speciation requires the evolution of reproductive isolation between initially compatible individuals .one of the most accepted mechanisms for the development of reproductive barriers is the geographic isolation of the populations , which can be complete ( allopatry ) or partial ( parapatry ) . in both casesthe reduction of gene flow between individuals inhabiting different regions facilitates the fixation of local adaptations .these , in turn , may lead to pre - zygotic or post - zygotic mating incompatibilities and can eventually develop into full reproductive isolation .sympatric speciation , where new species arise from a population inhabiting a single geographic region , has been highly debated and its occurrence in nature is documented in only a few cases .mathematical models have shown that sympatric speciation is theoretically possible and the model by dieckmann and doebeli , in particular , attracted a lot of attention . in their model , speciation results from the interplay between strong competition for resources and the way resources are distributed .it was shown that , under appropriate conditions , the population will evolve in such a way as to consume the most abundant type of resource , but it will arrive at the corresponding ` optimal ' phenotype in a minimum of the fitness landscape , causing it to split in two. the resulting system of two subpopulations with different phenotypes would then be at a fitness maximum .for sexually reproducing individuals the evolution of reproductive barriers between the two nearly split populations would need a further ingredient , assortative mating .this is tendency of individuals to mate with others that are similar to themselves and prevents the mixing of the two subpopulations . in the model assortative matingwas shown to evolve naturally because only then the fitness maximum could be achieved . in spite of its theoretical plausibility , this andother models have been criticized for their supposedly unrealistic assumptions .more surprising than sympatric speciation under strong selection is the possibility of sympatric speciation in a neutral scenario .it was shown by derrida and higgs ( dh ) that a population of initially identical individuals subjected only to mutations and assortative mating can indeed split into species in sympatry if the genomes are infinitely large .the hypothesis of genomes containing infinitely many loci , though not new , raises important questions about the meaning of a locus ( a gene , the position of a base - pair on a chromosome or other replicating molecule ) and the actual number of such loci that would be necessary for speciation to occur , since , after all , genes or base - pairs are always finite .moreover , the model assumes that all these infinitely many loci are involved in assortative mating , which also needs to be taken into account in the interpretation of the loci .  in this paperwe consider the dh model for finite genomes .we find that , for the parameters considered in the original paper , speciation occurs only for very large number of loci , of the order of .we compare the dynamics of the model for finite number of loci with the infinite loci model and show how the broadening of the genetic distribution of genotypes in the finite loci model prevents the onset of speciation .we also compare the finite loci dh model with a spatial version of the dynamics introduced in . when space is added and reproduction is restricted by assortative mating and spatial proximity , the minimum number of loci needed for speciation drops by several orders of magnitude and can occur even for as few as 100 loci .our simulations show that when the number of loci is small species form in well defined spatial regions , with little overlap at the boundaries .when the number of loci is large , on the other hand , space becomes less important and the species overlap considerably more in space .the model introduced by derrida and higgs considers a sympatric population of haploid individuals whose genomes are represented by binary strings of size , where , can assume the values . for simplicity , each locus of the genome will be called a gene and the values and the corresponding alleles .the number of individuals at each generation is kept constant and the population is characterized by an matrix measuring the degree of genetic similarity between pairs of individuals : if the genomes of and are identical whereas two genomes with random entries will have close to zero .alternatively , the genetic distance between the individuals , measuring the number of genes bearing different alleles , is .each generation is constructed from the previous one as follows : a first parent is chosen at random .the second parent has to be genetically compatible with the first , i.e. , their degree of similarity has to satisfy . in other wordsparents must have at least genes bearing the same allele to be compatible ( assortative mating ) .individuals are then randomly selected until this condition is met . if no such individual is found , is discarded and a new first parent is selected .the offspring inherits , gene by gene , the allele of either parent with equal probability ( sexual reproduction ) .the process is repeated until offspring have been generated .individuals are also subjected to a mutation rate per gene , which is typically small .the model is neutral in the sense that the probability that an individual is chosen as first parent ( ) does not depend on its genome .however , once the first parent has been selected , the chances of being picked as second parent and produce an offspring does depend on the genome and a fitness measure can actually be defined in association with assortativeness .  to understand how the similarity matrix changes through generations , consider first an asexual population where each individual has a single parent in previous generation .the allele will be equal to with probability and with probability , so that the expected value is for independent genes , the expected value of the similarity between and is , therefore , in sexual populations and have two parents each , , and , , respectively , and since each inherits ( on the average ) half the alleles from each parent , it follows that , on the average , with .  in the limit of infinitely many genes , , this expression becomes exact and the entire dynamics can be obtained by simply updating the similarity matrix . if there is no restriction on mating , , the overlaps converge to a stationary distribution centered at .the approximation holds for and much smaller than one , which is always the case for real populations .if the population splits into species formed by groups of individuals whose average similarity is larger than and such that interspecies similarity is smaller than , tending to zero with time .figure [ fighd](d ) shows the histogram of between all pairs of individuals for , ( ) and for 100 , 200 , 300 and 400 generations .the initial condition is for all and , representing genetically identical individuals at the start of the simulation . at and histogram displays peaks to right of showing that all pairs are still compatible . at , on the other hand , the peak at is a signature of speciation , showing that several pairs of individuals have become reproductively isolated .the corresponding species are represented by the smaller peaks to the right of .  , , and generations .in all cases , and .,title="fig : " ] , , and generations . in all cases , and .,title="fig : " ] , , and generations . in all cases , and .,title="fig : " ] , , and generations . in all cases , and .,title="fig : " ]  the only condition for speciation in the dh model is  . changing the mutation rate ( or the population size ) but keeping fixed only affects the time to speciation , which increases approximately linearly with , or , as shown in fig .[ time](a ) .this is consistent with eqs.([aveq ] ) and ( [ hdupdate ] ) , which indicade that the change in the similarity matrix per time step is proportional to for . increasing , on the other hand , decreases and increases the number of species formed .( stars , upper horizontal axis ) and ( squares , lower horizontal axis ) . the values and used in the other figures are marked with a larger ( red ) symbol . in both panels .,title="fig : " ]  ( stars , upper horizontal axis ) and ( squares , lower horizontal axis ). the values and used in the other figures are marked with a larger ( red ) symbol . in both panels .,title="fig : " ]an important feature of the dh model is that there is no need to describe the genomes explicitly : all it takes is the initial similarity matrix , which is set to 1 , and the update rule eq .