text:
## Introduction

A comprehensive understanding of population-specific demographics, life-history traits and behavioural ecology is essential for the effective management of long-lived, slow-reproducing species1,2. Such parameters are linked to morphological and age-specific processes, often regulated by physiological, ecological, evolutionary, and anthropogenic factors3. Morphometric data of cetaceans contribute to the assessment of individual- and population-based reproduction4,5, health6,7 and demography8,9,10 and in conjunction with genetic information, taxonomic status11,12. Despite its many applications, obtaining accurate morphometric data on free-ranging cetaceans is challenging.

Traditionally, three approaches have been used to obtain morphometric data on cetaceans: post-mortem specimens13, live captive study subjects14,15, and capture-release programs16,17. Post-mortem specimens are usually sourced from stranding events18 in addition to incidental19 and deliberate kills20. The dependence on post-mortem study subjects has the disadvantage of reliance on unpredictable access to animals and small sample sizes21. In addition, post-mortem specimens may provide a biased sample, if animals of specific age, sex, size or health are more likely to strand, be incidentally caught or killed22. Capture-release programs provide a unique opportunity to repeatedly measure individuals over time but come with both considerable ethical and logistical considerations17.

In recent decades, photogrammetry has emerged as an alternative morphometric technique, highlighted by its application in studies of animal populations in both terrestrial23,24,25 and marine environments26,27,28. Stereo-photogrammetry, where two parallel cameras capture a composite image simultaneously, is considered one of the earliest forms of photogrammetry. However, its practicality is limited by the cumbersome nature of the required hardware29,30. Stereo-laser photogrammetry is a popular alternative to stereo-photogrammetry, due to its simplicity. The technique consists of two perfectly parallel laser dots calibrated at a specific distance apart (e.g. 10 cm), thereby providing a known-length scale within a photograph which allows the size of animals in an image to be measured (i.e. by converting measured pixels to centimetres)8. The technique also allows for photo-identification data to be obtained simultaneously, so that a specific measurement can be linked to a particular individual31,32.

We used stereo-laser photogrammetry to obtain total length (TL) estimates of coastal Indo-Pacific bottlenose dolphins (Tursiops aduncus) from south-western (SW) and Shark Bay (SB), Western Australia. Laser-derived measurements, in conjunction with available long-term demographic records of individual dolphins, were applied to develop length-at-age (LaA) growth curves for each region (SW and SB). Growth parameter estimates derived from growth models were used to characterise and compare growth adaptations, with the aim of quantifying potential differences in the morphology of T. aduncus from two geographically separated regions.

## Results

### Relationship between blowhole-to-dorsal fin length and total length

Physical measurements of blowhole-to-dorsal fin length (BH-DF) and total length (TL) were obtained on 12 post-mortem individuals (males n = 6, females n = 6) stranded in SW Australia. The significant positive relationship between BH-DF and TL (in centimetres) was then used to estimate the TL of T. aduncus in both SW and SB regions (F1,10 = 1341, p < 0.001, R2 = 0.992, TL = 5.0583 + 3.17 × BH-DF, Supplementary Fig. S1). While our sample size was small (n = 12), there is evidence to suggest the relationship between BH-DF and TL can be generalized across both T. aduncus and Tursiops truncatus33 (Supplementary Fig. S1).

### Description of length-at-age data obtained in south-west and Shark Bay

For the SW region, laser-derived measurements were collected during 40 boat-based surveys in Bunbury (n = 28) and Mandurah (n = 12). Of the 2,103 photographs taken, 828 photographs were of sufficient quality for analyses. A total of 129 individual dolphins were identified and measured, including 56 females, 39 males, and 34 of unknown-sex. An average of 6.4 measurements were available for each individual (SE = 0.37), with a mean coefficient of variance (CV) of 1.9% estimated for repeated TL estimates of the same individuals across multiple photographs (range = 0.02–6.68%). Minimum age estimates for females ranged from three days old to 29 years (Supplementary Fig. S2a), with laser-derived TL estimates ranging between 106.1 cm and 256.8 cm (Supplementary Fig. S3a, Table S1). Minimum age estimates for males ranged from four days old to 29 years (Supplementary Fig. S2a), with a TL range of 105.7–254.4 cm (Supplementary Fig. S3a, Table S1). No significant differences in TL were observed between males (n = 10, mean = 243.7 cm, SE = 3.1 cm) and females (n = 13, mean = 242.9 cm, SE = 2.1 cm) over the age of 20 (p = 0.832).

In SB, stereo-laser photogrammetry data were collected during boat-based surveys (n = 11 days) and beach food-provisioning events (n = 10 days). Over this period, 732 photographs were taken, with 355 images (boat = 216, beach = 139) of sufficient quality to warrant inclusion for further analyses. Of the 74 individuals sampled, 42 were female, 24 were male, and 8 were of unknown sex, with an average of 4.8 measurements per individual (SE = 0.81). A mean CV of 1.7% was estimated for repeated TL estimates of the same individuals across multiple photographs, with a range of 0.03–8.8%. Minimum age estimates for females ranged from 1.7 to 44 years (Supplementary Fig. S2b), with TL estimates ranging from 139.9 to 210.5 cm (Supplementary Fig. S3b, Table S1). Minimum age estimates for males ranged between 3.8 and 41 years (Supplementary Fig. S2b), with a TL range of 157.8–209.9 cm (Supplementary Fig. S3b, Table S1). The youngest SB individual was < 2 weeks old, with a TL estimate of 102.8 cm (Table S1), and of unknown sex. Like the SW region, no sexual dimorphism was detected in SB individuals >20 years old (p = 0.084), with males averaging 202.9 cm (n = 11, SE = 1.25 cm) and females 198.7 cm (n = 13, SE = 1.92 cm).

### Selection of the best-fitting growth models

Of the four growth models fitted to the LaA data, Richards growth model (RGM) best described T. aduncus growth in both study regions. For the SW sample, the RGM provided the best fit (wi = 1.00, Supplementary Table S2). Visual inspection of the Typical von Bertalanffy (TvB), Original von Bertalanffy (OvB) and Gompertz function (GOM) growth curves highlighted the lack of model fit for younger individuals in the SW sample (Supplementary Fig. S4a), with the flexible RGM curve best fitting the observed data at younger ages. For the SB sample, the RGM received moderate support (wi = 0.49), suggesting the OvB (wi = 0.22) and TvB (wi = 0.22) models also fitted the data reasonably well (Supplementary Table S2). Despite stronger overlapping of candidate growth curves observed for T. aduncus in the SB region (Supplementary Fig. S4b), only the most parsimonious model (RGM) was used to infer estimates of growth.

### Estimation of biological parameters

All SW calves less than two-weeks old measured between 106.1–110.7 cm (mean = 107.6 cm, SD = 1.87 cm, n = 6). Both SW and SB growth curves were characterised by rapid early growth before slowing as juveniles approached maturity (Fig. 1a,b). Southwest calves exhibited a distinct period of accelerated growth during their first year (Fig. 1a), with the RGM predicting a median TL of 155.9 cm (95% CI = 155.3–156.5 cm) at age 1 year. By the end of their third year, SW calves were estimated to have a median TL of 187.1 cm (95% CI = 186.5–187.7 cm). Median TL values of 120.1 cm (95% CI = 118.4–122.2 cm) and 149.5 cm (95% CI = 148.2–151.1 cm) were predicted for SB calves aged 1 and 3 years, respectively.

