Herpetol.
4
,
141
-153.
Warburg, M. R. (
1965
). The influence of ambient temperature and humidity on the body temperature and water loss from two australian lizards, Tiliqua rugosa (Gray) (Scincidae) and Amphibolarus barbatus Cuvier (Agamidae).
Aust. J. Zool.
13
,
331
-350.
Webb, G. J. W., Firth, B. T. and Johnson, C. R.(
1972
). Head–body temperature differences in lizards.
Physiol. Zool.
45
,
130
-142.
Wood, S. C. and Gonzales, R. (
1996
). Hypothermia in hypoxic animals: mechanisms, mediators, and functional significance.
Comp. Biochem. Physiol. B
113
,
37
-43.
Wood, S. C. and Stabenau, E. K. (
1998
). Effect of gender on thermoregulation and survival of hypoxic rats.
Clin. Exp. Pharmacol. P
25
,
155
-158.


text:
## Lie algebra and differential equations

I am having trouble understanding a section in these notes: . It is on page 3. Section 3 -- Discretization of the Korteweg-de Vries equation. I don't understand why $$V_4=x∂_x+3t∂_t-2u∂_u$$ generates a symmetry group of the KdV. I see that it generates the transformation
$$(x',t',u')= (x\exp(\epsilon), 3t\exp(\epsilon), -2u\exp(\epsilon))$$
So $$u'_{t'}-6u'u'_{x'}+u'_{x'x'x'}=-{2\over 3}u_t-24\exp(\epsilon)uu_x-2\exp(-2\epsilon)u_{xxx}$$ How does this vanish (so that we get symmetry) given that $u$ satisfies the KdV?
 PhysOrg.com science news on PhysOrg.com >> Intel's Haswell to extend battery life, set for Taipei launch>> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens

 Tags diff equation, lie algebra

format: plain


text:
# Using (Euclidean) AD in Manopt.jl

Since Manifolds.jl 0.7, the support of automatic differentiation support has been extended.

This tutorial explains how to use Euclidean tools to derive a gradient for a real-valued function $f\colon \mathcal M → ℝ$. We will consider two methods: an intrinsic variant and a variant employing the embedding. These gradients can then be used within any gradient based optimization algorithm in Manopt.jl.

While by default we use FiniteDifferences.jl, you can also use FiniteDiff.jl, ForwardDiff.jl, ReverseDiff.jl, or Zygote.jl.

In this Notebook we will take a look at a few possibilities to approximate or derive the gradient of a function $f:\mathcal M \to ℝ$ on a Riemannian manifold, without computing it yourself. There are mainly two different philosophies:

1. Working instrinsically, i.e. staying on the manifold and in the tangent spaces. Here, we will consider approximating the gradient by forward differences.

2. Working in an embedding – there we can use all tools from functions on Euclidean spaces – finite differences or automatic differenciation – and then compute the corresponding Riemannian gradient from there.

## Setup

If you open this notebook in Pluto locally it switches between two modes. If the tutorial is within the Manopt.jl repository, this notebook tries to use the local package in development mode. Otherwise, the file uses the Pluto pacakge management version. In this case, the includsion of images might be broken. unless you create a subfolder optimize and activate asy-rendering.

Since the loading is a little complicated, we show, which versions of packages were installed in the following.

with_terminal() do
Pkg.status()
end
�[32m�[1mStatus�[22m�[39m /private/var/folders/_v/wg192lpd3mb1lp55zz7drpcw0000gn/T/jl_hpuuJD/Project.toml
�[90m [26cc04aa] �[39mFiniteDifferences v0.12.26
�[90m [af67fdf4] �[39mManifoldDiff v0.2.1
�[90m [0fc0a36d] �[39mManopt v0.4.0 ~/Repositories/Julia/Manopt.jl
�[90m [7f904dfe] �[39mPlutoUI v0.7.49
�[90m [37e2e46d] �[39mLinearAlgebra
�[90m [44cfe95a] �[39mPkg v1.8.0
�[90m [9a3f8284] �[39mRandom

## 1. (Intrinsic) Forward Differences

A first idea is to generalize (multivariate) finite differences to Riemannian manifolds. Let $X_1,\ldots,X_d ∈ T_p\mathcal M$ denote an orthonormal basis of the tangent space $T_p\mathcal M$ at the point $p∈\mathcal M$ on the Riemannian manifold.

We can generalize the notion of a directional derivative, i.e. for the “direction” $Y∈T_p\mathcal M$. Let $c\colon [-ε,ε]$, $ε>0$, be a curve with $c(0) = p$, $\dot c(0) = Y$, e.g. $c(t)= \exp_p(tY)$. We obtain

$$Df(p)[Y] = \left. \frac{d}{dt} \right|_{t=0} f(c(t)) = \lim_{t \to 0} \frac{1}{t}(f(\exp_p(tY))-f(p))$$

We can approximate $Df(p)[X]$ by a finite difference scheme for an $h>0$ as

$$DF(p)[Y] ≈ G_h(Y) := \frac{1}{h}(f(\exp_p(hY))-f(p))$$

Furthermore the gradient $\operatorname{grad}f$ is the Riesz representer of the differential, ie.

$$Df(p)[Y] = g_p(\operatorname{grad}f(p), Y),\qquad \text{ for all } Y ∈ T_p\mathcal M$$

and since it is a tangent vector, we can write it in terms of a basis as

$$\operatorname{grad}f(p) = \sum_{i=1}^{d} g_p(\operatorname{grad}f(p),X_i)X_i = \sum_{i=1}^{d} Df(p)[X_i]X_i$$

and perform the approximation from above to obtain

$$\operatorname{grad}f(p) ≈ \sum_{i=1}^{d} G_h(X_i)X_i$$

for some suitable step size $h$. This comes at the cost of $d+1$ function evaluations and $d$ exponential maps.

This is the first variant we can use. An advantage is that it is intrinsic in the sense that it does not require any embedding of the manifold.

