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# WikiJournal Preprints/Reinforcement (speciation)

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## Article information

Colvin, A; et al..

## Abstract

Reinforcement assists speciation by selecting against hybrids upon the secondary contact of two separated populations of a species.

Reinforcement is a process of speciation where natural selection increases the reproductive isolation between two populations of species. This occurs as a result of selection acting against the production of hybrid individuals of low fitness. The idea was originally developed by Alfred Russel Wallace and is sometimes referred to as the Wallace effect. The modern concept of reinforcement originates from Theodosius Dobzhansky. He envisioned a species separated allopatrically, where secondary contact of the two populations mate, producing hybrids with lower fitness. Natural selection results from the hybrid's inability to produce viable offspring; thus members of one species who do not mate with members of the other have greater reproductive success. This favors the evolution of greater prezygotic isolation (differences in behavior or biology that inhibit formation of hybrid zygotes). Reinforcement is one of the few cases in which selection can favor an increase in prezygotic isolation, influencing the process of speciation directly.[1] This aspect has been particularly appealing among evolutionary biologists.[2]

The support for reinforcement has fluctuated since its inception, and terminological confusion and differences in usage over history have led to multiple meanings and complications. Various objections have been raised by evolutionary biologists as to the plausibility of its occurrence. Since the 1990s, data from theory, experiments, and nature have overcome many of the past objections, rendering reinforcement widely accepted,[3]:354 though its prevalence in nature remains unknown.[4][5]

Numerous models have been developed to understand its operation in nature, most relying on several facets: genetics, population structures, influences of selection, and mating behaviors. Empirical support for reinforcement exists, both in the laboratory and in nature. Documented examples are found in a wide range of organisms: both vertebrates and invertebrates, fungi, and plants. The secondary contact of originally separated incipient species (the initial stage of speciation) is increasing due to human activities such as the introduction of invasive species or the modification of natural habitats.[6] This has implications for measures of biodiversity and may become more relevant in the future.[6]

## History

Reinforcement has had a complex history in that its popularity among scholars has changed over time.[7][8] Jerry Coyne and H. Allen Orr contend that the theory of reinforcement went through three phases of historical development:[3]:366

1. plausibility based on unfit hybrids
2. implausibility based on hybrids having some fitness
3. plausibility based on empirical studies and biologically complex and realistic models

Sometimes called the Wallace effect, reinforcement was originally proposed by Alfred Russel Wallace in 1889.[3]:353 His hypothesis differed markedly from the modern conception in that it focused on post-zygotic isolation, strengthened by group selection.[9][10][3]:353 Theodosius Dobzhansky was the first to provide a thorough description of the process in 1937,[3]:353 though the term itself was not coined until 1955 by W. Frank Blair.[11] In 1930, Ronald Fisher laid out the first genetic description of the process of reinforcement in The Genetical Theory of Natural Selection, and in 1965 and 1970 the first computer simulations were run to test for its plausibility.[3]:367 Later population genetic[12] and quantitative genetic[13] studies were conducted showing that completely unfit hybrids lead unequivocally to an increase in prezygotic isolation.[3]:367

Dobzhansky's idea gained significant support; he suggested that it illustrated the final step in speciation, for example after an allopatric population comes into secondary contact.[3]:353 In the 1980s, many evolutionary biologists began to doubt the plausibility of the idea,[3]:353 based not on empirical evidence, but largely on the growth of theory that deemed it an unlikely mechanism of reproductive isolation.[2] A number of theoretical objections arose at the time and are addressed in the Arguments against reinforcement section below.

By the early 1990s, reinforcement saw a revival in popularity among evolutionary biologists; due primarily from a sudden increase in data—empirical evidence from studies in labs and largely by examples found in nature.[3]:354 Further, computer simulations of the genetics and migration patterns of populations found, "something looking like reinforcement".[3]:372 The most recent theoretical work on speciation has come from several studies (notably from Liou and Price, Kelly and Noor, and Kirkpatrick and Servedio) using highly complex computer simulations; all of which came to similar conclusions: that reinforcement is plausible under several conditions, and in many cases, is easier than previously thought.[3]:374

### Terminology

Confusion exists around the meaning of the term reinforcement.[14] It was first used to describe the observed mating call differences in Gastrophryne frogs within a secondary contact hybrid zone.[14] The term secondary contact has also been used to describe reinforcement in the context of an allopatrically separated population experiencing contact after the loss of a geographic barrier.[15] The Wallace effect is similar to reinforcement, but is rarely used.[14] Roger Butlin demarcated incomplete post-zygotic isolation from complete isolation, referring to incomplete isolation as reinforcement and completely isolated populations as experiencing reproductive character displacement.[16] Daniel J. Howard considered reproductive character displacement to represent either assortive mating or the divergence of traits for mate recognition (specifically between sympatric populations).[14] Reinforcement, under his definition, included prezygotic divergence and complete post-zygotic isolation.[17] Servedio and Noor include any detected increase in prezygotic isolation as reinforcement, as long as it is a response to selection against mating between two different species.[4] Coyne and Orr contend that, "true reinforcement is restricted to cases in which isolation is enhanced between taxa that can still exchange genes".[3]:352

## Models

The four outcomes of secondary contact:
1. An extrinsic barrier separates a species population into two but they come into contact before reproductive isolation is sufficient to result in speciation. The two populations fuse back into one species
2. Speciation by reinforcement
3. Two separated populations stay genetically distinct while hybrid swarms form in the zone of contact
4. Genome recombination results in speciation of the two populations, with an additional hybrid species. All three species are separated by intrinsic reproductive barriers[18]

One of the strongest forms of reproductive isolation in nature is sexual isolation: traits in organisms involving mating.[19] This pattern has led to the idea that, because selection acts so strongly on mating traits, it may be involved in the process of speciation.[19] This process of speciation influenced by natural selection is reinforcement, and can happen under any mode of speciation[3]:355 (e.g. geographic modes of speciation or ecological speciation[20]). It necessitates two forces of evolution that act on mate choice: natural selection and gene flow.[21] Selection acts as the main driver of reinforcement as it selects against hybrid genotypes that are of low-fitness, regardless if individual preferences have no effect on survival and reproduction.[21] Gene flow acts as the primary opposing force against reinforcement, as the exchange of genes between individuals leading to hybrids cause the genotypes to homogenize.[21]

Butlin laid out four primary criteria for reinforcement to be detected in natural or laboratory populations:[16]

• Gene flow between two taxa exists or can be established to have existed at some point.
• There is divergence of mating-associated traits between two taxa.
• Patterns of mating are modified, limiting the production of low fitness hybrids.
• Other selection pressures leading to divergence of the mate-recognition system have not occurred.

After speciation by reinforcement occurs, changes after complete reproductive isolation (and further isolation thereafter) are a form of reproductive character displacement.[22] A common signature of reinforcement's occurrence in nature is that of reproductive character displacement; characteristics of a population diverge in sympatry but not allopatry.[6][5] One difficulty in detection is that ecological character displacement can result in the same patterns.[23] Further, gene flow can diminish the isolation found in sympatric populations.[23] Two important factors in the outcome of the process rely on: 1) the specific mechanisms that causes prezygotic isolation, and 2) the number of alleles altered by mutations affecting mate choice.[24]

In instances of peripatric speciation, reinforcement is unlikely to complete speciation in the case that the peripherally isolated population comes into secondary contact with the main population.[25] In sympatric speciation, selection against hybrids is required; therefore reinforcement can play a role, given the evolution of some form of fitness trade-offs.[1] In sympatry, patterns of strong mating discrimination are often observed—being attributed to reinforcement.[7] Reinforcement is thought to be the agent of gametic isolation.[26]

### Genetics

The underlying genetics of reinforcement can be understood by an ideal model of two haploid populations experiencing an increase in linkage disequilibrium. Here, selection rejects low fitness ${\displaystyle Bc}$  or ${\displaystyle bC}$  allele combinations while favoring combinations of ${\displaystyle BC}$  alleles (in the first subpopulation) and ${\displaystyle bc}$  alleles (in the second subpopulation). The third locus ${\displaystyle A}$  or ${\displaystyle a}$  (the assortive mating alleles) have an effect on mating pattern but is not under direct selection. If selection at ${\displaystyle B}$  and ${\displaystyle C}$  cause changes in the frequency of allele ${\displaystyle A}$ , assortive mating is promoted, resulting in reinforcement. Both selection and assortive mating are necessary, that is, that matings of ${\displaystyle A\times A}$  and ${\displaystyle a\times a}$  are more common than matings of ${\displaystyle a\times A}$  and ${\displaystyle A\times a}$ .[27] A restriction of migration between populations can further increase the chance of reinforcement, as it decreases the probability of the differing genotypes to exchange.[14]

