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## Abstract

The reinforcement of premating barriers due to reduced hybrid fitness in sympatry may cause secondary sexual isolation within a species as a by-product. Consistent with this, in the fly Drosophila subquinaria , females that are sympatric with D. recens mate at very low rates not only with D. recens , but also with conspecific D. subquinaria males from allopatry. Here, we ask if these effects of reinforcement cascade more broadly to affect sexual isolation with other closely related species. We assay reproductive isolation of these species with D. transversa and find that choosy D. subquinaria females from the region sympatric with D. recens discriminate strongly against male D. transversa , whereas D. subquinaria from the allopatric region do not. This increased sexual isolation cannot be explained by natural selection to avoid mating with this species, as they are allopatric in geographic range and we do not identify any intrinsic postzygotic isolation between D. subquinaria and D. transversa . Variation in epicuticular hydrocarbons, which are used as mating signals in D. subquinaria , follow patterns of premating isolation: D. transversa and allopatric D. subquinaria are most similar to each other and differ from sympatric D. subquinaria , and those of D. recens are distinct from the other two species. We suggest that the secondary effects of reinforcement may cascade to strengthen reproductive isolation with other species that were not a target of selection. These effects may enhance the divergence that occurs in allopatry to help explain why some species are already sexually isolated upon secondary contact.

Speciation is rarely an instantaneous event. When 2 populations have been diverging in allopatry, differences may accumulate so that when they come into secondary contact the hybrid offspring have reduced fitness ( Dobzhansky 1937 ; Coyne and Orr 2004 ). Such postmating isolation can occur because of genetic incompatibilities that affect hybrid fertility or viability and/or because hybrid phenotypes are a poor fit to local ecological conditions or make them poor competitors for mates. As a consequence, individuals that preferentially mate with members of their own species will have increased fitness relative to those individuals that mate indiscriminately. In this way, reduced hybrid fitness can select for increased premating isolation in a process termed reinforcement ( Butlin 1989 ; Howard 1993 ; Noor 1999 ; Servedio 2000 ). In addition to this classic or “narrow-sense” reinforcement, selection for increased sexual isolation may also arise from other consequences of heterospecific mating interactions such as efficient mate location or partitioning of mating signal space ( Servedio and Noor 2003 ; Groning and Hochkirch 2008 ; Pfennig and Pfennig 2012 ). Outside of the context of mating, ecological interactions between sympatric species (e.g., interspecific resource competition) may also drive ecological character displacement that strengthens sexual isolation as a by-product ( Rundle and Nosil 2005 ; Stuart and Losos 2013 ). Based on empirical, comparative, and theoretical studies, it is now generally accepted that the reinforcement of species boundaries is not uncommon in nature and may be a key step in the latter stages of many speciation events ( Coyne and Orr 2004 ).

Selection for increased premating isolation may not only help complete the speciation process for the now sympatric taxa, but it may also initiate secondary speciation events as a by-product in a process known as “cascade reinforcement” ( Howard 1993 ; Ortiz-Barrientos et al. 2009 ; Hoskin and Higgie 2010 ). Cascade reinforcement may occur if sympatric females become more choosy or are better able to discriminate between conspecific and heterospecific males. This increased mate discrimination arises from changes in female sexual preferences and the male sexual displays or signals they target ( Mendelson and Shaw 2012 ), for instance if sympatric females evolve to use population-specific rather than species-specific cues in mate discrimination ( Hoskin and Higgie 2010 ). If these changes to mate discrimination cause sympatric females to discriminate against their own (i.e., conspecific) males from allopatry as a side effect, then sexual isolation can arise and potentially initiate a secondary speciation event in the absence of any postmating isolation between the sympatric versus allopatric populations ( Howard 1993 ; Ortiz-Barrientos et al. 2009 ; Hoskin and Higgie 2010 ). Evidence for such cascading effects of reinforcement comes from several recent studies in insects, fish, and amphibians (e.g., Hoskin et al. 2005 ; Jaenike et al. 2006 ; Higgie and Blows 2007 ; Lemmon 2009 ; Porretta and Urbanelli 2012 ; Kozak et al. 2015 ; Pfennig and Rice 2015 ).

Although previously unconsidered, at a broader level the effects of reinforcement could also cascade further beyond the taxa involved. For example, if females evolve increased discrimination due to mate preferences that are more restrictive and/or target unique signals or displays that have evolved in males in sympatry, then this may also increase discrimination by these females against males from other closely related species. Reinforcement may thus have more widespread consequences via the strengthening of sexual isolation among multiple species, including those that were not the target of reinforcing selection and may not even currently be sympatric with the focal species. This may be particularly important in groups in which reproductive barriers between closely related taxa are not complete (e.g., Drosophila , Coyne and Orr 1989 , 1997 ; Yukilevich 2012 ), including those that have not yet come into secondary contact or in rapidly diversifying lineages.

In this study, we focus on the evolution of reproductive isolation and the cascading consequences of reinforcement among three closely related members of the quinaria group of Drosophila : D. subquinaria , D. recens , and D. transversa . The quinaria group is a holarctic species complex found in temperate and boreal forests and all of their life stages revolve around mushrooms. Drosophila subquinaria and D. recens occur in western and eastern North America, respectively, and their ranges overlap for about 1,200 km eastward from the Canadian Rockies. This region of sympatry is likely a secondary contact event that occurred within the last 12,000 years since the end of the Wisconsin glaciation ( Jaenike et al. 2006 ). Where these species co-occur, they can be found on the same mushrooms, and there are no known ecological differences between them. Drosophila transversa is allopatric to the other 2 species and has been found in northern and eastern Europe, Russia, Mongolia, and China ( Patterson and Stone 1952 ; Wheeler 1960 ; Sidorenko 2009 ). All 3 species are morphologically identical except for the internal male genitalia ( Wheeler 1960 ). A mtDNA phylogeny indicated that D. subquinaria and D. recens are most closely related and D. transversa is the outgroup ( Perlman et al. 2003 ), while an analysis that used a locus on the Y chromosome suggested D. recens was the outgroup to the other 2 species ( Dyer et al. 2011 ).

Patterns of reproductive isolation between D. subquinaria and D. recens are well characterized. When D. recens females mate with D. subquinaria males, the hybrids are viable and females are fertile ( Shoemaker et al. 1999 ). However, in the reciprocal cross in which D. subquinaria females mate with D. recens males, nearly all the hybrids die as embryos due to a Wolbachia infection that occurs in nearly all wild D. recens . If the Wolbachia infection is cured with antibiotics, then the production of fertile daughters is restored (hybrid sons are always sterile, Shoemaker et al. 1999 ). The cost of mating with the wrong species is, therefore, much greater for D. subquinaria females than for D. recens females, and the patterns of premating isolation are consistent with this asymmetry in postmating isolation ( Jaenike et al. 2006 ). Drosophila recens females from populations throughout the geographic range discriminate moderately against D. subquinaria males. In contrast, D. subquinaria females show a strong pattern of reproductive character displacement consistent with reinforcement: females from populations where both species occur discriminate strongly against mating with D. recens males, whereas females from outside this range will mate with D. recens at moderate rates ( Jaenike et al. 2006 ; Bewick and Dyer 2014 ). Occasional hybridization occurs between species as about 2.5% of wild D. subquinaria harbor a D. recens mtDNA haplotype (implying the mating of D. recens females with D. subquinaria males, Jaenike et al. 2006 ; Bewick and Dyer 2014 ).

Selection for heightened discrimination in sympatry may have cascading effects that initiate sexual isolation within one (or both) of the species, and this is exactly what we observe in D. subquinaria ( Jaenike et al. 2006 ; Bewick and Dyer 2014 ). The choosy sympatric D. subquinaria females not only discriminate against mating with heterospecific D. recens males, but they also mate at much reduced rates with conspecific males from allopatric populations. In contrast, allopatric D. subquinaria and both sympatric and allopatric populations of D. recens show no similar pattern. This sexual isolation within D. subquinaria occurs despite the absence of any detectable postzygotic barriers between conspecific crosses and the presence of some gene flow among populations that are sympatric and allopatric with D. recens ( Jaenike et al. 2006 ; Bewick and Dyer 2014 ). Furthermore, there is indirect evidence that discrimination against D. recens and allopatric D. subquinaria males has at least some shared genetic basis ( Bewick and Dyer 2014 ).

