during the process of selection, individuals having disadvantageous traits are weeded out. if the selection pressure isn't strong enough then mildly disadvantageous traits will continue to persist in the population. so the reasons for why a trait is not evolved even though it may be advantageous to the organism, are : there is no strong pressure against the individuals not having that trait. in other words lack of the trait is not strongly disadvantageous. the trait might have a tradeoff which essentially makes no change to the overall fitness. not enough time has elapsed for an advantageous mutation to get fixed. this doesn't mean that the mutation had not happened yet. it means that the situation that rendered the mutation advantageous had arisen quite recently. consider the example of a mutation that confers resistance against a disease. the mutation wouldn't be advantageous if there was no disease. when a population encounters the disease for the first time, then the mutation would gain advantage but it will take some time to establish itself in the population. the rate for that specific mutation is low and therefore it has not yet happened. mutation rates are not uniform across the genome and certain regions acquire mutations faster than the others. irrespective of that, if the overall mutation rate is low then it would take a lot of time for a mutation to arise and until then its effects cannot be seen. the specific trait is too genetically distant : it cannot be the result of a mutation in a single generation. it might, conceivably, develop after successive generations, each mutating farther, but if the intervening mutations are at too much of a disadvantage, they will not survive to reproduce and allow a new generation to mutate further away from the original population. the disadvantage from not having the trait normally arises only after the reproductive stage of the individual's lifecycle is mostly over. this is a special case of " no strong pressure ", because evolution selects genes, not the organism. in other words the beneficial mutation does not alter the reproductive fitness. koinophillia resulted in the trait being unattractive to females. since most mutations are detrimental females don't want to mate with anyone with an obvious mutation, since there is a high chance it will be harmful to their child. thus females instinctually find any obvious physical difference unattractive, even if it would have been beneficial. this tends to limit the rate or ability for physical differences to appear in a large & stable mating community. evolution is not a directed process and it does not


actively try to look for an optimum. the fitness of an individual does not have any meaning in the absence of the selection pressure. * if you have a relevant addition then please feel free to edit this answer. *


since this is physicsse, i am happy with an answer based purely on theoretical analysis of the forces involved. oh boy, time to spend way too much time on a response. lets assume the simple model of a peg that makes an angle $ \ alpha $ with the wall and ends in a circular cap of radius $ r $. then a towel of total length $ l $ and linear mass density $ \ rho $ has three parts : one part that hangs vertically, one that curves over the circular cap, and one that rests on the inclined portion like drawn. this is very simplistic, but it does encapsulate the basic physics. also, we ignore the folds of the towel. let $ s $ be the length of the towel on the inclined portion of the peg. i will choose a generalized $ x $ - axis that follows the curve of the peg. note this model works for both the front - back direction and side - side direction of the peg. in the side - side ( denoted $ z $ ) $ \ alpha $ is simply zero ( totally vertical ) : where $ \ eta $ is the fraction of the towel on the right side of the picture. then the total gravitational force $ f _ { g, x } $ will be : $ $ f _ { g, x } = \ rho g ( l - r ( \ pi - \ alpha ) - s ( 1 + \ cos ( \ alpha ) ) - \ int ^ { \ pi / 2 - \ alpha } _ { - \ pi / 2 } \ rho g r \ sin ( \ theta ) \, \ mathrm d \ theta $ $ $ $ f _ { g, x } = \ rho g ( l + r ( \ sin ( \ alpha ) - \ pi + \ alpha ) - s ( 1 + \ cos ( \ alpha ) ) $ $ the infinitesimal static frictional force will be $ \ mathrm df _ { s, x } = - \ mu _ s \, \ mathrm dn $. $ n $ is constant on the inclined part and varies with $ \ theta $ over the circular cap as $ \ mathrm dn = \ rho g r \ cos ( \ theta ) \, \ mathrm d \ theta $. then : $ $ f _ s = - \ mu _ s \ rho g s \ sin ( \ alpha ) - \ int ^ { \ pi / 2 - \ alpha } _ { - \