### An Example: The Rayleigh Quotient

The Rayleigh quotient is concerned with finding eigenvalues (and eigenvectors) of a symmetric matrix $A\in ℝ^{(n+1)×(n+1)}$. The optimization problem reads

$$F\colon ℝ^{n+1} \to ℝ,\quad F(\mathbf x) = \frac{\mathbf x^\mathrm{T}A\mathbf x}{\mathbf x^\mathrm{T}\mathbf x}$$

Minimizing this function yields the smallest eigenvalue $\lambda_1$ as a value and the corresponding minimizer $\mathbf x^*$ is a corresponding eigenvector.

Since the length of an eigenvector is irrelevant, there is an ambiguity in the cost function. It can be better phrased on the sphere $𝕊^n$ of unit vectors in $\mathbb R^{n+1}$, i.e.

$$\operatorname*{arg\,min}_{p \in 𝕊^n}\ f(p) = \operatorname*{arg\,min}_{\ p \in 𝕊^n} p^\mathrm{T}Ap$$

We can compute the Riemannian gradient exactly as

$$\operatorname{grad} f(p) = 2(Ap - pp^\mathrm{T}Ap)$$

so we can compare it to the approximation by finite differences.

begin
Random.seed!(42)
n = 200
A = randn(n + 1, n + 1)
A = Symmetric(A)
M = Sphere(n)
nothing
end
f1(p) = p' * A'p
f1 (generic function with 1 method)
gradf1(p) = 2 * (A * p - p * p' * A * p)
gradf1 (generic function with 1 method)

Manifolds provides a finite difference scheme in tangent spaces, that you can introduce to use an existing framework (if the wrapper is implemented) form Euclidean space. Here we use FiniteDiff.jl.

r_backend = ManifoldDiff.TangentDiffBackend(ManifoldDiff.FiniteDifferencesBackend())
ManifoldDiff.TangentDiffBackend{ManifoldDiff.FiniteDifferencesBackend{FiniteDifferences.AdaptedFiniteDifferenceMethod{5, 1, FiniteDifferences.UnadaptedFiniteDifferenceMethod{7, 5}}}, ExponentialRetraction, LogarithmicInverseRetraction, DefaultOrthonormalBasis{ℝ, ManifoldsBase.TangentSpaceType}, DefaultOrthonormalBasis{ℝ, ManifoldsBase.TangentSpaceType}}(ManifoldDiff.FiniteDifferencesBackend{FiniteDifferences.AdaptedFiniteDifferenceMethod{5, 1, FiniteDifferences.UnadaptedFiniteDifferenceMethod{7, 5}}}(FiniteDifferences.AdaptedFiniteDifferenceMethod{5, 1, FiniteDifferences.UnadaptedFiniteDifferenceMethod{7, 5}}([-2, -1, 0, 1, 2], [0.08333333333333333, -0.6666666666666666, 0.0, 0.6666666666666666, -0.08333333333333333], ([-0.08333333333333333, 0.5, -1.5, 0.8333333333333334, 0.25], [0.08333333333333333, -0.6666666666666666, 0.0, 0.6666666666666666, -0.08333333333333333], [-0.25, -0.8333333333333334, 1.5, -0.5, 0.08333333333333333]), 10.0, 1.0, Inf, 0.05555555555555555, 1.4999999999999998, FiniteDifferences.UnadaptedFiniteDifferenceMethod{7, 5}([-3, -2, -1, 0, 1, 2, 3], [-0.5, 2.0, -2.5, 0.0, 2.5, -2.0, 0.5], ([0.5, -4.0, 12.5, -20.0, 17.5, -8.0, 1.5], [-0.5, 2.0, -2.5, 0.0, 2.5, -2.0, 0.5], [-1.5, 8.0, -17.5, 20.0, -12.5, 4.0, -0.5]), 10.0, 1.0, Inf, 0.5365079365079365, 10.0))), ExponentialRetraction(), LogarithmicInverseRetraction(), DefaultOrthonormalBasis(ℝ), DefaultOrthonormalBasis(ℝ))
gradf1_FD(p) = ManifoldDiff.gradient(M, f1, p, r_backend)
gradf1_FD (generic function with 1 method)
begin
p = zeros(n + 1)
p[1] = 1.0
norm(M, p, X1 - X2)
end
1.0183547605045089e-12

We obtain quite a good approximation of the gradient.

## 2. Conversion of a Euclidean Gradient in the Embedding to a Riemannian Gradient of a (not Necessarily Isometrically) Embedded Manifold

Let $\tilde f\colon\mathbb R^m \to \mathbb R$ be a function on the embedding of an $n$-dimensional manifold $\mathcal M \subset \mathbb R^m$ and let $f\colon \mathcal M \to \mathbb R$ denote the restriction of $\tilde f$ to the manifold $\mathcal M$.

Since we can use the pushforward of the embedding to also embed the tangent space $T_p\mathcal M$, $p\in \mathcal M$, we can similarly obtain the differential $Df(p)\colon T_p\mathcal M \to \mathbb R$ by restricting the differential $D\tilde f(p)$ to the tangent space.

If both $T_p\mathcal M$ and $T_p\mathbb R^m$ have the same inner product, or in other words the manifold is isometrically embedded in $\mathbb R^m$ (like for example the sphere $\mathbb S^n\subset\mathbb R^{m+1}$), then this restriction of the differential directly translates to a projection of the gradient, i.e.

$$\operatorname{grad}f(p) = \operatorname{Proj}_{T_p\mathcal M}(\operatorname{grad} \tilde f(p))$$

More generally we might have to take a change of the metric into account, i.e.

$$\langle \operatorname{Proj}_{T_p\mathcal M}(\operatorname{grad} \tilde f(p)), X \rangle = Df(p)[X] = g_p(\operatorname{grad}f(p), X)$$

or in words: we have to change the Riesz representer of the (restricted/projected) differential of $f$ ($\tilde f$) to the one with respect to the Riemannian metric. This is done using change_representer.

### A Continued Example

We continue with the Rayleigh Quotient from before, now just starting with the defintion of the Euclidean case in the embedding, the function $F$.