An alternative model exists to address the antagonism of recombination, as it can reduce the association between the alleles that involve fitness and the assortive mating alleles that do not.[14] Genetic models often differ in terms of the number of traits associated with loci;[28] with some relying on one locus per trait[25][29][30] and others on polygenic traits.[22][21][31]

### Population structures

The structure and migration patterns of a population can affect the process of speciation by reinforcement. It has been shown to occur under an island model, harboring conditions with infrequent migrations occurring in one direction,[21] and in symmetric migration models where species migrate evenly back and forth between populations.[25][29]

A parameter space representing the conditions in which speciation by reinforcement can occur. Here, three outcomes can arise: 1) extinction of one of the initial populations; 2) the initial populations can hybridize; 3) the initial populations can speciate. The outcomes are determined by both initial divergence and level of fitness of the hybrids.[22]

Reinforcement can also occur in single populations,[28][22] mosaic hybrid zones (patchy distributions of parental forms and subpopulations),[30] and in parapatric populations with narrow contact zones.[32]

Population densities are an important factor in reinforcement, often in conjunction with extinction.[22] It is possible that, when two species come into secondary contact, one population can become extinct—primarily due to low hybrid fitness accompanied by high population growth rates.[22] Extinction is less likely if the hybrids are inviable instead of infertile, as fertile individuals can still survive long enough to reproduce.[22]

### Selection

Speciation by reinforcement relies directly on selection to favor an increase in prezygotic isolation,[1] and the nature of selection's role in reinforcement has been widely discussed, with models applying varying approaches.[28] Selection acting on hybrids can occur in several different ways. All hybrids produced may be equality low-fitness,[22] conferring a broad disadvantage. In other cases, selection may favor multiple and varying phenotypes[25] such as in the case of a mosaic hybrid zone.[30] Natural selection can act on specific alleles both directly or indirectly.[28][21][33] In direct selection, the frequency of the selected allele is favored to the extreme. In cases where an allele is indirectly selected, its frequency increases due to a different linked allele experiencing selection (linkage disequilibrium).[14]

The condition of the hybrids under selection can play a role in post-zygotic isolation, as hybrid inviability (a hybrid unable to mature into a fit adult) and sterility (the inability to produce offspring entirely) prohibit gene flow between populations.[7] Selection against the hybrids can even be driven by any failure to obtain a mate, as it is effectively indistinguishable from sterility—each circumstance results in no offspring.[7]

### Mating and mate preference

Some initial divergence in mate preference must be present for reinforcement to occur.[7][22][34] Any traits that promote isolation may be subjected to reinforcement such as mating signals (e.g. courtship display), signal responses, the location of breeding grounds, the timing of mating (e.g. seasonal breeding), or even egg receptivity.[14] Individuals may also discriminate against mates that differ in various traits such as mating call or morphology.[35] Many of these examples are described below.

## Evidence from nature

### Amphibians

The southern brown tree frog, Litoria ewingi

The two frog species Litoria ewingi and L. verreauxii live in southern Australia with their two ranges overlapping. The species have very similar calls in allopatry, but express clinal variation in sympatry, with notable distinctness in calls that generate female preference discrimination.[14] The zone of overlap sometimes forms hybrids and is thought to originate by secondary contact of once fully allopatric populations.[14]

Allopatric populations of Gastrophryne olivacea and G. carolinensis have recently come into secondary contact due to forest clearing.[36] The calls that the males make to attract females differ significantly in frequency and duration in the area where the two species overlap, despite them having similar calls where they do not.[3]:359 Further, the hybrids that form in sympatry have calls that are intermediate between the two.[36] Similar patterns of reproductive character displacement involving acoustic displays have been found in Hyla cinerea and H. gratiosa, with greater female preference for conspecific males in areas of sympatry.[37]

Three species of true frogs (Lithobates sphenocephalus, L. berlandieri, and L. blairi) are temporally isolated in that their breeding seasons are spaced out in areas where they live in sympatry, but not where they live in allopatry.[38] Selection against interspecific mating due to low hybrid fitness and low hybrid fertility has reinforced the observed character displacement of breeding times.[38]

The rainforests of northeast Queensland, Australia were separated into north and south refugia by climate fluctuations of the Pliocene and Pleistocene.[39] About 6500 years ago, the rainforests reconnected, bringing the diverged, incipient populations of Litoria genimaculata into secondary contact. The species contact zones exhibit, "strong postzygotic selection against hybrids" and enhanced isolation from differences in mating call.[35]

An alternative to detecting reproductive character displacement in populations that overlap in sympatry is measuring rates of hybridization in contact zones.[36] The frog species Anaxyrus americanus and Anaxyrus woodhousii have shown a decrease in hybridization from 9%–0% over approximately 30 years.[40][36] A similar pattern was detected in the sympatric spadefoot toads Spea multiplicata and S. bombifrons have hybridized with decreasing frequency over a 27-year period (about 13 generations).[41]

### Birds

The Ficedula flycatchers exhibit a pattern that suggests premating isolation is being reinforced by sexual selection.[42] The pied flycatcher (Ficedula hypoleuca) has brown females, brown males, and black-and-white males. The related collard flycatcher (Ficedula albicollis) has brown females and only black-and-white males. The two species exist in separate populations that overlap in a zone of sympatry.[42] In the range of overlap, only brown males of F. hypoleuca exist and are thought to have evolved the brown plumage to prevent hybridization.[43] Mating choice tests of the species find that females of both species choose conspecific males in sympatry, but heterospecific males in allopatry[42] (see conspecific song preference). The patterns could suggest mimicry, driven by interspecific competition;[3]:361 however, song divergence has been detected that shows a similar pattern to the mating preferences.[44]

Geospiza fulginosa and G. difficilis males on the Galápagos Islands show a noted preference for conspecific females where they meet in sympatry, but not in allopatry.[45] Other birds such as the dark and light subspecies of the western grebe show enhanced prezygotic isolation.[46] It has been argued that reinforcement is extremely common in birds and has been documented in a wide range of bird species.[47]

### Crustaceans

Reproductive character displacement in body size was detected in sympatric populations of Orconectes rusticus and O. sanbornii.[48]

### Echinoderms

An example of gametic isolation involves the allopatric sea urchins (Arbacia) have minimal bindin differences (bindin is a protein involved in the process of sea urchin fertilization, used for species-specific recognition of the egg by the sperm) and have insufficient barriers to fertilization.[3]:243 Comparison with the sympatric species Echinometra and Strongylocentrotus of the Indo-Pacific finds that they have significant differences in bindin proteins for fertilization and marked fertilization barriers.[49]

Laboratory matings of closely related sea urchin species Echinometra oblonga and E. sp. C (the species is unnamed, dubbed C) produce fertile and viable hybrids, but are unable to fertilize eggs of the parent species due to divergence of the alleles that code for bindin proteins: an example of post-zygotic isolation.[3]:343–344 Populations in sympatry manifest this difference in bindin protein versus those in allopatry.[3]:343–344 Selection actively acts against the formation of hybrids in both nature (as no documented cases of hybrids have been found) and in the laboratory.[50] Here, the evolution of female egg receptors is thought to pressure bindin evolution in a selective runaway process.[50] This example of reproductive character displacement is highly suggestive of being the result of—and has been cited as strong evidence for—reinforcement.[50][3]:343–344

### Fish

In British Columbia, benthic and limnetic morphs of Gasterosteus aculeatus exist together in sympatry in some lakes, while containing only one morph in other lakes.[51] Female benthic morphs in sympatric populations actively discriminate against limnetic males, resulting in low rates of crossing (some gene flow has occurred between the morphs) and low fitness hybrids.[3]:360 Both selection against hybrids and reproductive character displacement in egg fertilization is observed in Etheostoma lepidum and E. spectabile.[52]

### Fungi

A strong case of reinforcement occurring in fungi comes from studies of Neurospora.[53] In crosses between different species in the genera, sympatric pairs show low reproductive success, significantly lower than allopatric pairs.[53] This pattern is observed across small and large geographic scales, with distance correlating with reproductive success.[53] Further evidence of reinforcement in the species was the low fitness detected in the hybrids create from crosses, and that no hybrids have been found in nature, despite close proximity.[53]