Based on the evidence currently available, D. subquinaria is an example of reinforcement in the classic sense: it appears that interactions with D. recens have changed the mate recognition system of sympatric D. subquinaria , and as a side product, these females now discriminate against their own allopatric males. We have shown that contact pheromones, which consist of epicuticular hydrocarbons and their derivatives, are important male signal traits for mate choice in D. subquinaria ( Curtis et al. 2013 ; Giglio and Dyer 2013 ). These epicuticular hydrocarbons may be a target of reinforcing selection, as geographic variation in these compounds, as well as female preferences for them, show a pattern of reproductive character displacement that is largely consistent with the observed patterns of sexual isolation ( Dyer et al. 2014 ; Rundle and Dyer 2015 ).

In this study, we investigate whether these changes in mate recognition in sympatric D. subquinaria cascade more broadly to alter sexual isolation with a closely related allopatric species. Specifically, we characterize patterns of pre- and postmating isolation between D. transversa and both D. subquinaria (sympatric and allopatric) and D. recens . First, we ask whether allopatry of D. transversa from both of the other species corresponds with increased sexual isolation, or in contrast whether secondary contact and the subsequent reinforcement in sympatry have greater effects, as has been found in other Drosophila species (e.g., Coyne and Orr 1989 , 1997 ; Yukilevich 2012 ). Second, we ask whether the reinforced mate discrimination of sympatric D. subquinaria females has cascaded to cause increased sexual isolation with D. transversa relative to that of allopatric D. subquinaria females. Third, we assay divergence in epicuticular compounds, which are known mating signals for D. subquinaria and D. recens , and ask whether their divergence matches the patterns of sexual isolation we observe. Finally, we assay the rate of synonymous substitutions ( Ks ) at 20 autosomal loci to ask if patterns of genetic divergence among these 3 species mirror patterns of pre- and postmating isolation.

## Materials and Methods

### Drosophila strains and rearing

We created mixed stocks to provide healthy outbred populations from which to infer general insights into mating behaviors in these species. We created a mixed stock for D. transversa using 6 lines from Uppsala, Sweden, for allopatric D. subquinaria using two 2 from Portland, Oregon, and 4 lines from Seattle, Washington, and for sympatric D. subquinaria using 4 lines from Hinton, Alberta, and 2 lines from Kawtikh, Alberta. We also constructed 2 mixed stocks for sympatric D. recens , including 1 stock with naturally Wolbachia -infected lines using 1 line each from Hinton, Kawtikh, and Canmore in Alberta, and 1 stock with 3 naturally Wolbachia -uninfected lines from Kawtikh, Alberta. Only populations that had naturally high levels of gene flow and similar cuticular hydrocarbons (CHC) composition were combined in a mixed stock ( Bewick and Dyer 2014 ; Dyer et al. 2014 ). Each mixed stock was maintained for at least 3 generations before being used in experiments. Drosophila subquinaria and D. transversa are not known to be infected with Wolbachia in the wild, and we verified the Wolbachia infection status of each stock using Wolbachia -specific PCR.

Flies were reared at 20 °C on a 12-h light: 12-h dark schedule on Instant Drosophila food (Carolina Biological, Burlington, NC) supplemented with fresh commercial Agaricus bisporus mushroom. All flies used in experiments were reared at a controlled density. Virgins were collected using light CO 2 anesthesia within 24 h of emergence and stored 10–15 flies per vial on standard media.

### Premating isolation

We used no-choice mating trials to quantify patterns of premating isolation. We paired females and males from each of 4 mixed stocks ( D. transversa , allopatric D. subquinaria , sympatric D. subquinaria , Wolbachia -infected D. recens ) for a total of 16 combinations, and used a randomized block design to obtain an average of 32 trials per combination (range 25–36). Mating trials took place in 4.5 cm long x 1 cm diameter vials that contained a blended mushroom–agar medium, and commenced within an hour of the incubator lights turning on. All flies were virgin and 7 days post adult emergence, and were transferred into the mating vials by aspiration without the use of CO 2 . We observed each vial for 3 h, and if copulation occurred we noted the latency to copulation (the time from when the male was introduced to when copulation commenced) as well as the duration of the copulation.

For each species/type pair we used a logistic regression to assess the effect of female type, male type, female x male interaction, and block on mating rates. We calculated the Coyne and Orr sexual isolation index as 1 (frequency heterotypic matings/frequency homotypic matings)—for each species/type pair ( Coyne and Orr 1989 ), where homotypic matings are within a type (i.e., within a mixed stock) and heterotypic matings are between types. We also used Fisher’s exact tests (FET) to ask if females of each species/type mated with each opposite species/type less than with their own males. As a direct test of the broad-scale effects of reinforcement, we tested whether sympatric and allopatric D. subquinaria females mate at different rates with homotypic (i.e., from the same mixed stock) versus D. transversa males using a logistic regression, with female type (allopatric or sympatric), male type (homotypic or D. transversa ), and male x female type interaction as fixed effects in the model. We excluded the interaction effect in the final analysis because it was not significant ( $χ1,12$  = 0.07, P  = 0.8) in the initial analysis.

We tested whether copulation latency and duration differed between homotypic and heterotypic crosses using a Wilcoxon rank sum test, including only pairs that mated. We used nonparametric statistics here because the data violated the assumption of normality. Within each female type we tested whether the homotypic cross was different from each heterotypic cross using the Steel method for nonparametric comparisons. All statistical analyses were completed in JMP version 11 (SAS Institute, Cary NC).

### Postmating isolation

We tested whether D. transversa produced viable hybrids when paired with the other species by pairing 7-day-old virgin females and males and counting the number of offspring that resulted. We placed 1 female with 2 males in a standard food vial and then transferred them to a fresh vial 5 days later. After an additional 5 days the adults were removed, and then all of the offspring from both vials were counted. In this way, we paired D. transversa females and males reciprocally with allopatric and sympatric D. subquinaria and reciprocally with Wolbachia infected and uninfected D. recens . We set up 15 replicate vials of each type of cross. We tested for reduced production of hybrid offspring of D. transversa with each of the 3 other species/stocks using an analysis of variance, with female type, male type, female x male interaction as fixed effects in the model. We also used unpaired t -tests to ask specifically if male Wolbachia infection in D. recens had an effect on offspring production. Because we did not observe the vials to ensure that matings took place, vials with no offspring could result from flies that did not mate, females that ejected sperm, or from the death of the hybrid offspring. These factors may affect whether hybrids result at all as well as how many survive to adulthood. Thus, we did these analyses both with and without the vials that produced no offspring.

We tested whether the hybrids were fertile for each of the eight heterospecific crosses. To do this, we first intercrossed the F1 male and female hybrids together (5 flies of each sex per vial, 3–11 vials/cross) and assayed for the presence of offspring. When no F2 offspring were produced we backcrossed the F1 hybrids of each sex to D. transversa and assayed for the presence of offspring, and we also tested whether the sperm were motile using the methods of Coyne (1984) .

### Epicuticular hydrocarbons

The epicuticular compounds (often called cuticular hydrocarbons or CHCs) of 7- to 9-day-old virgin flies were extracted from each of 4 isofemale lines of D. transversa (three from Uppsala, Sweden and one from Lahti, Finland; the latter was kindly provided by J. Jaenike), and from the mixed stocks of D. transversa , allopatric D. subquinaria , sympatric D. subquinaria , and D. recens that were used in the reproductive isolation trials. The CHCs from 8-32 flies/sex/stock were extracted by placing an individual fly in 100 uL of hexane for 3 min and then vortexing the sample for 1 min, after which the fly was removed and discarded ( Dyer et al. 2014 ). CHCs were stored at −20 °C until they were shipped from Athens, GA, to Ottawa, ON, for analysis. Extractions were completed in a block design to minimize the effects of day, time of day, and order of extraction.