( [ hdupdate ] ) . to implement the model with a finite number of genesit is necessary to keep track of all the genomes at each generation and calculate the similarities from the definition eq .( [ qdef ] ) .offspring for the next generation are created by choosing ( gene by gene ) the allele of each parent with equal probability and letting the allele mutate ( from + 1 to -1 or vice - versa ) with probability .parents are chosen like in the original dh model , picking the first parent at random and a second parent compatible with it .if the number of genes is small the distribution of similarities starts from and moves to the left until , where it remains stationary and no speciation occurs .we found that speciation only occurs for very large values of , as shown in fig .[ fighd ] ( a)-(c ) . for the parameters used ( , ) we observed speciation only for larger than about .when compared to the dh model we see that the finite number of genes blurs the peaks of the distribution , preventing them from breaking up .the minimum number of genes required for speciation depends directly on the mutation rate and population size .figure [ time](b ) shows the minimum value of for speciation as a function of and for . for populations with only individuals drops to about 35000 whereas for it is necessary at least 430000 genes , showing that rapidly grows with population sizes . in all casesthe simulations were ran for at least twice the time to speciation of the dh model , fig .[ time](a ) to make sure the similarity distribution would either move to the left of and speciate ( for large enough ) or stay frozen close to .error bars for are of the order of symbols size : for and for .the importance of space in evolution has long been recognized and explicit empirical evidence of its role has been recently provided by ring species .the spatial model we discuss here is a simplified version of the model proposed in .the main additional ingredient with respect to the dh model with finite genomes is that the individuals are now distributed on a two - dimensional square area with periodic boundary conditions .mating is not only restricted by genetic similarity but also by spatial proximity , so that an individual can only choose as mating partner those inside a circular neighborhood of radius centered on its spatial location , called the _mating neighborhood_. we note that a number of other effects , such as demographic stochasticity , population expansions , costs of reproduction , and migration rates between subpopulations might also influence the outcome of speciation . herewe consider only the effect of finite mating neighborhoods and keep all the other ingredients as similar as possible to the original dh model .  .the top axis show the corresponding average number of individuals in the mating neighborhood ( , . ) ]  space represents an environment where resources are distributed homogeneously and the individuals should occupy it more or less uniformily so that enough resources would be available to all of them .the total carrying capacity is the population size and the average area needed per individual is .the dynamics is constructed in such a way that offspring are placed close to the location of the original parents and the approximately uniform distribution of the population is preserved at all times . however , the mechanism used in the dh model of picking a random individual from the population to be the first parent and then a second individual to be the second parent ( and repeating the process times ) promotes strong spatial clustering .  , and for , and generations .( b ) spatial distribution of the population at ; different colors ( shades of gray ) show different species .panels ( c ) and ( d ) show similar plots for and ; ( e ) and ( f ) show the results for and .,title="fig : " ] , and for , and generations .( b ) spatial distribution of the population at ; different colors ( shades of gray ) show different species .panels ( c ) and ( d ) show similar plots for and ; ( e ) and ( f ) show the results for and .,title="fig : " ] , and for , and generations .( b ) spatial distribution of the population at ; different colors ( shades of gray ) show different species .panels ( c ) and ( d ) show similar plots for and ; ( e ) and ( f ) show the results for and .,title="fig : " ] , and for , and generations .( b ) spatial distribution of the population at ; different colors ( shades of gray ) show different species .panels ( c ) and ( d ) show similar plots for and ; ( e ) and ( f ) show the results for and .,title="fig : " ] , and for , and generations .( b ) spatial distribution of the population at ; different colors ( shades of gray ) show different species .panels ( c ) and ( d ) show similar plots for and ; ( e ) and ( f ) show the results for and .,title="fig : " ] , and for , and generations .( b ) spatial distribution of the population at ; different colors ( shades of gray ) show different species .panels ( c ) and ( d ) show similar plots for and ; ( e ) and ( f ) show the results for and .,title="fig : " ]  in order to avoid clustering we implement the dynamics in a slightly different way ( see also section [ vlg ] ) : the initial population is randomly placed in the area .each one of the individuals has a chance of reproducing but there is a probability that it will not do so , accounting for the fact that not all individuals in the present generation will be first parents of the next . in the casethe _ focal _ individual does not reproduce , another one from its mating neighborhood is randomly chosen to reproduce in its place . in either case the offspring generated will be positioned exactly at the location of the focal individual or will disperse with probability to one of 20 neighboring sites .therefore , close to the location of every individual of the previous generation there will be an individual in the present generation , keeping the spatial distribution uniform and avoiding the formation of clusters .the first parent ( or a neighbor reproducing in its place ) chooses a compatible second parent within its mating neighborhood of radius .if is small , the number of individuals in the mating neighborhood can be close to zero due to fluctuations in the spatial distribution .to avoid this situation , and also to follow the procedure introduced in , if the number of compatible mates in the neighborhood is smaller than ( in the simulations ) , the individual expands the search radius to .if the number of compatible mates is still smaller than , the process is repeated twice more up to , and if there is still less than potential mates another neighbor is randomly selected to reproduce in its place .the probability was fixed in , which corresponds approximately to the probability that an individual is not selected in trials with replacement , , in accordance with the dh model .the size of mating neighborhood is a key extra parameter . if we recover the dh model with finite genomes .if is small speciation is strongly facilitated and can occur for much smaller values of for a given or .[ fig3 ] shows the minimum value of for which speciation happens as a function of the size of the mating neighborhood for .the figure also shows the average number of individuals inside the mating neighborhood , .the values were obtained by varying by units of when and by units of for larger s .populations evolved for generations .in some cases speciation did occur for slightly smaller values of , but it took much longer times . for , for instance , we observed speciation for for . comparing with the finite dh model we note that speciation can take place even for if .as increases the restriction in the size of the mating neighborhood required for speciation decreases , until it is unnecessary if is of the order of 110000 .figure [ fighds ] shows the histogram of similarity between pairs of individuals at three times and a snapshot of the population for , ; , and and .the peaks in the distribution are larger both because is small and because of the restriction in .for the species are well localized in space , with little overlap at their boundaries ( see ) . for and the spatial overlap between species is considerably larger .as discussed in section [ sm ] , the construction of spatial models that converge to the finite dh model as the parameter s becomes large is not straightforward .the problem resides in the way generations are constructed in the dh model , where a first parent is chosen at random from the population and then a second , genetically compatible , parent is also chosen at random to generate an offspring .the direct application of this procedure in the spatial model would consist in choosing a random first parent and picking the second parent from its mating neighborhood , an area centered on the individual .the offspring should be put close to the location of the first or second parent , or somewhere in between .this , however , leads to spatial clustering of the population , since a large fraction ( ) of the population is never chosen as first parents , leaving holes in these areas and overcrowding areas where individuals are picked twice or more . to avoid this situationwe have replaced the random choice of the first parent by going through the population one by one and giving each individual a chance ( ) to reproduce . when it did not we picked another individual from its mating neighborhood to reproduce in its place , instead of a random individual taken from the entire population as in the original dh or finite dh models .the offspring is always placed close to the location of the first parent , keeping the distribution spatially uniform .   