The RGM predicted a slightly larger L estimate for SW males (246.1 cm, 95% CI = 239.2–254.5 cm, Fig. 1a, Supplementary Table S1) relative to SW females (244.5 cm, 95% CI = 239.9–250.7 cm). Shark Bay males reached an asymptote at 201.9 cm (95% CI = 199.2–205.1 cm, Fig. 1b, Supplementary Table S1) and for SB females, the RGM produced a median L estimate of 200.5 cm (95% CI = 196.7–205.3 cm, Supplementary Table S1).

### Estimating age and length at independence and first reproduction

For SW and SB, age and length had a significant effect on the probability of a dolphin becoming independent and reproducing for the first time (p < 0.001, Supplementary Table S3). In the SW sample containing 45 dependent calves and 32 independent juveniles, 24 were female, 19 male and 34 of unknown sex. Fifty percent of SW calves were estimated to become independent by 3.0 years (95% CI = 2.5–3.3 years, Fig. 2b). The SB sample consisted of 11 dependent calves and 16 independent juveniles, including 11 females, nine males and seven dolphins of unknown sex. The A50 estimate for SB calves was 4.8 years (95% CI = 4.3–4.9 years, Fig. 2d). Despite becoming independent earlier, SW calves exhibited a larger L50 value (187.2 cm, 95% CI = 180.0–191.1 cm, Fig. 2a) relative to SB dolphins (162.4 cm, 95% CI = 151.9–168.2 cm, Fig. 2c).

For first reproduction, the SW sample consisted of 32 females with previous calving histories and 24 females never observed with a calf. In SB, 31 females had previous calving histories while 11 females exhibited no evidence of previous birth at the time of measurement. Southwest females attained A50 at a younger age (10.3 years, 95% CI = 9.6–11.3 years, Fig. 3b) than their SB conspecifics (11.9 years, 95% CI = 10.8–12.9 years, Fig. 3d). First reproduction L50 estimates calculated for SW and SB were 224.3 cm (95% CI = 220.9–226.4, Fig. 3a) and 185.4 cm (95% CI = 181.3–190.0 cm, Fig. 3c), respectively.

### Sensitivity analysis: accounting for measurement and age-estimation errors

We investigated the influence of error on our LaA estimates using maximum error values for both laser-derived measurements and individual age estimations. When stereo-laser images were obtained at 15° from perpendicular, a dolphin replica experiment yielded a mean measurement error of 1.27% (SD = 0.69, Supplementary Fig. S5) equating to 0.84 cm (range = 0.29–1.47 cm). At this angle, precision was high between length measurements from 45 non-sequential images (CV = 1.31%).

Age-estimation error distributions were greatest for SW individuals over the age of 20, with far greater age certainty for calves and juveniles (Supplementary Fig. S6a). Age estimations of SB dolphins were of better quality with no individuals assigned a maximum age of 45, reflected by reduced horizontal error distributions (Supplementary Fig. S6b). The profiles of both SW and SB RGM curves remained relatively unchanged following the resampling procedure, with little deviation from either of the original RGM curves (Supplementary Fig. S6). Each of the estimated LaA values displayed narrow distributions (Supplementary Fig. S7) and HPD intervals (Supplementary Table S4), indicating both SW and SB RGM models were robust to potential error present in this study.

### Regional differences in dolphin morphology

Bootstrapped TL estimates of dolphins differed significantly between the two study regions, with SB dolphins being significantly shorter across ages 1, 3, 12 and 25 years (p < 0.001 for all four tested ages, Fig. 4). Total-length differences across the four age classes ranged from 35.8 cm (age 1 year) to 39.2 cm (age 25 years), with a median of 36.8 cm (SD = 1.58 cm, Supplementary Table S4). This variation in age-specific growth can be visualised by the lack of overlapping distributions in each age class (Supplementary Fig. S8).

## Discussion

In this study we measured body morphometrics using non-invasive laser photogrammetry, to characterise and compare the growth of T. aduncus from two well-studied populations in Western Australia. We identified marked differences in the growth between the SW and SB, with T. aduncus in the SW being significantly longer in body length than their SB conspecifics. This difference in length was not caused by regional variation in birth size, but by a distinct difference in growth across all life stages (i.e. neonates, calves, juveniles and adults).

Latitudinal differences in body size have been well documented in Tursiops spp., with larger body sizes typically reported in cooler regions14,34,35,36. A recent study by Cheney et al.33 demonstrated the potential of laser photogrammetry by developing the first laser-derived LaA growth curves for a bottlenose dolphin population in a temperate environment. Using these measurements, the study reported adult T. truncatus inhabiting temperate waters were approximately 30% longer than estimates produced for adult T. truncatus in sub-tropical waters17. Using the same approach, we detected regional differences in body length across all life stages, with SW T.aduncus being up to 20% longer than SB dolphins. By identifying morphological differences between two populations of T. aduncus, we add further support for the accuracy and applicability of laser photogrammetry as a morphometric tool.

Coastal sea surface temperatures (SSTs) off Bunbury (in the SW) range between 14 °C (June to August, austral winter) and 23 °C (December-March, summer), with an annual mean SST of 18 °C37,38. In Monkey Mia, SSTs range between 16 °C (June to August, winter) and 30 °C (December-March, summer), with a mean annual SST of 22 °C39. While minimum SSTs are similar between study sites, long-term data39 (2009–2019) indicate the mean winter SST for Monkey Mia (19 °C) is greater than the mean annual SST reported for Bunbury (18 °C)37,38.

The inverse relationship between body size and surface water temperature is characteristic of Bergmann’s rule40, which describes a trade-off between surface area and volume. The surface area of an endothermic animal represents its ability to dissipate heat, while its volume serves as a measure of its heat generation capability41. A reduced surface-area-to-volume ratio is thus considered a selective advantage, enabling large-bodied animals residing in cooler environments to regulate body heat more efficiently42,43. While SST is likely the most dominant factor affecting body size between T. aduncus in SB and SW, our findings likely reflect a complex interplay of phenotypic and genotypic factors, which can only be elucidated using a combination of environmental, morphometric and genetic approaches11.

Total length estimates of all seven SW and SB neonate calves less than two-weeks old (2 females, 3 males, 2 unknown-sex) ranged between 102.8 cm and 110.7 cm (mean = 106.9 cm). These values complement birth length estimates of T. aduncus in Tanzania (103 cm)44, and T. truncatus in South Africa (103–111 cm)45,46,47 and the Gulf of Mexico (103–109 cm)48. Remote estimates of newborn length are rare, with Cheney et al.33 reporting laser-derived estimates for T. truncatus in Scotland of 128–188 cm for calves in their first three months. Length estimates of neonatal T. aduncus are typically limited to post-mortem observations with lengths between 93–121 cm recorded for South Australian T. aduncus younger than three months (n = 10, mean = 104.5 cm)49. While our TL estimates represent the first laser-derived length measurements of neonatal T. aduncus, repeat measurements of young calves over time will provide a more comprehensive profile of neonatal growth. Non-invasive morphometric techniques also reduce dependence on deceased specimens, which may not be representative of wild populations.

Using the mean birth lengths derived from this study, SW neonates (107.6 cm) were estimated to increase in length by 48.3 cm (44.8%) by the end of their first year, while SB calves (102.8 cm) only increased by 17.3 cm (16.8%). Although regional differences in asymptotic length are evident, caution is recommended when interpreting the lack of accelerated growth observed in SB calves. Additional sampling of SB calves would be beneficial in confirming whether such large differences in first year growth are robust, or due to limited calves sampled in SB (n = 11) relative to SW (n = 45).