Samples were analyzed on an Agilent 6890N dual channel “fast” (220 V oven) gas chromatograph (Agilent Technologies, Wilmington, Delaware) that used H 2 as the carrier gas and employed flame ionization detection. Preliminary analyses of D. transversa samples using a wide temperature ramp (20–310 °C) revealed a chromatographic profile that qualitatively matched that previously observed in both D. subquinaria and D. recens , uncovering no peaks with retention times outside of those seen in these other species. Subsequent gas chromatography of all samples (i.e., D. transversa , D. subquinaria , and D. recens ), therefore, used the method parameters previously optimized for D. subquinaria/D. recens , as described in Curtis et al. (2013) .

In both sexes, the pattern of peaks in D. transversa precisely matched that seen in both D. subquinaria and D. recens . Individual profiles were, therefore, determined by integration of the area under 13 peaks corresponding to those hydrocarbons previously identified and quantified in male and female D. subquinaria and D. recens ( Supplementary Figure S1 ). In D. subquinaria and D. recens , these consist of 13 long-chain hydrocarbons composed of odd carbon numbers (C 29 , C 31 , and C 33 ) that include several methyl-branched alkanes, alkenes, and alkadienes, all of which are shared between the sexes. The terminal group of five 35-carbon compounds (peaks 17–21 in Supplementary Figure S1 ) were not integrated because an unidentified problem with the gas chromatograph caused poor resolution of these peaks in the majority of samples. The male-specific 11- cis -Vaccenyl acetate (cVa) and five triacylglycerides were also integrated. However, our data analyses revealed very similar multivariate patterns in males whether or not these additional compounds were included (H.D. Rundle, unpublished results), so we present results for only the 13 CHCs that are shared between the sexes, thereby allowing species and sex-specific variation to be quantified in the same phenotypic space.

After integration, the relative concentration of each CHC was calculated by dividing the area under each peak by the total area of all peaks for a given individual. This corrects for technical error associated with quantifying absolute abundances via gas chromatography. Proportions such as these are a form of compositional data to which standard statistical methods should not be applied ( Aitchison, 1986 ). Following past studies ( Bonduriansky et al. 2015 ; Rundle and Dyer 2015 ), we calculated centered log-ratio (CLR) coefficients as:

(1)
$CLRn=ln(pn(∏n=1kpn)1/k),$
where p n is the relative concentration (i.e., proportional area) of CHC n and the divisor is the geometric mean of the proportions of all k  = 13 CHCs within an individual ( Aitchison 1986 ). A multivariate analysis of variance was then used to test for differences in CLR-transformed CHCs between the sexes, species/type (i.e., D. recens , allopatric D. subquinaria , sympatric D. subquinaria , and D. transversa ), and the interaction of these effects. Multivariate CHC variation between the sexes and among species/types was then visualized using the canonical variates from a discriminant function analysis that discriminated CHCs by all 8 combinations of sex and species/type. The discriminant function analysis was performed using all individuals of both sexes from all 4 species/types, and then repeated using only D. subquinaria and D. transversa males and females to depict finer-scale variation among these groups.

### Genetic divergence

To assess genetic divergence between strains/species, we sequenced 20 protein-coding autosomal loci from a single strain each of D. transversa , allopatric D. subquinaria , sympatric D. subquinaria , and allopatric D. recens . DNA sequencing used standard methods. These loci span all of the autosomal Muller Elements as characterized in D. melanogaster ( www.flybase.org ); there is a general conservation of locus to Muller Element across the Drosophila genus ( Bhutkar et al. 2008 ). In total, we sequenced 2,607 synonymous sites from each strain, with an average of 130 synonymous sites/locus ( Supplementary Table S2 ). For each locus we calculated synonymous divergence ( Ks ) between each pair of strains using a Jukes–Cantor correction, as implemented in MEGA ( Tamura et al. 2013 ).

We first used an analysis of variance to ask whether D. recens had higher genetic divergence relative to the other species. We included whether or not a species pair included D. recens and the species pair combination nested with this as fixed effects in the model. We also used an analysis of variance to ask whether divergence was different between the 3 pairwise combinations of sympatric D. subquinaria , allopatric D. subquinaria , and D. transversa .

## Results

### Premating isolation

For five of six species/type combinations we identified a highly significant female × male interaction effect due to the reduced mating of heterotypic pairs (each LRT $χ12$  > 10 and each P  < 0.0003; Supplementary Figure S2 ), indicating the presence of strong premating reproductive isolation. The exception was D. transversa and allopatric D. subquinaria , where interspecific pairs do not mate significantly less than conspecific pairs (female × male interaction LRT $χ12$  = 2.4, P  = 0.13). The Coyne and Orr index of isolation was very low for this pair (0.12), and was highest for D. recens and sympatric D. subquinaria (1.0), for which we never observed any heterospecific matings. For the remaining pairs the sexual isolation index was between 0.48 and 0.60 ( Supplementary Figure S2 ), with sympatric D. subquinaria approximately equally isolated from allopatric D. subquinaria (0.51) and D. transversa (0.48).

For each female type, we also compared the mating rate of each heterotypic cross to homotypic males ( Figure 1 ). For pairings that included D. transversa females, the mating rate was reduced only with D. recens males (FET P  < 0.0001); D. transversa females mated with allopatric and sympatric D. subquinaria about the same amount (FET P  = 0.3). Drosophila subquinaria allopatric females also had a reduced mating rate only when paired with D. recens males ( P  = 0.0014; Figure 1 ). In contrast, matings of sympatric D. subquinaria and D. recens females were reduced against all other male types except their own (FET P  < 0.004 for each; Figure 1 ). The frequency of matings between sympatric D. subquinaria females and allopatric D. subquinaria versus D. transversa males was similar (FET P  = 0.3; Figure 1 ), and it was also similar between D. recens females and D. transversa versus allopatric D. subquinaria males (FET P  = 0.6). We note that while males courted females no matter the cross (D. Humphreys, pers. obs.), in a no-choice mating scenario the mating rate may reflect not only female but also male choice.

Figure 1.

Premating isolation by female species/type. D. subquinaria allopatric and sympatric types are noted “subq allo” and “subq sym,” respectively. Homotypic pairings (i.e., within a type/strain) are shaded in dark gray and heterotypic pairings (i.e., between types/strains) are in light gray. The sample size for each cross type is 25–36 pairs, and the error bars indicate the 95% confidence intervals, as calculated using a binomial distribution. Each heterotypic pair was compared to the homotypic pair that used the same female type with a Fisher's Exact Test; significance is indicated as * ( P  < 0.004) and ** ( P  < 0.0001), where α = 0.05/12 = 0.0041 with a Bonferroni correction.

Figure 1.

Premating isolation by female species/type. D. subquinaria allopatric and sympatric types are noted “subq allo” and “subq sym,” respectively. Homotypic pairings (i.e., within a type/strain) are shaded in dark gray and heterotypic pairings (i.e., between types/strains) are in light gray. The sample size for each cross type is 25–36 pairs, and the error bars indicate the 95% confidence intervals, as calculated using a binomial distribution. Each heterotypic pair was compared to the homotypic pair that used the same female type with a Fisher's Exact Test; significance is indicated as * ( P  < 0.004) and ** ( P  < 0.0001), where α = 0.05/12 = 0.0041 with a Bonferroni correction.

As a direct test for the broader cascading effects of reinforcement, we compared the mating rate of sympatric and allopatric D. subquinaria females with D. transversa versus homotypic (i.e., their own) males. We found a significant male effect ( D. transversa versus the homotypic male;  = 11, P  = 0.0009), which is due to a high rate of mating of allopatric but not sympatric D. subquinaria with D. transversa males. This pattern is expected if changes in female mate preferences of sympatric D. subquinaria extend to D. transversa males. The female effect was also significant (  = 45, P  < 0.0001), which was due to the overall lower mating rates of sympatric D. subquinaria .