F(x) = x' * A * x / (x' * x);

The cost function is the same by restriction

f2(M, p) = F(p);

function grad_f2_AD(M, p)
M, F, p, Manifolds.RiemannianProjectionBackend(ManifoldDiff.FiniteDifferencesBackend())
)
end
grad_f2_AD (generic function with 1 method)
X3 = grad_f2_AD(M, p)
201-element Vector{Float64}:
0.0
0.14038510230588766
1.2081562751116681
-1.8650092022183193
-1.3296034534632897
0.8233596435551424
0.013307949254910802
⋮
-1.0895103349140873
2.423453243905215
2.349106449107357
0.5799736335284804
-0.07423232665910076
2.0859147085739465
norm(M, p, X1 - X3)
1.7211299825309384e-12

### An Example for a Nonisometrically Embedded Manifold

on the manifold $\mathcal P(3)$ of symmetric positive definite matrices.

The following function computes (half) the distance squared (with respect to the linear affine metric) on the manifold $\mathcal P(3)$ to the identity, i.e. $I_3$. Denoting the unit matrix we consider the function

$$G(q) = \frac{1}{2}d^2_{\mathcal P(3)}(q,I_3) = \lVert \operatorname{Log}(q) \rVert_F^2,$$

where $\operatorname{Log}$ denotes the matrix logarithm and $\lVert \cdot \rVert_F$ is the Frobenius norm. This can be computed for symmetric positive definite matrices by summing the squares of the logarithms of the eigenvalues of $q$ and dividing by two:

G(q) = sum(log.(eigvals(Symmetric(q))) .^ 2) / 2
G (generic function with 1 method)

We can also interpret this as a function on the space of matrices and apply the Euclidean finite differences machinery; in this way we can easily derive the Euclidean gradient. But when computing the Riemannian gradient, we have to change the representer (see again change_representer) after projecting onto the tangent space $T_p\mathcal P(n)$ at $p$.

Let's first define a point and the manifold $N=\mathcal P(3)$.

rotM(α) = [1.0 0.0 0.0; 0.0 cos(α) sin(α); 0.0 -sin(α) cos(α)]
rotM (generic function with 1 method)
q = rotM(π / 6) * [1.0 0.0 0.0; 0.0 2.0 0.0; 0.0 0.0 3.0] * transpose(rotM(π / 6))
3×3 Matrix{Float64}:
1.0  0.0       0.0
0.0  2.25      0.433013
0.0  0.433013  2.75
N = SymmetricPositiveDefinite(3)
SymmetricPositiveDefinite(3)
is_point(N, q)
true

We could first just compute the gradient using FiniteDifferences.jl, but this yields the Euclidean gradient:

FiniteDifferences.grad(central_fdm(5, 1), G, q)
([3.240417492806275e-14 1.1767558621318606e-14 2.075425785233536e-14; 0.0 0.35148121676557886 0.017000516835447246; 0.0 0.0 0.3612964697372195],)

Instead, we use the RiemannianProjectedBackend of Manifolds.jl, which in this case internally uses FiniteDifferences.jl to compute a Euclidean gradient but then uses the conversion explained above to derive the Riemannian gradient.

We define this here again as a function grad_G_FD that could be used in the Manopt.jl framework within a gradient based optimization.

function grad_G_FD(N, q)
N, G, q, ManifoldDiff.RiemannianProjectionBackend(ManifoldDiff.FiniteDifferencesBackend())
)
end
grad_G_FD (generic function with 1 method)
G1 = grad_G_FD(N, q)
3×3 Matrix{Float64}:
3.24042e-14  1.77319e-14  3.10849e-14
1.77319e-14  1.86368      0.826856
3.10849e-14  0.826856     2.81845

Now, we can again compare this to the (known) solution of the gradient, namely the gradient of (half of) the distance squared, i.e. $G(q) = \frac{1}{2}d^2_{\mathcal P(3)}(q,I_3)$ is given by $\operatorname{grad} G(q) = -\operatorname{log}_q I_3$, where $\operatorname{log}$ is the logarithmic map on the manifold.

G2 = -log(N, q, Matrix{Float64}(I, 3, 3))
3×3 Matrix{Float64}:
-0.0  -0.0       -0.0
-0.0   1.86368    0.826856
-0.0   0.826856   2.81845

Both terms agree up to $1.8×10^{-12}$:

norm(G1 - G2)
1.4601907912994879e-12
isapprox(M, q, G1, G2; atol=2 * 1e-12)
true

## Summary

This tutorial illustrates how to use tools from Euclidean spaces, finite differences or automatic differentiation, to compute gradients on Riemannian manifolds. The scheme allows to use any differentiation framework within the embedding to derive a Riemannian gradient.


text:
5
Q:

# If A1, A2, A3, A4, ..... A10 are speakers for a meeting and A1 always speaks after, A2 then the number of ways they can speak in the meeting is

 A) 9! B) 9!/2 C) 10! D) 10!/2

Explanation:

As A1 speaks always after A2, they can speak only in  1st  to 9th places and

A2 can speak in 2nd to 10 the places only when A1 speaks in 1st place

A2 can speak in 9 places the remaining

A3, A4, A5,...A10  has no restriction. So, they can speak in 9.8! ways. i.e

when A2 speaks in the first place, the number of ways they can speak is 9.8!.

When A2 speaks in second place, the number of ways they can speak is  8.8!.

When A2 speaks in third place, the number of ways they can speak is  7.8!. When A2 speaks in the ninth place, the number of ways they can speak is 1.8!

Therefore,Total Number of ways they can  speak = (9+8+7+6+5+4+3+2+1) 8! = $92(9+1)8!$ = 10!/2

Q:

If each of the vowels in the word 'MEAT' is kept unchanged and each of the consonants is replaced by the previous letter in the English alphabet, how many four-lettered meaningful words can be formed with the new letters, using each letter only once in each word?

 A) 3 B) 4 C) 1 D) 2

Explanation:

21 1709
Q:

In a bag containing red, green and pink tokens, the ratio of red to green tokens was 5 : 12 while the ratio of pink to red tokens was 7 : 15. What was the ratio of green to pink tokens?

 A) 25 : 28 B) 36 : 7 C) 8 :25 D) 12 : 7

Explanation:

6 1281
Q:

The ratio of the members of blue to red balls in abag is constant. When there were 44 red balls, the number of blue balls was 36. If the number of blue balls is 54, how many red balls will be in the bag?