Our age-at-independence estimates for both SW (3.0 years) and SB (4.8 years) fall within the range previously reported for T. aduncus (2.5–8.5 years)50,51,52,53. Our A50 estimate is slightly larger than the mean age at independence previously reported for SB dolphins (3.98 years)50,53, and is more reflective of the age at independence for last-born SB calves53 (4.86 years). Our study is the first to report laser-derived L50 estimates for T. aduncus, with SW dolphins becoming independent at a larger body length than SB dolphins (L50: SW = 187.2 cm and SB = 162.4 cm). The ability to quantify individual and population-level calf growth promotes the value of stereo-laser photogrammetry, with the potential to investigate the influence of maternal investment on calf size and associated fitness consequences in future studies.

Age at first reproduction estimates for female T. aduncus vary considerably across studies (7–15 years)14,44,45,50,54, with our SW A50 estimate (10.3 years) within the range of those previously reported. In SB, 42% of first births occur by age 1253,55, validating our SB A50 estimate of 11.9 years. Like independence, our estimates of age and length at first reproduction indicate SW females to be younger, but larger the first time they give birth (SW = 10.3 years and 224.3 cm, SB = 11.9 years and 185.4 cm). Our length-at-first reproduction estimates are the first to be derived from live, free-ranging dolphins using non-invasive laser photogrammetry. These estimates are similar to those reported for captive and post-mortem Tursiops spp. females from temperate (227–238 cm)14,34 and tropical waters globally (190–200 cm)44,54. In our study, SW mothers may be maximising the reproductive fitness of their offspring by providing the maternal investment necessary to attain optimal independence and first reproduction sizes at earlier ages, given faster growing mammals tend to mature earlier than slower-growing mammals18,56,57.

Stereo-laser photogrammetry shows promise in its ability to detect individual variations in growth, providing opportunities to elucidate the effects of maternal investment and experience on calf growth and survival. Cheney et al.33 recently demonstrated the efficacy of this technique by investigating the fitness implications of variable calf length, reporting first-born calves were shorter than calves of experienced mothers and, more importantly, that calf length was a significant predictor of first-year mortality. These results suggest calf growth may be a valuable proxy of maternal investment and condition, with future studies recommended to incorporate laser photogrammetry into long-term monitoring efforts33,58.

In both SW and SB, L estimates were slightly larger for males than females. Despite this, no significant differences in TL were detected in either SW or SB adults over the age of 20. While adult male T. aduncus are typically heavier than adult females45, little to no sexual dimorphism has been detected using TL14,35,45. Since we could not estimate mass using stereo-laser photogrammetry, we were unable to investigate regional growth characteristics and sexual dimorphism using a combination of TL and mass.

Our L estimates for T. aduncus SW males (246.1 cm) and females (244.5 cm) are slightly larger than those reported for male (243 cm) and female (238 cm) T. aduncus in South Africa45 and T. truncatus in near-shore waters off Perth, Western Australia (~240 cm)14. Interestingly, our L estimates for SB male (201.9 cm) and female (200.5 cm) T. aduncus are among the shortest reported for this species. Adults of similar size have been documented from deceased subjects in the Arafura Sea, Northern Australia (214 cm, 11°42′S, 137°14′E34, Spencer Gulf, South Australia (214 cm, 32°29′S, 133°17′E)54 and Zanzibar, Tanzania (222 cm, 06° S, 39° E)44. While a few qualitative accounts have briefly described the small size (~200 cm) of adult SB dolphins50,59,60, this study is the first to measure and estimate the size and growth of coastal dolphins in both SW and SB.

The mean measurement error in the dolphin replica experiment (1.27% at 15° horizontal angle) was within the error range previously reported for stereo-laser photogrammetry (1.2–3.5%)27,28,32,33. Precision was also high, with mean CV estimates achieved in both the error experiment (1.3%) and field-based laser photogrammetry (1.7–1.9%) comparing favourably to previous laser photogrammetry (0.7–3.7%)28,31,32, stereo-photogrammetry (4.3%)29 and aerial photogrammetry studies (<2%)61,62.

Our sensitivity analysis demonstrated all four LaA estimates (i.e. ages 1, 3, 12, 25 years) were robust to measurement and age estimation errors. Age estimations for all SB individuals were accurate to within three years. Age estimation for SW individuals under 12 years of age were also accurate (median = 2 months, range = 1 day to 4 years). The narrow density distributions produced by the sensitivity analysis showed the SW RGM model remained robust with LaA values easily within the typical length variation reported in mature Tursiops spp.17 (Supplementary Fig. S8). This is because all SW individuals first sighted as adults were positioned at asymptotic regions, as opposed to earlier regions, of the RGM. This suggests any influence on the RGM output would have been minimal, as T. aduncus growth usually ceases by the age of 1545,50. This approach, however, may not be appropriate for immature individuals who are still subject to growth. Nonetheless, results from our sensitivity analysis propose that it may be possible to investigate individual and population-specific growth using incomplete demographic data on older individuals. The sensitivity analysis also confirms the high accuracy of our stereo-laser photogrammetry approach in obtaining accurate body morphometric measurements of T. aduncus and consequent growth curves.

Repeated measurements of known-age individuals over time will yield more comprehensive information on the variability of individual and population-level growth rates, and establish an ideal platform for the investigation of genetic, biological, ecological and anthropogenic factors influencing growth. Our ability to quantify differences in growth over a relatively small geographical distance demonstrates the value of using this technique to investigate body size at various ages and life history stages (birth, independence, first reproduction and physical maturity). Stereo-laser photogrammetry, therefore, provides a valuable opportunity to collect morphometric data on free-ranging cetacean populations in an accurate, non-invasive manner, which can ultimately inform conservation management strategies.

## Methods

### Study locations and dolphin populations

Between May 2016 and March 2017, three study sites along the Western Australian coast were used to collect stereo-laser photogrammetry data on T. aduncus (Fig. 5). The SW region comprised of dolphins sampled in both Bunbury (33°32′S, 115°63′E, Fig. 5C) and Mandurah (32°32′S, 115°44′E, Fig. 5B). The distance between these locations is approximately 95 km. Both locations exhibit an identical Mediterranean climate with temperate coastal environments38,63. The SB region comprised of dolphins sampled off Monkey Mia (25°47′S, 113°43′E, Fig. 5A), located in the Eastern Gulf of SB, approximately 860 km north-west of Bunbury. This study was approved by the Western Australian Department of Biodiversity, Conservation and Attractions (SF010738, CE005422), with all fieldwork conducted in accordance to standards set by the Murdoch University Ethics Committee (R2649/14).

### Age estimates for south-west and Shark Bay individuals

#### Bunbury

Since 2007, dedicated year-round photo-identification surveys have been conducted off Bunbury as part of the South West Marine Research Program (SWMRP). The SWMRP has identified approximately 500 individual dolphins, facilitating the development of a long-term demographic dataset comprising of photo-identification64, demographic (age and sex)65 and sighting-history information (Supplementary Table S5). Additional photo-identification, demographic and sighting data have been obtained on food-provisioned dolphins since 1989, following the establishment of a not-for-profit dolphin-tourism centre conducting dolphin-provisioning and eco-tours66.

#### Mandurah

The Mandurah Dolphin Research Project (MDRP) commenced in 2016, with approximately 500 dolphins subsequently identified (Supplementary Table S5). Additional long-term demographic are available through collaborative citizen science and historic live-stranding records dating back to 1987 (Supplementary Table S5).