### Insects

Ethological isolation has been observed between some mosquito species in the Southeast Asian Aedes albopictus group, suggesting—from laboratory experiments of mating trials—that selection against hybrids is occurring, in the presence of reproductive character displacement.[54]

Female mate discrimination is increased with intermediate migration rates between allopatric populations of Timema cristinae (genus Timema) compared to high rates of migration (where gene flow impedes selection) or low rates (where selection is not strong enough).[55][56]

Distribution of the periodical cicadas in the U.S. The yellow area corresponds to the sympatric overlap of Magicicada neotredecim (blue) with Magicicada tredecim (red)

Where the ranges of the cicada species Magicicada tredecim and M. neotredecim overlap (where they are sympatric), the pitch of M. neotredecim male calling songs is roughly 1.7 kHz compared to 1.1 kHz for those of M. tredecim, with corresponding female song pitch preference differences.[57] In allopatric M. neotredecim populations, the mating call pitch is 1.3–1.5 kHz.[57] The biogeography of the cicadas suggests that M. neotredecim originated after the retreat of the last glacial advance in North America.[57]

The song differences of Laupala crickets on the Hawaiian Islands appear to exhibit patterns consistent with character displacement in sympatric populations.[16] A similar pattern exists with Allonemobius fasciatus and A. socius, species of ground crickets in eastern North America.[58]

Males in sympatric populations of the damselflies Calopteryx maculata and C. aequabilis are able to discriminate between females of different species better than those in allopatric populations; with females of C. aequabilis in sympatric populations exhibiting lighter wing colors compared to allopatric females—an illustration of reproductive character displacement.[59][60]

Fifteen species of sympatrically distributed Agrodiaetus butterflies with pronounced differences in wing color pattern likely arose as a result of speciation by reinforcement.[61] Phylogenetic patterns indicate the differences arose in allopatry and were reinforced when the distributions came into secondary contact.[62]

Drosophila is one of the most studied species in speciation research.[63] Dobzhansky and Koller were the first to study isolation between Drosophila species.[3]:358 Since then, other studies of natural populations such as the D. paulistorum races exhibiting stronger isolation in sympatry versus allopatry,[64] or the enhanced isolation found in sympatric populations of D. mojavensis and D. arizonae in southwest America.[65] Rare, sterile hybrids form between D. pseudoobscura and D. persimilis, with sympatric D. pseudoobscura females discriminating against D. persimilis males; more so than allopatric populations.[66] Other Drosophila research on reinforcement has been from laboratory experiments and is discussed below. On the east coast of Australia, D. serrata shares a zone of sympatric overlap with the closely relates species D. birchii.[67] The species exhibits reproductive character displacement, with sexual selection operating on the hydrocarbons of the flies cuticle.[68] Reinforcement appears to be driving their speciation in nature, supported by simulated experimental laboratory populations.[69][19]

### Mammals

The deer mice Peromyscus leucopus and P. gossypinus exhibit reproductive character displacement in mating preferences, with heterospecific matings taking place between the species.[70]

### Molluscs

Some gastropod shells, like those of Partula can coil in sinistral and dextral directions such as these, Neptunea angulata (left) and N. despecta (right) shells.

Partula suturalis is polymorphic for shell chirality in that it has two forms: sinistral (left-handed) and dextral (right-handed) shells, unlike other monomorphic species on the island of Mo'orea which have only one form (with the exception of P. otaheitana).[71] This polymorphic trait has a direct effect on mate choice and mating behavior; as shown in laboratory mating tests that opposite-coil pairs mate much less often.[71] In areas where P. suturalis lives sympatrically with other sinistral and dextral Partula species, the opposite P. suturalis morph is typically present.[36] Butlin succinctly describes one example of this unique pattern:

P. suturalis is sympatric with the dextral P. aurantia and sinistral P. olympia, whose ranges abut but do not overlap; P. suturalis is sinestral in the range of P. aurantia and dextral in the range of P. olympia and does not normally hybridize with either species. However, where their ranges meet there is a sharp transition in the coil of P. suturalis and in this transition zone it hybridizes with both P. aurantia and P. olympia.[36]

The reversal in chirality to sinistrality must have evolved as an isolating mechanism,[72] with patterns of reproductive character displacement suggesting speciation by reinforcement.[71]

Satsuma largillierti lives on the western half of Okinawa Island while Satsuma eucosmia lives on the eastern half. Both populations overlap in sympatry along the middle of the island, where the penis length of the species differs significantly in sympatry (a case of reproductive character displacement[73]), but not in allopatry.[74] A similar pattern in snails is found with Lymnaea peregra and L. ovata in the Swiss lake Seealpsee; with mating signal acting as the sympatrically displaced trait.[75]

The abalone genus Haliotis has 19 species that occur in sympatry and one that occurs in allopatry. Of the sympatric species, they all contain sperm lysin that drives gamete isolation, but the allopatric species does not.[76][3]:343 A similar pattern of sperm lysin differentiation is found in the mussel species Mytilus galloprovincalis and M. trossulus and has likely occurred within the last 200 years due to human-mediated distribution by ships.[3]:343

### Plants

Plants are thought to provide suitable conditions for reinforcement to occur.[5] This is due to a number of factors such as the unpredictability of pollination, pollen vectors, hybridization, hybrid zones, among others.[5] The study of plants experiencing speciation by reinforcement has largely been overlooked by researchers;[3]:364 however, there is evidence of its occurrence in them.[77]

In the Texas wildflower Phlox drummondii, cis-regulatory mutations of genes that code for anthocyanin pigmentation have caused genetic divergence of two populations.[78] Hybrids (between P. drummondii and P. cuspidata) with maladaptive, intermediate characteristics are under-pollinated; increasing reproductive isolation through reinforcement.[78] The maintenance of the ancestral flower color in the allopatric population is favored weakly by selection, where the derived color in the sympatric population is being driven by strong selection.[79] Similarly, in P. pilosa and P. glaberrima, character displacement of petal color has been driven by selection, aided by pollen discrimination.[80] Displacement in flower size has also been observed in the nightshade species Solanum grayi and S. lymholtzianum in sympatry as well as S. rostratum and S. citrullifolium.[81]

Green southern population foliage of Pinus muricata
Blue northern population

The bishop pine is divided into two populations distinguished by monoterpene, stomata, and alloenzyme differences; flowering time; and needle color: blue foliage in the northern population and green foliage in the southern populations in California.[82] A small region exists where the species meet in a cline—sustained by selection due to a flowering time divergence, thought to represent reinforcement taking place.[36]

Similar patterns of both character displacement in sympatric populations of species have been documented in:[36][3]:361

## Evidence from comparative studies

Prezygotic isolation in allopatric (red) and sympatric (blue) species pairs of Drosophila. Gradients indicate the predictions of reinforcement for allopatric and sympatric populations.[88]

Assortive mating is expected to increase among sympatric populations experiencing reinforcement.[14] This fact allows for the direct comparison of the strength of prezygotic isolation in sympatry and allopatry between different experiments and studies.[3]:362 Jerry Coyne and H. Allen Orr surveyed 171 species pairs, collecting data on their geographic mode, genetic distance, and strength of both prezygotic and post-zygotic isolation; finding that prezygotic isolation was significantly stronger in sympatric pairs, correlating with the ages of the species.[3]:362 Additionally, the strength of post-zygotic isolation was not different between sympatric and allopatric pairs.[14]

This finding lends support the predictions of speciation by reinforcement and correlates well with another later study by Daniel J. Howard.[3]:363 In his study, 48 studies with observed reproductive character displacement (including plants, insects, crustaceans, molluscs, fish, amphibians, reptiles, birds, and mammals) were analyzed.[17] The cases met several criteria such as the trait in question serving as a reproductive barrier and if there existed clear patterns of sympatry versus allopatry.[17] Out of the 48 candidates, 69 percent (33 cases) found enhanced isolation in sympatry, suggesting that the pattern predicted by reinforcement is common in nature.[17] In addition to Howard's comparative study, he guarded against the potential for positive-result publication bias by surveying 37 studies of hybrid zones. A prediction of reinforcement is that assortive mating should be common in hybrid zones; a prediction that was confirmed in 19 of the 37 cases.[17]

A survey of the rates of speciation in fish and their associated hybrid zones found similar patterns in sympatry, supporting the occurrence of reinforcement.[89] One study in the plants Glycine and Silene; however, did not find enhanced isolation.[90]

## Evidence from laboratory experiments

Laboratory studies that explicitly test for reinforcement are limited.[3]:357 In general, two types of experiments have been conducted: using artificial selection to mimic natural selection that eliminates the hybrids (often called "destroy-the-hybrids"), and using disruptive selection to select for a trait (regardless of its function in sexual reproduction).[3]:355–357 Many experiments using the destroy-the-hybrids technique are generally cited as supportive of reinforcement; however, some researchers such as Coyne and Orr and William R. Rice and Ellen E. Hostert contend that they do not truly model reinforcement, as gene flow is completely restricted between two populations.[91][3]:356 The table below summarizes some of the laboratory experiments that are often cited as testing reinforcement in some form.