Across all species/type combinations, homotypic pairs that mated had a shorter latency to copulation than did the heterotypic pairs (homotypic median = 6 min, heterotypic median = 15 min; Wilcoxon rank sum test $χ12$  = 23, P  < 0.0001; Supplementary Figure S3 ). The homotypic crosses also copulated for a longer duration than did heterotypic crosses (homotypic median = 11 min, heterotypic median = 10 min; Wilcoxon rank sum test  = 7.9, P  = 0.005; Supplementary Figure S4 ). With respect to cascade reinforcement of sympatric D. subquinaria with D. transversa , the copulation latency of sympatric and allopatric D. subquinaria females was not significantly different when paired with homotypic versus D. transversa males (Wilcoxon rank sum test, each P  > 0.1). The copulation duration of allopatric D. subquinaria with D. transversa males was significantly longer than with homotypic males (Wilcoxon rank sum test $χ12$  = 9.4, P  = 0.002), but there was no statistical difference in the copulation duration of sympatric D. subquinaria males with D. transversa versus homotypic males (Wilcoxon rank sum test $χ12$  = 1.2, P  = 0.28). We note that the sample sizes of sympatric D. subquinaria that mated with heterotypic males are small, which limits the power of these analyses.

### Postmating isolation

There was no evidence of post-mating isolation between D. transversa and D. subquinaria . When D. transversa is paired with allopatric D. subquinaria , hybrid crosses in both directions produced a similar number of offspring as the parental crosses (male × female interaction: F1,55  = 0.18, P  = 0.67 using all replicate vials; F1,48  = 0.05, P  = 0.81 including only vials with offspring; Figure 2 ). When D. transversa was paired with sympatric D. subquinaria , there was no evidence of postmating isolation (male × female interaction: F1,56  = 1.6, P  = 0.22 using all replicate vials; F1,37  = 0.03, P  = 0.85 including only vials with offspring), but sympatric D. subquinaria females produced fewer offspring overall (female type: F1,56  = 13, P  = 0.0005 including vials without offspring; F1,37  = 11, P  = 0.0016 including only vials with offspring). Furthermore, with both allopatric and sympatric types, and in both reciprocal directions, the male and female hybrid offspring were fertile, as every F1 intercross produced an abundance of offspring ( n  = 6–11 vials/cross).

Figure 2.

Mean offspring counts from crosses with D. transversa , separated by female type. D. subquinaria allopatric and sympatric types are noted “subq allo” and “subq sym,” respectively. Drosophila recens stocks with and without Wolbachia infection are noted W+ and W-, respectively. Homotypic crosses (i.e., within a type) have grid shading, and heterotypic crosses (i.e., between types) are solid in color. Samples sizes are shown within each bar; white bars include the vials that did not produce any offspring, and grey bars exclude these vials.

Figure 2.

Mean offspring counts from crosses with D. transversa , separated by female type. D. subquinaria allopatric and sympatric types are noted “subq allo” and “subq sym,” respectively. Drosophila recens stocks with and without Wolbachia infection are noted W+ and W-, respectively. Homotypic crosses (i.e., within a type) have grid shading, and heterotypic crosses (i.e., between types) are solid in color. Samples sizes are shown within each bar; white bars include the vials that did not produce any offspring, and grey bars exclude these vials.

Post-mating isolation between D. transversa and D. recens depends on the direction of the cross and the Wolbachia infection status of D. recens . For either Wolbachia -infected or uninfected D. recens crossed with D. transversa , the only significant effect is the male x female interaction when all vials are included ( Wolbachia infected: F1,56  = 28, P  < 0.0001; Wolbachia uninfected: F1,56  = 5.4, P  = 0.003). The low mating rate between these species likely affected the results: when only vials that produced offspring are included, this effect (and the general model) is no longer significant (whole model F3,26  = 1.8, P  = 0.18). However, for infected D. recens crossed with D. transversa , both the male and interaction terms are significant (male: F1,39  = 6.9, P  = 0.012; interaction: F1,39  = 29, P  < 0.0001). This is due to lower hybrid production, which is especially pronounced when D. transversa females mate with Wolbachia -infected D. recens males ( Figure 2 ). This is suggestive of cytoplasmic incompatibility due to Wolbachia infection. Additional observations support the presence of unidirectional cytoplasmic incompatibility due to Wolbachia infection in D. recens . First, we find that many fewer offspring were produced from D. transversa females ×  Wolbachia -infected D. recens males compared to Wolbachia -uninfected D. recens (t-test t12  = 3.8, P  = 0.0028 excluding vials without offspring; t28  = 1.7, P  = 0.09 including all replicate vials; Figure 2 ). Furthermore, this pattern is asymmetric, as the offspring produced by the reciprocal cross ( D. recens female ×  D. transversa male) did not differ significantly depending on whether the female was infected with Wolbachia ( t -test t11  = 1.2, P  = 0.3 excluding vials without offspring; t28  = 0.7, P  = 0.5 including all replicate vials).

Finally, when hybrids are produced between D. transversa and D. recens , the males are sterile. When we intercrossed the hybrid males and females no offspring were produced from any of the four cross types ( n  = 3–10 vials/cross). The hybrid females produced many offspring when backcrossed to pure D. transversa males and were thus fertile. The hybrid males produced no offspring, and dissection showed they lack any mature motile sperm ( > 10 F1 males dissected/cross). This pattern of unidirectional cytoplasmic incompatibility and hybrid male sterility is the same that is seen between D. subquinaria and D. recens ( Shoemaker et al. 1999 ).

### Epicuticular hydrocarbons

The chromatographic profiles of male and female D. transversa corresponded qualitatively to that of both D. subquinaria and D. recens ( Supplementary Figure S1 ), strongly suggesting that the same epicuticular compounds are all shared among these three species. Despite qualitative similarity of CHC identity, the relative concentrations varied significantly between the sexes and among the species (MANOVA, sex × type interaction: Wilks’ lambda = 0.208, F26,414  = 18.95, P  < 0.0001). The dominant axis of among sex and species/type variation distinguished D. recens individuals (males and females) from the other 2 species ( Figure 3 A; Supplementary Table S1 ). Within D. subquinaria and D. transversa , the first canonical variate largely reflected the effect of sex, with males of all three types having lower values than females, although it also separated D. transversa from allopatric D. subquinaria ( Figure 3 B; Supplementary Table S1 ). Sympatric D. subquinaria differed from allopatric D. subquinaria and D. transversa along the second canonical variate, with males being more divergent than females.

Figure 3.

Among-individual variation in epicuticular compounds (CHCs). Analyses include females (filled symbols/solid lines) and males (open symbols/dashed lines) of (A) all four species/types and (B) only D. subquinaria (allopatric and sympatric) and D. transversa . Axes are the first and second canonical variates from separate discriminant function analyses that discriminated among individuals according to species/type and sex. Lines are 90% bivariate normal density ellipses.

Figure 3.

Among-individual variation in epicuticular compounds (CHCs). Analyses include females (filled symbols/solid lines) and males (open symbols/dashed lines) of (A) all four species/types and (B) only D. subquinaria (allopatric and sympatric) and D. transversa . Axes are the first and second canonical variates from separate discriminant function analyses that discriminated among individuals according to species/type and sex. Lines are 90% bivariate normal density ellipses.