pi / 2 } \ mu _ s \ rho g r \ cos ( \ theta ) \, \ mathrm d \ theta $ $ $ $ f _ s = - \ mu _ s \ rho g ( s \ sin ( \ alpha ) + r ( \ cos ( \ alpha ) + 1 ) ) $ $ now we can set the frictional force equal to the gravitational force and solve for what values of $ \ mu _ s $ will satisfy static equilibrium. you get : $ $ \ mu _ s = \ frac { l + r ( \ sin ( \ alpha ) + \ alpha - \ pi ) - s ( \ cos ( \ alpha ) + 1 ) } { r ( \ cos ( \ alpha ) + 1 ) + s \ sin ( \ alpha ) } $ $ $ $ \ mu _ s = \ frac { 1 + \ gamma ( \ sin ( \ alpha ) + \ alpha - \ pi ) - \ eta ( \ cos ( \ alpha ) + 1 ) } { \ gamma ( \ cos ( \ alpha ) + 1 ) + \ eta \ sin ( \ alpha ) } $ $ where the second line $ \ gamma = r / l $ and $ \ eta = s / l $, the fraction of the towel on the peg's cap and incline, respectively. thus $ \ mu _ s $ depends on three factors : the angle of the peg, $ \ alpha $ the fraction of the towel past the cap of the peg, $ \ eta $. the fraction of the towel on the circular cap, $ \ gamma $. lets make some graphs : the above graph shows what $ \ mu _ s $ would have to be with a $ \ gamma = 0 $ ( no end cap, just a 1d stick ). the above graph shows what $ \ mu _ s $ would have to be with a $ \ eta = 0 $ ( no stick, just a circular cap that the towel drapes over. the above graph shows what $ \ mu _ s $ would have to be when the angle is fixed $ \ alpha = \ pi / 4 $ and the length of the peg ( $ \ eta $ ) is varied. summary what all the graphs above should show you is that the coefficient of static friction has to be enormous ( $ \ mu _ s > 50 $ - - most $ \ mu _ s $ are close to 1 ) unless the fraction of the towel on the peg ( $ \ eta $ and $ \


gamma $ ) is large, like over 50 % combined. the large values for $ \ eta $ can only be accomplished when you put the towel at approximately position $ \ mathbf { a } $, whereas its very difficult to hang a towel from position $ \ mathbf { b } $ because it reduces $ \ eta $ in both the $ z $ and $ x $ - directions. 3 ) the towel has a center of mass below the peg this isn't a sufficient condition for static equilibrium ; a towel isn't a rigid object. as a counter - example, see an atwood's machine. the block - rope system has a center of mass below the pulley, but that doesn't prevent motion of the blocks.


brain, indeed, cannot feel pain, as it lacks pain receptors ( nociceptors ). however, what you feel when you have a headache is not your brain hurting - - there are plenty of other areas in your head and neck that do have nociceptors which can perceive pain, and they literally cause the headaches. in especially, many types of headaches are generally thought to have a neurovascular background, and the responsible pain receptors are associated with blood vessels. however, the pathophysiology of migraines and headaches is still poorly understood.


an excerpt from history of lambda - calculus and combinatory logic by f. cardone and j. r. hindley ( 2006 ) : by the way, why did church choose the notation “ $ \ lambda $ ”? in [ church, 1964, § 2 ] he stated clearly that it came from the notation “ $ \ hat { x } $ ” used for class - abstraction by whitehead and russell, by first modifying “ $ \ hat { x } $ ” to “ $ \ wedge x $ ” to distinguish function abstraction from class - abstraction, and then changing “ $ \ wedge $ ” to “ $ \ lambda $ ” for ease of printing. this origin was also reported in [ rosser, 1984, p. 338 ]. on the other hand, in his later years church told two enquirers that the choice was more accidental : a symbol was needed and “ $ \ lambda $ ” just happened to be chosen.