at for ( a ) and ( b ) .small mating neighborhoods favors spatial isolation of species.,title="fig : " ]  at for ( a ) and ( b ) .small mating neighborhoods favors spatial isolation of species.,title="fig : " ]  in order to compare the spatial model with an equivalent sympatric system we implemented a variation of the finite dh model in which , similarly to the spatial model , we give each individual a chance ( ) to reproduce and , when it does not , we pick another random individual from the entire population to reproduce in its place .the results obtained with this variation are qualitatively identical to those described in sections [ dh ] and [ dhf ] .this validates the comparison of the dh finite genome model with the spatial model introduced in section [ sm ] .finally we consider the process of speciation in the spatial model for genome sizes above the threshold where speciation occurs in the sympatric model . in this case speciationoccurs for any value of , small or large . for small local matingcreates a spatial distribution of genotypes where nearby individuals tend to be similar but different from others located far away . for finite populationsthis genetic gradient is not smooth , but step - like , and prone to break up into genetically isolated groups that are spatially correlated .speciation happens not because is large , but because is small . for large ,on the other hand , the mechanism is very different and takes place on the global scale of the population .the balance between mutations , which generally increases the average genetic distance between two individuals , and sexual reproduction , which mixes the genomes and has the opposite effect , leads to a distribution of genetic distances in the population .only when this distribution becomes wide enough , as compared with the criterion of assortative mating , the population splits into species .figure [ fig5 ] shows the resulting populations in each case , where a signature of is clearly seen in the spatial organization of the species .the mechanisms responsible for the origin of species remain controversial . among the important questions yet to be fully understood is the role of geography ( allopatric , parapatric and sympatric modes ) and the number of genes involved in the evolution of reproductive isolation .it has been recently argued that these two points are closely related and that detailed molecular analysis might reveal the geographic mode behind specific speciation events .sympatric speciation has been thought to be the most unlikely of the modes , since the possibility of constant gene flow would keep the populations mixed and prevent the evolution of reproductive barriers .dieckmann and collaborators have argued that this is not so if competition for resources is strong enough . in that casereproductive isolation would not only evolve naturally but would _ require _ sympatry , otherwise competition for local resources would not be intense .  surprisingly , derrida and higgs have shown that sympatric speciation may occur even without competition , in a totally neutral scenario , if mating is assortative and if the number of loci in the genome can be considered infinite .when loci are interpreted as genes the hypothesis of an infinitely large genome becomes rather unrealistic .however , at the molecular level , where nucleotide sequences are the units to be considered , infinite loci models become attractive .nevertheless , nucleotides can hardly be considered independent and certainly do not segregate in the way as assumed by the dh models .understanding how many independently segregating loci are necessary for neutral sympatric speciation is , therefore , an important question .the main parameters controlling the dynamics in the dh model are and the assortativity measure .the combination also appears in hubbell s neutral theory of biodiversity and is the _fundamental biodiversity number _ , since it controls the number of species in a community and the abundance distribution .  in this paperwe have revisited the dh model and simulated it for finite numbers of loci .we found that , for typical parameters used in the original paper , speciation happens only for very large number of loci , of the order of . when the number of loci is small , the genetic variability within the population is large , hindering speciation .the histogram of genetic similarity between individuals evolves into a broad peak instead of multiple sharp peaks that represent groups of very similar individuals that are dissimilar among different groups .as the number of genes increase the peaks become thiner , secondary structures appear and eventually turn into species .increasing the mutation rate and keeping fixed decreases the minimum number of loci needed and also the time to speciation .in contrast with the sympatric dh model , the spatial model introduced in displays speciation with much smaller number of loci . the model considers a population that is uniformly distributed in space , without any explicit separation into demes or subpopulations .although it falls into the class of parapatric models , it has been termed topopatric to distinguish it from models involving demes or metapopulations .mating is restricted not only by genetic similarity but also by spatial proximity , allowing gene flow across the entire population but substantially reducing the speed of the flow .mutations are transmitted diffusively across the population , and not instantaneously like in the sympatric model , largely facilitating speciation .previous studies with metapopulation models ( and infinitely large genomes ) observed similar effects , with speciation occurring for smaller mutation rates due to the isolation of subpopulations . for small number of genes , of the order of 100 ,speciation occurs only with severe spatial mating restrictions .accordingly , the species that form display strong spatial segregation , with little overlap between adjacent species ( figs . [ fighds](b ) and [ fig5](a ) ) .this type of geographic distribution leads back to question of modes of speciation and suggests that it might happen that species appeared not because the populations were geographically separated ( as in allopatry ) but rather they are geographically separated because they emerged in a homogeneous environment with slow gene flow .if the number of loci participating in the assortative process is large the spatial restriction on mating can be relaxed . as a consequence , the spatial segregation decreases and the overlap among species increases .+  + acknowledgments :  it is a pleasure to thank david m. schneider , ayana martins and blake stacey for their critical reading of this paper and many suggestions .this work was partly supported by the brazilian agencies fapesp ( grant 2016/06054 - 3 ) and cnpq ( grant 302049/2015 - 0 ) .marcus a. m. de aguiar , david m. schneider , eduardo do carmo , paulo r. a. campos , and ayana b. martins .error catastrophe in populations under similarity - essential recombination ., 374:4853 , june 2015 .devin w. goodsman , barry cooke , david w. coltman , and mark a. lewis . the genetic signature of rapid range expansions : how dispersal , growth and invasion speed impact heterozygosity and allele surfing . , 98:1 10 , 2014 .carlos a machado , richard m kliman , jeffrey a markert , and jody hey . inferring the history of speciation from multilocus dna sequence data : the case of drosophila pseudoobscura and close relatives . , 19(4):47288 , apr 2002 .