For SW (Bunbury and Mandurah) calves (0–3.0 years) and juveniles (3.5–10 years), minimum and maximum age estimates were established based on long-term sighting records of both the individual dolphin in question and its mother. For example, the minimum age of an individual was derived from the date it was first sighted, with its maximum age calculated using the date the mother was last sighted with either a sibling calf or no calf present. For some SW individuals first sighted as adults, minimum age was calculated from the date the dolphin was first sighted, with a maximum age of 45 assigned corresponding to the accepted life expectancy of T. aduncus45. While minimum age may underestimate the true age of an individual first sighted as an adult, physical maturity (cessation of growth due to fusing of vertebral epiphyses) usually occurs between 10–15 years45,67,68. This ultimately supports the notion that growth in length would be minimal (or non-existent) for individuals first identified as adults.

#### Shark Bay

Dolphin food-provisioning has occurred in Monkey Mia since 1964, with some demographic data recorded prior to the commencement of governmental monitoring in the 1980s (Supplementary Table S5). The Shark Bay Dolphin Project (SBDP) has identified over 1,600 individual dolphins since 198469, making it the second-longest-running dolphin research program worldwide70. The estimated birth date of each SB dolphin was assigned to one of four accuracy categories, including day, week, month and year estimates. These categories represent age estimates accurate to within seven days, four weeks, one year and three years, respectively. For each SB dolphin sampled, minimum and maximum age estimates were calculated using the accuracy category assigned to that individual.

#### Stereo-laser photogrammetry system

Total length (tip of rostrum to tail notch) is a fundamental morphometric parameter in marine mammal life history studies71. With direct estimates of TL being difficult to acquire on free-ranging dolphins33, morphometric indices such as BH-DF are instead frequently employed to estimate TL, using well-established allometric relationships derived through stranding and post-mortem subjects15,29,33. Since the BH-DF region is regularly visible when a dolphin surfaces, the BH-DF measurement is the most practical means of estimating TL from boat-based platforms.

To estimate the BH-DF length of T. aduncus, a stereo-laser system was used in combination with a 12.3 MP Nikon D300s camera body equipped with a Nikon 80–400 mm f/4.5–5.6D ED lens. The laser system consisted of a custom-made aluminium block housing two Beamshot (Quarton USA Inc, USA; 5 mW; 532 nm) laser modules separated by a 10 cm distance and attached to the camera lens using a tripod mount (Fig. 6a)33. To improve safety around research personnel and dolphins, the laser system was activated and deactivated using an electronic switch box (see Cheney et al.33 for a detailed description of the laser system). The laser system was designed, manufactured and supplied by Barnacle Electronics, Scotland33. To ensure the lasers remained parallel at 10 cm apart, calibration photographs were taken at five incremental distances (5–25 m) before and after each boat survey. If the laser dots did not align with the reference points on the calibration board (Fig. 6b), adjustments were made by rotating the vertical and horizontal-configured grub screws on the laser modules.

#### Field methods

Laser-derived data were collected during boat-based photo-identification surveys, conducted in Bunbury and Mandurah between 2016 and 2017. In Shark Bay, laser-derived data were collected over a three-week period in October 2016, using both boat-based surveys and beach food-provisioning events. All boat-based surveys were conducted using small research vessels less than 5.5 m in length, powered by 60–100 hp outboard engines. Once a dolphin group was sighted, the research vessel was positioned parallel to the targeted group, at distances of 5–25 m. Photographs were taken of surfacing dolphins with the photographer using the camera autofocus point within the viewfinder to place the laser sights on the dorso-lateral surface of the dolphin. All sampled individuals were subsequently identified using dorsal fin photo-identification72 records specific to each population.

#### Image processing

To date, no means of quantifying the horizontal angle within single-camera photographs have been developed. Consequently, rigorous image selection was considered best practice in reducing horizontal angle error. To be included in this study, photographs were required to be in-focus, displaying both laser dots clearly, with the dolphin positioned as close to parallel to the camera as possible (Fig. 7; Supplementary Figs. S5 and S9). Additionally, the blowhole and dorsal fin of the dolphin needed to be unobstructed for both measurement and identification purposes.

All measurements were made using the free image processing software ImageJ73. The number of pixels between the medial point of the blowhole (BH) and the anterior origin of the dorsal fin (DF, BH-DF) were converted into centimetres, using the 10 cm scale signified by the distance between the two laser dots. The BH-DF lengths of individuals were converted into TL estimates using the linear relationship between BH-DF and TL derived from post-mortem individuals (Supplementary Fig. S1).

#### Growth curve analysis and estimation of growth parameters

To describe the growth of T. aduncus, four common non-linear growth functions were applied to derived LaA data. Length-at-age data for each region (SW and SB) were subset to estimate male, female and combined-sex asymptotic lengths (L) separately. The first two candidate models consisted of the Original (OvB, Eq. 1) and Typical (TvB, Eq. 2) forms of the von Bertalanffy growth function74. The OvB is considered more appropriate for marine mammals as length-at-birth (L0) can be estimated following a gestational period75. The final two candidate models included the Gompertz function76 (GOM, Eq. 3), and the Richards growth model77 (RGM, Eq. 4). The shape parameter (p) within the RGM enables the inflection point of the curve to be set anywhere between the range of minimum and maximum asymptote values, providing additional flexibility. The respective equations for these growth functions are as follows:

$${\rm{Original}}\,{\rm{von}}\,{\rm{Bertalanffy}}\,({\rm{OvB}}):\,{{\rm{L}}}_{t}={{\rm{L}}}_{\infty }-({{\rm{L}}}_{\infty }-{{\rm{L}}}_{0})\ast {{\rm{e}}}^{(-K\ast t)}$$
(1)
$${\rm{Typical}}\,{\rm{von}}\,{\rm{Bertalanffy}}\,({\rm{TvB}}):\,{{\rm{L}}}_{t}={{\rm{L}}}_{\infty }\,(1-{{\rm{e}}}^{(-K(t-{t}_{0}))})$$
(2)
$${\rm{Gompertz}}\,({\rm{GOM}}):\,{{\rm{L}}}_{t}={{\rm{L}}}_{\infty }\ast \,{\rm{e}}{(}^{-e({gi}(t-{t}_{0}))})$$
(3)
$${\rm{Richards}}\,({\rm{RGM}}):\,{{\rm{L}}}_{t}={{\rm{L}}}_{\infty }{(1-{{\rm{e}}}^{(-K(t-{t}_{0}))})}^{p}$$
(4)

where Lt denotes length-at-age t,

L is the asymptotic average length,

K is the Brody growth rate coefficient,

t0 is an artificial modelling artefact representing time when average length is zero,

t is a theoretical function of time and age,

L0 is the mean length at time zero (birth),

gi is the instantaneous growth rate at the inflection point; and

p is an artificial modelling artefact determining the shape of the curve.

To reduce the likelihood of overparameterization within the RGM, some starting values for the non-biological parameters ‘t0’ and ‘p’ were fixed to constant values78 (Supplementary Table S6). All candidate models were fit using the ‘FSA’79 and ‘nlstools’80 packages in R version 3.2.481 with median L estimates and 95% confidence intervals obtained through bootstrap resampling (1,000 iterations). Models were compared using Akaike Information Criterion (AIC)82, where each model was ranked relative to the best fitting model using corrected delta AIC values (AICc) in the ‘AICmodavg’ package83. Akaike model weights (wi) were used to determine which of the four growth functions provided the best fit for LaA data obtained on dolphins in the SW and SB regions.