 A) 68 B) 32 C) 64 D) 66

Explanation:

26 2421
Q:

If each vowel of the word NICELY is changed to the next letter in the English alphabetical series and each consonant is changed to the previous letter in the English alphabetical series and then the alphabets are arranged in alphabetical order, which of the following will be fourth from the left?

 A) M B) J C) B D) O

Explanation:

Thus after arranging the letters as per English alphabetical series; we get; Thus 4th letter from the left end will be K.

16 2461
Q:

How many such pair of letters are there in the word ‘TROUBLED’ which have as many letters between them in the word as they have between them in the English alphabet?

 A) 2 B) 3 C) 4 D) 5

Explanation:

Hence there are two pairs.

13 1947
Q:

If it is possible to make a meaningful word with the first, the seventh, the ninth and the tenth letters of the word RECREATIONAL, using each letter only once, which of the following will be the third letter of the word? If more than one such word can be formed, give ‘X’ as the answer. If no such word can be formed, give ‘Z’ as the answer.

 A) T B) X C) N D) R

Explanation:

The first, the seventh, the ninth and the tenth letters of the word RECREATIONAL are R, T, O and N respectively. Meaningful word from these letters is only TORN. The third letter of the word is ‘R’.

6 1788
Q:

16 persons are participated in a party. In how many differentways can they host the seat in a circular table, if the 2particular persons are to be seated on either side of the host?

 A) 16! × 2 B) 14! × 2 C) 18! × 2 D) 14!

Explanation:

(16 – 2)! × 2 = 14! × 2

3 2579
Q:

From a pack of 52 cards, 3 cards are drawn together atrandom, What is the probability of both the cards are king?

 A) 1/5225 B) 1/5525 C) 5525 D) 1/525


text:
## College Algebra (10th Edition)

$n=13$ or $n=27$
The sum $S_{n}$ of the first $n$ terms of $\left\{a_{n}\right\}$: $S_{n}=\displaystyle \frac{n}{2}\left[2a_{1}+(n-1)d\right]$ $702=\displaystyle \frac{n}{2}\left[2(78)-4(n-1)\right]\quad$ ... solve for n $1404=n[156-4n+4]$ $1404=n[160-4n]$ $4n^{2}-160n+1404=0$ $n^{2}-40n+351=0$ ... use the quadratic formula $n=\displaystyle \frac{40\pm\sqrt{40^{2}-4(1)(351)}}{2(1)}$ $n=\displaystyle \frac{40\pm 14}{2}$ $n=13$ or $n=27$


text:
Biodiv Sci ›› 2022, Vol. 30 ›› Issue (7): 21445.

• Original Papers: Animal Diversity •

### The spatial-temporal patterns of bird diversity and its determinants in the small wetlands in Hefei City

Ting Wang1,2, Lizhi Zhou1,2,*()

1. 1. School of Resources and Environmental Engineering, Anhui University, Hefei 230601
2. Anhui Province Key Laboratory of Wetland Ecosystem Protection and Restoration, Anhui University, Hefei 230601
• Received:2021-11-08 Accepted:2022-03-06 Online:2022-07-20 Published:2022-03-21
• Contact: Lizhi Zhou

Abstract:

Aims: Small wetlands are an important part of urban ecosystems and important shelters for wildlife. Bird diversity as an indicator of urban small wetland ecosystems, its spatial and temporal patterns are affected by many environmental factors. In this study, we studied urban small wetlands and their birds to identify the key environmental factors that affect the spatial and temporal patterns of bird diversity in small wetlands. We then make suggestions on the construction and management of urban small wetlands to promote the protection of urban birds and their habitat restoration.
Methods: A total of 45 small wetlands in Hefei City were taken as research units, and we used the fixed radius method to investigate the distribution of bird species and their abundance from August 2020 to July 2021. A number of environmental factors were also measured, including wetland area, wetland shape, proportion of construction area, proportion of vegetation area, environmental noise, man-made disturbance, and urbanization index. The spatial and temporal characteristics of bird diversity in urban small wetlands were studied by α and β diversity analyses. Then, the model selection and model averaging based on information theory were used to determine the main environmental factors affecting α diversity of bird communities, and the multiple regression model based on distance matrix were used to determine the main environmental factors affecting β diversity of bird communities and their components.
Results: A total of 13 orders, 39 families, and 102 bird species were found in the surveyed wetlands, including 31 waterbird species. There were 2 species in the List of National Second-Class Key Protected Wild Animals, 17 species in the List of Key Protected Wild Animals of Anhui Province, and 1 Vulnerable (VU) species on the IUCN Red List of Species. The results indicated that the α diversity and β diversity of terrestrial birds and waterbirds in small wetlands were both significantly affected by wetland area and urbanization index. The proportion of vegetation area had a strong effect on terrestrial bird diversity, while the proportion of construction area had a significant effect on waterbird diversity. Meanwhile, the species richness of waterbirds and terrestrial birds were both positively correlated with wetland area. The species richness of waterbirds was negatively correlated with the urbanization index, while the species richness of terrestrial birds increased initially and then decreased as urbanization increased. The species richness of terrestrial birds peaked at the small wetlands between intermediate urbanization intensities and low urbanization intensities. When the area of the wetland was about 4 ha, the wetlands could maintain more waterbird species. In addition, terrestrial bird species richness was positively correlated with the proportion of vegetation area, while waterbird species richness was negatively correlated with the proportion of construction area. Beta diversity and species nestedness increased with the increase of key environmental factors, while species turnover exhibited an opposite trend. Furthermore, the results of total β diversity and its compositions indicated that the turnover component of species was predominant, signifying that small wetlands have important conservation significance as a meta-ecosystem.
Conclusion: Our study indicates that wetland area, urbanization index, the proportion of construction area, and the proportion of vegetation area were the key environmental factors affecting bird diversity in small wetlands. On the premise of protecting all small wetlands, we suggest that small wetlands with large areas should be retained and constructed. We must pay more attention to protect small wetlands with intermediate and low urbanization intensities. In addition, the area of natural vegetation should be retained and increased during the management and construction of small wetlands, and the number of artificial structures should be reduced.


text:
## anonymous 4 years ago radical 40 plus radical 100 how to do??

1. amistre64

you have to see if you can rewrite them with like radical parts

2. anonymous

...

