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Social interaction with conspecifics can influence the developing brain and behaviour of the exposed animal. This experience can involve the exchange and retention of visual, chemical, acoustic and tactile signals. When several Drosophila melanogaster male flies are associated with mated females in the presence of food, they show frequent aggressive interactions. To measure the role of social experience on male–male interaction, two tester males – naïve or exposed to sibling(s) during a variable period of their adult development – were confronted in the absence of female and food. The two males displayed homosexual courtship and aggressive behaviours, the frequency, intensity and directionality of which varied according to their experience. The effect of social experience was greatly enhanced between transgenic males partially defective for pheromonal perception, indicating that male inhibitory pheromones are normally used to repress male–male interaction.

Social experience can change the development of the nervous system and the behaviour of vertebrates and invertebrates alike(Shors et al., 2001; van Praag et al., 2002; Lledo and Gheusi, 2003; Eliassen et al., 2003; Fahrbach et al., 2003; McRobert et al., 2003; Wommack et al., 2003). Interaction with conspecifics involves multiple sensory signals that need to be retained by individuals who can predict the significance of these stimuli in relation with their reproduction, feeding and survival(Engel and Hoy, 1999; Schaal et al., 2000; Wilson and Stevenson, 2003). Experience-dependent change in social behaviour has been studied in the fruitfly Drosophila melanogaster, with a particular emphasis on courtship behaviour (Siwicki and Ladeswki,2003). Male courtship of a virgin female can decline after conditioning with a mated female. Conversely, young males that are exposed to older mature males during early adulthood show increased heterosexual courtship and mating propensity once they became mature(McRobert and Tompkins, 1988). Conversely, mature males presented during 30 min to young immature males will learn to suppress this misdirected courtship for up to 4 h(Gailey et al., 1982). Early imaginal visual deprivation has been shown to affect mate choice(Hirsch et al., 1995). However, no other sensory modality has been shown to be involved in the modulation of experience-dependent behaviour although several experiments suggest that exposure to pheromonal and acoustic cues during development can change sexual behaviour (Gailey et al.,1982; McRobert and Tompkins,1988).

Apart from sexual behaviour, Drosophila females can change their interactions after encountering other females and show aggressive behaviour(Kamyshev et al., 2002; Nilsen et al., 2004), whereas males can display territorial and aggressive behaviours. The occurrence of these two male behaviours overlap because territoriality often results in aggressive interaction, and both are preferentially induced in the presence of mated females with food (Hoffmann and Cacoyianni, 1990; Chen et al.,2002). Furthermore, during aggressive interactions, males very often display behavioural elements similar to those observed during heterosexual courtship (Hoffmann,1987; Chen et al.,2002). Territorial behaviour has been shown to change after experience with conspecifics because males held in isolation act more aggressively than males held in groups(Hoffmann, 1990). However, the interpretation of these data was complicated by the presence of multiple flies tested during an extended period of observation. Recently, the measure of aggression was standardized with only two males allowed to fight(Chen et al., 2002).

Here, we tested pairs of transgenic D. melanogaster male flies defective for one set of sensory structures potentially involved in pheromonal perception (Xu et al., 2002). In the absence of food and of mated females, we found that the variation of social experience (e.g. exposure to one or several siblings) during adult life affected the nature and strength of the behavioural interaction between two transgenic mature males. Control males tested in similar conditions showed a much weaker interaction.

### Strains and crosses

D. melanogaster (Meigen) strains were kept in glass vials (150 ml)containing roughly 30 cm3 standard cornmeal and yeast medium at 25°C under a 12 h:12 h dark:light cycle, with 65±5% humidity. Canton-S (Cs) is a widely used wild-type strain. Dijon 2000 (Di2),another wild-type strain, was initiated with five pairs of flies caught during the year 2000 in an orchard at Dijon, France. CheB42a-Gal4; UAS-GFPis a transgenic strain homozygous for the two transgenes inserted on chrosomosome 3: CheB42a-Gal4 contains the promoter of CheB42a, a male-specific gene expressed only in few taste sensilla of the tarsa (Xu et al., 2002)fused to the yeast Gal4 sequence; UAS-GFP is the reporter transgene specifically targeted by Gal4 and used to visualize directly the expression of CheB42a. For the sake of clarity, this strain and the CheB42a-Gal4 transgene are named B42'. Two other strains containing the CheB42a-Gal4 transgene (but not UAS-GFP) inserted in different chromosomal positions were also tested(the three B42 lines were kindly provided by C. Pikielny). To produce B42/Di2 (or B42/Cs) males, five B42 virgin females were mated with five Di2 (or Cs) males, and their F1 male progeny collected. We also tested two other male genotypes: B42 homozygotes and B42 ×UAS-grim (grim is a pre-apoptosis gene that deletes B42-Gal4-expressing cells; Wing et al., 1998).

### Grouping procedure

After a light CO2 anesthesia, 1–3 h after eclosion, tester male flies were kept either isolated or grouped with varying numbers of same-age siblings in a fresh food vial for a controlled period of time(×2=with one; ×5=with four; ×10=with nine). At the end of the grouping period, which generally took place at 14:00 h, tester males were individually aspirated into a fresh food vial. Males grouped until 5 days old were isolated 1 h before the test. In Figs 1, 2, 3, 4, the grouping period' is shown by a black-filled bar above the histogram, and the open bar represents the isolation period'. To distinguish males, wing clipping – equally distributed for each treatment – was performed with a small pair of iris scissors (#14558-11; FST, Heidelberg, Germany). This operation induced no detectable effect on male behaviour (data not shown).

Fig. 1.

The strength and directionality of male–male interaction varies with male social experience. In each test, two tester males of similar age (5 days old) and genotype (B42/Di2) were paired, and their reciprocal behavioural interaction noted during 10 min. For each test, the duration and nature of social experience of each of the two males is shown above the histograms (the experience line', which is shown as a bar). Each experience line is divided into five segments representing the 5 days of adult life (from eclosion to the test); a black-filled bar represents grouping with four other same-age and genotype siblings; an open bar represents the period during which the tester male was kept in isolation. The numbers shown above each experience line indicate the mean of the behavioural index (BI ± s.e.m.) for each treatment, and the statistical significance between treatments is represented above the middle bar(P<0.001; *P<0.05; NS,not significant). The histograms indicate the frequency for pairs of males according to their BI difference: the two bars on the left represent the cases of interactions where the left' male directed a higher BI (differences >10 and >55) towards the right' male than the reciprocal situation; and vice versa for the two bars shown on the right (see also Materials and methods). The middle bar (difference <10) represents the cases with low or no male–male interaction. N=30–54 except in (C)(120).

Fig. 1.

The strength and directionality of male–male interaction varies with male social experience. In each test, two tester males of similar age (5 days old) and genotype (B42/Di2) were paired, and their reciprocal behavioural interaction noted during 10 min. For each test, the duration and nature of social experience of each of the two males is shown above the histograms (the experience line', which is shown as a bar). Each experience line is divided into five segments representing the 5 days of adult life (from eclosion to the test); a black-filled bar represents grouping with four other same-age and genotype siblings; an open bar represents the period during which the tester male was kept in isolation. The numbers shown above each experience line indicate the mean of the behavioural index (BI ± s.e.m.) for each treatment, and the statistical significance between treatments is represented above the middle bar(P<0.001; *P<0.05; NS,not significant). The histograms indicate the frequency for pairs of males according to their BI difference: the two bars on the left represent the cases of interactions where the left' male directed a higher BI (differences >10 and >55) towards the right' male than the reciprocal situation; and vice versa for the two bars shown on the right (see also Materials and methods). The middle bar (difference <10) represents the cases with low or no male–male interaction. N=30–54 except in (C)(120).

Fig. 2.

The effect of social experience is not affected by the number of siblings,but changes with exposure during the days preceding the test. Four same-age siblings were grouped with the tester male during social exposure unless noted×2 (with one sibling; A–D), or ×10 (with nine siblings;C,D). All tested flies were 5 days old B42/Di2 males. N=22–35, except B (76), C (73) and F (16). For all other parameters, refer to Fig. 1. P<0.01.

Fig. 2.

The effect of social experience is not affected by the number of siblings,but changes with exposure during the days preceding the test. Four same-age siblings were grouped with the tester male during social exposure unless noted×2 (with one sibling; A–D), or ×10 (with nine siblings;C,D). All tested flies were 5 days old B42/Di2 males. N=22–35, except B (76), C (73) and F (16). For all other parameters, refer to Fig. 1. P<0.01.

Fig. 3.

The isolation period, but not the duration of social exposure, influences the strengh of male–male interaction. All tester males were grouped for different periods of their imaginal development with four other same-age siblings. All flies were 5 days old B42/Di2 males. N=56–82, except C (101), and D (35). For all other parameters,refer to Fig. 1.

Fig. 3.

The isolation period, but not the duration of social exposure, influences the strengh of male–male interaction. All tester males were grouped for different periods of their imaginal development with four other same-age siblings. All flies were 5 days old B42/Di2 males. N=56–82, except C (101), and D (35). For all other parameters,refer to Fig. 1.