#### Estimating age and length at independence and first reproduction

We estimated the mean length (L50) and age (A50) at which 50% of dolphins were predicted to be (a) independent, and (b) reproductive for the first time (females only). These estimates were fit separately for SW and SB regions using binomial generalized linear models (GLMs), with a logit link function. All GLMs were fit using the ‘FSA’79 and ‘car’84 packages in R. Logistic regression was used to calculate the respective L50 and A50 estimates, by applying Eq. (5) 85,86:

$${{\rm{A}}}_{\mathrm{50}}\,{\rm{and}}\,{{\rm{L}}}_{{\rm{50}}}=\frac{\mathrm{log}(\frac{p}{1-p})-{\rm{\alpha }}}{{{\rm{\beta }}}_{1}}$$
(5)

where p is the probability of ‘success’, 1 − p is the probability of ‘failure’, and; α and β are fitting constants85. To calculate L50 and A50 at independence, ‘success’ and ‘failure’ were defined as being ‘independent’ or ‘dependent’, respectively. For L50 and A50 at first reproduction, ‘success’ and ‘failure’ were defined as being ‘mature’ or ‘immature’, respectively. To quantify the uncertainty around the estimates, 95% confidence intervals were produced through bootstrap resampling (1,000 iterations).

Independence: Every individual with a laser-derived TL estimate and minimum age was assigned a binary value based on their dependence (‘0’) or independence (‘1’) status at the time of sampling. The status of each individual was determined using long-term demographic and sighting information specific to each population, including repeated sightings of calves in infant position, mother-calf association patterns and birth records of the mother. For example, individuals were assigned a value of ‘0’ if they were consistently sighted in infant position or sighted in close association with their mother (i.e. no repeated observations of separation between mother and calf). Individuals assigned a value of ‘1’ displayed marked changes in mother-calf association patterns, were sighted >5 times on their own or with other juveniles or their mother was repeatedly observed with a new sibling calf.

First reproduction: Combined samples of reproductively immature and mature female individuals were used to estimate the mean length and age at which 50% of females were predicted to reproduce for the first time. Using long-term demographic and sighting records, females with confirmed calving histories (i.e. repeated sighting of the female with a dependent calf) were assigned a value of ‘1’ and individuals never observed in close association with a calf were given a value of ‘0’.

#### Sensitivity analysis: accounting for measurement and age-estimation errors

We investigated the extent to which errors influenced the predicted LaA values. Errors were attributed to two factors: errors in laser-derived length measurements and uncertainty in age estimation of individual dolphins. Multiple laser photogrammetry studies have highlighted the potential influence that the horizontal angle between the photographer and the subject (i.e. dolphin) could have on measurement accuracy8,27,31. Thus, to examine the extent to which horizontal angle influenced laser-derived length estimates, we conducted a measurement error experiment using a three-dimensional dolphin replica model (T. aduncus courtesy of the Western Australian Department of Biodiversity, Conservation and Attractions). The model was rotated in 15° horizontal angle increments, ranging from perpendicular (hereafter referred to as 0°) to 75° from perpendicular to the camera (see Supplementary Fig. S9a). Eighteen photographs (with the laser dots positioned on the dolphin model) were taken at each angle increment, enabling the known BH-DF length of the replica model to be measured using nine non-sequential images. This process was repeated utilising five-metre distance increments between 5 and 25 m (Supplementary Fig. S9b). The laser-derived length measurement error estimate was calculated as the mean percentage error obtained at 15° from perpendicular (the greatest horizontal angle permissible in image selection). To assess the influence of age estimation error, we used the minimum and maximum age values assigned to each individual sampled in the study.

Bootstrap resampling was used to calculate median TL at ages 1, 3, 12, and 25 years, by resampling the laser-derived measurement (in cm) and age estimate (in years) of each individual by 1,000 iterations, and then re-fitting the chosen growth model. These simulated values were visually displayed in a square-shaped error distribution around each true data point, with the height and width representing maximum measurement and age-estimation error for each dolphin, respectively. From the output density distributions, 95% highest posterior density (HPD) intervals were calculated.

#### Regional differences in dolphin morphology

Intra-specific differences in total length were investigated using median LaA estimates obtained from the sensitivity analysis procedure. Using years 1, 3, 12, and 25, we compared dolphin length across periods of both rapid growth and growth approaching an asymptote, while also accounting for regions of the curve where physical maturity has been reached. Years 3 and 12 also coincide with major life history events for T. aduncus: independence (~3–4 years)50 and the attainment of sexual maturity (~10–12 years)45,50,71, respectively. The HPD distributions obtained from the sensitivity analyses were plotted against each other, as the degree of overlapping between the LaA distributions would indicate whether significant differences were present in each age class. Welch two sample t-tests were carried out on each age class (α = 0.05), using the median LaA estimates as the dependent variable.


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# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Definition 8.5.2.1. Let $\operatorname{\mathcal{C}}$ be a category. An idempotent endomorphism in $\operatorname{\mathcal{C}}$ is a pair $(X,e)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $e: X \rightarrow X$ is an endomorphism of $X$ which satisfies the identity $e = e \circ e$, so that we have a commutative diagram

$\xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{e} & \\ X \ar [ur]^{e} \ar [rr]^{e} & & X. }$

In this situation, we will also say that $e$ is an idempotent endomorphism of $X$.

format: plain


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• July 25th 2012, 05:51 AM
subhan2010
Hi all(Hi)(Hi), can someone please give description about the use of composite and inverse functions with examples...? any help will be appreciated (Clapping)
• July 25th 2012, 02:49 PM
pickslides
I like to think of composit functions as one function inside another.

Consider $f(x)= x^2$ and $g(x) = \cos x$ then $f(g(x)) = g(x)^2 = (\cos x)^2$ is a composite function

For an inverse function, think about what the inverse of an operation is. If I square something the inverse operation is a square root. Therefore when thinking about functions i.e. $f(x) =x^2$ then the inverse is $f^{-1}(x) = \sqrt{x}$
• July 25th 2012, 05:17 PM
HallsofIvy
"Square" and "square root" are not good examples here. For example, if x= -2, then $x^2= 4$ and then $\sqrt{4}= 2$, not -2. That's because $f(x)= x^2$ is not "one-to-one"- both $2^2$ and $(-2)^2$ give four- so it does not have a true "inverse". A better example would be "cube" and "cube root". For any real number x, $\sqrt[3]{x^3}= \left(\sqrt[3]{2}\right)^3$.


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## College Algebra (10th Edition)

RECALL: Absolute value is defined as: $|x| =x$ if $x \ge 0$ $|x| = -x$ if $x \lt 0$ Note that: (1) The absolute value of a positive number is positive. (2) The absolute value of 0 is 0. (3) The absolute value of a negative number is positive. Thus, for any real number $x$, $|x| \ge 0$ The given statement is true.


text:
Zum Inhalt springen
Autor
Rusty Laracuenti
Letzte Aktualisierung
7 years ago
Lizenz
Creative Commons CC BY 4.0
AbstraktAn attempt to create a compact introduction to Overleaf and LaTeX in general. Topics and examples are tailored to meet the needs of The Hudson School. This is a work in progress. The document works well as a PDF and most examples are included verbatim.


text:
# New paper in eLife

Kinetic competition during the transcription cycle results in stochastic RNA processing

Matthew L FergusonValeria de TurrisMurali PalangatCarson C ChowDaniel R Larson

Abstract

Synthesis of mRNA in eukaryotes involves the coordinated action of many enzymatic processes, including initiation, elongation, splicing, and cleavage. Kinetic competition between these processes has been proposed to determine RNA fate, yet such coupling has never been observed in vivo on single transcripts. In this study, we use dual-color single-molecule RNA imaging in living human cells to construct a complete kinetic profile of transcription and splicing of the β-globin gene. We find that kinetic competition results in multiple competing pathways for pre-mRNA splicing. Splicing of the terminal intron occurs stochastically both before and after transcript release, indicating there is not a strict quality control checkpoint. The majority of pre-mRNAs are spliced after release, while diffusing away from the site of transcription. A single missense point mutation (S34F) in the essential splicing factor U2AF1 which occurs in human cancers perturbs this kinetic balance and defers splicing to occur entirely post-release.