3. anonymous

step by step?

4. amistre64

rewrite the given terms in a new way such that they each have the same radicals

5. amistre64

$\sqrt{40}=\sqrt{4*10}=\sqrt{4}\sqrt{10}=2\sqrt{10}$

6. amistre64

you can do the same methods for the 100

7. amistre64

maybe

8. anonymous

im still confused...

9. amistre64

really? sqrt(100) = 10 :)

10. amistre64

theres no good way to combine them i spose

11. amistre64

2sqrt(10) + 10 is the best you can get if im seeing it right

12. anonymous

im gonna ask someone else, thanks anyway.


text:
# Chapter 2, Problem 2/174 Determine the velocity of cart A if cylinder B has a downward...

###### Question:

Chapter 2, Problem 2/174 Determine the velocity of cart A if cylinder B has a downward velocity of 2.6 ft/sec at the instant illustrated. The two pulleys at C are pivoted independently. The velocity of A is positive if down the slope, negative if up the slope. UB B Answer: VA = ft/sec the tolerance is +/-2%

#### Similar Solved Questions

##### PLEASE SHOW ALL WORK THANKS Microphone - - - - Speakers Click to go forward, hold...
PLEASE SHOW ALL WORK THANKS Microphone - - - - Speakers Click to go forward, hold to see history Two small speakers are separated by a distance of 6 cm, as shown. The speakers are driven in phase with a sine wave signal of frequency 14 kHz. A small microphone is placed a distance 14 m away from the ...
Quiz: Chapter 4 homework: optional and due 5/4 at 11 p.m. This Quiz: 10 pts This Question: 1 pt $1,250,000$1,300,000 $1,500,000 The new Learn... 1 answer ##### Part C What is the conjugate acid of HPO42- ? Express your answer as a chemical... Part C What is the conjugate acid of HPO42- ? Express your answer as a chemical formula. View Available Hint(s) Submit Previous Answers Request Answer According to the Brønsted-Lowry theory, an acid is any substance (molecule or ion) that can transfer a proton (H" ion) to another substanc... 1 answer ##### Consider a mixture of 2.00 g of helium and 10.0 g of xenon atoms in a... Consider a mixture of 2.00 g of helium and 10.0 g of xenon atoms in a container with a volume of 1 L. Assume that helium and xenon are ideal gases. (a) If the root mean square speed (vrms or urms in Chapter 9) of helium is 1500 m/s, what is the temperature of the gas? (b) Assuming that the helium an... 1 answer ##### Consumers spend an average of$13.80 on a meal at a restaurant. Assume that the amount...
Consumers spend an average of $13.80 on a meal at a restaurant. Assume that the amount spent on a restaurant meal is normally distributed and the standard deviation is$2.00. What is the probability that a randomly selected person spent between $9.00 -$11.50? 0.037 0.117 0.214 0.135...
how come there is a eight-year lease from 2021 to 2027? Liabilities and Shareholders' Equity (Note: This problem is a variation of P 15–19, modified to cause the lease to be a sales-type lease with a selling profit.) Bidwell Leasing purchased a single-engine plane for $400,000 and leased i... 1 answer ##### XIncorrect; Try Again; 4 attempts remaining Part C How many grams of CO2 are produced from... XIncorrect; Try Again; 4 attempts remaining Part C How many grams of CO2 are produced from the complete reaction of 51.4g of C4H10? Express your answer with the appropriate units. ? Value m(CO2) Units Request Answer Submit Part D If the reaction in part e produces 145 g of CO2, what is the percent y... 1 answer ##### A system consists of two positive point charges, q1 and q2>q1. The total charge of the... A system consists of two positive point charges, q1 and q2>q1. The total charge of the system is 62.0 μC , and each charge experiences electrostatic force of magnitude 82.0 N when the separation between them is 0.270 m. Find q1 Find q2... 1 answer ##### DISCUSSION 1. Using what you learned about the structures of soap and micelles, explain what would... DISCUSSION 1. Using what you learned about the structures of soap and micelles, explain what would map if soap molecules were mixed into a sample of pure vegetable oil. You may provide an illustration to help support your explanation. 2. Using what you've learned previously about light's int... 1 answer ##### Given the following custom function, what is the result of the function call: modify (0) function... Given the following custom function, what is the result of the function call: modify (0) function [y] =modify (x) Smodify changes the input value based on the range in which it falls if x<0 y=x+10; else y=x-5; end end... 1 answer ##### On July 1, 2020, Swifty Company purchased for$6,120,000 snow-making equipment having an estimated useful life...
On July 1, 2020, Swifty Company purchased for $6,120,000 snow-making equipment having an estimated useful life of 5 years with an estimated salvage value of$255,000. Depreciation is taken for the portion of the year the asset is used. Complete the form below by determining the depreciation expense ...


text:
# Clever Algorithms: Nature-Inspired Programming Recipes

By Jason Brownlee PhD

This is the ad-supported version of the book. Buy it now if you like it.

# Extremal Optimization

Extremal Optimization, EO.

## Taxonomy

Extremal Optimization is a stochastic search technique that has the properties of being a local and global search method. It is generally related to hill-climbing algorithms and provides the basis for extensions such as Generalized Extremal Optimization.

## Inspiration

Extremal Optimization is inspired by the Bak-Sneppen self-organized criticality model of co-evolution from the field of statistical physics. The self-organized criticality model suggests that some dynamical systems have a critical point as an attractor, whereby the systems exhibit periods of slow movement or accumulation followed by short periods of avalanche or instability. Examples of such systems include land formation, earthquakes, and the dynamics of sand piles. The Bak-Sneppen model considers these dynamics in co-evolutionary systems and in the punctuated equilibrium model, which is described as long periods of status followed by short periods of extinction and large evolutionary change.

## Metaphor

The dynamics of the system result in the steady improvement of a candidate solution with sudden and large crashes in the quality of the candidate solution. These dynamics allow two main phases of activity in the system: 1) to exploit higher quality solutions in a local search like manner, and 2) escape possible local optima with a population crash and explore the search space for a new area of high quality solutions.