Fig. 4.

Male genotype influences the effect of social experience. All tester males were 5 days old, and were from (A) the Canton-S strain (Cs), (B) the Dijon2000(Di2) strain, or (C,D) from the cross between a B42 female and either a Cs male (B42/Cs; C,D left), or a Di2 male(B42/Di2; D, right). N=28–57. For all other parameters, refer to Fig. 1.‡ P<0.01.

Fig. 4.

Male genotype influences the effect of social experience. All tester males were 5 days old, and were from (A) the Canton-S strain (Cs), (B) the Dijon2000(Di2) strain, or (C,D) from the cross between a B42 female and either a Cs male (B42/Cs; C,D left), or a Di2 male(B42/Di2; D, right). N=28–57. For all other parameters, refer to Fig. 1.‡ P<0.01.

### Behavioural tests

All assays were performed on 5-day-old males, 1–5 h after lights on(between 9:00 h and 13:00 h). In all tests, a male was aspirated into the observation chamber (2.8 cm diameter, 0.5 cm high). 10 min later, a second male (for male–male interaction), or two decapitated objects (for discrimination tests) was (were) introduced. The observation period lasted 600 s. The order in which the two intact males were introduced was randomized. We noted the occurrence and duration of various male behaviours that included sequences very similar to those shown during heterosexual courtship (tapping,wing vibrating, licking and rare attempted copulation, for the two types of tests; O'Dell, 2003) as well as chasing and aggressive behaviours (tussling, lunging; Hoffman, 1987; Chen, 2002) between the two intact males. Aggressive episodes were noted in about 5% of cases.

The total percentage of time that each male spent directing a behaviour(courtship or aggressive) is the behavioural index (BI). However, we did not take into account refusal behaviours like wing flicking and jumps that were often performed by courted intact males. For male–male interactions, we calculated the BI difference as follows: if an isolated A' male behaved towards a grouped B' male during a total of 120 s (BI=20), and if B acted towards A during 30 s (BI=5), the BI difference (15, in favour of A) is represented under the isolated males, between 10 and 55 (as the second bar from the left shown on Fig. 1A). When both males showed similar BIs, their BI difference was very low (or equal to 0'), and these values made up for the central bar of the histogram. For most pairs represented in the central bars of all figures(N=538), males showed either no BI (0' for both males=67%), or very low BI (0 < BI × 10; 29%). In only 4% of the cases, at least one of the two males showed a substantial behaviour (BI>10). Males of the control Di2 and Cs strains generally showed very little physical contact,with the exception of a few brief tapping episode that occurred at the beginning of the observation period. This also explains the high central histogram bar noted for pairs of wild-type males(Fig. 4A,B). In many cases,males jumped away to avoid physical contact.

For the discrimination test, we measured the duration of courtship behaviour directed towards each of the two decapitated sex-objects (BI1/BI2). Decapitation of sex objects allowed us to standardize behavioural observations and to prevent flies producing most of their acoustic signals(Ferveur et al., 1995). Some observations were carried out under a red light (25 W with a Kodak Safe-light filter no. 1) to remove visual stimuli(Boll and Noll, 2002).

### Statistics

To test for the directionality of male interactions, the BI of the two series of confronted males were compared using a Mann Whitney test (Statview 4.0). The BIs of the two confronted flies can be considered as independent parameters given that each male can direct or receive a behavioural action, or not interact. The comparison between two experimental sets (each one involving two males) was carried out with a two-factor ANOVA to evaluate the role of social experience, and of the number of siblings or of male genotype. The significance of the probability shown here represents the interaction between these two factors.

For the discrimination experiment, a Student's t-test was used to compare the intensity of behaviour directed towards each sex-object, after testing the normality of the data distribution with an F test.

In a preliminary observation, we found that naive 5-day-old B42/Di2 male flies carrying one copy of the B42 transgene(B42/Di2 males result from the cross between B42 mothers and Di2 fathers) showed high levels of homosexual behaviour(CI=38.5±2.3; N=94). Males of two other related genotypes also showed high levels of homosexual courtship: B42 homozygotes(26.4±7.1; N=12), and B42/UAS-grim where B42 drives the expression of grim, a death cell gene(19.3±3.9; N=16). We chose to use B42/Di2 males because (1) of their heterozygous genotype combining the wild-type Di2 genetic background, and (2) because they showed a stronger male homosexual CI than B42/UAS-grim (d.f.=119, 2; F=6.207; P=0.0027). All results shown on Figs 1, 2, 3 were obtained with B42/Di2 males.

### The perception of male pheromones is altered in transgenic males

The B42 transgene contains the promoter of the CheB42amale-specific factor, which is secreted by a small number of non-neuronal cells associated with several chemosensory hairs carried by the male frontal legs (Xu et al., 2002). Because B42 is potentially involved in pheromonal perception, the discrimination of B42/Di2 males was measured towards two decapitated control flies – a female and a male – that were simultaneously presented for 10 min (Table 1). Discrimination experiments were performed either in red light to remove all visual stimuli or in white light. In red light, B42/Di2 males could not discriminate the sex of control flies (P=0.19), whereas sexual discrimination was possible in white light (P=0.015). This contrasts with wild-type subject males, which were able to discriminate sex-object both in red light and in white light (P<0.001).

Table 1.

Pheromonal discrimination is affected in B42/Di2 males

Genotype of tester male
B42/Di2
Cs
Decapitated object flyRed lightWhite lightRed lightWhite light
Cs female 41.3 52.3 43.3 46.8
BI1 (5.5) (6.0) (4.4) (4.2)
Cs male 31 31.6 6.9 7.3
BI2 (5.6) (5.7) (1.8) (1.7)
t 1.31 2.5 7.67 8.68
P 0.19 0.015 <0.001 <0.001
Genotype of tester male
B42/Di2
Cs
Decapitated object flyRed lightWhite lightRed lightWhite light
Cs female 41.3 52.3 43.3 46.8
BI1 (5.5) (6.0) (4.4) (4.2)
Cs male 31 31.6 6.9 7.3
BI2 (5.6) (5.7) (1.8) (1.7)
t 1.31 2.5 7.67 8.68
P 0.19 0.015 <0.001 <0.001

Single tester males of transgenic (B42/Di2) and control (Cs)genotype were simultaneously presented to two decapitated objects flies– a female and a male – of the Cs strain, and the duration of their behavioural index was measured towards each object fly (BI1, BI2). The observation, carried out in red light or in white light, lasted 10 min. Data shown are mean BIs (±s.e.m.) for N=42–56 tests. The statistical difference between BI1 and BI2 was tested with a Student's t-test, and the level of significance is shown below.

### Social experience decreases male–male interaction

The aim of this study was to determine whether social experience during adult development can influence behavioural interactions between differently exposed male flies. Social experience with sibling male(s) clearly changed the pattern of interaction between two hybrid male flies of similar genotype(B42/Di2) and age (5 days old). The effect of experience is revealed by the comparison of two extreme situations (shown on the top and bottom of Fig. 1). Fig. 1A represents the confrontation between a male kept isolated throughout his adult life (left)and a second male that was held grouped during his adult life with four same-age sibling males (right). The left two bars of the histogram indicate that most isolated males directed an intense behaviour towards grouped males(the reciprocal situation was not observed) during the 10 min observation period. Isolated males frequently showed behavioural sequences similar to those displayed during heterosexual courtship: sustained unilateral wing vibration while chasing, and attempts to lick the genitalia. In response,grouped males often flicked both wings simultaneously and jumped away indicating that they refused the advance of the chaser(Paillette et al., 1991). The central bar of the histogram indicates the absence of a strong directional behaviour, caused by a mutual neutralization, in 27% of the pairs (see Material and methods). The total amount of time spent in directional behaviour(behavioural index = BI, with its mean ± s.e.m. shown above each treatment) indicates the high activity of isolated males (56±6)and the absence of activity in grouped males. In this case, the difference between the BIs of the two males was highly significant (P<0.001;see † between treatments above the central bar). By contrast, Fig. 1Frepresents the confrontation between two males held in isolation during all adult development(from eclosion to test). These naive males showed strong and reciprocal interactions consisting of courtship elements (wing extension and flickering)sometimes combined with sequences characteristic of aggressive behaviour(lunging and tussling; Hoffman, 1987). In this case, the BIs shown by the two males (31±6 and 37±6) were not significantly different(P>0.05). No difference of directionality was detected between males and only 19% pairs showed a mutual neutralization.

To outline the period of imaginal development during which social experience with siblings can influence homosexual interaction, tester males were held with siblings for various periods of adult development. The experiments shown on Fig. 1always involved a male raised in total isolation (left) with a second male held grouped between eclosion and 1–4 days of adult development. Isolated males directed a strong directional BI towards males that were grouped at least 2 days after eclosion. The behavioural directionality and the difference of BIs increased in favour of the isolated male as a function of the duration of grouping.