# New Papers

Two new papers are now in print:
The first is on applying compressed sensing to genomics is now published in Gigascience. The summary of the paper is here and the link is here.
The second is on steroid-regulated gene induction and can be obtained here.
Biochemistry. 2014 Mar 25;53(11):1753-67. doi: 10.1021/bi5000178. Epub 2014 Mar 11.

### Abstract

A gene induction competition assay has recently uncovered new inhibitory activities of two transcriptional cofactors, NELF-A and NELF-B, in glucocorticoid-regulated transactivation. NELF-A and -B are also components of the NELF complex, which participates in RNA polymerase II pausing shortly after the initiation of gene transcription. We therefore asked if cofactors (Cdk9 and ELL) best known to affect paused polymerase could reverse the effects of NELF-A and -B. Unexpectedly, Cdk9 and ELL augmented, rather than prevented, the effects of NELF-A and -B. Furthermore, Cdk9 actions are not blocked either by Ckd9 inhibitors (DRB or flavopiridol) or by two Cdk9 mutants defective in kinase activity. The mode and site of action of NELF-A and -B mutants with an altered NELF domain are similarly affected by wild-type and kinase-dead Cdk9. We conclude that Cdk9 is a new modulator of GR action, that Ckd9 and ELL have novel activities in GR-regulated gene expression, that NELF-A and -B can act separately from the NELF complex, and that Cdk9 possesses activities that are independent of Cdk9 kinase activity. Finally, the competition assay has succeeded in ordering the site of action of several cofactors of GR transactivation. Extension of this methodology should be helpful in determining the site and mode of action of numerous additional cofactors and in reducing unwanted side effects.

PMID: 24559102 [PubMed – indexed for MEDLINE]
PMCID: PMC3985961 [Available on 2015/2/21]

# New paper on genomics

James Lee and I have a new paper out: Lee and Chow, Conditions for the validity of SNP-based heritability estimation, Human Genetics, 2014. As I summarized earlier (e.g. see here and here), heritability is a measure of the proportion of the variance of some trait (like height or cholesterol levels) due to genetic factors. The classical way to estimate heritability is to regress standardized (mean zero, standard deviation one) phenotypes of close relatives against each other. In 2010, Jian Yang, Peter Visscher and colleagues developed a way to estimate heritability directly from the data obtained in Genome Wide Association Studies (GWAS), sometimes called GREML.  Shashaank Vattikuti and I quickly adopted this method and computed the heritability of metabolic syndrome traits as well as the genetic correlations between the traits (link here). Unfortunately, our methods section has a lot of typos but the corrected Methods with the Matlab code can be found here. However, I was puzzled by the derivation of the method provided by the Yang et al. paper.  This paper is our resolution.  The technical details are below the fold.

# New paper on neural networks

Michael Buice and I have a new paper in Frontiers in Computational Neuroscience as well as on the arXiv (the arXiv version has fewer typos at this point). This paper partially completes the series of papers Michael and I have written about developing generalized activity equations that include the effects of correlations for spiking neural networks. It combines two separate formalisms we have pursued over the past several years. The first was a way to compute finite size effects in a network of coupled deterministic oscillators (e.g. see here, herehere and here).  The second was to derive a set of generalized Wilson-Cowan equations that includes correlation dynamics (e.g. see here, here, and here ). Although both formalisms utilize path integrals, they are actually conceptually quite different. The first formalism adapted kinetic theory of plasmas to coupled dynamical systems. The second used ideas from field theory (i.e. a two-particle irreducible effective action) to compute self-consistent moment hierarchies for a stochastic system. This paper merges the two ideas to generate generalized activity equations for a set of deterministic spiking neurons.

# Paper on compressed sensing and genomics

New paper on the arXiv. The next step after the completion of the Human Genome Project, was the search for genes associated with diseases such as autism or diabetes. However, after spending hundreds of millions of dollars, we find that there are very few common variants of genes with large effects. This doesn’t mean that there aren’t genes with large effect. The growth hormone gene definitely has a large effect on height. It just means that variations of genes that are common among people have small effects on the phenotype. Given the results of Fisher, Wright, Haldane and colleagues, this was probably expected as the most likely scenario and recent results measuring narrow-sense heritability directly from genetic markers (e.g. see this) confirms this view.

Current GWAS microarrays consider about a million or two markers and this is increasing rapidly. Narrow-sense heritability refers to the additive or linear genetic variance, which means the phenotype is given by the linear model $y= Z\beta + \eta$, where $y$ is the phenotype vector, $Z$ is the genotype matrix, $\beta$ are all the genetic effects we want to recover, and $\eta$ are all the nonadditive components including environmental effects. This is a classic linear regression problem. The problem comes when the number of coefficients $\beta$ far exceeds the number of people in your sample, which is the case for genomics. Compressed sensing is a field of high dimensional statistics that addresses this specific problem. People such as David Donoho, Emmanuel Candes and Terence Tao have proven under fairly general conditions that if the number of nonzero coefficients are sparse compared to the number samples, then the effects can be completely recovered using L1 penalized optimization algorithms such as the lasso or approximate message passing. In this paper, we show that these ideas can be applied to genomics.

Here is Steve Hsu’s summary of the paper

Application of compressed sensing to genome wide association studies and genomic selection

(Submitted on 8 Oct 2013)

We show that the signal-processing paradigm known as compressed sensing (CS) is applicable to genome-wide association studies (GWAS) and genomic selection (GS). The aim of GWAS is to isolate trait-associated loci, whereas GS attempts to predict the phenotypic values of new individuals on the basis of training data. CS addresses a problem common to both endeavors, namely that the number of genotyped markers often greatly exceeds the sample size. We show using CS methods and theory that all loci of nonzero effect can be identified (selected) using an efficient algorithm, provided that they are sufficiently few in number (sparse) relative to sample size. For heritability h2 = 1, there is a sharp phase transition to complete selection as the sample size is increased. For heritability values less than one, complete selection can still occur although the transition is smoothed. The transition boundary is only weakly dependent on the total number of genotyped markers. The crossing of a transition boundary provides an objective means to determine when true effects are being recovered. For h2 = 0.5, we find that a sample size that is thirty times the number of nonzero loci is sufficient for good recovery.

 Comments: Main paper (27 pages, 4 figures) and Supplement (5 figures) combined Subjects: Genomics (q-bio.GN); Applications (stat.AP) Cite as: arXiv:1310.2264 [q-bio.GN] (or arXiv:1310.2264v1 [q-bio.GN] for this version)

# New paper on population genetics

James Lee and I just published a paper entitled “The causal meaning of Fisher’s average effect” in the journal Genetics Research. The paper can be obtained here. This paper is the brainchild of James and I just helped him out with some of the proofs.  James’s take on the paper can be read here. The paper resolves a puzzle about the incommensurability of Ronald Fisher’s two definitions of the average effect noted by population geneticist D.S. Falconer three decades ago.