## Strategy

The objective of the information processing strategy is to iteratively identify the worst performing components of a given solution and replace or swap them with other components. This is achieved through the allocation of cost to the components of the solution based on their contribution to the overall cost of the solution in the problem domain. Once components are assessed they can be ranked and the weaker components replaced or switched with a randomly selected component.

## Procedure

Algorithm (below) provides a pseudocode listing of the Extremal Optimization algorithm for minimizing a cost function. The deterministic selection of the worst component in the SelectWeakComponent function and replacement in the SelectReplacementComponent function is classical EO. If these decisions are probabilistic making use of $\tau$ parameter, this is referred to as $\tau$-Extremal Optimization.

Input: ProblemSize, $iterations_{max}$, $\tau$
Output: $S_{best}$
$S_{current}$ $\leftarrow$ CreateInitialSolution(ProblemSize)
$S_{best}$ $\leftarrow$ $S_{current}$
For ($i=1$ To $iterations_{max}$)
For ($Component_{i}$ $\in$ $S_{current}$)
$Component_{i}^{cost}$ $\leftarrow$ Cost($Component_{i}$, $S_{current}$)
End
RankedComponents $\leftarrow$ Rank($Si_{components}$)
$Component_{i}$ $\leftarrow$ SelectWeakComponent(RankedComponents, $Component_{i}$, $\tau$)
$Component_{j}$ $\leftarrow$ SelectReplacementComponent(RankedComponents, $\tau$)
$S_{candidate}$ $\leftarrow$ Replace($S_{current}$, $Component_{i}$, $Component_{j}$)
If (Cost($S_{candidate}$) $\leq$ Cost($S_{best}$))
$S_{best}$ $\leftarrow$ $S_{candidate}$
End
End
Return ($S_{best}$)
Pseudocode for Extremal Optimization.

## Heuristics

• Extremal Optimization was designed for combinatorial optimization problems, although variations have been applied to continuous function optimization.
• The selection of the worst component and the replacement component each iteration can be deterministic or probabilistic, the latter of which is referred to as $\tau$-Extremal Optimization given the use of a $\tau$ parameter.
• The selection of an appropriate scoring function of the components of a solution is the most difficult part in the application of the technique.
• For $\tau$-Extremal Optimization, low $\tau$ values are used (such as $\tau \in [1.2,1.6]$) have been found to be effective for the TSP.

## Code Listing

Listing (below) provides an example of the Extremal Optimization algorithm implemented in the Ruby Programming Language. The algorithm is applied to the Berlin52 instance of the Traveling Salesman Problem (TSP), taken from the TSPLIB. The problem seeks a permutation of the order to visit cities (called a tour) that minimizes the total distance traveled. The optimal tour distance for Berlin52 instance is 7542 units.

The algorithm implementation is based on the seminal work by Boettcher and Percus [Boettcher1999]. A solution is comprised of a permutation of city components. Each city can potentially form a connection to any other city, and the connections to other cities ordered by distance may be considered its neighborhood. For a given candidate solution, the city components of a solution are scored based on the neighborhood rank of the cities to which they are connected: $fitness_k \leftarrow \frac{3}{r_i + r_j}$, where $r_i$ and $r_j$ are the neighborhood ranks of cities $i$ and $j$ against city $k$. A city is selected for modification probabilistically where the probability of selecting a given city is proportional to $n_i^{-\tau}$, where $n$ is the rank of city $i$. The longest connection is broken, and the city is connected with another neighboring city that is also probabilistically selected.

def euc_2d(c1, c2)
Math.sqrt((c1[0] - c2[0])**2.0 + (c1[1] - c2[1])**2.0).round
end

def cost(permutation, cities)
distance =0
permutation.each_with_index do |c1, i|
c2 = (i==permutation.size-1) ? permutation[0] : permutation[i+1]
distance += euc_2d(cities[c1], cities[c2])
end
return distance
end

def random_permutation(cities)
perm = Array.new(cities.size){|i| i}
perm.each_index do |i|
r = rand(perm.size-i) + i
perm[r], perm[i] = perm[i], perm[r]
end
return perm
end

def calculate_neighbor_rank(city_number, cities, ignore=[])
neighbors = []
cities.each_with_index do |city, i|
next if i==city_number or ignore.include?(i)
neighbor = {:number=>i}
neighbor[:distance] = euc_2d(cities[city_number], city)
neighbors << neighbor
end
return neighbors.sort!{|x,y| x[:distance] <=> y[:distance]}
end

def get_edges_for_city(city_number, permutation)
c1, c2 = nil, nil
permutation.each_with_index do |c, i|
if c == city_number
c1 = (i==0) ? permutation.last : permutation[i-1]
c2 = (i==permutation.size-1) ? permutation.first : permutation[i+1]
break
end
end
return [c1, c2]
end

def calculate_city_fitness(permutation, city_number, cities)
c1, c2 = get_edges_for_city(city_number, permutation)
neighbors = calculate_neighbor_rank(city_number, cities)
n1, n2 = -1, -1
neighbors.each_with_index do |neighbor,i|
n1 = i+1 if neighbor[:number] == c1
n2 = i+1 if neighbor[:number] == c2
break if n1!=-1 and n2!=-1
end
return 3.0 / (n1.to_f + n2.to_f)
end

def calculate_city_fitnesses(cities, permutation)
city_fitnesses = []
cities.each_with_index do |city, i|
city_fitness = {:number=>i}
city_fitness[:fitness] = calculate_city_fitness(permutation, i, cities)
city_fitnesses << city_fitness
end
return city_fitnesses.sort!{|x,y| y[:fitness] <=> x[:fitness]}
end

def calculate_component_probabilities(ordered_components, tau)
sum = 0.0
ordered_components.each_with_index do |component, i|
component[:prob] = (i+1.0)**(-tau)
sum += component[:prob]
end
return sum
end

def make_selection(components, sum_probability)
selection = rand()
components.each_with_index do |component, i|
selection -= (component[:prob] / sum_probability)
return component[:number] if selection <= 0.0
end
return components.last[:number]
end

def probabilistic_selection(ordered_components, tau, exclude=[])
sum = calculate_component_probabilities(ordered_components, tau)
selected_city = nil
begin
selected_city = make_selection(ordered_components, sum)
end while exclude.include?(selected_city)
return selected_city
end