### The number of siblings has no effect on experience

To determine whether the frequency or intensity of social experience can influence male–male interaction, we varied the number of siblings in theconditioning groups' (×2, ×10; Fig. 2A–D) and compared the effect that they induced with that of ×5 groups(Fig. 1). Isolated tester males similarly behaved towards ×2 or ×5 males grouped for 5 days(Fig. 2A and Fig. 1A; d.f.=159; F=0.183; P=0.67). Isolated males also behaved very similarly towards ×10 and ×5 males grouped for 3 days after eclosion(Fig. 2B and Fig. 1C; d.f.=391; F=0.034; P=0.86). The confrontation between ×2 and×10 males grouped for either 3 or 5 days after eclosion induced weak male–male interaction without directionality(Fig. 2C,D). Instead, the frequency of mutual neutralization increased with the duration of grouping(59% for 3 days; 91% for 5 days).

### The effect of experience varies with the exposure period...

The experiment shown in Fig. 2D suggests that two males grouped with siblings until the day of the test very frequently neutralized their mutual interactions. To measure the effect of social exposure during the days preceding the test, one tester male grouped ×5 during all adulthood was confronted with a male grouped×5 that was isolated either 1, 2 or 3 days before the test (respectively shown on the right and left sides of the histograms of Fig. 2E–G). Social experience induced significant effect during the 2 days preceding the test because 36% males isolated 1 day before the test chased constantly grouped males. This frequency only slightly increased, together with the BIs of the males that were isolated earlier. In these experiments, constantly grouped males rarely chased temporarily grouped males.

### ...but not with its duration

To distinguish the effect of the isolation period (the lap between isolation and test) from the effect of experience duration, males to be confronted were grouped at different ages but isolated on the same day(Fig. 3A–E). The frequency of mutual neutralization increased when the isolation period decreased: neutralization was very frequent when the isolation period was shorter than 3 days. These data also indicate that the total duration of grouping had no influence on the directionality of male–male interaction. Interestingly, males held during either one or 3 days after eclosion showed low BIs that were not significantly different even if a tendency was noted in favour of males isolated earlier(Fig. 3F).

### The effect of experience varies with male genotype

To test the incidence of the B42 transgene on the intensity and directionality of male–male interaction, various male genotypes were compared. Unlike B42/Di2 males (Figs 1A, 2A), no or very little interaction was observed between grouped and isolated males of the two wild-type strains Canton-S (Cs) and Dijon2000 (Di2; Fig. 4). However, a significant effect (P=0.0018) was detected between isolated and grouped Cs males. When wild-type males were paired, they were generally very active and strongly avoided mutual physical contact with the exception of rare and brief tapping episodes.

B42/Cs males (containing the B42 transgene in the Cs genetic background) showed a high behavioural directionality between isolated and grouped males. However, the directionality of this interaction was weaker than that shown by B42/Di2 males raised in the same social conditions(Figs 1A, 4C; d.f.=193; F=9.773; P=0.002). Moreover, the pairing of one naive B42/Di2 male with a naive B42/Cs male induced an interaction that was similar to that observed between two naive B42/Di2 males (Figs 1F, 4D; d.f.=175; F=2.296; P=0.131). Finally, two independent transgenic lines carrying a single copy of the B42–Gal4 transgene also induced a strong directional behaviour between isolated and grouped males (data not shown).

We found that the nature and strength of the first behavioural interaction between two males depends upon both their genotype and experience. D. melanogaster male and female flies have a relatively limited social repertoire that mostly consists of aggressive and sexual interactions(Dow and von Schilcher, 1975; Chen et al., 2002; Nilsen et al., 2004). Environmental parameters can also change social interaction, because inter-male aggressive behaviour is particularly strong in the presence of a mated female and food (Skrzipek et al.,1979; Hoffman, 1987). When two males of similar age and genotypes were paired in a small chamber without mated females nor food, they induced a mixture of homosexual and aggressive behaviours. The proportion and intensity of these behaviours varied according to their mutual social experience: a strong courtship-like behaviour was always directed by the naive male to the experienced male, and aggressive sequences were mostly observed between two naive males.

Among the parameters that contributed to social experience, the most significant was the exposure period during adult development. The highest proportion of mutual neutralization occurred when both tester males were isolated <3 days before the test. When the two males were isolated earlier,their interactions were strong but non-directional. Male–male behaviour was directional only between (1) an isolated male towards a male grouped at least during the 3 days after eclosion, and (2) a male isolated at least 1 day before the test towards a constantly grouped male. However, the total duration of the social experience and the size of the group (2–10 siblings) had no effect.

The fact that social exposure during different developmental periods induced a strong behavioural directionality (between B42/Di2 males)suggests that associative and/or non-associative learning processes could be involved. Previous studies revealed that a 30 min exposure period induced an experience-dependent courtship modification (EDCM) that lasted for 4 h(Gailey et al., 1982), whereas a 5 h longassociative training induced a change that lasted >9 days(McBride et al., 1999). Given that with our procedure the exposure period lasted more than 24 h and its effect was still detected 3–4 days later, both learning mechanisms,as well as a remodelling of the nervous system, could be involved. We observed that naive B42/Di2 males strongly and mutually interact during the grouping period (data not shown). It is possible that, after at least 24 h of this conditioning, exposed males became habituated and avoid the sensory signals produced by sibling males. The hypothesis of a reinforced EDCM is consistent with the fact that the frequency of mutual neutralization increased with the age of exposure. However, the habituation process by itself cannot explain why exposed males showed no directional interaction and relatively low BIs, 3–4 days after their isolation(Fig. 3E,F). This relatively long from-isolation-to-test lapse of time suggests that the quality of male–male interactions could be affected by long-term memory involving associative learning.

Why did social exposure produce a strong effect in males carrying a copy of the B42 transgene, but not in wild-type males? If the two naive wild-type males confronted in a small observation chamber systematically avoided physical contact, it is probably because they constantly exchanged strong aversive stimuli. The moderate but significant difference detected between the two control strains, and also between B42/Di2 and B42/Cs males, indicates that one or several modifier gene(s) of the background can modulate the effect of experience. We believe that B42/Di2 males showed much stronger physical interaction probably because of their defective perception of inhibitory male pheromones. This interpretation is supported by the fact that the sensory structures, where B42 is expressed, are required to detect the pheromones of conspecific flies (Robertson,1983; Balakireva et al.,1998; Xu et al.,2002; Bray and Amrein,2003). The altered behaviour of B42/Di2 males could result from the toxic effect of Gal4, or GFP, or of both products, in the B42–Gal4-expressing cells, causing a similar effect to the grim gene that killed the same cells. If it is true, it means that the B42 transgene affects a function of the targeted subset of tarsal hairs that normally allow male flies to detect inhibitory pheromones of conspecific males (Ferveur and Sureau,1996). Alternatively, could the interaction between males be the result of their pheromonal difference after the passive transfer of cuticular hydrocarbons during conditioning? This hypothesis can be ruled out because (1)most hydrocarbons transferred by rub-off are eliminated from the cuticle after few hours (Scott and Jackson,1990), and (2) males that were either kept with one or with nine siblings induced no behavioural difference. Therefore, if social experience can theoretically affect all males, its effect is better measured in males with defective sensory perception.

Male inhibitory pheromones, which strongly prevent homosexual interaction between wild-type males, can lead to intense fighting episodes in the presence of mated females and food (Hoffman, 1987; Chen et al., 2002). This shows that male inhibitory pheromones act in a context-dependent manner, and we hypothesize that when these substances are perceived simultaneously with chemical signals inducing rewarding effects (female pheromone, food odors),they can elicit male–male antagonistic behaviours. Although it remains to be shown that sensory integration of different pheromonal inputs can change the release of substance(s) that affect neural function, certain neuromodulators present in the central nervous system can precisely control male courtship and aggressive behaviours(Yellman et al., 1997; Neckameyer, 1998; Lee and Hall, 2001; Baier et al., 2002). We note that the experimental manipulation of β-alanine changes male aggressivity(Jacobs, 1978), and this effect could be partly caused by male defective visual acuity(Baier et al., 2002). This makes an interesting analogy with our data because both observations suggest that the behavioural interaction can be changed between males that are defective for either modality of sensory perception.

In conclusion, the strong directional interaction observed between two B42 transgenic males – a naive male chasing an experienced male– may result from their very different experience: the naive male courts because he is not (very) inhibited by the pheromones of the exposed male who escapes because he retains a negative experience of the repulsive signals produced by sibling males.

We wish to thank C. Pikielny for the generous gift of several CheB42a lines before publication, M. Cobb, C. Pikielny, one reviewer and members of the lab for fruitful discussions and comments on the manuscript. Funding was partly provided by the Burgundy Region and the Centre National pour la Recherche Scientifique (CNRS).

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text:
## torsdag den 10. januar 2013

### Möbius ring

My wife asked for a ring shaped like a Möbius band. I thought it would be a good first try at making something, so I started making it in openSCAD.

On left: Prototype 1: The first thing I have designed and printed myself! On right: Prototype 2.

All prototypes was printed with 0,27 mm layer thickness.

The first prototype had a radius of 1 cm, and was a bit too narrow for her finger. It was increased to 1.2 cm in prototype 2. Prototype 1 was 3 mm high, which was doubled in prototype 2.