Abstract: the speciation model proposed by derrida and higgs demonstrated that a sexually reproducing population can split into different species in the absence of natural selection or any type of geographic isolation , provided that mating is assortative and the number of genes involved in the process is infinite . here we revisit this model and simulate it for finite genomes , focusing on the question of how many genes it actually takes to trigger neutral sympatric speciation . we find that , for typical parameters used in the original model , it takes of the order of genes . we compare the results with a similar spatially explicit model where about 100 genes suffice for speciation . we show that when the number of genes is small the species that emerge are strongly segregated in space . for larger number of genes , on the other hand , the spatial structure of the population is less important and the species distribution overlap considerably .


rayleigh - bnard convection , the buoyancy - driven flow of a fluid heated from below and cooled from above , is a fundamental paradigm of complex nonlinear dynamics , pattern formation , and turbulence .an ongoing challenge for analysis , theory , computation , and experiment is to ascertain how the heat transport depends on the thermal forcing as gauged by a nondimensional rayleigh number and the fluid s material properties ( typically the prandtl number , the ratio of kinematic viscosity to thermal diffusivity ) . rigorous bounds on heat transport in rayleigh - bnard convection within the boussinesq approximation were pioneered by howard and subsequently elaborated by busse .later , following the motivational work of hopf , an alternative variational framework for bounds on turbulent transport of momentum , mass , and in the case of rayleigh - bnard convection , heat , known as the ` background method ' was formulated .this is the approach we adopt here .  in this paperwe consider an infinite prandtl number boussinesq fluid contained between two rigid isothermal boundaries thermally driven by constant internal heating .this model is inspired by convection in the earth s mantle where the prandtl number is and the motion is predominantly driven by a semi - uniform heating from radioactive decay . for definitenesswe consider an isoviscous fluid subject to no - slip isothermal vertical boundary conditions ( without loss of generality on the boundaries ) with uniform heating in the bulk , an idealization of the actual geophysical conditions . at low heating rates as measured by the dimensionless ` heat rayleigh number ' , the fluid remains at rest and heat is transported to the boundaries by conduction with a parabolic temperature profile across the layer . at high convection sets in ,increasing the effective thermal conductivity of the layer and consequently lowering the bulk averaged temperature .the enhancement of heat transport is indicated by , the ratio of the bulk averaged temperature to that of the parabolic profile resulting from conduction alone .the challenge is to determine how varies with .the no - convection conduction solution exists ( be it stable or unstable ) for all and realizes the upper limit on the bulk averaged temperature , . at high question is , how low can go ?this problem was previously considerd by lu _ who used estimates originally derived for boundary driven convection and a simple piecewise linear background profile to produce a lower bound on the space and time averaged temperature relative to that of the conduction state .that result was because , this is not inconsistent with a scaling law measured from direct numerical simulations suggesting that we note , however , that those computations employed ` free - slip ' boundary conditions on the velocity field rather than the no - slip conditions employed in the analysis .boundary conditions for traditional rayleigh benard convection can drastically affect the dynamics so the comparison must be taken with a degree of caution .more recent developments indicate that background profiles including some stable stratification in the bulk may be optimal for infinite prandtl number convection suggesting room for improvement for rigorous results . in particular , a singular integral analysis was performed to obtain a key estimate that was then utilized in the background method to produce an upper bound on the nusselt number , the dimensionless measure of the enhancement of heat transport in boundary - driven rayleigh - bnard convection , in terms of the traditional rayleigh number of the form ^{1/3} ] .previous analysis of this problem and boundary driven convection considered background profiles constant in the bulk of the layer so the only possible negative contribution to relied on the product of and in boundary layers where both are constrained to be relatively small in magnitude .as was shown for boundary driven convection , however , a stably stratified ( i.e. ) profile in the bulk can be exploited to utilize the positive weighted correlation between and resulting from the slaving and improve the positivity of , allowing for sharper estimates .consider the family of background profiles illustrated in figure [ fig : tau_final ] , & -\delta \leq z\leq 0 , \end{array}\right.\ ] ] where the positive parameters , , and will be chosen to optimize the bound .the logarithmic term enhances the positivity of , and hence leads to an improved scaling of the boundary layer with , while the quadratic term is meant to increase the integral of sufficiently to offset the slow logarithmic growth near and lessening the negative impact of the dirichlet integral in .it is easily verified that +b\left[\frac{2}{3}-\frac{\delta}{2}-\frac{\delta^3}{6}\right]\\ \label{integral2 } \int_{-1}^0\left(\tau'(z)\right)^2dz & = & a^2\left(\frac{1}{\delta}-1+\frac{\left[\log(1/\delta)\right]^2}{\delta}\right)\\ & + & ab\left(2\frac{\log(1/\delta)}{\delta}+4 - 2\delta\log(1/\delta)-4\delta\right)\\ & + & b^2\left(\frac{1}{\delta}+\frac{4}{3}-2\delta-\frac{\delta^3}{3}\right)\end{aligned}\ ] ] producing the lower bound on the average temperature given by when the positivity of is maintained .an appropriate choice of the scaling of the parameters and with respect to will allow us to determine both the ` correct ' boundary layer scaling , and to maximize the lower bound on .the two key inequalities required for the analysis are the first inequality ( [ new_estimate ] ) is an exercise in integration by parts the details of which are left to the reader .the second inequality is a restatement and slight improvement in the prefactor of the key result previously derived for rayleigh - bnard convection .while a prefactor improvement may be considered minor , our approach to prove it is simplified significantly and embeds the problem in the familiar context of a generalized hardy - rellich inequality .the proof , provided in the appendix , also indicates that the estimate with this prefactor is sharp .  