Laboratory studies of reinforcement[91][3]:354–357
Species Experimental design Result Year
Drosophila paulistorum Destroyed hybrids Prezygotic isolation 1976[92]
D.pseudoobscura & Prezygotic isolation; reproductive character displacement 1950[93]
Zea mays 1969[94]
D. melanogaster 1974[95]
1956[96]
No prezygotic isolation detected 1970[97]
Prezygotic isolation 1953[98]
1974[99]
1971[100] 1973[101] 1979[102]
Allopatric populations in secondary contact 1982[103]
1991[104]
No prezygotic isolation detected 1966[105][106]
Allowed gene flow between populations 1969[107]
1984[108]
Destroyed some hybrids 1983[109]
Disruptive selection Prezygotic isolation; assortive mating; all later replications of the experiment have failed 1962[110]
1997[111]

## Alternative hypotheses

Various alternative explanations for the patterns observed in nature have been proposed.[3]:375 There is no single, overarching signature of reinforcement; however, there are two proposed possibilities:[3]:379 that of sex asymmetry (where females in sympatric populations are forced to become choosy in the face of two differing males)[112] and that of allelic dominance: any of the alleles experiencing selection for isolation should be dominate.[7] Though this signature does not fully account for fixation probabilities or ecological character displacement.[3]:380 Coyne and Orr extend the sex asymmetry signature and contend that, regardless of the change seen in females and males in sympatry, isolation is driven more by females.[3]:380

### Ecological or ethological influences

Ecology can also play a role in the observed patterns—called ecological character displacement. Natural selection may drive the reduction of an overlap of niches between species instead of acting to reduce hybridization[3]:377 Though one experiment in stickleback fish that explicitly tested this hypotheses found no evidence.[51]

Species interactions can also result in reproductive character displacement (in both mate preference or mating signal).[19] Examples include predation and competition pressures, parasites, deceptive pollination, and mimicry.[19] Because these and other factors can result in reproductive character displacement, Conrad J. Hoskin and Megan Higgie give five criteria for reinforcement to be distinguished between ecological and ethological influences:

(1) mating traits are identified in the focal species; (2) mating traits are affected by a species interaction, such that selection on mating traits is likely; (3) species interactions differ among populations (present vs. absent, or different species interactions affecting mating traits in each population); (4) mating traits (signal and/or preference) differ among populations due to differences in species interactions; (5) speciation requires showing that mating trait divergence results in complete or near complete sexual isolation among populations. Results will be most informative in a well-resolved biogeographic setting where the relationship and history among populations is known.[19]

### Fusion

It is possible that the pattern of enhanced isolation could simply be a temporary outcome of secondary contact where two allopatric species already have a varying range of prezygotic isolation: with some exhibiting more than others.[113] Those that have weaker prezygotic isolation will eventually fuse, losing their distinctiveness.[7] This hypothesis does not explain the fact that individual species in allopatry, experiencing consistent gene flow, would not differ in levels of gene flow upon secondary contact.[7][66] Furthermore, patterns detected in Drosophila find high levels of prezygotic isolation in sympatry but not in allopatry.[114] The fusion hypothesis predicts that strong isolation should be found in both allopatry and sympatry.[114] This fusion process is thought to occur in nature, but does not fully explain the patterns found with reinforcement.[3]:376

### Sympatry

Phylogenetic signature to distinguish sympatric speciation from reinforcement. Stronger prezygotic isolation (indicated by the red boxes and associated arrows) should be detected between Z and Y and between Z and X if species Z sympatrically speciated (green) from the common ancestor of species Y and X. If Z, Y, and X speciated allopatrically (blue), with Z and Y experiencing secondary contact, strong prezygotic isolation should be found between Z and Y, but not between Z and X.[115]

It is possible that the process of sympatric speciation itself may result in the observed patterns of reinforcement.[3]:378 One method of distinguishing between the two is to construct a phylogenetic history of the species, as the strength of prezygotic isolation between a group of related species should differ according to how they speciated in the past.[115] Two other ways to determine if reinforcement occurs (as opposed to sympatric speciation) are:

• if two recently speciated taxa do not show signs of post-zygotic isolation of both sympatric and allopatric populations (in sympatric speciation, post-zygotic isolation is not a prerequisite);[116]
• if a cline exists between two species over a range of traits (sympatric speciation does not require a cline to exist at all).[117]

### Sexual selection

In a runaway process (not unlike Fisherian runaway selection), selection against the low-fitness hybrids favors assortive mating, increasing mate discrimination rapidly.[7][114] Additionally, when there is a low cost to female mate preferences, changes in male phenotypes can result, expressing a pattern identical to that of reproductive character displacement.[118] Post-zygotic isolation is not needed, initiated simply by the fact that unfit hybrids cannot get mates.[7]

## Arguments against reinforcement

A number of objections were put forth, mainly during the 1980s, arguing that reinforcement is implausible.[7][19][3]:369 Most rely on theoretical work which suggested that the antagonism between the forces of natural selection and gene flow were the largest barriers to its feasibility.[3]:369–372 These objections have since been largely contradicted by evidence from nature.[17][3]:372

### Gene flow

Concerns about hybrid fitness playing a role in reinforcement has led to objections based on the relationship between selection and recombination.[5][3]:369 That is, if gene flow is not zero (if hybrids aren't completely unfit), selection cannot drive the fixation of alleles for prezygotic isolation.[27] For example: If population ${\displaystyle X}$  has the prezygotic isolating allele ${\displaystyle A}$  and the high fitness, post-zygotic alleles ${\displaystyle B}$  and ${\displaystyle C}$ ; and population ${\displaystyle Y}$  has the prezygotic allele a and the high fitness, post-zygotic alleles ${\displaystyle b}$  and ${\displaystyle c}$ , both ${\displaystyle ABC}$  and ${\displaystyle abc}$  genotypes will experience recombination in the face of gene flow. Somehow, the populations must be maintained.[3]:369

In addition, specific alleles that have the selective advantage within the overlapped populations are only useful within that population.[119] However, if they are selectively advantageous, gene flow should allow the alleles to spread throughout both populations.[119] To prevent this, the alleles would have to be deleterious or neutral.[3]:371 This is not without problems, as gene flow from the presumably large allopatric regions could overwhelm the area when two populations overlap.[3]:371 For reinforcement to work, gene flow must be present, but very limited.[25][30]

Recent studies suggest reinforcement can occur under a wider range of conditions than previously thought[28][116][3]:372–373 and that the effect of gene flow can be overcome by selection.[120][121] For example, the two species Drosophila santomea and D. yakuba on the African island São Tomé occasionally hybridize with one another, resulting in fertile female offspring and sterile male offspring.[120] This natural setting was reproduced in the laboratory, directly modeling reinforcement: the removal of some hybrids and the allowance of varying levels of gene flow.[121] The results of the experiment strongly suggested that reinforcement works under a variety of conditions, with the evolution of sexual isolation arising in 5–10 fruit fly generations.[121]

### Rapid requirements

In conjunction with the fusion hypothesis, reinforcement can be thought of as a race against both fusion and extinction.[113] The production of unfit hybrids is effectively the same as a heterozygote disadvantage; whereby a deviation from genetic equilibrium causes the loss of the unfit allele.[109] This effect would result in the extinction of one of the populations.[122] This objection is overcome by when both populations are not subject to the same ecological conditions.[3]:370 Though, it is still possible for extinction of one population to occur, and has been shown in population simulations.[123] For reinforcement to occur, prezygotic isolation must happen quickly.[3]:370