Fisher was well known for both brilliance and obscurity and people have long puzzled over the meaning of some of his work.  The concept of the average effect is extremely important for population genetics but it is not very well understood. The field of population genetics was invented in the early twentieth century by luminaries such as Fisher, Sewall Wright, and JBS Haldane to reconcile Darwin’s theory of evolution with Mendelian genetics. This is a very rich field that has been somewhat forgotten. People in mathematical,  systems, computational, and quantitative biology really should be fully acquainted with the field.

For those who are unacquainted with genetics, here is a quick primer to understand the paper. Certain traits, like eye colour or the ability to roll your tongue, are affected by your genes. Prior to the discovery of the structure of DNA, it was not clear what genes were, except that they were the primary discrete unit of genetic inheritance. These days it usually refers to some region on the genome. Mendel’s great insight was that genes come in pairs, which we now know to correspond to the two copies of each of the 23 chromosomes we have.  A variant of a particular gene is called an allele.  Traits can depend on genes (or more accurately genetic loci) linearly or nonlinearly. Consider a quantitative trait that depends on a single genetic locus that has two alleles, which we will call a and A. This means that a person will have one of three possible genotypes: 1) homozygous in A (i.e. have two A alleles), 2) heterozygous (have one of each), or 3) homozygous in a (i.e. have no A alleles). If the locus is linear then if you plot the measure of the trait (e.g. height) against the number of A alleles, you will get a straight line. For example, suppose allele A contributes a tenth of a centimetre to height. Then people with one A allele will be on average one tenth of a centimetre taller than those with no A alleles and those with two A alleles will be two tenths taller. The familiar notion of dominance is a nonlinear effect. So for example, the ability to roll your tongue is controlled by a single gene. There is a dominant rolling allele and a recessive nonrolling allele. If you have at least one rolling allele, you can roll your tongue.

The average effect of a gene substitution is the average change in a trait if one allele is substituted for another. A crucial part of population genetics is that you always need to consider averages. This is because genes are rarely completely deterministic. They can be influenced by the environment or other genes. Thus, in order to define the effect of the gene, you need to average over these other influences. This then leads to a somewhat ambiguous definition of average effect and Fisher actually came up with two. The first, and as James would argue the primary definition, is a causal one in that we want to measure the average effect of a gene if you experimentally substituted one allele for another prior to development and influence by the environment. A second correlation definition would simply be to plot the trait against the number of alleles as in the example above. The slope would then be the average effect. This second definition looks at the correlation between the gene and the trait but as the old saying goes “correlation does not imply causation”. For example, the genetic loci may not have any effect on the trait but happens to be strongly correlated with a true causal locus (in the population you happen to be examining). Distinguishing between genes that are merely associated with a trait from ones that are actually causal remains an open problem in genome wide association studies.

Our paper goes over some of the history and philosophy of the tension between these two definitions. We wrote the paper because these two definitions do not always agree and we show under what conditions they do agree. The main reason they don’t agree is that averages will depend on the background over which you average. For a biallelic gene, there are 2 alleles but 3 genotypes. The distribution of alleles in a population is governed by two parameters. It’s not enough to specify the frequency of one allele. You also need to know the correlation between alleles. The regression definition matches the causal definition if a particular function representing this correlation is held fixed while the experimental allele substitutions under the causal definition are carried out. We also considered the multi-allele and multi-loci case in as much generality as we could.

# New paper on childhood growth and obesity

Kevin D Hall, Nancy F Butte, Boyd A Swinburn, Carson C Chow. Dynamics of childhood growth and obesity: development and validation of a quantitative mathematical model. Lancet Diabetes and Endocrinology 2013 .

You can read the press release here.

In order to curb childhood obesity, we need a good measure of how much food kids should eat. Although people like Claire Wang have proposed quantitative models in the past that are plausible, Kevin Hall and I have insisted that this is a hard problem because we don’t fully understand childhood growth. Unlike adults, who are more or less in steady state, growing children are a moving target. After a few fits and starts we finally came up with a satisfactory model that modifies our two compartment adult body composition model to incorporate growth. That previous model partitioned excess energy intake into fat and lean compartments according to the Forbes rule, which basically says that the ratio of added fat to lean is proportional to how much fat you have so the more fat you have the more excess Calories go to fat. The odd consequence of that model is that the steady state body weight is not unique but falls on a one dimensional curve. Thus there is a whole continuum of possible body weights for a fixed diet and lifestyle. I actually don’t believe this and have a modification to fix it but that is a future story.

What puzzled me about childhood growth was how do we know how much more to eat as we grow? After some thought, I realized that what we could do is to eat enough to maintain the fraction of body fat at some level, using leptin as a signal perhaps, and then tap off the energy stored in fat when we needed to grow. So just like we know how much gasoline (petrol) to add by simply filling the tank when it’s empty, we simply eat to keep our fat reserves at some level. In terms of the model, this is a symmetry breaking term that transfers energy from the fat compartment to the lean compartment. In my original model, I made this term a constant and had food intake increase to maintain the fat to lean ratio and showed using singular perturbation theory that his would yield growth that was qualitatively similar to the real thing. This then sat languishing until Kevin had the brilliant idea to make the growth term time dependent and fit it to actual data that Nancy Butte and Boyd Swinburn had taken. We could then fit the model to normal weight and obese kids to quantify how much more obese kids eat, which is more than previously believed. Another nice thing is that when the child stops growing the model is automatically the adult model!


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# dimod.generators.random_nae3sat¶

random_nae3sat(variables: , num_clauses: int, *, plant_solution: bool = False, seed: = None) [source]

Generate a random not-all-equal 3-satisfiability problem as a binary quadratic model.

Not-all-equal 3-satisfiability (NAE3SAT) is an NP-complete problem class that consists in satisfying a number of conjunctive clauses of three literals (variables, or their negations). For valid solutions, the literals in each clause should be not-all-equal; i.e. any assignment of values except (+1, +1, +1) or (-1, -1, -1) are valid for each clause.

Each clause contributes -1 to the energy when the clause is satisfied, and +3 when unsatisfied. The energy $$H(s)$$ for a spin assignment $$s$$ is thus lower bounded by $$E_{SAT}=-$$num_clauses, this lower bound matches the ground state energy in satisfiable instances. The number of violated clauses is $$(H(s) - E_{SAT})/4$$.

NAE3SAT problems have been studied with the D-Wave quantum annealer 1.

Parameters
• num_variables – The number of variables in the problem.

• num_clauses – The number of clauses. Each clause contains three literals.

• plant_solution – Create literals uniformly subject to the constraint that the all 1 (and all -1) are ground states satisfying all clauses.

• seed – Passed to numpy.random.default_rng(), which is used to generate the clauses and the variable negations.

Returns

A binary quadratic model with spin variables.

Example

Generate a NAE3SAT problem with a given clause-to-variable ratio (rho).

>>> num_variables = 75
>>> rho = 2.1
>>> bqm = dimod.generators.random_nae3sat(num_variables, round(num_variables*rho))

Note

The clauses are randomly sampled from the space of 3-variable clauses with replacement which can result in collisions. However, collisions are allowed in standard problem definitions, are absent with high probability in interesting cases, and are almost always harmless when they do occur.

Planting of a not all 1 solution state can be achieved with a spin-reversal transform without loss of generality. Planting can significantly modify the hardness of optimization problems.