def vary_permutation(permutation, selected, new, long_edge)
perm = Array.new(permutation)
c1, c2 = perm.rindex(selected), perm.rindex(new)
p1,p2 = (c1<c2) ? [c1,c2] : [c2,c1]
right = (c1==perm.size-1) ? 0 : c1+1
if perm[right] == long_edge
perm[p1+1..p2] = perm[p1+1..p2].reverse
else
perm[p1...p2] = perm[p1...p2].reverse
end
return perm
end

def get_long_edge(edges, neighbor_distances)
n1 = neighbor_distances.find {|x| x[:number]==edges[0]}
n2 = neighbor_distances.find {|x| x[:number]==edges[1]}
return (n1[:distance] > n2[:distance]) ? n1[:number] : n2[:number]
end

def create_new_perm(cities, tau, perm)
city_fitnesses = calculate_city_fitnesses(cities, perm)
selected_city = probabilistic_selection(city_fitnesses.reverse, tau)
edges = get_edges_for_city(selected_city, perm)
neighbors = calculate_neighbor_rank(selected_city, cities)
new_neighbor = probabilistic_selection(neighbors, tau, edges)
long_edge = get_long_edge(edges, neighbors)
return vary_permutation(perm, selected_city, new_neighbor, long_edge)
end

def search(cities, max_iterations, tau)
current = {:vector=>random_permutation(cities)}
current[:cost] = cost(current[:vector], cities)
best = current
max_iterations.times do |iter|
candidate = {}
candidate[:vector] = create_new_perm(cities, tau, current[:vector])
candidate[:cost] = cost(candidate[:vector], cities)
current = candidate
best = candidate if candidate[:cost] < best[:cost]
puts " > iter #{(iter+1)}, curr=#{current[:cost]}, best=#{best[:cost]}"
end
return best
end

if __FILE__ == \$0
# problem configuration
berlin52 = [[565,575],[25,185],[345,750],[945,685],[845,655],
[880,660],[25,230],[525,1000],[580,1175],[650,1130],[1605,620],
[1220,580],[1465,200],[1530,5],[845,680],[725,370],[145,665],
[415,635],[510,875],[560,365],[300,465],[520,585],[480,415],
[835,625],[975,580],[1215,245],[1320,315],[1250,400],[660,180],
[410,250],[420,555],[575,665],[1150,1160],[700,580],[685,595],
[685,610],[770,610],[795,645],[720,635],[760,650],[475,960],
[95,260],[875,920],[700,500],[555,815],[830,485],[1170,65],
[830,610],[605,625],[595,360],[1340,725],[1740,245]]
# algorithm configuration
max_iterations = 250
tau = 1.8
# execute the algorithm
best = search(berlin52, max_iterations, tau)
puts "Done. Best Solution: c=#{best[:cost]}, v=#{best[:vector].inspect}"
end

Extremal Optimization in Ruby

## References

### Primary Sources

Extremal Optimization was proposed as an optimization heuristic by Boettcher and Percus applied to graph partitioning and the Traveling Salesman Problem [Boettcher1999]. The approach was inspired by the Bak-Sneppen self-organized criticality model of co-evolution [Bak1987] [Bak1993].

A number of detailed reviews of Extremal Optimization have been presented, including a review and studies by Boettcher and Percus [Boettcher2000], an accessible review by Boettcher [Boettcher2000a], and a focused study on the Spin Glass problem by Boettcher and Percus [Boettcher2001].

## Bibliography

 [Bak1987] P. Bak and C. Tang and K. Wiesenfeld, "Self-organized criticality: An explanation of the 1/f noise", Physical Review Letters, 1987. [Bak1993] P. Bak and K. Sneppen, "Punctuated equilibrium and criticality in a simple model of evolution", Physical Review Letters, 1993. [Boettcher1999] S. Boettcher and A. G. Percus, "Extremal Optimization: Methods derived from Co-Evolution", in Proceedings of the Genetic and Evolutionary Computation Conference, 1999. [Boettcher2000] S. Boettcher and A. Percus, "Nature’s Way of Optimizing", Artificial Intelligence, 2000. [Boettcher2000a] S. Boettcher, "Extremal optimization: heuristics via coevolutionary avalanches", Computing in Science & Engineering, 2000. [Boettcher2001] S. Boettcher and A. G. Percus, "Optimization with extremal dynamics", Phys. Rev. Lett., 2001.

### Free Course

Get one algorithm per week...
• ...described in detail
Sign-up Now

### Own A Copy

This 438-page ebook has...
• ...45 algorithm descriptions
• ...best practice usage
• ...pseudo code
• ...Ruby code
• ...primary sources

Please Note: This content was automatically generated from the book content and may contain minor differences.

Do you like Clever Algorithms?


text:
# Image spaces of $X^2$

Let $(X,\tau)$ be a topological space and let $$\Omega(X)=\{A\subseteq X: \text{there is surjective continuous }f:X\to A\}.$$ Can we express $\Omega(X\times X)$ in terms of $\Omega(X)$? (We assume $X\times X$ to carry the product topology.)

• You don't assume anything regarding connectivity of $X$? – H. H. Rugh Aug 15 '16 at 10:42
• How do you formalize the verb "to express"? One possible way is that you are given $X$ and $\Omega(X)$ but not the topology of $X$ and the question is whether you can find $\Omega(X\times X)$ but I'm not sure if that's what you had in mind. – fedja Aug 16 '16 at 15:47
• What's the motivation for this question? – Todd Trimble Oct 14 '16 at 11:43
• @fedja We don't need both $X$ and $\Omega(X)$ as input, since $X$ is determined by $\Omega(X)$, as the largest element of it. I interpret the question as: can we calculate the continuous images of the product $X\times X$ from the set of continuous images of $X$ in $X$? In other words, is $\Omega(X\times X)$ a function of $\Omega(X)$? For a negative answer, you need to find a set $X$ with two topologies on it, such that they have the same set of continuous images $X\to A\subset X$, but different continuous images of the product $X\times X$ in itself. – Joel David Hamkins Oct 14 '16 at 13:10

## 1 Answer

First it is clear that $$\{ A \times B: A, B\subset \Omega (X) \} \subset \Omega(X \times X) \}$$ However, just thinking about $(0,1)$, it is easy to see that this is not an equality, and that, in fact, it is much larger. For example $$\Omega( (0,1) ) = \{ I : I \textrm{ interval} \}$$ whereas $$\Omega( (0,1)^2 ) \supset \{ U: U \textrm{ homeomorphic to a square } \}$$ which includes not only squares, but ellipses and whichever other shape you can think of.