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107. B. S. Grant and L. E. Mettler (1969), "Disruptive and stabilizing selection on the" escape" behavior of Drosophila melanogaster", Genetics, 62 (3): 625–637, PMC 1212303, PMID 17248452
108. E. B. Spiess and C. M. Wilke (1984), "Still another attempt to achieve assortive mating by disruptive selection in Drosophila", Evolution, 38 (3): 505–515, doi:10.1111/j.1558-5646.1984.tb00316.x, PMID 28555983
109. A. A. Harper and D. M. Lambert (1983), "The population genetics of reinforcing selection", Genetica, 62 (1): 15–23, doi:10.1007/BF00123305
110. J. M. Thoday and J. B. Gibson (1962), "Isolation by disruptive selection", Nature, 193 (4821): 1164–1166, Bibcode:1962Natur.193.1164T, doi:10.1038/1931164a0
111. Ellen E. Hostert (1997), "Reinforcement: a new perspective on an old controversy", Evolution, 51 (3): 697–702, doi:10.1111/j.1558-5646.1997.tb03653.x, PMID 28568598
112. L. Partridge and G. A. Parker. (1999). Sexual conflict and speciation. In A. E. Magurran and R. M. May (eds) Evolution of Biological Diversity. Oxford University Press, pp.130–159
113. Alan R. Templeton (1981), "Mechanisms of Speciation – A Population Genetic Approach", Annual Review of Ecology, Evolution, and Systematics, 12: 23–48, doi:10.1146/annurev.es.12.110181.000323
114. Jerry A. Coyne; H. Allen Orr (1989), "Patterns of speciation in Drosophila", Evolution, 43 (2): 362–381, doi:10.1111/j.1558-5646.1989.tb04233.x, PMID 28568554
115. Mohamed A. F. Noor (1997), "How often does sympatry affect sexual isolation in Drosophila?", The American Naturalist, 149 (6): 1156–1163, doi:10.1086/286044, PMID 18811269
116. Mark Kirkpatrick and Virginie Ravigné (2002), "Speciation by Natural and Sexual Selection: Models and Experiments", The American Naturalist, 159: S22–35, doi:10.1086/338370, PMID 18707367
117. N. H. Barton and G. M. Hewitt (1989), "Adaptation, speciation and hybrid zones", Nature, 341 (6242): 497–503, doi:10.1038/341497a0, PMID 2677747
118. Troy Day (2000), "Sexual Selection and the Evolution of Costly Female Preferences: Spatial Effects", Evolution, 54 (3): 715–730, doi:10.1554/0014-3820(2000)054[0715:SSATEO]2.3.CO;2
119. J. A. Moore. (1957). An embryologist's view of the species concept. In Ernst Mayr (eds) The Species Problem, American Association for the Advancement of Science, pp. 325–338.
120. Daniel R. Matute (2010), "Reinforcement Can Overcome Gene Flow during Speciation in Drosophila", Current Biology, 20 (24): 2229–2233, doi:10.1016/j.cub.2010.11.036, PMC 3019097, PMID 21129972
121. Jerry A. Coyne (2010), "Reinforcement" and the origin of species, Wordpress
122. H. E. H. Paterson (1978), "More evidence against speciation by reinforcement", South African Journal of Science, 74: 369–371
123. Hamish G. Spencer, Brian H. McArdle, and David M. Lambert (1986), "A Theoretical Investigation of Speciation by Reinforcement", The American Naturalist, 128 (2): 241–262, doi:10.1086/284557CS1 maint: multiple names: authors list (link)


text:
## LATEX-L@LISTSERV.UNI-HEIDELBERG.DE

#### View:

 Message: [ First | Previous | Next | Last ] By Topic: [ First | Previous | Next | Last ] By Author: [ First | Previous | Next | Last ] Font: Proportional Font

Subject:

Re: Robustness (was Re: Expandable templates)

From:

Date:

Fri, 22 Oct 1999 18:25:59 +0100

Content-Type:

text/plain

Parts/Attachments:

 text/plain (40 lines)
 Frank, >Bernard, > > > >The new definition of add@accent is supposed to keep correct > > >spacefactor's after accented uppercase letters. The cost is > > >that a command like \edef\agrave{\a} will not work as > > >expected (with respect to spacefactors). > > >More importantly, this makes the babel package fail, > > >and possibly others. [...] >within standard LaTeX the new defnition of \ and friends still work >perfectly >in all circumstances. the fact that inside babel \edef was used in a place >where \protected@edef should have been used was an oversight and is >corrected. The "expandability" of \add@accent has not changed. It is still expandable, if its definition is: \def\add@accent#1#2{{%    \setbox\@tempboxa\hbox{#2%       \global\mathchardef\accent@spacefactor\spacefactor}%    \accent#1 #2}\spacefactor\accent@spacefactor} The problem is that \accent@spacefactor is not yet defined in \edef's prior to an actual use of any accent. That is a LaTeX bug which can be fixed by letting \accent@spacefactor to \relax or by defining it with \mathchardef (in both cases it expands into itself). Example: \documentclass{minimal} \edef\agrave{\a} % Doesn't work \setbox0\hbox{\a} % We use an accent \edef\agrave{\a} % Now it does work (but \agrave looks horrible! :-) ) Regards Javier Bezos`

format: plain


text:
# How do I find and classify the critical points of $x^2 \sin (\frac 1x)$

Find the non-zero critical points of $f$ and the signs of the values of $f$ at these non zero critical points.

Determine the behaviour of $f$ between the non zero critical points and the types of non-zero critical points

I understand that this function is oscillating in between $(-0.3183, 0)$ and $(0.3183,0)$ and have also proven that $f'(0) = 0$ using the formal derivative definition as well as the squeeze theorem.

I also found $f'(x) = 2x\sin \frac 1x - \cos \frac 1x$ but am having difficulty finding a way to list and classify the critical points in a general manner.

Any help is appreciated!

• Nothing like trying to describe infinitely many critical points on a finite interval, what form of solution are you hoping for. – David Jun 8 '16 at 3:35
• Haha sure sounds like a bit of a feat, I'm honestly not too sure myself. After reading the question, I assumed there would be some sort of general solution to express all the critical points (or perhaps rearranging the subject for x in the derivative?). But yeh, still trying to get my head around analysis and approaching questions. – Astro551 Jun 8 '16 at 3:56
• Interesting, I'll give it a thought. To me, stating all solutions to $xtan(1/x) = 1/2$ would be a satisfactory description of the set, but maybe there is something better we can do... – David Jun 8 '16 at 3:58


text:
# Set up the differential equation showing the voltage V(t) for this RC circuit

spsch
Homework Statement:
A capacitor with Capacity C, a Battery with Voltage ## V_0 ## and Resistance R are connected.
C = 1mF, R = 500 Ohms, V_0 = 10V

a) Set up a differential equation showing the change in voltage over time.
b) draw a slope field of the equation
c) solve the equation
d) show that ## V(t) =V_0(1-e^(\frac{-t}{RC})) is a solution
e) draw the function
Relevant Equations:
C = Q/V, The voltage difference around a closed loop is zero.
Hi all.

I have another exam question that I am not so sure about. I've solved similar problems in textbooks but I have a feeling once again that the correct way to solve this problem is much simpler and eluding me.
Especially because my answer to a) is already the solution to c) and d) (I did my way before looking at c and d) and I can't figure out b with my equation.

So I set up ## V_0 -IR -\frac{Q}{C} = 0 ##
I separated and got to ## Q = V_0C(1-e^(\frac{-t}{RC}))##
Then because Q/C = V I divided both sides by the Capacitance and got:
## V = V_0(1-e^(\frac{-t}{RC}))##
(Solution to c and d)

Now for b If I take the time derivative again I end up with ## \frac{dv}{dt} = -V_0(RC)(1-e^(\frac{-t}{RC})) ##
Which I guess I could draw but it seems to complicate to be correct?

If there is another way to start, a better way, could someone point me in that direction without giving the answer away?
Thank you all!

Last edited by a moderator:

doggydan42
So I set up ## V_0 -IR -\frac{Q}{C} = 0 ##

Using ##V(t) = \frac{Q}{C}##, you could have rewritten the differential equation in terms of V(t).

It looks like you should draw the slope field before solving the differential equation. You can use the rewritten version, the one in terms of V(t), then use your R, C, and ##V_0## values to make approximations to draw the slope field.