### Genetic divergence

Average pairwise synonymous divergence ( Ks ) was highest between pairs involving D. recens , suggesting D. recens is the outgroup to D. subquinaria and D. transversa (mean Ks(R-T)  = 0.081, Ks(R-SA)  = 0.079, Ks(R-SS)  = 0.069; Figure 4 ; Supplementary Table S2 ). This higher divergence of D. recens is statistically significant ( F1,114  = 8.0, P  = 0.0055), and variation among species pairs within this effect were not significantly different ( F4,114  = 0.4; P  = 0.8). We also found that the pairwise divergence among sympatric D. subquinaria , allopatric D. subquinaria , and D. transversa were not statistically different from each other (mean Ks(T-SA)  = 0.053, Ks(T-SS)  = 0.060, Ks(SA-SS)  = 0.050; F2,57  = 0.24, P  = 0.48; Figure 4 ).

Figure 4.

Pairwise synonymous divergence ( Ks ) across 20 autosomal loci. Notched Tukey boxplot where the bold horizontal line indicates the median, the box indicates the 25th and 75th percentiles, the whiskers indicate 1.5 * the interquartile range, and the open circles indicate outliers. Species are abbreviated D. transversa (T), allopatric D. subquinaria (SA), sympatric D. subquinaria (SS), and D. recens (R).

Figure 4.

Pairwise synonymous divergence ( Ks ) across 20 autosomal loci. Notched Tukey boxplot where the bold horizontal line indicates the median, the box indicates the 25th and 75th percentiles, the whiskers indicate 1.5 * the interquartile range, and the open circles indicate outliers. Species are abbreviated D. transversa (T), allopatric D. subquinaria (SA), sympatric D. subquinaria (SS), and D. recens (R).

## Discussion

The reinforcement of premating barriers in sympatry may cause secondary premating isolation as a byproduct ( Howard 1993 ; Ortiz-Barrientos et al. 2009 ; Hoskin and Higgie 2010 ). In D. subquinaria , females that are sympatric with D. recens mate at very low rates not only with D. recens , but also with conspecific D. subquinaria males from allopatry ( Jaenike et al. 2006 ; Bewick and Dyer 2014 ). Here, we find evidence that the effects of reinforcement between D. subquinaria and D. recens cascade more broadly to affect sexual isolation with the closely related species D. transversa . Specifically, allopatric D. subquinaria females mate with D. transversa males as readily as with their own males, while sympatric D. subquinaria females discriminate as strongly against D. transversa males as they do against males from allopatric D. subquinaria and nearly as strong as against D. recens . Since D. transversa is completely allopatric to sympatric populations of D. subquinaria , this increased discrimination is not due to natural selection on these females to avoid mating with D. transversa males. In addition, the hybrids of crosses between D. transversa and D. subquinaria are abundant and both sexes are fertile, suggesting the absence of intrinsic postzygotic isolation required for reinforcement (although ecologically dependent forms have not been tested).

We also characterized the patterns of reproductive isolation between D. transversa with D. recens , which were not previously known. We find that D. transversa females mate at a low frequency with D. recens males , whereas they mate with allopatric and sympatric D. subquinaria at a rate only slightly reduced relative to conspecific males ( Figure 1 ). In the reciprocal direction, D. recens females mate a moderate amount with D. transversa males, similar to their acceptance of allopatric D. subquinaria males. At the postmating level, D. transversa and D. recens show a pattern of hybrid male sterility and asymmetric hybrid death consistent with Wolbachia -induced cytoplasmic incompatibility (though egg hatching rates have not been measured). In contrast, there appears to be no intrinsic genetic postzygotic isolation between D. transversa and D. subquinaria ( Figure 2 ).

Male epicuticular compounds (CHCs) are a mating signal used by female D. subquinaria ( Curtis et al. 2013 ; Giglio and Dyer 2013 ; Dyer et al. 2014 ) and thus we hypothesized that variation in CHCs should largely reflect patterns of premating isolation. Two observations are consistent with this. First, the dominant axis of CHC variation among sexes and types distinguished between D. recens and D. transversa/D. subquinaria ( Figure 3 A). This is consistent with the reduced mating observed between D. transversa and D. subquinaria females and D. recens males ( Figure 1 ). Second, males of D. transversa and allopatric D. subquinaria are more similar to each other than they are to sympatric D. subquinaria males, especially for the second canonical variate ( Figure 3 B). This is consistent with our finding that sympatric D. subquinaria females mate at reduced rates with both of these types of males, and would be expected if these choosy females used the same male signal trait to discriminate against each type of male. However, variation in CHCs does not explain all the patterns of premating isolation that we see. For instance, matings between D. recens females and allopatric D. subquinaria will occasionally occur in spite of the divergence in their CHCs. It is, therefore, likely that mating signals in these species involve traits beyond just CHCs (see Giglio and Dyer 2013 ; Rundle and Dyer 2015 ).

Reinforcement may not only select for changes in female preferences (e.g., Hoskin et al. 2005 ; Pfennig and Ryan 2006 ; Kozak et al. 2015 ), but may also result in changes in the male mating signals which females use to choose mates. Sexual isolation may thus also be strengthened if allopatric females reject the altered male signal traits of sympatric males (e.g., Higgie and Blows 2007 ; Higgie and Blows 2008 ). Our behavioral and CHC data support a scenario in which the mate preferences of sympatric D. subquinaria females have become more specific or restrictive, but that the sympatric D. subquinaria males have not changed their mating signals in a way that renders them highly unattractive to allopatric D. subquinaria or D. transversa females. Thus, in this system, the effects of reinforcement have cascaded only in one direction: sympatric D. subquinaria females have narrowed their preferences substantially, and while sympatric D. subquinaria males have altered their sexual displays ( Dyer et al. 2014 ), they are still within the range that is accepted by the other types of females in no-choice situations (this study; Bewick and Dyer 2014). The topic of male mate choice has not been investigated in this system, but it may also contribute to patterns of isolation. While D. subquinaria males will court and mate with females from any other population in no choice trials, it would be interesting to ask if males prefer to mate with females from their own population and also if males differ in the suite of female CHCs they prefer. No-choice trials as conducted here reflect both male and female mate preferences, and whether more sensitive multiple-choice mating trials may detect any additional isolation remains to be determined.

Our study used strains of D. transversa that are allopatric and very geographically distant from D. subquinaria , but the ranges of these 2 species are not fully described and the potential for sympatry exists . D. subquinaria is thought to extend northward into Alaska ( Wheeler 1960 ), but to our knowledge there has been no recent sampling north of Vancouver, British Columbia. D. transversa has been described as being present in eastern Russia ( Sidorenko 2009 ), although again there is a lack of recent sampling in this region. It will be important to survey these regions, as well as around the Bering Sea and the Aleutian Islands, to determine if there are areas of sympatry and if so, how much gene flow occurs and whether a pattern of reproductive character displacement is present. These were initially described as different species based on the morphology of the internal genitalia (specifically the shapes of the aedeagus and hypandrium shelf, Wheeler 1960 ). At this time, the possibility exists that allopatric D. subquinaria and D. transversa are a single biological species with a continuous range over which CHCs show some variation.

Patterns of genetic divergence have not been well characterized in these species, but may shed light on patterns of reproductive isolation and the species status of allopatric D. subquinaria and D. transversa . We sequenced 20 autosomal loci from each strain/species, and the results follow the patterns of postzygotic isolation and variation in CHCs discussed above. First, the data support a model where D. recens is the outgroup to D. transversa and D. subquinaria . This pattern is consistent with data from the Y chromosome ( Dyer et al. 2011 ) but contrary to previous results from the mtDNA ( Perlman et al. 2003 ). However, mtDNA has been shown to introgress repeatedly from D. recens into D. subquinaria and thus it may not reveal the true species history ( Jaenike et al. 2006 ). Second, we find that the heightened mate discrimination of female sympatric D. subquinaria is not simply due to increased genetic divergence. Further work on the demographic history of these species is necessary, as patterns of divergence are likely complicated by ongoing and past hybridization as well as incomplete lineage sorting due to their generally large effective population sizes. An investigation that incorporates divergence between species and polymorphism within species across regions of the genome that vary in effective population size is therefore underway.