Cheers

• Both displayed equations are wrong: no reason for $U$ to be open. – Adam Przeździecki Sep 14 '16 at 10:51
• Correct. Edited – D G Sep 14 '16 at 11:06
• Do you mean $A,B\in\Omega(X)$ rather than $\subset$ in the first line? – Joel David Hamkins Oct 14 '16 at 13:02


text:
# After four half-lives have elapsed, the amount of a radioactive sample that has decayed is which of the following? a. 1/4 the original amount b. 3/4 the original amount c. 1/16 the original amount d. no correct response

### General, Organic, and Biological C...

7th Edition
H. Stephen Stoker
Publisher: Cengage Learning
ISBN: 9781285853918

### General, Organic, and Biological C...

7th Edition
H. Stephen Stoker
Publisher: Cengage Learning
ISBN: 9781285853918

#### Solutions

Chapter
Section
Chapter 11.4, Problem 3QQ
Textbook Problem

## After four half-lives have elapsed, the amount of a radioactive sample that has decayed is which of the following?a. 1/4 the original amountb. 3/4 the original amountc. 1/16 the original amountd. no correct response

Expert Solution
Interpretation Introduction

Interpretation:

After four half-lives, the amount of decayed sample of the radioactive sample has to be chosen from the given options.

Concept Introduction:

Radioactive nuclides undergo disintegration by emission of radiation.  All the radioactive nuclide do not undergo the decay at a same rate.  Some decay rapidly and others decay very slowly.  The nuclear stability can be quantitatively expressed by using the half-life.

The time required for half quantity of the radioactive substance to undergo decay is known as half-life.  It is represented as t1/2.

### Explanation of Solution

Reason for correct option:

A radioactive sample will disintegrate into half of its initial amount during its half-life period.  The next half-life period decays half of the amount that is present after the first half-life period.  Therefore, 1/4th of the initial amount will be present after two half-lives.  During the third half-life, another half amount of the radioactive nuclide that is present after two half-lives will be decayed.  Therefore, 1/8th of the initial amount will be present after three half-lives elapsed.  During the fourth half-life, another half amount of the radioactive nuclide that is present after three half-lives will be decayed.  Therefore, 1/16th of the initial amount will be present after four half-lives elapsed.  The sample that has decayed is 15/16.

Hence, option (d) is the correct one.

Reason for incorrect options:

Option (a) tells that 1/4 the original amount of radionuclide will be decayed after four half-lives...

### Want to see the full answer?

Check out a sample textbook solution.See solution

### Want to see this answer and more?

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

See solution


text:
# minted environment inside subcaptionbox is not working

I've seen some similar questions here and here, but none mention the \subcaptionbox command. Here is my MWE:

\documentclass{article}
\usepackage{subcaption}
\usepackage{minted}

\begin{document}
\begin{figure}[htb]
\subcaptionbox{One subfigure.}{%
\begin{minted}{c}
printf("hello, world\n");
\end{minted}
}
\end{figure}
\end{document}

This fails to compile with a Paragraph ended before \FV@BeginScanning was complete. } error.

I could use a minipage environment and the \subcaption command instead, but \subcaptionbox has some advantages that I'd like to use.

Is there any way to make these two work together?

• A verbatim environment cannot go in the argument to another command. Mar 14 '19 at 7:38
• use the minted form that inputs a file and put your code fragments in external files so they are not contained in the argument of another command. Mar 14 '19 at 7:52
• @DavidCarlisle I tried using \subcaptionbox{One subfigure.}{\inputminted{c}{helloWorld.c}} but still get errors: Something's wrong--perhaps a missing \item. \begin{Verbatim}[commandchars=\\\{\}]. Mar 15 '19 at 1:46

## 1 Answer

Here's a crude implementation, with just the standard form for \subcaptionbox (no optional arguments).

\documentclass{article}
\usepackage{subcaption}
\usepackage{minted}

\newsavebox{\mintedbox}
\newenvironment{mintedsubcaptionbox}[2]
{%
\VerbatimEnvironment
\def\mscbcaption{#1}%
\RecustomVerbatimEnvironment{Verbatim}{BVerbatim}{}%
\begin{lrbox}{\mintedbox}%
\begin{minted}{#2}%
}
{%
\end{minted}%
\end{lrbox}%
\subcaptionbox{\mscbcaption}{\usebox{\mintedbox}}%
}

\begin{document}

\begin{figure}[htb]

\begin{mintedsubcaptionbox}{One subfigure.}{c}
printf("hello, world\n");
\end{mintedsubcaptionbox}

\end{figure}

\end{document}

I used BVerbatim because it's quite unlikely that you want a listing to be full width inside a subfloat.

A fuller support with syntax

\begin{mintedsubcaptionbox}[<list entry>]{<heading>}[<minted options>]{<language>}

can be

\documentclass{article}
\usepackage{subcaption}
\usepackage{minted}
\usepackage{xparse}

\newsavebox{\mintedbox}

\NewDocumentEnvironment{mintedsubcaptionbox}{O{#2}mO{}m}
{%
\VerbatimEnvironment
\RecustomVerbatimEnvironment{Verbatim}{BVerbatim}{}%
\begin{lrbox}{\mintedbox}%
\begin{minted}[#3]{#4}%
}
{%
\end{minted}%
\end{lrbox}%
\subcaptionbox[#1]{#2}{\usebox{\mintedbox}}%
}

\begin{document}

\begin{figure}[htb]

\begin{mintedsubcaptionbox}{One subfigure.}{c}
printf("hello, world\n");
\end{mintedsubcaptionbox}

\end{figure}

\end{document}

• Thank you. Is this solution too hacky to be included in the minted package out-of-the-box? Maybe I could suggest it. Mar 15 '19 at 1:48