Last edited:
spsch
spsch
Using ##V(t) = \frac{Q}{C}##, you could have rewritten the differential equation in terms of V(t).

It looks like you should draw the slope field before solving the differential equation. You can use the rewritten version, the one in terms of V(t), then use your R, C, and ##V_0## values to make approximations to draw the slope field.
Hi doggydan42,

thank you very much. I thought of this, but for some reason I thought it's not ok to just use V in there. But it makes sense. Let me try again, now I'm also curious if I'll get a similar derivative to the one I got doing my long and tedious way!

thanks so much!

spsch and DaveE
Gold Member
One easy way to check your results is with dimensional analysis. If you find something like [meters]/[sec] = [meters], then you know something is wrong. Note that this doesn't tell you the answer is correct, it won't tell you about factors of -2 or π. But it's a quick way of spotting bad formulas.
https://en.wikipedia.org/wiki/Dimensional_analysis

For example, if you asked what sort of phenomena is measured in (bushells)/(fortnight-acre), then I can substitute the units:
bushell = volume = m3
fortnight = time = seconds
acre = area = m2
so, bushells/(fortnight-acre) = m3/(seconds*m2) = m/sec which is a velocity.
Note that I never cared about how many m3 are in a bushell. It's only about the type of dimension, not the conversion constants.

spsch
Staff Emeritus
Homework Helper
Now for b If I take the time derivative again I end up with ## \frac{dv}{dt} = -V_0(RC)(1-e^(\frac{-t}{RC})) ##
Which I guess I could draw but it seems to complicate to be correct?
This expression for dV/dt isn't the one you want because it comes from a specific solution to the differential equation. What you want to do is follow @doggydan42's suggestion and write the original differential equation in terms of V. You'll end up with something of the form
$$\frac{dV}{dt} = f(t,V)$$ for some function ##f##. For various point on the tV-plane, you can calculate what dV/dt is equal to and draw a short line segment with that slope at that point.

spsch and doggydan42
spsch
Hi all, thank you very much for your help.
At first I felt confident after@doggydan42's suggestion and tried last night, but I had problem identifying my ## \frac {dV}{dt} ## part.
I did a dimensional analysis as suggested by @DaveE of my solution and got Coloumbs * Ohms over seconds which seems to be correct at least.
I tried again this morning and I've been thinking where the Voltage changes.
The voltage over the resistor changes as the current is changing when the capacitors is discharging or charging.
I found in a book that R = dV/dI but R isn't changing in this case, right?
the only varying voltage part i'd have would be the capacitor.
But if I insert dV/dt there and just consider the current as the current I get ##V(t) =(V_0 -IR)*t##
:-(

EDIT: writing it out here made me see something, that's probably wrong, but may I ask:
Could I use ##R = \frac{dV}{dI}## and set I to ##\frac{dI}{dt}## and then I'd have ## V_0=\frac{dV}{dt} + V##
Feels completely wrong, but, I'm at my wit's end here. sorry..

Staff Emeritus
Homework Helper
The ##I## in ##IR## is the current through the resistor. How is the current through the resistor related to the current through the capacitor? What is the relationship between the current through the capacitor and the voltage across the capacitor?

spsch
spsch
The ##I## in ##IR## is the current through the resistor. How is the current through the resistor related to the current through the capacitor? What is the relationship between the current through the capacitor and the voltage across the capacitor?
Hi Vela, thank you very much for helping.
I thought about your comment and tried a few ways earlier. Didn't quite seem to find a solution.
I came back after reviewing basics but I'm still stuck.

The current decreases as the Voltage (Electric PE) increases in the capacitor.
So I thought that maybe ## dI = -\frac {dV}{dt}## I put that into IR and I ended up with just V_0 = - V in the end.
It's nonsense but interestingly one of my first results was V = V_0 but I can't find the page where I did that anymore. I think there and then I just replaced IR with dV/dt.

edit: I have another question to your comment also @vela, I thought there is no current through the capacitor?

Staff Emeritus
Homework Helper
In your original attempt, you went from
$$V_0 - IR - \frac QC = 0$$ to
$$Q=CV_0(1-e^{-t/RC}).$$ How did you deal with ##I## there to get everything in terms of ##Q##? And what was your reasoning?

spsch
spsch
In your original attempt, you went from
$$V_0 - IR - \frac QC = 0$$ to
$$Q=CV_0(1-e^{-t/RC}).$$ How did you deal with ##I## there to get everything in terms of ##Q##? And what was your reasoning?
Hi @vela thank you very much.

the current was more intuitive, I knew it's dq/dt so it just came.
I did some algebra (Q to the dq/dt side) and then I integrated and then raise to the power of e to get Q out of the natural log to get the Result.
dq/dt is changing due to the amount of charge on the capacitor.

Staff Emeritus
Homework Helper
##I## in the term ##IR## represents the current flowing through the resistor. ##dQ/dt## is the current flowing into the capacitor. How do you know they are equal?

You have the relation ##Q=CV## for the capacitor. Try using that.

spsch
spsch
Hi vela,
thank you very much. It seems like your earlier comment put me on the right track but I messed up somehow. I tried replacing I in IR through and the Resistance canceled and I ended up with the V_0 = V expression.

I will try to think about it some more and see if I can't come up with it.

spsch
##I## in the term ##IR## represents the current flowing through the resistor. ##dQ/dt## is the current flowing into the capacitor. How do you know they are equal?

You have the relation ##Q=CV## for the capacitor. Try using that.
I think I might be making an integration mistake.

Is ## V_0 = CR*\frac{dV}{dt} +V ## the correct Differential equation?
edit: I am definitely making integration mistakes. Sorry! I thought I had to use the integrating factor. And then used u sub. But with some algebra now, I could do without and I almost got the correct solution.
I ended up with ##V_0 - V_0 e^(\frac{t}{RC})## instead of with e raised to the negative (t/RC).
I think I'll better review some calculus and come back soon!

I still end up with ##V_0 = V## or ## V_0 = -V ## depending on which way I believe the current goes.

Last edited:
doggydan42
Is ## V_0 = CR*\frac{dV}{dt} +V ## the correct Differential equation?
edit: I am definitely making integration mistakes. Sorry! I thought I had to use the integrating factor. And then

You have the right differential equation. You could use an integration factor to solve it, or you could use separation of variables. Try solving it again, and pay careful attention to the sign in the exponent. Please show all your work in solving the differential equation next time.

I still end up with ##V_0 = V## or ## V_0 = -V ## depending on which way I believe the current goes.
How are you getting this?

spsch
spsch
Hi @doggydan42 thank you.
I tried to make a clean version of what happened. I didn't upload the work previously because once i get confused it becomes a mess that only I can read.
First is the integrating factor.

I was able to solve the latter for the right solution, though if I use -dV/dt for the current in the loop.
instead of dV/dt in the equation, you mentioned was correct.
I feel really silly, I know I'm making a sign mistake, but I just seem unable to locate where.
thank you for helping!

#### Attachments

• IMG_20190806_173739.jpg
68.4 KB · Views: 242
doggydan42
I tried to make a clean version of what happened. I didn't upload the work previously because once i get confused it becomes a mess that only I can read.
You're missing a constant that you need to solve for using an initial or boundary condition.

Edit:
It looks like you are incorporating the initial condition into the integral as one of the bounds. But you have to use that as a bound instead of ignore it.

Edit 2:
In your latter solution, watch what you are integrating over, and look at the signs. Try a u-substitution to make it clear.

spsch
spsch
You're missing a constant that you need to solve for using an initial or boundary condition.

Edit:
It looks like you are incorporating the initial condition into the integral as one of the bounds. But you have to use that as a bound instead of ignore it.

Edit 2:
In your latter solution, watch what you are integrating over, and look at the signs. Try a u-substitution to make it clear.

Hi doggydan42, could it be that the differential equation should be : ## V_0 = -\frac{dV}{dt}RC + V ## ?

doggydan42
Hi doggydan42, could it be that the differential equation should be : ## V_0 = -\frac{dV}{dt}RC + V ## ?
I don't believe so. Solve the earlier differential equation using your first method again. It looks like your mistake is setting ##e^0=0## when evaluating the integral over the bounds.

berkeman and spsch
spsch
Omg! You're right doggydan42, I evaluated e^0 as 0 instead of 1. I made this mistake each time I solved it. It's so frustrating, these simple things seem to still happen to me so often. (This morning I was working on another problem where I kept 2*1 = 3 and couldn't figure out what I was doing wrong.)