The second prototype was a bit to wide, so the third prototype had 1.1 cm radius. I also disliked the uneveneness of the turn, so I made it more even. However, in order for the openSCAD rendering not to take FOREVER I could not just increase the segment number. I now had to make it in three parts. There must be an easier way, but I can't find it right now. openSCAD doesen't seem to like for(i=XXX), where XXX is a vector equation, and I can't define the segment size for each loop, as variables are global. I suppose I could make a vector of segment sizes, but I cant find how to get an arbitrary value out of a vector.

Third prototype was beautiful, according to both me and my wife, so I rendered it again with 0.1 mm layer thickness.

Final ting would not stick to build surface, so a raft was added.

Left: Prototype 3. Right: Final ring

format: plain


text:
# Exercise 11.3.5

$\star$ Define a family $\mathscr{H}$ of hash functions from a finite set $U$ to a finite set $B$ to be $\epsilon$-universal if for all pairs of distinct elements $k$ and $l$ in $U$,

$$\Pr\{h(k) = h(l)\} \le \epsilon$$.

where the probability is over the choice of the hash function $h$ drawn at random from the family $\mathscr{H}$. Show that an $\epsilon$-universal family of hash functions must have

$$\epsilon \ge \frac{1}{|B|} - \frac{1}{|U|}$$


text:
# Finding $\lim_{x \to \infty} \left(\frac{a_{1}^{1/x}+a_{2}^{1/x}+\cdot\cdot\cdot{a_{n}}^{1/x}}{n}\right)^{nx}$

For non zero positive reals $a_{1},a_{2}\cdot\cdot\cdot a_{n}$ how to find $$\lim_{x \to \infty} \left(\frac{a_1^{1/x} +a_2^{1/x}+\ldots +a_n^{1/x}}{n}\right)^{nx}?$$

It becomes indeterminant form $1^{\infty}.$ But difficult to solve by L'Hospital's Rule. By using A.M.-G.M. inequality it comes that limit is $\geq a_{1}a_{2}\cdot\cdot\cdot a_{n}.$ I also tried by using Squeeze. Please help.Thanks.

• Is it $a_1...a_n$ – Archis Welankar Sep 12 '16 at 16:59
• take ln of the expression would help. – Qing Zhang Sep 12 '16 at 17:00
• @ArchisWelankar That is not really helpful, is it? Answering "How to find the limit" by "here is the limit." – Clement C. Sep 12 '16 at 17:12
• As you have given the answer so i didnt give it – Archis Welankar Sep 13 '16 at 5:21
• You can given another short and nice answer... – neelkanth Sep 13 '16 at 5:42

For $a_1,\dots, a_n$ positive numbers, and $x>0$, $$\lim_{x\to\infty}\left(\frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\cdots+{a_{n}}^{\frac{1}{x}}}{n}\right)^{nx} = \prod_{k=1}^na_{k}$$
Proof. Detailed approach, where we use the (low-order) Taylor expansions $e^u=1+u+o(u)$ and $\ln(1+u)=u+o(u)$ when $u\to 0$.
You can rewrite \begin{align} \left(\frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\cdots+{a_{n}}^{\frac{1}{x}}}{n}\right)^{nx} &= \exp\left(nx \ln \frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\cdots+{a_{n}}^{\frac{1}{x}}}{n}\right) = \exp\left(nx \ln \frac{1}{n}\sum_{k=1}^na_{k}^{\frac{1}{x}}\right)\\ &= \exp\left(nx \ln \frac{1}{n}\sum_{k=1}^n e^{\frac{1}{x}\ln a_{k}}\right) \end{align} Since $n$ is a constant and $\frac{1}{x}\ln a_{k}\xrightarrow[x\to\infty]{} 0$ for each $k$, we have $$e^{\frac{1}{x}\ln a_{k}} = 1 + \frac{1}{x}\ln a_{k} + o\left(\frac{1}{x}\right)$$ for each $k$ as $x\to\infty$, and therefore \begin{align} \left(\frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\cdots+{a_{n}}^{\frac{1}{x}}}{n}\right)^{nx} &= \exp\left(nx \ln \frac{1}{n}\sum_{k=1}^n e^{\frac{1}{x}\ln a_{k}}\right)\\ &= \exp\left(nx \ln\left( 1+ \frac{1}{nx}\sum_{k=1}^n\ln a_{k}+ o\left(\frac{1}{x}\right)\right)\right) \\ &= \exp\left(nx \left( \frac{1}{nx}\sum_{k=1}^n\ln a_{k}+ o\left(\frac{1}{x}\right)\right)\right) \\ &= \exp\left(\sum_{k=1}^n\ln a_{k}+ o\left(1\right)\right) \\ &= \exp\left(\ln \prod_{k=1}^na_{k}+ o\left(1\right)\right) = e^{o(1)} \prod_{k=1}^na_{k} \\ & \xrightarrow[x\to\infty]{} \prod_{k=1}^na_{k}. \end{align}


text:
# Factorization formula

1. Jul 28, 2007

### ehrenfest

1. The problem statement, all variables and given/known data

Is there like a formula with a name easy to remember of which the following is a specific instance:

x^200 -y^200 = (x-y)(x^199+x^198y+... + y^198*x + y^199)

?

2. Relevant equations

3. The attempt at a solution

2. Jul 28, 2007

### Feldoh

Not that I've ever heard...

$$(x+y)(x-y)(x^2+y^2)(x^4+y^4)(x^4+x^3y+x^{2}y^2+xy^3+y^4)(x^4-x^3y+x^2y^2-xy^3+y^4)(x^8-x^6y^2+x^4y^4-x^2y^6+y^8)(x^{16}-x^{12}y^4+x^8y^8-x^4y^{12}+y^{16})$$
$$(x^{20}+x^{15}y^5+x^{10}y^{10}+x^5y^{15}+y^{20})(x^{20}-x^{15}y^5+x^{10}y^{10}-x^5y^{15}+y^{20})(x^{40}-x^{30}y^{10}+x^{20}y^{20}-x^{10}y^{30}+y^{40})$$
$$(x^{80}-x^{60}y^{20}+x^{40}y^{40}-x^{20}y^{60}+y^{80})$$

Sorry wouldn't fint onto one line...

Maybe to can find a pattern to that mess though. If not, well, it took my calculator about 3 second to find the answer, might want to use one the next time you need to factor that...

Last edited: Jul 28, 2007
3. Jul 28, 2007

### rocomath

which calculator do you have? i been thinking about upgrading to a TI-89 ...

4. Jul 28, 2007

### Feldoh

Ti-89 Titanium

5. Jul 28, 2007

### HallsofIvy

Staff Emeritus
Do you mean
$$x^n- y^n= (x-y)(x^{n-1}+ x^(n-2)y+ /cdot/cdot/cdot+ xy^{n-2}+ y^{n-1}$$?
I've never worried about it having a name!

6. Jul 28, 2007

### ehrenfest

Yes. I just forget identities so easily when I have to remember them by their statement not some label. However, you're right, this is a pretty simple one that is easily verifiable and probably doesn't warrant a name!


text:
# Is there any significance to the logarithm of a sum?

Many years ago, while working as a computer programmer, I tracked down a subtle bug in the software that we were using. Management had dispaired of finding the bug, but I pursued it in odd moments over a period of a few days, and finally found that the problem was that, in computing the geometric mean, the program was taking the log of the sum instead of the sum of the logs. When thinking back on that, I always wonder whether there is any situation in which taking the log of a sum would be of interest. The only case I can think of is that it is often convenient to shift the logarithm to the left by 1 unit, which is done by adding 1 to the argument, that is, one often wishes to deal with log(1 + x) instead of log(x), so that one has the convenient situation of f(0) = 0. Let’s call this the trivial scenario.

So, can anyone think of any non-trivial scenario in which taking the log of a sum is the thing to do?

-

The log of a sum has been a topic in the time of Gauss; I'll see whether I can find an article which I do recall in the german math newsgroup at about 2002. So far here is a link to tables of such logarithms of Tafeln der Additions- und Subtractions-Logarithmen für sieben Stellen. by Christoph Zech.
As far as I remember it was an approach to simplify computations in astronomy (where Gauss has been involved)

A longer article which explains the rationale and the use of the "Gauss'sche Additionslogarithmen", unfortunately only in german, is at this online archiv The article contains some references which might be helpful even if german is foreign language to you. Sorry I can't be of more help here...

-
–  Ilmari Karonen Aug 30 '11 at 18:30
"only in German"? Just one more hint that all mathematics materials ought to exist also in parallel Esperanto translation. –  Mike Jones Sep 1 '11 at 3:04
This answer seems closest to what I was looking for, and so I have up-voted it and accepted it. –  Mike Jones Sep 3 '11 at 8:08

The logarithm is a concave function. This means that Jensen's inequality can be applied to it, giving the Log sum inequality, an useful lemma in information theory.

Lemma (Log sum inequality) Let $a_i\dots a_n$ and $b_1\dots b_n$ be nonnegative reals. Then we have $$\sum_{i=1}^n a_i\log{\frac{a_i}{b_i}} \ge \left(\sum_{i=1}^n a_i\right)\log{\frac{\sum_{i=1}^n a_i}{\sum_{i=1}^n b_i}}$$

On the right hand side we recognize two logarithms of a sum.