to determine conditions guaranteeing the positivity of we reformulate it neglecting much of the norm of as well as the term and use to observe }{z}dz\\ & - & \int_{-\delta}^0\left(\frac{2a\log(1/\delta)}{\delta}-\frac{2a}{z}-4bz+\frac{2b(1-\delta^2)}{\delta}\right)re[\theta w^*]dz.\end{aligned}\ ] ] in the above we added the bulk terms to the boundary layer in order to apply and to the entire interval . applying then implies .\ ] ] bound the magnitude of the last integral in as follows : \right|\\ & \leq & \int_{-\delta}^0 \left(\frac{2a\log(1/\delta)}{\delta}-\frac{2a}{z}-4bz+\frac{2b(1-\delta^2)}{\delta}\right)z^2\frac{|\theta|}{|z|^{1/2}}\frac{|w|}{|z|^{3/2}}dz\\ \label{sup_first_appears } & \leq & 2\left(\sup_{-\delta\leq z\leq 0}\frac{|\theta(z)|}{|z|^{1/2}}\right)\left(\int_{-\delta}^0z^4\left[\frac{a\log(1/\delta)}{\delta}-\frac{a}{z}-2bz+\frac{b(1-\delta^2)}{\delta}\right]^2dz\right)^{1/2}\\ & \quad & \times \left(\int_{-1}^0\frac{|w|^2}{|z|^3}dz\right)^{1/2}.\end{aligned}\ ] ] the homogeneous boundary conditions on that for , hence we can bound the supremum in and apply young s inequality to see that \right|\\\label{final_inner_integral } & \leq & \int_{-\delta}^0\left|\frac{d\theta}{dz}\right|^2dz + \int_{-\delta}^0z^4\left[\frac{a\log(1/\delta)}{\delta } - \frac{a}{z}-2bz+\frac{b(1-\delta^2)}{\delta}\right]^2dz \times \int_{-1}^0\frac{|w|^2}{|z|^3}dz.\end{aligned}\ ] ] inserting into we see that ^ 2dz\right\}\int_{-1}^0\frac{|w|^2}{|z|^3}dz.\ ] ]  the integral about the boundary layer in can be computed exactly .at this point we choose and where and are absolute constants .then ^ 2dz\\ & = & a'^2\frac{\delta^5}{5}+2a'b'\frac{\delta^5}{5}+b'^2\frac{\delta^5}{5}+o\left(\delta^5\log(1/\delta)\right)\end{aligned}\ ] ] as . comparing this with we see that the minimal requirement for to remain positive in the or limit is }{5}+2b'\frac{\delta^4\log(1/\delta)}{5}+\frac{b'^2}{a'}\frac{\delta^4\log(1/\delta)}{5}\\ & \rightarrow & \frac{1}{{\mbox{r } } } \sim \frac{\xi(a',b')}{4}\delta^4\log(1/\delta)\end{aligned}\ ] ] where this yields the scaling of the boundary layer thickness as ^{-1/4}.\ ] ]  the average temperature is bounded by two times minus implying that , asymptotically , to obtain the ` best ' prefactor, we maximize over and to achieve where the optimal prefactor is obtained for and .the background profile can be considered the sum of a singular logarithmic profile and a smooth conduction - like quadratic profile .if the logarithmic term only is considered , i.e. , , then the profile would be this would be analogous to the approach taken in for boundary driven convection . however if the same steps are followed one sees that the optimal bound occurs for ^ 2} ] and . in this context , is recognized as a factor of two improvement on the original proof . as in the original proof we will prove the following proposition :   in order to see the connection between and hardy - rellich inequalities , make the change of variables .it follows that also satisfies . inserting this change of variables into the fourth order term that results from the definition of , we see that a similar calculation leads to putting and together , we see that can be restated as   traditionally a hardy - rellich inequality is formulated in terms of the norms of the operator where and possibly higher orders ( see for example ) . is , with the appropriate integrations by parts , nothing else than the norm of the differential operator acting on .the inclusion of the wave number here causes us to refer to this inequality as a generalized hardy - rellich inequality .  to prove the lemma ,consider the following one - parameter family of integrals , ^ 2dz,\ ] ] where satisfies the homogeneous boundary conditions . expanding and integrating by parts multiple times leads to the following identity : letting produces which is easily rearranged to establish the lemma .the methodology employed above is not restrictive to this particular situation , and lends itself immediately to extension to higher order operators , and possibly higher dimensions as well .the free parameter can be adjusted as desired , indicating a significant utility to this method of producing hardy - rellich type inequalities .hardy - rellich inequalities with remainder terms can also be computed by optimizing over the wave - number ( for an example of other hardy - rellich type inequalities with remainder terms see the work of evans and lewis ) .the strictness of the inequality derived here can be verified by considering functions that saturate , that is those functions satisfying the boundary conditions together with solutions of are linear combinations of modified bessel functions : \ ] ] where . just as the original hardy inequality is not saturated for any nontrivial analytic functions, the functions can not satisfy all the boundary conditions simultaneously so there is no analytic solution to that saturates .howeverm regularizing appropriately at the boundaries will produce a sequence of functions that satisfy the boundary conditions and , in the unregularized limit , satisfy .hence while is never saturated , there can be no improvement on the prefactor derived by this method , i.e. , the approach outlined here is not only robust and amenable to adaptation , but also produces sharp estimates .