Thank you very much. I'm so sorry, I really worked through it about 10 times and I seem to have always automatically discarded e^0 as 0.

berkeman


text:
# Changing the color of environments in beamer

I have just started lerning Latex. So, it is possible that the following doubt is very easy.

I have to do a presentation with Beamer. I know that there exist specific comands to theorems, definitions, examples... I wonder if it is possible to modify those environments and change its default color. That is can I use the Theorem environment in such a way that the color of the block is green instead of blue? In the following example I wanto to change the color of the block to red.

\documentclass{beamer}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts}
\usepackage{amsmath, amsthm, amssymb}
\usepackage{enumerate}
\usepackage{graphicx}
\usepackage[english]{babel}
\begin{document}
\begin{frame}{Test}
\begin{theorem}[Pythagoras]
$a^+b^2=c^2$
\end{theorem}
\end{frame}
\end{document}

\documentclass{beamer}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts}
\usepackage{amsmath, amsthm, amssymb}
\usepackage{enumerate}
\usepackage{graphicx}
\usepackage[spanish]{babel}

\AtBeginEnvironment{theorem}{%
\setbeamercolor{block title}{use=example text,fg=white,bg=example text.fg!75!black}
\setbeamercolor{block body}{parent=normal text,use=block title example,bg=block title example.bg!10!bg}
}

\begin{document}
\begin{frame}{Test}
\begin{theorem}[Pythagoras]
$a^+b^2=c^2$
\end{theorem}
\end{frame}
\end{document}

For a yellow colored "Teorema" block, please see the following MWE:

\documentclass{beamer}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts}
\usepackage{amsmath, amsthm, amssymb}
\usepackage{enumerate}
\usepackage{graphicx}
\usepackage[spanish]{babel}

\newtheorem{theo}{Teorema}
\AtBeginEnvironment{theo}{%
\setbeamercolor{block title}{use=example text,fg=black,bg=yellow!75!black}
\setbeamercolor{block body}{parent=normal text,use=block title example,bg=yellow!10}
}

\begin{document}
\begin{frame}{Test}
\begin{theo}[Pythagoras]
$a^+b^2=c^2$
\end{theo}
\end{frame}
\end{document}

• Thanks, but I have a doubt. How could I change the color to yellow? I don't see where is the specific command to change the color – mathlife Jun 16 '19 at 16:28
• @mathlife texdoc xcolor or osl.ugr.es/CTAN/macros/latex/contrib/xcolor/xcolor.pdf – Rmano Jun 16 '19 at 16:31
• Finally, my last doubt: How can I change in that environment the word "Theorem" by "Teorema"? – mathlife Jun 16 '19 at 16:50
• @mathlife: I have edited my answer according to your requests. – leandriis Jun 16 '19 at 16:58
• @leandriis A lot of thanks!!!!!!! – mathlife Jun 16 '19 at 17:03


text:
×

# Triple backslashes in my LaTeX?

What the heck? For some reason, every time I use LaTeX, it works fine, and then I click "Edit", and it generates all these wacky symbols, like triple slashes, double brackets, etc. It's permanent, too! It doesn't revert back, and it weirdly spaces every command. Arg!! Somebody please help me!

Note by Finn Hulse
4 years ago

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
2 \times 3 $$2 \times 3$$
2^{34} $$2^{34}$$
a_{i-1} $$a_{i-1}$$
\frac{2}{3} $$\frac{2}{3}$$
\sqrt{2} $$\sqrt{2}$$
\sum_{i=1}^3 $$\sum_{i=1}^3$$
\sin \theta $$\sin \theta$$
\boxed{123} $$\boxed{123}$$

Sort by:

I had told the Brilliant staff as well.

- 4 years ago

Refer here.

- 4 years ago

Thank you. I had only tried \ [ ... \ ], not \ ( ... ).

- 4 years ago

How do you use LaTeX? $\LaTeX$

- 4 years ago

Update: You can now view Latex codes by hovering over the equation. Read Seeing actual $$\LaTeX$$ for more details!

Staff - 3 years, 3 months ago

Howdy David,

Check out the formatting guide. It tells you everything you need to know about using LaTex on Brilliant.

Staff - 4 years ago

Sorry, what's LaTeX?

- 4 years ago

The thing where you make pretty math symbols. :D

- 4 years ago

We are aware of this bug. We will be fixing it soon. It should only be happening in a few places.

Staff - 4 years ago


text:
by Alex Mennen 25 days ago | Patrick LaVictoire, Scott Garrabrant and Stuart Armstrong like this | link | parent This notion of dependency seems too binary to me. Concretely, let’s modify your example from the beginning so that $$P_3$$ must grant an extra $$10^{-10}$$ utility to either $$P_1$$ or $$P_2$$, and gets to decide which. Now, everyone’s utility depends on everyone’s actions, and the game is still zero-sum, so again, so any strategy profile with $$p=q$$ will be a stratified Pareto optimum. But it seems like $$P_1$$ and $$P_2$$ should ignore still ignore $$P_3$$.

 by Scott Garrabrant 25 days ago | link I agree with this. I think that the most interesting direction of future work is to figure out how to have better notions of dependency. I plan on writing some on this in the future, but basically we have not successfully figured out how to deal with this. reply

### NEW DISCUSSION POSTS

The AI defers to anything
 by Paul Christiano on Corrigibility thoughts II: the robot operator | 0 likes

Thus anything that can
 by Stuart Armstrong on Corrigibility thoughts II: the robot operator | 0 likes

Ah, thanks! That seems more
 by Stuart Armstrong on Loebian cooperation in the tiling agents problem | 0 likes

It doesn't mean computation
 by Vladimir Slepnev on Loebian cooperation in the tiling agents problem | 1 like

I'm not sure this would work,
 by Stuart Armstrong on Loebian cooperation in the tiling agents problem | 0 likes

>How can the short term
 by Stuart Armstrong on Humans are not agents: short vs long term | 0 likes

I expect a workable approach
 by Paul Christiano on Corrigibility thoughts II: the robot operator | 0 likes

 by Stuart Armstrong on Corrigibility thoughts II: the robot operator | 0 likes

It is ‘a preference for
 by Stuart Armstrong on Humans are not agents: short vs long term | 0 likes

Note that we don't need to
 by Paul Christiano on ALBA requires incremental design of good long-term... | 0 likes

If I want my boat to travel
 by Paul Christiano on Corrigibility thoughts II: the robot operator | 0 likes

I don't think it's much like
 by Abram Demski on An Approach to Logically Updateless Decisions | 0 likes

Yeah, I like tail dependence.
 by Sam Eisenstat on An Approach to Logically Updateless Decisions | 0 likes

This is basically the
 by Paul Christiano on Cooperative Oracles: Stratified Pareto Optima and ... | 1 like

I think AsDT has a limited
 by Abram Demski on An Approach to Logically Updateless Decisions | 2 likes


text:
# Find the value of the constant k for which the line y = 2x + 1 is a tangent to the curve y = x^3 + kx + 3

1
oleh vienvienliquisa
Komentar sudah dihapus
hei, risep. show to me the right answer ? can you ?
Yeah but maybe tomorrow i'll answer
good, and can you come back mines which deleted ? it would be funny to compare betwen mine and yours, thanks !

## Jawabanmu

2014-07-07T07:31:54+07:00


text:
@Title("EM (Gaussian Mixture Modeling with Expectation Maximization)")

Package elki.clustering.em

Expectation-Maximization clustering algorithm for Gaussian Mixture Modeling (GMM).
• Class Summary
Class Description
BetulaGMM
Clustering by expectation maximization (EM-Algorithm), also known as Gaussian Mixture Modeling (GMM), with optional MAP regularization.
BetulaGMM.Par
Parameterizer
BetulaGMMWeighted
Clustering by expectation maximization (EM-Algorithm), also known as Gaussian Mixture Modeling (GMM), with optional MAP regularization.
BetulaGMMWeighted.Par
Parameterizer
EM<O,M extends MeanModel>
Clustering by expectation maximization (EM-Algorithm), also known as Gaussian Mixture Modeling (GMM), with optional MAP regularization.
EM.Par<O,M extends MeanModel>
Parameterization class.
KDTreeEM
Clustering by expectation maximization (EM-Algorithm), also known as Gaussian Mixture Modeling (GMM), calculated on a kd-tree.
KDTreeEM.KDTree
KDTree class with the statistics needed for EM clustering.
KDTreeEM.Par
Parameterization class.


text:
# Agrihance-S

### Seed Starter. The best possible start for your crop.

Agrihance-S is a seed starter that is applied on seed in row. It provides key nutrients needed for germination and early growth. Agrihance-S is designed to enhance plant emergence and root development.