Remark By convention, $0\log{0} = 0\log{\frac{0}{0}}=0$ and $a\log{\frac{a}{0}}=\infty$ for $a>0$. All these are justified by continuity.

-
You mean, "we recognize two logarithms of a sum"? –  Slaviks Aug 30 '11 at 17:50
Definitely! Corrected my mistake. –  Jacopo Notarstefano Aug 30 '11 at 22:47
It seems like "nonnegative reals" should be "positive reals"; otherwise, you're dividing by 0 or taking the log of 0. –  Mike Jones Sep 1 '11 at 3:02
Ah, yes, there are some conventions about the behavior at zero. I'll add them to the answer. –  Jacopo Notarstefano Sep 1 '11 at 22:48
@Jacopo Notarstefano: Thanks for the clarification about the behavior at zero. I'm up-voting your answer now. –  Mike Jones Sep 3 '11 at 8:13

One plausible scenario is this:

Say you want a soft maximum function. The usual function $\max(x,y)$ is not "soft" (try visualizing its graph).

So we define the "soft maximum function" $\mathrm{smax}(x,y)=\log(e^x+e^y)$. The idea is that if $x \gg y$, then $\log(e^x+e^y) \approx x$.

-
Nice, but as someone pointed out, there is the overflow problem. –  Mike Jones Sep 3 '11 at 8:09
This reminds me that I have actually have seen logs of sums of exponentials many times in my research. I can't believe I didn't think of that right away when I read this question... Details can be found here. –  Hans Lundmark Sep 3 '11 at 10:12
+1 this is useful in number of places where a differentiable maximum function is needed (optimization algorithms) and even in computer graphics: soft edges dramatically improve realism in raytraced scenes. –  orion May 17 at 8:49

I don't know if this is anything like what you had in mind, but when you integrate rational functions (by partial fractions) you may get terms like $\ln|x+a|$ and $\ln((x+a)^2+b^2)$ in the result.

-
Thanks. I remember now seeing that, but I had forgotten it. –  Mike Jones Sep 3 '11 at 8:10

The logarithm of a sum also occurs in information theory, when one wants to compute the differential entropy $$H = - \int p(x) \log p(x) ~ \mathrm d x$$ or related measure for a distribution whose density is defined by a sum, e.g. a Gaussian mixture, $$p(x) = \sum_i \frac1{\sigma_i}\phi \left ( \frac{x - \mu_i}{\sigma_i} \right),$$ where $\phi$ is the density of the standard normal distribution.

-

For $a\ge b$, $\log _2(2^a + 2^b) = b+\log_2(2^{a-b}+1)$, where $\log_2$ is the base $2$ logarithm.

For $a\ge b$, $\log(e^a + e^b) = b+\log(e^{a-b}+1)$, where $\log$ is natural logarithm.

Express the terms in the sum as powers of the base of the log to get $a,b$ values.

This was developed to perform calculations with very large numbers that exceeded the MaxNumber on my Mathematica system (had to use logs).

-
Nice. (The editor won't let me just say'Nice.', so, hmmm, Let me also say, Three cheers for Esperanto!) –  Mike Jones Aug 24 at 8:00

The logarithm is the variance-stabilizing transformation of the chi-squared distribution, and a chi-squared-distributed random variable arises as the sum of squared standard-normally-distributed random variables.

One example of a (scaled) chi-squared-distributed random variable is the power spectral density estimate, obtained as a mean of squared frequency-domain coefficients, and the logarithm is routinely used on spectral density estimates, particularly to express it in dB units.

-
I have up-voted your answer. (I’m not claiming to understand it - I’m just giving you the benefit of the doubt, namely, that your answer is indeed as astute as it seems.) Since you say in your profile that you are a non-native speaker interested in writing and speaking decent English, you might be interested in the learner-friendly front-end to the (free, online) Merriam-Webster dictionary that I am creating, at prof-vortaro-de-la-merk-angla.weebly.com –  Mike Jones Aug 24 at 8:01
@MikeJones, I (once more) stumbled across your question late at night, and I just wrote a short answer. Is there something I can do to make it better understandable? – Thanks for the tip with the website. –  A. Donda Aug 26 at 0:03


text:
# Q. 6 Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from $60\degree$, what is A′?

A' is the set of all triangles whose angle is $60\degree$ in other words A' is set of all equilateral triangles.


text:
# Are soft questions allowed on MSE? I see there is a tag for it but I'm scared I will be downvoted for it.

What I was really looking for was a SE but for "soft questions mathematics" - the closest thing I found was the History of Mathematics and Physics SE, which is unlucky considering the example question I wanted to ask:

$$\text{Can the Leibniz Integral rule be used for indefinite integrals?}$$

And other questions of similar nature. Also simple "identifying" questions eg.

$$\text{What is the name of the following property (so that I can look into it more):}$$ $$\int_a^b \int_c^d f(x)g(y) \, dx \, dy = \left(\int_a^b g(y) \, dy \right) \left(\int_c^d f(x) \, dx\right)$$

I could ask on Reddit r/learnmath but it doesn't support MathTex, which I've grown quite fond of.

Thanks for any suggestions

• My personal reaction is that your first isn't a case of a soft question, but rather "didn't bother to do any research", and I'd probably vote to close it, unless is came with evidence of more thought or context. – JonathanZ supports MonicaC Apr 3 at 14:00
• For the second one, it's not soft but it's simple, which is fine by me. It can be hard to search for some math stuff without a name. (The only question I've posted would have been unnecessary if I'd know it was called the "central binomial coefficient"!). It might get deleted as a duplicate (as mine was), but I don't think people would begrudge your asking. – JonathanZ supports MonicaC Apr 3 at 14:04
• I'm inclined to require even less context for terminology or "what's this called" questions than my already de minimus standards. Certainly where you encountered a formula, phrase, or notation should be mentioned, but it is typical that one would have a limited approach to solving it yourself. – hardmath Apr 3 at 16:36
• As long as you provide context, then you shouldn't be scared about what could happen. For the former, you need context: For example "I was reading this book... I think can be used because... it seems...but this contradicts... so idk, etc" And for the later, that might need the reference-request or notation tag. – Verónica Rmz. Apr 3 at 17:38
• Some questions with the soft-questions tag have been very well-received on mathstackexchange, e.g., math.stackexchange.com/questions/20555/… and math.stackexchange.com/questions/21324/… and math.stackexchange.com/questions/1041623/… and math.stackexchange.com/questions/505367/… – others, not so well, e.g., math.stackexchange.com/questions/2422689/… – Gerry Myerson Apr 3 at 22:48
• @GerryMyerson It's worth noting that a lot of the well-received soft questions are from when this site was fairly new. The site culture has changed massively in the intervening years and so the OP is right to be worried. This question was still closed despite receiving many upvotes, and I reckon that a lack of mathematical experience stops users from being taken seriously when more experienced mathematicians do not share their concerns. – Toby Mak Apr 7 at 6:24
• There are soft questions about the mathematics itself, and there are other questions about meta-mathematics, for instance studying mathematics. Perhaps it would be useful to make a distinction between the two? – Toby Mak Apr 7 at 6:32
• @TobyMak There are several tags that have not kept up with the culture, and many in worse shape than soft-question. Certain tags, when they fit the question, seem to reliably indicate that a question is about to accrue close votes, such as advice, self-learning, math-software, and even education and math-history due to the MESE and HSM sites. – Theo Bendit Apr 11 at 19:48
• Your fear is quite understandable. But what do you care more about: having lots of Internet points, or getting your question answered? Besides, if your question gets downvoted into negative territory and no answers nor even useful comments, you can withdraw it. – Robert Soupe Apr 14 at 3:29
• @RobertSoupe +1: being `scared' to ask is the last thing we may want I believe. Considering before asking how to make the question relevant and interesting for the general readership is good, and that's the purpose of this reward-based system... which has unwanted side-effects. – Joce Apr 14 at 7:53
• @RobertSoupe yes ok you are right. It's not even the internet points but a badge which I was after - asking questions on separate days maintaining a positive question record. Pretty silly actually for me to put this badge before my questions lol. I'm a relatively new user but its clear that the site culture has indeed changed massively compared to 8+ years ago. As long as questions are crafted really well and appropriately, any question should be fine right? – user71207 Apr 14 at 10:04
• Speaking only for myself, the thing I care about is that I can see that you have made a good faith effort to figure out the answer yourself. But of course that applies more to a question like "how do I solve a quadratic equation?" than to a question that might be off-topic, like "how do I increase numeric precision in MATLAB?" I think your question about Leibniz integrals would be on-topic, but others here might disagree with me. – Robert Soupe Apr 15 at 2:04


text:
812 US dry barrels in US customary cups

Conversion

812 US dry barrels is equivalent to 396846.545454545 US customary cups.[1]

Conversion formula How to convert 812 US dry barrels to US customary cups?

We know (by definition) that: $1\mathrm{drybarrel}\approx 488.727272727273\mathrm{uscup}$

We can set up a proportion to solve for the number of US customary cups.