Abstract: we derive an improved rigorous lower bound on the space and time averaged temperature of an infinite prandtl number boussinesq fluid contained between isothermal no - slip boundaries thermally driven by uniform internal heating . a novel approach is used wherein a singular stable stratification is introduced as a perturbation to a non - singular background profile , yielding the estimate ^{-1/4}$ ] where is the heat rayleigh number . the analysis relies on a generalized hardy - rellich inequality that is proved in the appendix .


observations of the anisotropy and polarization of the cosmic microwave background ( cmb ) provide a unique information about the primordial inhomogeneity of the universe . since detection by cobe ( smoot etal , 1992 , bennett et al , 1996 ) of the cmb anisotropy several groups have reported on high angular resolution observational data at the angular scales in the vicinity of the so called doppler peak in the power spectrum ( hancock , et al , 1994 ; gundersen , 1993 ; de bernarais , et al . , 1994 ; masi , et al , 1996 ; tanaka , et al . , 1995 ; cheng , et al ., 1994 ; netterfield et al . , 1996 ; scott , et al , 1996 ) .determination of the spectrum of the primordial anisotropy on scales will yield valuable clues to the formation of the large scale structure of the universe and the most important parameters of the universe : the total and baryonic densities at the present time ( and ) ; the hubble constant , ionization history etc .however , the interpretation of these experimental results as well as the comparison with the expected power spectra of anisotropy in different cosmological models is complicated .future experiments ( map and planck ) will construct the map of the cmb with high level of resolution and sensitivity .the temperature distribution in the sky also contains the information beyond power spectrum .gaussianity of the primordial signal is well motivated by the inflationary cosmology and has been adopted by many authors ( see for review starobinsky 1982 , bardeen et al .1983 , bardeen et al . 1986 ) . in this casethe distribution of the cmb temperature in the sky is in the form of the two - dimensional scalar random gaussian field .this field can be completely characterized by its power spectrum .many authors proposed to calculate various statistical characteristics of the cmb anisotropy as tests of gaussianity .all these techniques provide information beyond power spectrum :  \1 .statistic of peaks in a random gaussian fields .following classical papers of doroshkevich 1970 and bardeen , j.m.,bond , j.r.,kaiser , n., szalay , 1986 - ( bbks ) ) , this approach has been developed by ( bond and efstathiou 1987 , p.coles 1988 ) for cmb anisotropy .higher - order correlations - 3 , 4 ets.(luo and schramm 1994 , smoot et al .1994 , kogut et al .1996 ) ;  \3 .minkowski functionals as a morphological descriptors of the cmb anisotropy maps ( shmalzing and gorski 1997 , winitzki and kosowsky 1997 ) . as it was mentioned by shmalzing and gorski 1997, minkowski functionals are very sensitive to non - gaussianity .this approach is very effective because these functionals are additive with respect to the isolated regions in the sky and they have simple analytical form in the case of the gaussian field .this approach can be used to test predictions of gaussianity .percolation and cluster analysis .this statistical method based on a very attractive idea : if the experimental signal is a sum of the primordial signal and non - gaussian noise ( for example foreground sources , dust emission and so on ) , then statistical properties of the pure gaussian signal could be distorted .this effect provides a basis for the investigation of the characteristics of the non - gaussian noise in the experimental data .note , that percolation has become a popular term among cosmologists .the percolation technique has been successfully applied for investigation of the evolution of the spatial density distribution in the universe due to gravitational instability ( see for review zeldovich 1982 , shandarin 1983 , dominik , shandarin 1992 ) . for cmbanisotropy this technique has been developed by ( naselsky and novikov d. 1995 , novikov d. and jorgensen 1996 ) .therefore , the data analysis of the cmb anisotropy can be divided into two parts : power spectrum estimation with subsequent cosmological parameters extraction , and investigation of the nature of the observed signal .there is another important characteristic of the distribution of the cmb on the sky : this is the cmb polarization .the idea that polarization provides important information about the primordial cosmic plasma was pointed out by rees ( 1968 ) .the properties of the power spectrum of the cmb polarization field were analyzed in for example ( basco and polnarev 1979 , polnarev 1985 , bond and efstathiou 1987 , coulson et al .1994 , crittenden et al .1995 , zaldarriaga and harari 1995 , ng k.l .and ng k.w . , 1995 , kosowsky 1996 ; kamionkowski et al .1996 , jungman et al .1996 , naselsky and polnarev 1987 , ng k.l . and ng k.w .1996 , hu and white , 1997 ) .the polarization field also contains information beyond power spectrum which also can be used for investigation of the nature of the primordial inhomogeneity in the universe .statistical properties of the polarization field caused by gaussian fluctuations was partly discussed by bond and efstathiou 1987 , arbuzov et al .1997a , arbuzov et al .it is important to note , that polarization contains more information about nature of the primordial signal than the anisotropy ( the polarization field is a combination of two random independent gaussian fields ( bond and efstathiou 1987 , while anisotropy of the cmb is only one ) .  in this paperwe focus attention on the general statistical properties of the cmb polarization field .this is not a scalar field ( unlike the anisotropy ) and can be completely described in terms of stokes parameters - , and . since thomson scattering does not produce circular polarization , we can consider the level of polarization , which depends only on two parameters - , where i is the total intensity , and is the polarized intensity . therefore , polarization field can be described in terms of the angle of polarization and polarized intensity - p. since polarization of the radiation does not have any direction ( it has only the orientation and intensity p ) , it can not be formally interpreted as a vector field .nevertheless , below we use the term `` vector of polarization '' ( so that ) for simplicity , taking into account that this `` vector '' is not directed .we assume that and components of the autocorrelated pseudo - vector are statistically independent ( bond and efstathiou 1987 ) and have a gaussian distribution on the sky .we are interested in the general statistical properties of the distribution such as surface density and classification of the non - polarized points in the sky , minkowski functionals for the value of and the percolation of the relatively strongly polarized spots .in this section we discuss very specific features of the polarization pattern of the cosmic microwave background .all results were obtained under the assumption that the polarization field is the result of a random gaussian process .we describe the statistical properties of the two dimensional vector field of the polarization such as the surface density of the singular points ( section 2.1 ) , genus curve for the two dimensional scalar field and the level of percolation through the relatively strongly polarized spots ( section 2.2 ) . in this sectionwe consider small angular parts of the sky without loss of generality .thus the geometry is approximately flat and the vector of polarization can be described in the following form :    where and are the unit vectors of the cartesian coordinate system on the small angular part of the unit sphere and components and can be expressed in terms of stokes parameters and :    where is the orientation of polarization and :    therefore , components and are also independent random two - dimensional gaussian fields with the same parameters as and .it means , that the statistical properties of the vector are equivalent to the properties of the vector .it allows us to use the usual terms and instead of and .first , we are interested in the statistics of the singular points of the vector : .this condition means , that both components and are equal to zero in such points simultaneously : the surface density of these points can easily be computed analytically .points are the points of the intersection of the lines of zero level of the and surfaces . the angular density of such points can be found by using the properties of the joint probability function for distribution of , , , , , .here and are the first derivatives of and respectively in the point : these 6 different values are independent ( bardeen et al . 