Based on results from comparative studies, premating isolation is expected to be strong when a species pair is sympatric or when genetic divergence is high ( Coyne and Orr 1989 , 1997 ; Yukilevich 2012 ). Here we show that mate discrimination can be high even when a species pair is allopatric and genetic divergence is low. Previous studies suggest that selection arising from the presence of D. recens has caused changes within D. subquinaria such that females sympatric with D. recens now differ in species discrimination, sexual displays, and mate preferences ( Jaenike et al. 2006 ; Bewick and Dyer 2014 ; Dyer et al. 2014 ; Rundle and Dyer 2015 ). This selection has also likely caused these females to discriminate against their own allopatric males as a side effect. We show that these changes in the mate recognition system appear to cascade more broadly to affect sexual isolation with the closely related and allopatric species D. transversa . In sum, we suggest that the secondary effects of reinforcement could cascade more broadly than previously appreciated, especially in rapidly diversifying groups, to include not only diversification within the focal species but also strengthening of isolation with other species that were not a target of selection. This process may augment the basal divergence that occurs in allopatry and explain why some species are already reproductively isolated upon secondary contact.

## Supplementary Material

Supplementary material can be found at http://www.cz.oxfordjournals.org/

## Acknowledgments

We are grateful to Brooke White for laboratory assistance, to Jake Russell-Mercier for help with the gas chromatography and sample integration, and to Nick Arthur, Becky Fuller, and 3 anonymous reviewers for suggestions to improve the manuscript. Funding was provided by National Science Foundation grant DEB-1149350 to KAD, a grant from Natural Sciences and Engineering Research Council of Canada to HDR, and a fellowship from the Center for Undergraduate Research Opportunities at the University of Georgia to DPH.

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text:
2018
Том 70
№ 9

# Quasi-Frobenius Rings and Nakayama Permutations of Semiperfect Rings

Abstract

We say that ${\mathcal{A}}$ is a ring with duality for simple modules, or simply a DSM-ring, if, for every simple right (left) ${\mathcal{A}}$ -module U, the dual module U* is a simple left (right) ${\mathcal{A}}$ -module. We prove that a semiperfect ring is a DSM-ring if and only if it admits a Nakayama permutation. We introduce the notion of a monomial ideal of a semiperfect ring and study the structure of hereditary semiperfect rings with monomial ideals. We consider perfect rings with monomial socles.

English version (Springer): Ukrainian Mathematical Journal 54 (2002), no. 7, pp 1112-1125.

Citation Example: Dokuchaev M. A., Kirichenko V. V. Quasi-Frobenius Rings and Nakayama Permutations of Semiperfect Rings // Ukr. Mat. Zh. - 2002. - 54, № 7. - pp. 919-930.

Full text

format: plain


text:
# Solving this Linear Differential Equation?

1. Nov 23, 2011

### τheory

1. The problem statement, all variables and given/known data
Find the general solution to this differential equation:
$\frac{dy}{dx}+18x^{17}y=x^{18}$
Use the variable $I$ for replacing $\int e^{x^{18}} dx$

2. The attempt at a solution
I've solved the differential equation using integrating factors and obtained the following result:
$y=\frac{\int e^{x^{18}}x^{18} dx + C}{e^{x^{18}}}$

My problem is using the variable $I$ for replacing the $\int e^{x^{{18}}} dx$. How am I suppose to do this with $x^{18}$ inside the integral?

Last edited: Nov 24, 2011
2. Nov 24, 2011

### vela

Staff Emeritus
You need to group characters using { } in LaTeX.
You can't. You want to integrate by parts now.

3. Nov 24, 2011

### flyingpig

Are you familiar with the product rule?

(f*g)' = f'g + g'f

I am not following what you are doing at all...

You already got the integrating factor $$e^{x^18}$$, why do you have $$\int e^{x^18} dx$$?

4. Nov 24, 2011

### τheory

Okay well I did try using integration by parts after I obtained the general solution, but didn't get far as I got this:
$\int e^{x^{18}}x^{18}dx$

$u = x^{18}$
$du = 18x^{17} dx$
$dv = e^{x^{18}} dv$
$v = \int e^{x^{18}} dv$

$uv - \int v du$

$x^{18}\int e^{x^{18}} dv - \int[ \int e^{x^{18}} dv] 18x^{17} dx$

At this point, how am I suppose to utilize the substitution of $\int e^{x^{18}}dx$ with $I$? In other words, how do I input the general solution into the website without inputting integrals? Since the website doesn't recognize integral signs, which is why it's asking me to use $I$ for every $\int e^{x^{18}}dx$ I encounter.

To the flyingpig's question, I need to use the variable "I" because it says to do so in the online problem that I'm doing.

5. Nov 24, 2011

### vela

Staff Emeritus
Try using u=x and $dv=x^{17}e^{x^{18}}\,dx$.


text:
# Which experiments confirm that the Born rule is $|\psi|^2$ rather than $|\psi|$?

It seems like some experiments on quantum systems, like the electron $$g-2$$ measurement, do not rely directly on the Born rule, since they are more so measuring inherent characteristics of the evolution of the wavefunction.

Whereas, an experiment that obviously does rely on the Born rule is the single-electron double-slit experiment, since in principle you could tell whether the distribution of electrons looks more like $$|\psi|^2$$ versus, let's say, just $$|\psi|$$. However, I've watched a couple videos of that experiment, and to me it doesn't look precise enough to distinguish between $$|\psi|^2$$ and $$|\psi|$$. (I mean, it's a mammoth achievement just to get that experiment working at all)

So maybe it's just my utter lack of experimental knowledge, but I'm wondering, can we really be sure the probability goes as $$|\psi|^2$$ and not $$|\psi|$$?

• some people will rush to reply that "but only the squared version preserves probability", but this is simply fixed by diving by the evolving norm. Also not a strong theoretical argument as particle conservation is a low energy illusion. True quantum mechanics must predict rates of events, not specific probabilities. Good question +1 Sep 12 at 19:21
• The TDSE implies a continuity equation, equivalent to conservation of total probability, in which the probability density is $|\psi|^2$. So empirical tests of the TDSE fit the bill.
– J.G.
Sep 12 at 22:15
• "can we really be sure the probability goes as $|\psi|^2$ and not $|\psi|$ ?" Why not $|\psi|^3$? Why not $|\psi|^{1.42}$? Why not $|\psi|^{2.001}$?
– hft
Sep 12 at 22:32

I might be missing the depth somewhere, but I would suggest the electron double-slit experiment first: $$|\sin(kx)|$$ clearly looks different as compared to $$\sin^2(kx)$$.

The second example is the Malus' law in optics. Polarizers are performing the quantum measurement on photons' polarization state. The fact that the measured intensity goes as $$\cos^2(\theta)$$ seems like a proof enough to me of the "squared version" of the Born rule.

• This is the correct answer Sep 12 at 23:07

Any photon-counting diffractive spectroscope. We see a rate of photon arrival proportional to the intensity of the diffracted wave, that is the square of the amplitude.

So maybe it's just my utter lack of experimental knowledge, but...

It's effectively unrelated to your knowledge (or lack thereof) of experiments.

...I'm wondering, can we really be sure the probability goes as $$|\psi|^2$$ and not $$|\psi|$$?

By definition, given a wavefunction solution to the Schrodinger equation $$\psi(x_1, x_2, \ldots)$$, the probability density is (see, e.g., Messiah, Quantum Mechanics, Volume 1, Chapter IV, Section 2): $$|\psi(x_1, x_2, \ldots)|^2$$

You could re-write a bunch of assumptions and conventions and thereby replace $$|\psi|^2$$ with some other monotonic function of $$|\psi|^2$$ and call that a probability density instead.

But, all the the conventions we use in practice are already directed at using $$|\psi|^2$$ as the probability density, not some other function.