1

Adam Douglass, Andrew D. King & Jack Raymond, “Constructing SAT Filters with a Quantum Annealer”, https://link.springer.com/chapter/10.1007/978-3-319-24318-4_9


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a field-theory motivated approach to computer algebra

## LaTeXForm

Changes the way in which expressions are displayed.
Changes the way in which expressions are displayed in the graphical interface. Example:
\del{#}::LaTeXForm("\partial"). ex:=\del_{m}(A);
$$\displaystyle{}\partial_{m}\left(A\right)$$
\del_{m}(A)
This prints $\partial_{m}(A);$ in the notebook, despite the fact that \del is not a LaTeX{} command. If you use this property to make a symbol printable, make sure to declare it before any other properties are declared, otherwise the notebook will not know how to display the symbol and produce an error message. Note that the property is attached to a pattern (\del{#} in this case) which matches the expression in which the replacement has to be made. If the pattern matches, the replacement will be done on the head symbol (\del in this case). A pattern \del without the argument wildcard # would only replace when \del occurs without any arguments (as in e.g. \del + A).
If you want more control over the way in which elements of your original expression are printed, you can use patterns like in the example below:
\ket{A??}{q??}::LaTeXForm("|", A??, "\rangle_{", q??, "}").
ex:=\ket{n+1}{a};
$$\displaystyle{}|n+1\rangle_{a}$$
\ket(n + 1 , a)
The comma-separated list of arguments to LaTeXForm will be concatenated into a single LaTeX output string, with patterns replaced by their normal output form.


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Thread: 0 divided by 0 View Single Post
Sci Advisor
P: 6,077
 Quote by HallsofIvy We might note that many text refer to "0/0" as "undetermined" rather than "undefined" because if you have a limit of a fraction where the numerator and denominator both go to 0, the actual limit itself can exist and can be anything.
There is a distinction between (1) lim f/g, where f -> 0 and g -> 0 and (2) 0/0. Case (1) is undetermined, case (2) is undefined.


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# [SVN] Passing variables to osascript, or other workaround?

Allan Odgaard allan at macromates.com
Thu May 12 03:33:23 UTC 2005

On May 12, 2005, at 0:53, Charilaos Skiadas wrote:

> Neat! It works great if I remove the ': $2' part above. If I have it > in though, it produces: > 200:201: syntax error: Expected “"” but found unknown token. (-2741) > > The part matched by$2 is basically the rest of the line where the
> \label occurs, so that the user has a context and can figure out
> better what label they want. So it is a whole LaTeX line, with lots of
> unescaped \'s. So I added the line:
> list="${list//\\\\/\\\\}" > above the res=osascript... > Now it works properly, but it took unacceptably long, I'm guessing > because the lines might be pretty long. Is there any way to speed it > up? Well, you could use sed to do the replacement, I did this benchmark (also for perl, ruby, and bash): time { for (( i = 0; i < 100; i++ )); do list='abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\a bed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abe d\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\ abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\ab ed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed\abed' list=$(sed -e <<<$list 's|\\|\\\\|g') done; } And you're certainly right about the shell being slow, here's the times on my 2.5 GHz G5: sed 1.020s perl 1.300s ruby 2.191s bash 11.008s Btw: if you want to experiment with this, just paste the loop into TM, select it, and press ctrl-R, then it'll run it within TM and output the result -- a bit easier than to play with this in a terminal window :) > I tried adding a line: list="${list:0:80}" before the above assignment
> to show only a reasonable part of the rest of the line (btw, can I
> combine them in one step? the obvious way didn't work.)

Bash rarely allow nested statements, what you can do is use commands
list=$(cut <<<$list -b-80|sed -e 's|\\|\\\\|g')

From the above benchmark, it seems that this also has better
performance ;)

Btw: I switched from ... to \$(...) because the former requires
escaping of \, and I switched from s/.../.../ to s|...|...| in the sed
substitution for cosmetic reasons.

> but that took me back to the previous error. Is there something else I
> might need to escape, maybe braces or something? (i.e. when I cut at
> the 80 characters, maybe I've left some braces unended?)

I don't think so -- but if you did the cut after escaping escapes, you
could cut in the middle of such an escape pair, and thus have a bad
string again.

> In terms of scope, this is to be used inside a \ref{} or an \eqref{}
> (can't remember if there are others). would the proper scope be
> text.latex, declaration.ref-or-label.latex or
> keyword.control.ref-or-label.latex ?

declaration.ref-or-label is most appropriate IMHO. The
keyowrd.control.ref-or-label would work, but only because the start
bracket is currently part of the keyword, so placing the caret just
after { would still give a scope of being in the keyword, but that
could change.

This of course assuming that the command is only useful inside a
\(eq)?ref, otherwise the scope should be text.latex. Limiting a scope
should mainly be done when it's an advantage, e.g. if you bind it to
'escape' then it's required that you put it inside the
declaration.ref-or-label scope, so that escape doesn't lose its value
outside this scope. But if you put it on some unused (for latex) key,
then there's probably little reason to not provide the feature in the
entire text.latex scope.

`


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Back to Course

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# Water Fundamentals Sample Lesson

## Sample Content

Rapid mixing is the instant and thorough dispersal of coagulants added to raw water stream to start the coagulation process.  Rapid mixing is very fast, with detention time as short as a few seconds, hence it is also called flash mixing.  Typical detention time in the rapid mix chamber could be as short 10 seconds and may not exceed 5 minutes…

Exercise 4.1

Find detention time for a 6 ft long, 6 ft wide, and 7 ft deep flash mix operating at 10 mgd flow rate.

V, Flash mix volume = 6 ft × 6 ft × 7 ft = 252 ft3

Q, Flow rate = 10 mgd × 1.547 $$\frac{ft^{3}}{s}$$$$\frac{1}{mgd}$$

= 15.47 $$\frac{ft^3}{s}$$

Detention time = $$\frac{\text{V}}{\text{Q}}$$ = $$\frac{\text{252 ft}^{3}}{\text{15.47 ft}^{3}}$$s = 16.3 s

You can compute flash mix detention time by entering values for parameters in red texts below:

 #fbuilder input[type=number], #fbuilder textarea, #fbuilder select { color: red; }
Lesson Content


text:
## 110.1 Short introductory articles

• Barbara Fantechi: Stacks for Everybody

• Dan Edidin: What is a stack?

• Dan Edidin: Notes on the construction of the moduli space of curves

• Angelo Vistoli: Intersection theory on algebraic stacks and on their moduli spaces, and especially the appendix.

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text:
# SSS Similarity

## Triangles are similar if their corresponding sides are proportional.

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Geometry Similarity

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When you compare two triangles, if the corresponding sides are in ratio of $\frac{5}{10}$$\dfrac{5}{10}$ and $\frac{8}{20}$$\dfrac{8}{20}$, then the triangles are similar by the SSS similarity test.

qid: 100274
Reviews


text:
# n lines in a plane

• Jun 21st 2008, 06:45 AM
anujgargcool
n lines in a plane
find the maximum number of points of intersection and number of regions formed by n lines in a plane(n=0 to 6)
• Jun 21st 2008, 10:59 AM
Soroban
Hello, anujgargcool!

Quote:

Find the maximum number of points of intersection and
number of regions formed by $n$ lines in a plane $(n=0\text{ to }6)$

. . $\begin{array}{c|c|c}
n & \text{Points} & \text{Regions} \\ \hline
0 & 0 & 1 \\ 1 & 0 & 2 \\ 2 & 1 & 4 \\ 3 & 3 & 7 \\ 4 & 6 & 11 \\ 5 & 10 & 16 \\ 6 & 15 & 22 \\ \vdots & \vdots & \vdots \end{array}$

See the pattern?

. . $\begin{array}{ccc}n(\text{Points}) &=& \dfrac{n(n-1)}{2} \\ \\[-3mm] n(\text{Regions}) &=& \frac{1}{2}(n^2+n+2) \end{array}$