$1  drybarrel 812  drybarrel ≈ 488.727272727273  uscup x  uscup$

Now, we cross multiply to solve for our unknown $x$:

$x\mathrm{uscup}\approx \frac{812\mathrm{drybarrel}}{1\mathrm{drybarrel}}*488.727272727273\mathrm{uscup}\to x\mathrm{uscup}\approx 396846.54545454565\mathrm{uscup}$

Conclusion: $812  drybarrel ≈ 396846.54545454565  uscup$

Conversion in the opposite direction

The inverse of the conversion factor is that 1 US customary cup is equal to 2.51986570490265e-06 times 812 US dry barrels.

It can also be expressed as: 812 US dry barrels is equal to $\frac{1}{\mathrm{2.51986570490265e-06}}$ US customary cups.

Approximation

An approximate numerical result would be: eight hundred and twelve US dry barrels is about three hundred and ninety-six thousand, eight hundred and forty-six point five four US customary cups, or alternatively, a US customary cup is about zero times eight hundred and twelve US dry barrels.

Footnotes

[1] The precision is 15 significant digits (fourteen digits to the right of the decimal point).

Results may contain small errors due to the use of floating point arithmetic.

Was it helpful? Share it!


text:
## Selection on floral traits through male fertility in a natural plant population

2008
##### Authors
Burczyk, Jaroslaw
Journal article
##### Published in
Evolutionary Ecology ; 22 (2008), 1. - pp. 39-54. - ISSN 0269-7653. - eISSN 1573-8477
##### Abstract
Most studies on selection in plants estimate female fitness components and neglect male mating success, although the latter might also be fundamental to understand adaptive evolution. Information from molecular genetic markers can be used to assess determinants of male mating success through parentage analyses. We estimated paternal selection gradients on floral traits in a large natural population of the herb Mimulus guttatus using a paternity probability model and maximum likelihood methods. This analysis revealed more significant selection gradients than a previous analysis based on regression of estimated male fertilities on floral traits. There were differences between results of univariate and multivariate analyses most likely due to the underlying covariance structure of the traits. Multivariate analysis, which corrects for the covariance structure of the traits, indicated that male mating success declined with distance from and depended on the direction to the mother plants. Moreover, there was directional selection for plants with fewer open flowers which have smaller corollas, a smaller anther–stigma separation, more red dots on the corolla and a larger fluctuating asymmetry therein. For most of these traits,however, there was also stabilizing selection indicating that there are intermediate optima for these traits. The large number of significant selection gradients in this study shows that even in relatively large natural populations where not all males can be sampled, it is possible to detect significant paternal selection gradients, and that such studies can give us valuable information required to better understand adaptive plant evolution.
##### Subject (DDC)
570 Biosciences, Biology
##### Keywords
Isozymes,Natural selection,Mating system,Paternity,Pollen dispersal,Reproduction
##### Cite This
ISO 690VAN KLEUNEN, Mark, Jaroslaw BURCZYK, 2008. Selection on floral traits through male fertility in a natural plant population. In: Evolutionary Ecology. 22(1), pp. 39-54. ISSN 0269-7653. eISSN 1573-8477. Available under: doi: 10.1007/s10682-007-9157-9
BibTex
@article{vanKleunen2008Selec-12545,
year={2008},
doi={10.1007/s10682-007-9157-9},
title={Selection on floral traits through male fertility in a natural plant population},
number={1},
volume={22},
issn={0269-7653},
journal={Evolutionary Ecology},
pages={39--54},
author={van Kleunen, Mark and Burczyk, Jaroslaw}
}

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No


text:
Home > CALC > Chapter 12 > Lesson 12.1.2 > Problem12-19

12-19.

Given $f ( x ) = \frac { 1 } { 1 - x }$:

1. Find the Maclaurin polynomial, $p_n (x)$, which approximates $f(x)$ near $x = 0$.

The original function and first few derivatives are listed below. Do you see a pattern?

Evaluate the original function and each of the derivatives at $x = 0$.

2. Write $p_n (x)$ using sigma notation.

Simplify the terms in the equation given in Step 2 (a). Then write an expression for the general term. Use this in your summation.


text:
In conversation with Bernard Silverman

We chat to the chief scientific advisor to the Home Office about the role of scientists and mathematicians in politics

It’s been said that a degree in mathematics opens many doors, but to many this might seem a slight exaggeration. Bernard Silverman, however, is an excellent example of a mathematics graduate who has indeed done it all. Silverman is currently the chief scientific advisor to the Home Office, a statistician, and an Anglican priest. These are just a few examples of his many achievements, starting from the gold medal he won at the 1970 International Mathematics Olympiad—the only person to do so from the western side of the iron curtain—at the beginning of his mathematical career. He went on to read mathematics at university, and eventually obtained a PhD in data analysis in 1977. “I was always interested in maths, but as time went on I became keen on doing it in a way that has applications in different things, and that is what drew me to statistics.” He jokingly adds that he felt he was never good enough to be a pure mathematician. In the course of our conversation with him, he took us on a journey through the diverse areas in which he has applied his statistical approach.

Data analysis

With a PhD in data analysis, we felt that Silverman was the perfect person to ask about how big data has changed the landscape of statistics. At the start of his career, “the towering figure was John Tukey, who wrote a book called Exploratory Data Analysis. At the time of its publication, it was an enormous advance because it had the idea of letting the data tell us the story, instead of fitting it to completely specified parametric models.” These ideas are still present in big data, he says, but “of course the challenge has gone from dealing with just a small set of numbers to enormously large data sets”. Silverman describes some of his work as seeking needles in haystacks or, in other words, “dealing with the problem of understanding when what we see is real, rather than just noise. All of this arises from handling these enormously large data sets: that is something that wasn’t there before.”

During his academic career, Silverman has written several books. His work varies from the theoretical to more practical applications of statistics. His work, described in his book Functional Data Analysis, enabled advances in spatial statistics and image recognition. “In functional data analysis,” he says, “the data is not just numbers but curves and surfaces, and this will continue to be the case in the future, with our data consisting of a lot of pictures or sounds, which we need to make statistical sense of.” Understanding this more complicated statistical situation is something we should look forward to, along with machine learning. Silverman predicts a “closer melding of machine learning and big data. The data will set its own questions as well as answer them.”

Calculator development & the next technological breakthrough

The Sinclair Cambridge calculator, a close relative of Silverman’s calculator, as seen in the Science Museum

Although Silverman has always been interested in computational statistics, when he began his statistical journey during his master’s at Cambridge, there were no computers as we know them today. This meant that it would take him “weeks to write a simple programme to draw a density curve from some data”. His desire to develop the technology that would overcome this difficulty led him to move into industry, working for Sinclair Radionics on the Sinclair Cambridge Programmable calculator. First released in 1975, it was marketed as an affordable pocket programmable calculator and Silverman proudly recalls that “you could keep 36 programme steps in its memory”. At the other end of the spectrum, he expects the next technological development to be in quantum computing, which “if quantum computers ever get going—which is an ‘if’, we don’t know for sure—would be a complete game changer”. And, on a more immediately practical day-to-day level: “you just turn on your new computer and it recognises you and logs you in. This is again an application of functional data analysis, using your face as the data.”

Climate change and genetics

Silverman’s attitude is to take a statistical approach to many different subjects, especially those of enormous global importance. For example, some of his collaborative work is in climate change with the Smith School of Enterprise and Environment and in human genetics with the Wellcome Trust Centre for Human Genetics. Climate change modelling is different from standard statistical modelling: “it is not done by conducting experiments or taking measurements, because the models are all computer models. Rather, it is a question of making sense of all the different trajectories of temperature that you can possibly get under given scenarios.” In genetics, however, the challenge is different. In genomes and DNA, the idea is to “use data analysis to see what information there is in the very high-dimensional data that is generated”.

Home Office

When people say that “all the evidence points in this direction”, you should be worried.

Silverman leads a team of researchers at the Home Office. The Centre for Applied Science and Technology is a branch of the Home Office that focuses on serving a range of the government’s national interests: anything from crime prevention and community safety to security and identity assurance. Protective security is an example where maths, specifically finite element analysis, plays an important role. Finite element analysis is a method used to simulate complex physical systems: how explosions and earthquakes affect infrastructure, or the effect of wind on skyscrapers, among many other things. Take, for example, a model of an explosion in a building. We first consider the structure of the building as being fixed. The structure can be split up into units, and the blast will move back and forth through these units. When the explosion is triggered, waves propagate outwards, and initially the pressure of the system is very high, but as time goes on the range of pressures decreases.

The Home Office, Marsham Street (Steve Cadman CC-BY 2.0)

This work, carried out (some years ago, now) at the Centre for Applied Science and Technology thus involves modelling data, without actually carrying out experiments with physical resources—rather problematic in this example!

It is not surprising that political decisions are not based solely on science, although it plays an important role: ultimately, a politician’s job is not the same as that of a scientist and priorities sometimes differ. However, on issues that really require scientific input and where evidence is needed (for example, when a policy is to be drafted regarding driving under the influence of drugs), it is the role of scientists to determine how to carry out any experiments or surveys that may be required. And, of course, although policies may be based on politics, underneath it all is how they will actually be carried out: this is where science decides what will and will not work.