• That waves diffract according to amplitude but deposit energy according to its square was understood a century before Born. Adam Herbst's question is "why not linear in amplitude?". You're taking the "standard/typical meaning of $|\psi|^2$" as an axiom. But that emerged from the requirement to match long-held experimental knowledge. Physics doesn't come from definitions and axioms. Sep 12 at 21:21
• @JohnDoty I am taking it as an axiom. And I really don't care where it came from. It has been a standard axiom for almost a hundred years, and I don't feel a great need to justify it by saying that ultimately it must be checked against experiment.
– hft
Sep 12 at 21:42


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# How to Calculate Earnings Before Interest and Tax (EBIT)

This video will show you how to use the EBIT calculating system to figure out what a company's profit will be before tax and interest. EBIT is an indicator of accomplished profit ability and it's calculated as revenue minus expenses excluding tax and interest.

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# Banach-Tarski vs von Neumann

While not so well known, the von Neumann paradox is built among the same lines, in dimension 2 and with transforms within the special linear group. But what is wrong with the following "proof" : it is not very hard to show that the group generated by the two rotations of angle $\alpha$ around $O$ and $C$, with $OC<1/100$, say, and $\alpha/\pi$ irrational and small (say $<1/100$ too), is non-abelian and free. The von Neumann construction applied to two disjoint unit discs then split them in four sets (plus a few fixed points) and sends those sets injectively, by rotations and translations, to disjoint sets included in the union of four copies of the unit disc, this union having something like 1.04 area of the disc. Obviously, this is very wrong, as the Banach-Tarski construction of a full additive measure invariant by isometries of the plane preclude such a thing. Where am I mistaken?

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The group of motions of the plane is solvable of class two, so your two rotations can't generate a free group (and satisfy an explicit relation, e.g. $[[u,v],u[u,v]u^{-1}]=1$). But you have a variant of the paradox, using a free semigroup, implying, if I remember well that there is no finitely additive (unbounded!) measure on the bounded Borel sets, for which the unit disc has measure 1 (amenability of the group of motion provides an invariant mean on subset of the plane, which necessarily takes value 0 on the bounded subsets. – YCor Oct 3 '12 at 14:44
Sorry, I don't understand your notation. If $u$ and $v$ are the two rotations, $v=tu$, with $t$ translation by the vector $OA$, with $A=v(O)$, $v^{-1}=t'u$ ($t'$ similarly translation by $OB$, with $B=v^{-1}(O)$) ; then $ut=Tu$ (the new translation vector being rotation of the first one by $alpha$). So how can $u$ and $v$ related, especially by a short relation ? – Feldmann Denis Oct 3 '12 at 15:27
@FeldmannDenis Brackets usually denote the commutator $[g, h] = g^{−1}h^{−1}gh.$ – Trevor Wilson Oct 3 '12 at 17:04
Oups... Sorry for my misunderstanding. And yes, the problem stems indeed from the existence of this relation. Next, I have to find the mistake in my "proof" that no such relation exists (even when the two angles of rotation are distinct), but this has no interest for MathOverfloxw, I am afraid :-( – Feldmann Denis Oct 3 '12 at 17:46


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# Is this a coincidence?

The smallest distance probed in any experiment so far conducted is $$10^{-18}$$m and the largest distance we can have is the radius of the universe, nearly $$10^{26}$$m. The ratio is $$10^{44}$$.

In another case, if we think about time, the shortest time studied is about $$10^{-26}$$ sec and the longest time is "the lifetime of the universe", which is estimated to be $$13.7$$ billion years, or nearly $$10^{18}$$ sec. The ratio is again $$10^{44}$$!

Is this(the ratio belongs is of same order) a coincidence or there is some deeper explanation(probably space and time are related etc.)? I am an undergrad student, don't know much about astronomy. I wondering about this question while studying quantum mechanics from the book The Quantum World, which claims that it is not a coincidence.

The book explains it in the following way -

"This is not a coincidence. The outermost regions of the universe are moving at a speed near the speed of light, and the particles flying about in the subatomic world are also moving at such speed. On both subatomic and cosmological frontiers, the speed of light is the natural link between distance and time measurements."

I haven't posted exactly what the book is saying because I wanted to hear what other people are saying about this and also because I haven't understood what the book is saying.

• BTW, the radius of the observable universe is roughly 46.6 billion light-years, or $4.4\times 10^{26}$ metres. – PM 2Ring Jan 20 '19 at 16:00
• The only thing which makes sense in this whole SE page is that time and distance are directly proportional for anything moving at constant speed. Hope we don't waste time on here anymore. Starting myself, of course. – Alchimista Jan 21 '19 at 13:56

Giving it some thought, I think this is a question worth some examination, so I voted your question up.

The largest distance we can see and the amount of time we can see by looking as far in the past as we can are directly related. We can see $$13.8$$ billion light years away, though we can infer that those most distant objects are currently 44 billion light years away (a variation of 3.2), and we can also infer that the diameter of the observable universe is twice that, but let's ignore the expansion and just use the farthest distant object we can observe $$- 13.8$$ billion light years and $$13.8$$ billion years. Using those numbers doesn't significantly change your question and those two numbers are clearly related.

Translating to seconds and meters, we get $$4.35 \times 10^{17}$$ seconds (let's not call that nearly $$10^{18}$$, it's less than half).

And translating light years to meters, we get $$1.31 \times 10^{26}$$ meters. About 300 million times the other, which makes sense cause that's how many meters light travels in 1 second.

OK, let's look at small and see if we get the same.

The Large Hadron Collider gives a pretty precise measurement of the smallest distance we can measure, see this question for details. That number is what you said, about $$10^{-18}$$ meters.

The shortest time we can measure isn't quite as clear cut. This article gives a time of 250 atto seconds ($$250 \times 10^{-18}$$ seconds):

Most recently, the techniques of pulse generation and measurement have been extended to the ultraviolet and soft x-ray region of the spectrum where pulses on the order of 250 attoseconds have been created and used to study the motion of electrons around the nucleus of a neon atom.

But it mentions that quantum models can infer time periods much shorter. The shortest is the estimated average lifespan of the top quark at about 0.4 yoctoseconds or about $$4 \times 10^{-25}$$ seconds (40 times your estimate of the 1 to the power of -26).

Events as short as about $$10^{-25}$$ second have been indirectly inferred in extremely energetic collisions in the largest particle accelerators. For example, the mean lifetime of the top quark, the most massive elementary particle so far observed, has been inferred to be about 0.4 yoctosecond.

So, when I run the numbers, I get the shortest estimated time and the age of the universe have a ratio of about $$1.1 \times 10^{42}$$ and the shortest measured distance and the longest observable distance have a ratio of about $$1.3 \times 10^{44}$$. A variation of more than 100. That's pretty close, but nothing remarkable.

Also, I can't for the life of me see what the lifespan of the top quark and the energy they can generate per collision in the LHC would have to do with each other, so I would say that there is no relation at all, and at risk of taking on your author, I would disagree with his claim.

• On discussing what we can see you cuts the legs of the chair you are sitting on. I didn't read the rest but I wonder the question didn't get close and further got voted and accepted. Going to up vote the one brlow that basically say "forget this question". – Alchimista Jan 21 '19 at 10:26
• I just see the very last part now and I feel better ;) – Alchimista Jan 21 '19 at 10:27
• @Alchimista I'll see what I can do about tidying up my writing style. – userLTK Jan 21 '19 at 17:25
• Dear userLTK my comment was mostly on the fact that the Q doesn't deserve a deep discussion. At least if no reason are given for the assertion "is not a coincidence". There is no link between the minimum measurable distance and the maximum one. Who say the opposite should explain why, else. I an afraid the book linked space and time by speed. So what? Your question is ok as you point out expansion but neglect it according to the Q. – Alchimista Jan 22 '19 at 8:27

It's indeed a coincidence. The human limits on observables cannot have some deeper explanation.

In fact, we can claim many things like this, without having some sort of deep meaning.

The other thing is they are not "exactly" equal. Their ratios just happens to be in the same order. There are also uncertainties in all these measurements and the ratios are never equal to each other. If they were exactly equal in some sense then we may thought there is some deeper meaning.