• Nutrient management program
• Provides key nutrients needed for germination and early growth
• Seed starter applied on seed in-row
• Designed to enhance plant emergence and root development
• Helps maximize yield

# Featured Video

Total Nitrogen (N). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4%
-1.4% Ammoniacal Nitrogen (N)
-2.6% Urea Nitrogen (N)
Available Phosphate (P2O5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5%
Soluble Potash (K2O). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7%
Iron (fe) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2%
Derived from Ammonium Hydroxide, Urea, Phosphoric Acid, and Potassium Hydroxide and Iron EDTA.

Non-plant food ingredients
Humic Acid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1%

Agrihance is a registered trademark of Monty’s Plant Food Company.

Apply Monty’s Agrihance-S in furrow on the seed, or as a 2×2 at a rate of ½ to 1 gallon mixed in 3 to 8 gallons of a carrier (water) per acre. May be applied with most chemical applications.

# Product Information

VIDEOS  ~  PHOTOS  ~  TESTIMONIALS  ~

In years past we used starter fertilizer of a competing brand. We did some trials with Agrihance-S last year – and we saw a noticeable yield difference at the end of the year. This year, we planted half with Agrihance-S and half without. What we see is more root mass and more feeder roots in the corn plant – the more root mass you have in the corn plant getting started, the healthier the start we are going to have.We’ve seen more even emergence vs not using a starter and also over competing brands.
I could tell the treated areas were not only getting more nutrients, but that the traditional fertilizer that I had applied was breaking down better and being absorbed more by the crop. The treated beans just utilized the nutrients more efficiently.


text:
# Find $zw, \frac{z}{w},\frac{1}{z}$ for $z=2\sqrt{3}-2i, w=-1+i$

Find $zw, \frac{z}{w},\frac{1}{z}$ for $z=2\sqrt{3}-2i, w=-1+i$

I went wrong somewhere, this is what I have so far (this is in polar):

$z=4\left(\cos\left(\frac{11\pi}{6}\right)+\sin\left(\frac{11\pi}{6}\right)\right)$

$w=\sqrt2\left(\cos\left(\frac{7\pi}{4}\right)+\sin\left(\frac{7\pi}{4}\right)\right)$

Now my setup should be:

$zw=4\sqrt2\left(\cos\left(\frac{11\pi}{6}+\frac{7\pi}{4}\right)+\sin\left(\frac{11\pi}{6}+\frac{7\pi}{4}\right)\right)$

The common denominator is $12$ so

$zw=4\sqrt2\left(\cos\left(\frac{22\pi}{12}+\frac{21\pi}{12}\right)+\sin\left(\frac{22\pi}{12}+\frac{21\pi}{12}\right)\right)$

which then should equal out to

$zw=4\sqrt2\left(\cos\left(\frac{43\pi}{12}\right)+\sin\left(\frac{43\pi}{12}\right)\right)$

The answer in the book says:

$zw=4\sqrt2\left(\cos\left(\frac{7\pi}{12}\right)+\sin\left(\frac{7\pi}{12}\right)\right)$

Where did I go wrong?

I haven't even tried the other problems yet.

• What do you mean by find zw, z/w, and 1/z? you want the result of the form a+ib? – Jika Jun 23 '14 at 20:35
• @jika zw is a form, z1z2. complex numbers of polar form. – Joshhw Jun 23 '14 at 20:37

First, notice that the argument of $w$ is $\frac{3 \pi}{4}$, not $\frac{7\pi}{4}$. And you forgot to put the "$i$"'s together with the sines. Just a little distraction. The other argument, and the absolute values are ok. You had setup everything else correctly. Using the right value above, we get: \begin{align}zw &=4\sqrt2\left(\cos\left(\frac{11\pi}{6}+\frac{3\pi}{4}\right)+i\sin\left(\frac{11\pi}{6}+\frac{3\pi}{4}\right)\right) \\ &= 4\sqrt2\left(\cos\left(\frac{22\pi + 9\pi}{12}\right)+i\sin\left(\frac{22\pi + 9\pi}{12}\right)\right) \\ &= 4\sqrt2\left(\cos\left(\frac{31\pi}{12}\right)+i\sin\left(\frac{31\pi}{12}\right)\right)\end{align} Normally, we would stop here, but, we can always reduce the argument so it is between $0$ and $2\pi$, and use that $\sin$ and $\cos$ have both period $2\pi$. Notice that: $$\frac{31\pi}{12} = \frac{7\pi}{12} + \frac{24\pi}{12} = \frac{7\pi}{12} + 2\pi$$ This way, we obtain: $$zw = 4\sqrt2\left(\cos\left(\frac{7\pi}{12}\right)+i\sin\left(\frac{7\pi}{12}\right)\right)$$ as desired. Ok?
• how is it $\frac{3\pi}{4}$? is it because the circle goes counterclockwise and I should've found the that angle before the one I used? – Joshhw Jun 23 '14 at 22:21
• Plot the point in the complex plane to see it better. Yes, we count it counterclockwise. The point in the line $y = -x$. – Ivo Terek Jun 23 '14 at 22:23
You have the angle wrong for $w$. It should be $3\pi/4$, right?


text:
Yampa-0.10.5.1: Library for programming hybrid systems.

Copyright (c) Antony Courtney and Henrik Nilsson Yale University 2003 BSD-style (see the LICENSE file in the distribution) ivan.perez@keera.co.uk provisional non-portable (GHC extensions) None Haskell98

FRP.Yampa.Hybrid

Description

Synopsis

# Discrete to continuous-time signal functions

## Wave-form generation

hold :: a -> SF (Event a) a Source #

Zero-order hold.

dHold :: a -> SF (Event a) a Source #

Zero-order hold with delay.

Identity: dHold a0 = hold a0 >>> iPre a0).

trackAndHold :: a -> SF (Maybe a) a Source #

Tracks input signal when available, holds last value when disappears.

!!! DANGER!!! Event used inside arr! Probably OK because arr will not be !!! optimized to arrE. But still. Maybe rewrite this using, say, scan? !!! or switch? Switching (in hold) for every input sample does not !!! seem like such a great idea anyway.

dTrackAndHold :: a -> SF (Maybe a) a Source #

## Accumulators

accum :: a -> SF (Event (a -> a)) (Event a) Source #

Given an initial value in an accumulator, it returns a signal function that processes an event carrying transformation functions. Every time an Event is received, the function inside it is applied to the accumulator, whose new value is outputted in an Event.

accumHold :: a -> SF (Event (a -> a)) a Source #

Zero-order hold accumulator (always produces the last outputted value until an event arrives).

dAccumHold :: a -> SF (Event (a -> a)) a Source #

Zero-order hold accumulator with delayed initialization (always produces the last outputted value until an event arrives, but the very initial output is always the given accumulator).

accumBy :: (b -> a -> b) -> b -> SF (Event a) (Event b) Source #

Accumulator parameterized by the accumulation function.

accumHoldBy :: (b -> a -> b) -> b -> SF (Event a) b Source #

Zero-order hold accumulator parameterized by the accumulation function.

dAccumHoldBy :: (b -> a -> b) -> b -> SF (Event a) b Source #

Zero-order hold accumulator parameterized by the accumulation function with delayed initialization (initial output sample is always the given accumulator).

accumFilter :: (c -> a -> (c, Maybe b)) -> c -> SF (Event a) (Event b) Source #

Accumulator parameterized by the accumulator function with filtering, possibly discarding some of the input events based on whether the second component of the result of applying the accumulation function is Nothing or Just x for some x.


text:
# timerstop

Version:

Stops the stopwatch timer and outputs the elapsed time in seconds. Use timerstop with timerstart to determine the computation time of a specific code segment.

## Syntax

timerstop(t)
Legacy name: toc

## t

Stopwatch timer which is returned by timerstart. t is a uint64.

t = timerstart
for i = 1:100000
a = sqrt(2);
end
timerstop(t)

Where This Node Can Run:

Desktop OS: Windows

FPGA: This product does not support FPGA devices