1986 ) for an arbitrary point of the map and have zero average and the following variances : where and are the spectral parameters , as they were defined by bond and efstaphiou 1987 .the joint probability for these values is :  in the vicinity of the singular point ( ) , the value of and can be described by the following expression : the substitution of and integration over gives us the number density of the singular points : , .the analytical calculation of the surfase density of the singular points can be found in appendix a. here we present the main results and conclusions only .* a. classification of singular points *  we investigate the polarization vector field around the singular points in the following way .let us imagine that a point in the vicinity of the singular point is moving along the lines of the vector field .in this case the investigation is similar to that for the singular points of linear differential equations . following eq.(5 ) the field in the small vicinity of the point , where can be described in terms of the matrix of first derivatives of the field and , and we can consider the following equation where , , and the characteristic equation for eq.(10 ) is : this equation has two roots : and , ordered by , and the classification of the singular point depends on their values .if , then this is a focus and the vector field will spiral toward the point ( fig.1 left ) .+ 2 . if and are real , then the matrix has two eigenvectors which correspond to different values and , and we can consider two different cases :  . - both values are positive or - both values are negative .this means , that the point is a knot and the lines of the vector field tend to be aligned to the direction of the eigenvector with maximal value of , ( fig.1 middle ) .  . and - values with opposite signs . in this casethe point is a saddle ( fig.1 right ) .singular points of different types determine the behavior of the vector field in their vicinities .the distribution of the singular points on the map of cosmic microwave background polarization determines the topology of the relatively small polarized zones fig.2 .* b. surface density of the singular points .*  detailed calculation of the surfase density of the singular points is in the appendix a. the surface density of singular points of different kinds are : where , , are the number densities of focuses , knots and saddles respectively and is the correlation radius .the total number density of non - polarized points is : density of the singular points depends essentially on correlation radius ( eq .12 , 13 ) and therefore on the spectral parameters , .these spectral parameters depend on the spectra of polarization and on the device resolution ( bond and efstathiou , 1987 ) ( see also fig.3 ) .the ratios are the spectral independent constants determined only by the gaussian nature of the primordial inhomogeneity in the universe .these ratios are a characteristic feature of the cmb polarization vector field in the inflationary cosmology .note , that for example in the two - dimensional potential vector field ( ) the number density of foci is equal to zero since this field does not have a rotational component .the joint probability for the distribution of the eigenvalues of and in the singular points which can be either knots or saddles ( not foci ) is : note , that this distribution is universal for all kinds of spectra of polarization as well as ratios ( 15 ) .   herewe present our results for the genus statistics of the value .we can divide the map of polarization of the cmb into two parts : regions with relatively strong polarization ( `` strongly polarized zones '' ) and regions with relatively weak polarization ( `` weakly polarized zones '' ) .below we find the value where percolation through the `` strongly polarized zones '' changes to percolation through the `` weakly polarized zones '' ( fig .let us suppose that we can measure only a signal with polarized intensity , where is the threshold which determines by the sensitivity of the device .if we can measure only `` strongly polarized '' signal - , then we can see only the separated polarized spots which do not percolate .therefore , the percolation trough polarized zones can be reached only by the device with the sensitivity .  the value can be found analytically in the following way .we consider the value as a two - dimensional random scalar field with a rayleigh distribution ( coles and barrow , 1987 ) .this field can be imagined as a two - dimensional surface in a three - dimensional space .this surface has extreme points such as maxima , minima , saddle points and singular points .the last ones have been considered in the previous subsection .the densities of maxima , minima and saddle points have some distributions with : where , , - are the number densities of maxima , minima and saddle points respectively on some interval - ( p , p+dp ) , and , , are the number densities of maxima , minima and saddle points respectively above some level .( we note , that here saddle points are the saddle points of the two - dimensional surfase of p(x , y ) .these points are not the same as saddle ( kind of singular points ) in the previous section . )+ the definition of the genus is : the integrated genus is then : the level of percolation has to be found from the condition .we recognize that this condition does not automatically mean that is the level of percolation for an arbitrary scalar field .it is well - known that for the gaussian random field the percolation level corresponds to the level where genus curve intersects the zero .we have checked this condition for the rayleigh distribution by simulating a large number of realizations for a two - dimensional field . in the case of rayleigh distributionthis condition also mean that level corresponds to the percolation contour .the detailed calculation of the genus can be found in appendix b. formally the steps of its calculation are as follows : + 1 .the value is a combination of the independent random values and .the first and second derivatives of them are : , , , , ( ) , , where is also the spectral parameter as it was defined by bond and efstaphiou 1987 : .these values obey the following conditions :   \2 .the joint probability f of the gaussian distribution for the values , , , , , is : where is the covariance matrix and a is the quadratic form of the 12-dimensional vector .the substitution of , , in eq.(20 ) from eq.