• Thanks for your answer. I am getting that they are not exactly equal and the ratio is of same order. But the book also claims that it is not a coincidence. – tarit goswami Jan 20 '19 at 18:34
• What the book claims ? – Layla Jan 20 '19 at 18:44
• It says - "this(what I have explained in question) is not a coincidence".. after reading it I also thinking in same way you are saying now.. that is, human limits on observables can't have any deeper explanation.. – tarit goswami Jan 20 '19 at 18:46
• Indeed, thats what I am thinking – Layla Jan 20 '19 at 19:10
• The only thing that I can consider not coincidence is the fine-structure constant – Layla Jan 20 '19 at 19:15

Actually, I favor the idea that it is not a coincidence, at least it's not a coincidence that those two ratios bear some strong resemblance. The reason is because at the large scale, as seen already, the largest time is we can detect is the age of the universe T, and the largest distances L we can measure are going to be directly related to that age by L~cT, at least in rough order (the expansion of the universe makes distance L a rather nonunique issue). On the smallest scales, the energies E we require to probe the shortest times t obey Et ~ h by the uncertainty relation, but if we assume those energies will be manifested in relativistic particles, the best we can localize those particles in a length scale l obeys E/c ~ h/l, also by the uncertainty principle. Eliminating E gives l ~ ct, and so forming the ratio of the largest and smallest scales gives L/l ~ (cT)/(ct) ~ T/t. So whatever that ratio comes out, the two should not be all that much different.

So that those ratios are similar is not nearly as surprising as that they come out so incredibly huge. Why does our universe exhibit such incredible contrasts of times and distances? One might look to the anthropic principle there, in the sense that life requires both long timescales to appear and short timescales to support the required complexity, but then there's still the issue of whether or not one regards anthropic explanations as explanations at all. Also, even if we take anthropic thinking, it seems likely the universe could have produced intelligent life even if it was much smaller, and could have supported such life using much slower processes than those we can detect in our supercolliders. It appears our universe is built with a significant degree of "overkill" in regard to both the volume available for events to happen, and the smallest possible distance scales on which they do happen.


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# Perform a simulation with a C-alpha Structure-Based Model using OpenSMOG¶

This tutorial should take between 5 to 15 minutes to complete.

Input files for this tutorial can be found here

The first step is to import the OpenSMOG module

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from OpenSMOG import SBM

SBM class sets the parameters for the simulation:

name="2ci2" Sets the name of each simulation (this name is used as prefix for the outputs). time_step=0.0005 (reduced time unit) Sets the time step used in integration. collision_rate=1.0 (reduced inverse-time unit) Sets the collision rate for the Langevin integrator. r_cutoff=3.0 (nanometers) Sets the non-bonded cutoff. temperature=0.5 (reduced temperature unit) Sets the temperature in the simulation.

sbm_CA is an arbitrarily chosen variable name for the SBM object

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sbm_CA = SBM(name='2ci2', time_step=0.0005, collision_rate=1.0, r_cutoff=3.0, temperature=0.5)

There are three hardware platform options to run the simulations:

platform="cuda" platform="HIP" platform="opencl" platform="cpu"

if cuda, opencl or HIP is choosen the GPUindex can be defined as “0”. If two GPUs are used, one may give “0,1”

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sbm_CA.setup_openmm(platform='cuda',GPUindex='default')

Sets the directory name where to save the simulation outputs

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sbm_CA.saveFolder('output_2ci2_CA')

Load the gro, top and xml files into the sbm_CA object

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sbm_CA_grofile = 'SMOG2_CA_CI2/2ci2.CA.gro'
sbm_CA_topfile = 'SMOG2_CA_CI2/2ci2.CA.top'
sbm_CA_xmlfile = 'SMOG2_CA_CI2/2ci2.CA.xml'

This function returns the name of each contact potential that is being used in the current model. In this example, only a Lennard-Jones-style 10-12 potential is being applied.

The simulation context is created with all information given in the previous steps.

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sbm_CA.createSimulation()

Create the reporters that will save the simulation data in an output folder.

trajectory=True Save the trajectory in .dcd format. energies=True Save the energy in text format separated by a comma. interval=10**3 The interval (in steps) at which the trajectory and energies are saved.

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sbm_CA.createReporters(trajectory=True, energies=True, energy_components=True, interval=10**3)

The run function receives the following parameters:

nsteps=10**6 Number of steps to be performed in the simulation. report=True Show the simulation details (Progress (%), Step and Time Remaining) interval=10**3 The step interval to show the details

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sbm_CA.run(nsteps=10**6, report=True, interval=10**3)

The output files are located in the output folder


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Theoretical Computer Science
it.information-theory coding-theory st.statistics
Updated Thu, 02 Jun 2022 06:18:38 GMT

Notation of sequences in rate distortion theory

I have been reading whatever sources I could get my hands on today, regarding this problem. Most notes online about rate distortion theory come from the book Elements of Information Theory by Thomas M. Cover and Joy A. Thomas. The book seems well regarded so I assume i am misunderstanding something.

I.e the following slides. I am confused by the notation - which comes from the book. If we look at slides 14 and 16 in the slides deck linked. Firstly on slide 14 they denote a sequence of random variables (i.e a sequence of bits, as far as I understand) as $$X^n$$ $$X^n = (X_{1}, \ldots, X_{n}), \quad X_{n} \sim p(x)$$ and the codeword for that sequence as $$\hat{X}^n$$. Now moving on to slide 16 they seem to mix between $$X^n$$ and $$x^n$$ i.e they mention that the distortion between sequences $$d\left(x^{n}, \hat{x}^{n}\right)=\frac{1}{n} \sum_{i=1}^{n} d\left(x_{i}, \hat{x}_{i}\right)$$

And distortion for a $$\left(2^{n R}, n\right)$$ code: $$D=E d\left(X^{n}, g_{n}\left(f_{n}\left(X^{n}\right)\right)\right)=\sum_{x^{n}} p\left(x^{n}\right) d\left(x^{n}, g_{n}\left(f_{n}\left(x^{n}\right)\right)\right)$$

but as far as i can $$x^n$$ and $$X^n$$, and consequently $$\hat{x}^n$$ and $$\hat{X}^n$$ denote the same thing? What is the idea behind this, if any? It would make more sense to me to keep the capitalization consistent unless I am missing something? This notation is consistent across all sources I could find.

Solution

In information theory notation, capital letters such as $$X$$ denote random variables, and lowercase letters such as $$x$$ mean their possible outcomes (i.e., fixed values). For example you can write the definition of expectation as $$E[X] = \sum p(x) \cdot x$$. Moreover, vector values are denoted using length as superscripts, for example $$X^n$$ means $$(X_1, \ldots, X_n)$$ (similarly for $$x^n$$). Moreover a substring $$(X_i, \ldots, X_j)$$ is often denoted by $$X_i^j$$. These slides seem to be consistent with this convention.


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# ideal(LieAlgebra) -- get the relations in a Lie algebra

## Synopsis

• Function: ideal
• Usage:
x=ideal(L)
• Inputs:
• Outputs:
• x, a list, $x$ may also be of type LieIdeal

## Description

The output is either the list of generators of the ideal or the ideal itself of type LieIdeal.

 i1 : L=lieAlgebra{a,b}/{a a a b,b b b a} o1 = L o1 : LieAlgebra i2 : ideal L o2 = { - (a a b a), (b b b a)} o2 : List i3 : describe L o3 = generators => {a, b} Weights => {{1, 0}, {1, 0}} Signs => {0, 0} ideal => { - (a a b a), (b b b a)} ambient => LieAlgebra{...10...} diff => {} Field => QQ computedDegree => 0 i4 : F=lieAlgebra{a,b} o4 = F o4 : LieAlgebra i5 : f=map(L,F) o5 = f o5 : LieAlgebraMap i6 : J=kernel f o6 = J o6 : LieIdeal i7 : N=F/J o7 = N o7 : LieAlgebra i8 : ideal N o8 = J o8 : LieIdeal