Silverman emphasises the importance of “making sure that policymakers are aware of evidence and the objective”. Presenting this evidence is very different from presenting a research paper: simplicity is often preferred over detail (although presenting detail in a simple manner is even better!) and information needs to be incredibly clear. “It should empower your audience and make them feel comfortable about understanding it”. After that, though, “what they do is up to them”.

Modern slavery

There are between 10,000 and 13,000 victims of modern slavery in the UK.

Silverman is part of the prime minister’s task force on tackling modern slavery. In particular, he managed to work out a way of using the techniques of multiple systems estimation and generalised mark and recapture to estimate the size of a hidden population: the slaves of today’s society. He explains the basics with a simple example from his paper Modern slavery: an application of multiple systems estimation. Suppose that we want to estimate the number of fish in a pond. You first catch 100 fish, mark them and release them back into the pond. After letting the fish swim around for a bit, you catch another batch of 100 fish and count how many are marked. If, for example, 20 are marked (one fifth of the second batch), the natural estimate for the whole population size would be $100 \div 1/5 = 500$. This is the approach of mark and recapture. Multiple systems estimation extends that idea to situations where there are more than two lists. You can use it to estimate the number of individuals that are not on any list, giving what is known as the dark figure.

By using this approach, Silverman concluded that, for a confidence of 95%, there are between 10,000 and 13,000 victims of modern slavery in the UK. To underline the importance of research in the decision-making process of the Home Office, he adds that “the whole government strategy was built on this analysis. The paper was launched at the same time as the Modern Slavery Act was launched, but we are still trying to understand the scale and nature of this problem.”

Improving the image of maths and science

As a man who has always been interested in science, Silverman believes that “people are very odd about it. On the one hand, everybody wants all the latest gadgets; but on the other, they are suspicious about scientists and don’t like them very much.” As someone who has had first-hand experience of science in both academia and public service, why does Silverman think that people perceive science in the way they do? He notes that science isn’t perfect, but the way it is taught at school does not suggest this. At school, we are only taught facts, ideas and methods that have been around for centuries, which have been perfected time and time again before they have reached our blackboards. Modern science, however, is experimental and, for those who have not yet understood this, “it feels either very boring or irrelevant”. The truth is that it is neither: “it is very exciting. Communicating the excitement is important.”

Women in mathematics

In recent years, there has been an increased awareness of the lack of females in mathematics and science. We were interested to hear Silverman’s opinion on why that is so, and what we can do to reduce the gender imbalance. He admits that “it has been an enormous struggle to get girls interested in science. I have always wondered to myself why that is. We have to say what it is about the way we do science that women find off-putting. Because I don’t think women are any worse at it or men any better.” He believes that the culture of science is “still very macho and very male-dominated”, which is why some women might find it off-putting. To move forward, our research into how to fix this needs to be more subtle than the current penchant for asking women in the system why they stayed in it.

Priesthood

The observations of modern science are incontrovertible. They are the world that we live in; they are the world as we have it.

What people may find surprising about Silverman is that aside from his extraordinary work in science, he also takes an interest in religion. He earned a bachelor’s degree in theology from the Southern Theological Education and Training Scheme, and is currently a practising Anglican priest. So what about his views regarding the relationship between science and religion? He says that although he is not always certain about his faith, it remains an influence on everything that he does.

However, “any view of religion that doesn’t take into account modern science must be wrong, because the observations of modern science are incontrovertible. They are the world that we live in; they are the world as we have it.” He ends with the comment that “it is not a way of thinking I’ve always had, but it is what makes sense to me. It is part of your whole existence”.

Something, we suppose, that could also be said about mathematics and statistics.

Niki is a PhD student at UCL, working in analytic number theory.
+ More articles by Nikoleta

TD is an undergraduate at UCL who actually understands the 1967 James Bond spoof Casino Royale, starring David Niven, Peter Sellers, Woody Allen, and Orson Welles as Le Chiffre.
@televisionduck    + More articles by TD

• In conversation with Matt Parker

Interviewing Matt was a mistake


text:
## How to delete users with no order history through phpmyadmin sql?

Question

I have a wordpress ecommerce website which I have begun to update after not touching it for over a year. The first thing I did was implement reCaptcha into all inquiry and contact forms to reduce the amount of spam that is sent to my business’ inbox. I have also got it working for login and user account creation, to hopefully reduce the amount of bot user accounts registering on my website.

I think I am in a good place to now to tackle the users side of this cleanup, which I have roughly 14,000 in my database. Of these users, there is probably only ~100-200 that have actually placed orders on my ecommerce website. I have been searching for a way to delete users with no order history, and I came across this post: https://wordpress.org/support/topic/how-do-i-delete-customers-with-no-orders/

Under phpmyadmin, I have navigated to SQL tab to run a query, but am not having success with the code provided by the above link. Here is a screenshot of what my SQL window looks like when selecting the wp_tom2users database. https://ibb.co/8488Zyk

If I try to hit update, without making any changes, I get this https://ibb.co/LrJcBmF
Same goes for if I try and paste the below code into the textbox and hit update.

SELECT * from wp_tom2users where wp_tom2users.ID not in (
SELECT meta_value FROM wp_tom2postmeta WHERE meta_key = '_customer_user'
) AND wp_tom2users.ID not in (
select distinct(post_author) from wp_tom2posts
)

I was hoping someone could help me with this, as I would prefer to leave the accounts with order history rather than bulk delete all users. I also want to mention, I have three users with administrator privileges which I would like to prevent from deleting as well.

Thank you for taking the time to read the essay I typed into my issue.

0
3 years 2019-10-22T13:08:30-05:00 0 Answers 110 views 0


text:
# All Questions

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### The modulus of RSA public key

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### Is pairing based cryptography ready for productive use?

I'm currently testing one among those many interesting cryptographic protocols based on bilinear maps. It's quite hard to understand the underlying fundamentals, especially since there are several ...
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### RSA Signature Weak Padding Attack

Assume that we have a message $m$ of size $n$, and it is padded with two 01 bytes in front. Then the signature $s$ is computed using a private key $ks$. Can we ...
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I noticed recently that a couple of pieces of encryption software (TrueCrypt being one of them) don't directly use a hash of the password as the key for the block cipher. Instead, they generate a ...
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### What is Identity-Based Encryption (IBE) and why is it “better”?

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text:
### Home > CC2 > Chapter 3 > Lesson 3.2.1 > Problem3-38

3-38.

Simplify the following fraction expressions. Show all of your work. Homework Help ✎

1. $\frac{3}{4}+\frac{2}{3}$

Draw diagrams to represent each fraction.

Now change the diagram so that each whole is divided into equal portions.

1. $\frac{7}{8}-\frac{1}{4}$

Follow the steps in part (a) to solve the problem.

1. $\frac{3}{5}\cdot\frac{1}{3}$

Draw a rectangle.

1. $\frac{4}{7}\cdot\frac{2}{3}$
$\frac{8}{21}$


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# Is deep learning appropriate to approximate dynamic programming problems?

I have a problem which can be completely solved using dynamic programming, but in a very intractable way (On^4, where n is around 1000). I won't get into the details of the problem since it's a bit complicated, but it involves comparing properties of subsequences of a single string, where the property of subsequence (xi... xj) is related in a complex way to the property of subsequence (xi... xj + 1). I am fine with getting an approximation of the correct answer, but obviously would like to maximize correctness and minimize compute.

I have a well-defined metric for how good an answer is, but I have no idea what approach I should be using to solve it. I was thinking about turning it into a reinforcement learning problem for a deep neural network, as a way of avoiding having to find the best algorithm myself. I realized that I have no intuition for the kind of "algorithmic" problems for which deep reinforcement learning fits well. For instance, playing Go well was considered an "algorithmic" problem and has been advanced by using deep reinforcement learning, but the fact that it is nontrivial to teach multiply makes me think that certain seemingly simple problems might be out of reach. Is there a good way to think about what problems are well suited for deep reinforcement learning?

• As a general comment, even if $f(x_i \ldots x_j)$ is related in a complex way to $f(x_i \ldots x_{j+1})$, the fact that they are related suggests that there is some kind of locality. So without knowing anything about the problem, I wouldn't rule out the more well-understood class of Monte Carlo algorithms (e.g. Metropolis-Hastings, simulated annealing, evolutionary algorithms, etc). – Pseudonym Jul 29 '18 at 23:45


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## Orthogonal stability of an Euler-Lagrange-Jensen $$(a, b)$$-cubic functional equation.(English)Zbl 07558439

Summary: In this paper, we introduce a new generalized $$(a, b)$$-cubic Euler-Lagrange-Jensen functional equation and obtain its general solution. Furthermore, we prove the Hyers-Ulam stability of the new generalized $$(a, b)$$-cubic Euler-Lagrange-Jensen functional equation in orthogonality normed spaces.