(19 ) and integration over 6 variables gives us the joint probability for values , , to be in the range from , , to , , .the differential density of the extreme points obeys the equation : where is the density of the extreme points .these extreme points can be maxima , minima or saddle points depending on the limits of the integration over .these limits determine by the values of and of the second derivatives matrix ( see appendix b ) .the genus curve obeys the equation : after integrating this , we have the integrated genus curve is condition gives us the value of : taking into account that random value has distribution we can obtain that percolation through the `` strongly polarized '' zone when a part of the map is detected as a `` strongly polarized '' .this corresponds to of the map s area .when in eq.(25 ) we have this value exactly coincides with in eq .( 13 ) with the opposite sign .these null - points are the non - smooth minima of the surface .the non - smooth minima have not been taken into account in eqs.(16)-(25 ) .therefore , the total number of minima per unit area is , where are minima , if p=0 and are minima , if p>0 .therefore the total number of extreme points per unit area are : taking into account equations ( 13,18,25,27 ) we obtain : as it should be .as it was mentioned above , the cmb polarization at any point of the map can be characterized by the orientation angle and polarized intensity - p. this intensity has the random rayleigh distribution on the sky - .therefore , the value of can be considered as a two - dimensional random raleigh field .it is well - known , that two - dimensional field has only three minkowski functionals which satisfy additivity and translational invariance ( minkowski 1903 , hadwiger 1959 ) .  geometrical interpretation of the minkowsky functionals on the two - dimensional map is essentially easy .analogously to the previous section , we consider polarized intensity as a two - dimensional surface in a three - dimensional space .if we cut this surface at the different levels , then the area of the map will be divided into two parts : the area , where polarization is above the threshold and the area , where .for a two - dimensional distribution , minkowski functionals correspond to the following values : + 1 .a - fraction of the area of the map , where ; + 2 .l - length of the boundary between fractions , where and per unit area ; + 3 . - euler characteristic ( equivalent to the genus ) per unit area .  therefore , threshold is the independent variable on which these functionals depend .the third functional has already been considered in the previous section .the obvious first one is .the second one can be obtained in the same way as it was done for the the gaussian field ( adler 1981 ) . herewe present the result without derivation :    the comparison of the minkowski functionals for the cmb polarization field with these functionals for cmb anisotropy is in the fig .functionals for rayleigh distribution are equal to zero for . the third functional should be described together with the number of non - polarized points ( see previous section ) .these functionals can be used as a morphological descriptor of the cmb polarization field in a similar manner as for the cmb anizotropy ( winitski and kosowski 1997 ) .in this paper we have presented calculations of the statistical properties cmb polarization maps .we believe that these statistical properties can be useful for checking the polarization patterns for presence of the non - gaussian noise ( for example , confusion signal from sources which can have the same spectral parameters as the polarization of the cmb ). if an observational signal is free from non - gaussian noise and is gaussian itself ( due to inflation ) then the topological approach is not necessary , because the correlation function or equivalent its power spectrum contains all information about the polarization signal .on the other hand , if the signal is a sum of polarization of the cmb ( which is gaussian ) and unresolved foreground sources ( which are non - gaussian ) , then a detailed topological picture of the polarization field around the non - polarization points will be distorted comparative the predictions of the theory for the gaussian distributions .  on the other hand ,the investigation of the nature of the primordial polarized signal is a test of the inflationary model of the evolution of the universe .therefore , the investigation of the minkowski functionals together with the non - polarized points on the observational data and comparison with the theoretical predictions can be used as the test on gaussianity of the primordial inhomogeneity .we would like to emphasize that the regions with strong polarization will be detected easier than the regions with weak polarization . as we demonstrated in the paper ( see section 2.2 ) these region occupy an essential part of a whole map . from this point of viewit is interesting to study the statistical properties of these regions .it is worth also mentioning that it is interesting to investigate the dependence of the spectral parameters of polarization in various cosmological models on the resolution of the detector and related statistical properties of the maps of polarization of the cmb .it is also very interesting to stady the cross - correlations between anisotropy and polarization on the sky map and make some theoretical predictions from the geometrical point of view .these quations will be considered in a separate paper .we would like to thank a. melott , i. novikov and s.shandarin for stimulating discussion . at the university of kansasthe research was supported by nsf - nato fellowship ( dge-9710914 ) and by the nsf epscor program .we are grateful to the staff of tac , and the copenhagen university observatory for providing excellent working conditions .this investigation was supported in part by the russian foundation for fundamental research ( code 96 - 02 - 17150 ) and by a grant isf mez 300 as well as by the danish natural science research council through grant no .9401635 and also in part by danmarks grundforskningsfond through its support for the establishment of the theoretical astrophysics center .below we describe the density of foci , knots and saddles .the total density of singular ( non - polarized ) points is : where the substitution and integration over gives us : the next substitution allows us to rewrite ( a3 ) in the following form : where , , . in terms of and the eigenvalues of the matrix are from ( a4 , a6 ) we can obtain the density of focuses , saddles and knots : using ( a4 ) , ( a7 ) we obtain using ( a4 , a6-a8 ) the joint probability for values , in the peculiar points which can be knots or saddles ( not focuses ) is      according to section 2.2 the value is the non - 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Abstract: the distribution of the polarization of the cosmic microwave background ( cmb ) in the sky is determined by the hypothesis of random gaussian distribution of the primordial density perturbations . this hypotheses is well motivated by the inflationary cosmology . therefore , the test of consistency of the statistical properties of the cmb polarization field with the gaussianity of primordial density fluctuations is a realistic way to study the nature of primordial inhomogeneities in the universe .   this paper contains the theoretical predictions of the general statistical properties of the cmb polarization field . all results obtained under assumption of the gaussian nature of the signal . we pay the special attention to the following two problems . first , the classification and statistics of the singular points of the polarization field where polarization is equal to zero . second , the topology of contours of the value of the degree of polarization . we have investigated the percolation properties for the zones of `` strong '' and `` weak '' polarization . we also have calculated minkowski functionals for the cmb polarization field . all results are analytical .  _ subject headings : _ cosmic microwave background , cosmology , statistics , observations .