### MSC:

 39B52 Functional equations for functions with more general domains and/or ranges 39B72 Systems of functional equations and inequalities 39B82 Stability, separation, extension, and related topics for functional equations 46B03 Isomorphic theory (including renorming) of Banach spaces
Full Text:


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# 4-Bit ripple down counter using negative edge-triggered J-K flip flops

Ok, so as the title says im wanting to build a 4-bit ripple down counter on logisim so that I can find what 15 in binary is along with what 9 in binary is to make a mod-10 ripple down counter. But how exactly would I connect up a display or counter on logisim to record/see these results?

Edited section below.

So from what you said Jonk I came up with this circuit if you could have a quick look over it, the top four Q0 Q1 Q2 Q3 links which will be connected to the decoder then to the clock when using the Logic board, just as a heads up the board i will be using is a Logic Tutor LT345 Mk2 board so I can just make the usual connections in my lab session. Also doubt I would need to but I just connected J and K for my lab session as both J K = 1. Hopefully im close at least.

• See this for a start.
– jonk
Nov 13 '19 at 16:48
• @jonk Appreciate the link it helps a lot, I will have a read through the post and see what I come up with
– Ryan
Nov 13 '19 at 16:52
• If you only need count-down and don't need to count in both directions, the logic is a lot simpler and there are many fewer tables to consider.
– jonk
Nov 13 '19 at 16:59
• @jonk Yes for my project ill only be counting down, my lessons for logic gates start this week but currently im just trying to get a head start. On the booklet it says to design the mod-10 counter which should count down from 9 to 0 then reset, previously as a hint it says to design a 4bit counter to find out what the binary is but I already know that 9 in binary is 0111 and 0 being 0000 so I just need to create the circuit. which i feel would be 10x easier doing in a lab on a logic gate board but i dont have the equipment until tomorrow.
– Ryan
Nov 13 '19 at 17:33
• If it starts at 9, then you may want to use the /Q, Q, Q, and /Q outputs. This way, the FFs start out at reset with the value you want. It may twist your mind a little to do that, but it's really all just the same process. So nothing new, really.
– jonk
Nov 13 '19 at 19:46

I understand you want to power-up with 9 (0b1001) as the initial output. Your counter is supposed to count downward, in binary, until it reaches 0 (0x0000), after which the next CLK event causes it to return to 9.

There is more than one way to do this, but probably the easiest for me to explain here is to configure all of your J-K (yes, those are my initials but I didn't invent the idea -- that was Jack Kilby -- but if anyone wants to send me donations as thanks to Jack Kilby's work, please feel free) flip-flops as toggle types (TFFs.) Just wire the J and K inputs together and tie them HIGH.

So let's take a look at the up/down counter table with the starting states, ending states, and the transitions needed in each case:

$$\begin{array}{c|c|c|c|c} \text{State} & \text{Next} & \text{Excite}\\\\ {\begin{smallmatrix}\begin{array}{cccc} \overline{Q_D} & Q_C & Q_B & \overline{Q_A}\\\\ 1&0&0&1\\ 1&0&0&0\\ 0&1&1&1\\ 0&1&1&0\\ 0&1&0&1\\ 0&1&0&0\\ 0&0&1&1\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\\\\ 1&0&1&0\\ 1&0&1&1\\ 1&1&0&0\\ 1&1&0&1\\ 1&1&1&0\\ 1&1&1&1\\ \end{array}\end{smallmatrix}} & {\begin{smallmatrix}\begin{array}{cccc} \overline{Q_D} & Q_C & Q_B & \overline{Q_A}\\\\ 1&0&0&0\\ 0&1&1&1\\ 0&1&1&0\\ 0&1&0&1\\ 0&1&0&0\\ 0&0&1&1\\ 0&0&1&0\\ 0&0&0&1\\ 0&0&0&0\\ 1&0&0&1\\\\ x&x&x&x\\ x&x&x&x\\ x&x&x&x\\ x&x&x&x\\ x&x&x&x\\ x&x&x&x\\ \end{array}\end{smallmatrix}} & {\begin{smallmatrix}\begin{array}{cccc} T_D & T_C & T_B & T_A\\\\ 0&0&0&1\\ 1&1&1&1\\ 0&0&0&1\\ 0&0&1&1\\ 0&0&0&1\\ 0&1&1&1\\ 0&0&0&1\\ 0&0&1&1\\ 0&0&0&1\\ 1&0&0&1\\\\ x&x&x&x\\ x&x&x&x\\ x&x&x&x\\ x&x&x&x\\ x&x&x&x\\ x&x&x&x\\ \end{array}\end{smallmatrix}} \end{array}$$

The above table should be pretty easy to follow. The left column just shows the current state of your TFF outputs. The middle column show you the next state that you want. The right column shows you which of the FF will need to be toggled (0 in the positions where there is no change in the bit value and 1 in the positions where there is a change.)

There are then four K-map tables drawn from the right column above:

$$\begin{array}{rl} \begin{smallmatrix}\begin{array}{r|cccc} T_D&\overline{Q_B}\:\overline{Q_A}&\overline{Q_B}\: Q_A&Q_B \:Q_A&Q_B \:\overline{Q_A}\\ \hline \overline{Q_D}\:\overline{Q_C}&0&1&x&x\\ \overline{Q_D}\:Q_C&x&x&x&x\\ Q_D\: Q_C&0&0&0&0\\ Q_D\:\overline{Q_C}&0&1&0&0 \end{array}\end{smallmatrix} & \begin{smallmatrix}\begin{array}{r|cccc} T_C&\overline{Q_B}\:\overline{Q_A}&\overline{Q_B}\: Q_A&Q_B \:Q_A&Q_B \:\overline{Q_A}\\ \hline \overline{Q_D}\:\overline{Q_C}&0&1&x&x\\ \overline{Q_D}\:Q_C&x&x&x&x\\ Q_D\: Q_C&0&1&0&0\\ Q_D\:\overline{Q_C}&0&0&0&0 \end{array}\end{smallmatrix}\\\\ \begin{smallmatrix}\begin{array}{r|cccc} T_B&\overline{Q_B}\:\overline{Q_A}&\overline{Q_B}\: Q_A&Q_B \:Q_A&Q_B \:\overline{Q_A}\\ \hline \overline{Q_D}\:\overline{Q_C}&0&1&x&x\\ \overline{Q_D}\:Q_C&x&x&x&x\\ Q_D\: Q_C&0&1&1&0\\ Q_D\:\overline{Q_C}&0&0&1&0 \end{array}\end{smallmatrix} & \begin{smallmatrix}\begin{array}{r|cccc} T_A&\overline{Q_B}\:\overline{Q_A}&\overline{Q_B}\: Q_A&Q_B \:Q_A&Q_B \:\overline{Q_A}\\ \hline \overline{Q_D}\:\overline{Q_C}&1&1&x&x\\ \overline{Q_D}\:Q_C&x&x&x&x\\ Q_D\: Q_C&1&1&1&1\\ Q_D\:\overline{Q_C}&1&1&1&1 \end{array}\end{smallmatrix} \end{array}$$

Assuming I didn't make any errors above, you can now use those tables to develop the reduced logic required for each TFF toggle input.

For example, take a look at both $$\T_A\$$ tables above. There are some "don't care" values (indicated by x), but all the rest are just 1s. By substituting in 1 for all of the x (it's a "don't care" so it doesn't matter what we do in those cases), the two tables become trivial and also identical. This means we can simply state:

\begin{align*} T_A &= 1 \end{align*}

Or, put another way, the toggle-input for the $$\Q_A\$$ TFF is always 1. No logic required. Just nail it to 1 and that's done.

The initial circuit, before detailed consideration of what logic to add, will now look like this:

simulate this circuit – Schematic created using CircuitLab

Note that TFF BIT 0 has its toggle-input set to 1, per the above trivial analysis for its table.

Can you come up with the logic required for the remaining three toggle inputs? (Just as a clue, you might notice that if you can develop the logic required for $$\T_\text{C}\$$, then you can OR that with $$\Q_\text{A}\cdot Q_\text{B}\$$ to get the logic required for $$\T_\text{B}\$$.)

Don't peek below the line here until you've tried your hand at it.

simulate this circuit

The above is only one of many different implementations, all of which derive from analyzing the K-maps.

To show the counter value for example on an seven segment LED you need a decoder. One commonly used IC is the 74xx47, there are lots of data sheets in the web. It can decode any binary value from decimal 0 (0000 binary) to 9 (1001 binary).

I remember that there used to be a 74xx IC library for Logisim but will leave the search for it as an exercise for you. In this circuit I used an additional circuit in Logisim:

The outputs of the decoder are "a" to "g" from top to bottom and correspond to the equally named segments of the LED device.


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# Among the following molecules,the dipole moment is the highest for

$\begin{array}{1 1}(a)\;NF_3 &(b)\; NH_3\\(c)\;CO_2&(d)\;BF_3\end{array}$

Can you answer this question?

The dipolemoment of $NF_3$ is 0.23
The dipolemoment of $NH_3$ is 1.47
The dipolemoment of $BF_3$ is 0
The dipolemoment of $CO_2$ is 0
Hence (b) is the correct answer.
answered Mar 10, 2014