if the goal of the standard deviation is to summarise the spread of a symmetrical data set ( i. e. in general how far each datum is from the mean ), then we need a good method of defining how to measure that spread. the benefits of squaring include : squaring always gives a non - negative value, so the sum will always be zero or higher. squaring emphasizes larger differences, a feature that turns out to be both good and bad ( think of the effect outliers have ). squaring however does have a problem as a measure of spread and that is that the units are all squared, whereas we might prefer the spread to be in the same units as the original data ( think of squared pounds, squared dollars, or squared apples ). hence the square root allows us to return to the original units. i suppose you could say that absolute difference assigns equal weight to the spread of data whereas squaring emphasises the extremes. technically though, as others have pointed out, squaring makes the algebra much easier to work with and offers properties that the absolute method does not ( for example, the variance is equal to the expected value of the square of the distribution minus the square of the mean of the distribution ) it is important to note however that there's no reason you couldn't take the absolute difference if that is your preference on how you wish to view'spread'( sort of how some people see 5 % as some magical threshold for $ p $ - values, when in fact it is situation dependent ). indeed, there are in fact several competing methods for measuring spread. my view is to use the squared values because i like to think of how it relates to the pythagorean theorem of statistics : $ c = \ sqrt { a ^ 2 + b ^ 2 } $ … this also helps me remember that when working with independent random variables, variances add, standard deviations don't. but that's just my personal subjective preference which i mostly only use as a memory aid, feel free to ignore this paragraph. an interesting analysis can be read here : revisiting a 90 - year - old debate : the advantages of the mean deviation - stephen gorard ( department of educational studies, university of york ) ; paper presented at the british educational research association annual conference, university of manchester, 16 - 18 september 2004


whenever you want to save space ( this can be a substantial savings ). until quite recently ( samtools / htslib 1. 7 ), only cram supported long cigar strings. if you need to guarantee that any random obscure downstream program will be able to handle it. uptake of cram has been pretty slow. java programs using htsjdk ( e. g., picard, igv and gatk ) have only relatively recently added support for cram. if you need to use an old version of those for some very odd reason then cram may not be supported. there are a lot of programs written in python that use pysam to open bam files and these should, theoretically, support cram. the issue is that some of the functions may fail and one can't assume that authors will have always written the code needed to handle this. i'll use deeptools as an example, since i'm one of its developers. one of the things about cram files is that they ( by default ) are made such that you require a reference genome in order to construct the sequence field in each alignment. this works fine if you're using a standard genome ( htslib, via pysam, can fetch many standard genomes from the web automatically ), but if you're not, then you need to specify a fasta file to use for decompression. every tool, then, needs to add an option for this. with pysam 0. 14 and htslib 1. 7 this can be circumvented by not decompressing the sequence, but behavior has to be explicitly requested. another issue is that many tools will use features from the file index, such as the. mapped accessor, to get the number of mapped reads in a file. cram files contain very very little information, so this then fails. consequently, tool authors need to check for cram files and both derive and propagate this information through their functions if it's needed. this can be a time - consuming task ( e. g., it took me a couple days to get this implemented in deeptools ). relatedly, samtools idxstats is useless on cram files, since there are no statistics stored in the index. that having been said, it's likely that crams slowly gaining acceptance will eventually make it the standard. it's already a convenient archival format, it '


s just a matter of time before users can assume that most analysis programs are written to handle it.


consider the set of keys $ k = \ { 0, 1,..., 100 \ } $ and a hash table where the number of buckets is $ m = 12 $. since $ 3 $ is a factor of $ 12 $, the keys that are multiples of $ 3 $ will be hashed to buckets that are multiples of $ 3 $ : keys $ \ { 0, 12, 24, 36,... \ } $ will be hashed to bucket $ 0 $. keys $ \ { 3, 15, 27, 39,... \ } $ will be hashed to bucket $ 3 $. keys $ \ { 6, 18, 30, 42,... \ } $ will be hashed to bucket $ 6 $. keys $ \ { 9, 21, 33, 45,... \ } $ will be hashed to bucket $ 9 $. if $ k $ is uniformly distributed ( i. e., every key in $ k $ is equally likely to occur ), then the choice of $ m $ is not so critical. but, what happens if $ k $ is not uniformly distributed? imagine that the keys that are most likely to occur are the multiples of $ 3 $. in this case, all of the buckets that are not multiples of $ 3 $ will be empty with high probability ( which is really bad in terms of hash table performance ). this situation is more common that it may seem. imagine, for instance, that you are keeping track of objects based on where they are stored in memory. if your computer's word size is four bytes, then you will be hashing keys that are multiples of $ 4 $. needless to say that choosing $ m $ to be a multiple of $ 4 $ would be a terrible choice : you would have $ 3m / 4 $ buckets completely empty, and all of your keys colliding in the remaining $ m / 4 $ buckets. in general : every key in $ k $ that shares a common factor with the number of buckets $ m $ will be hashed to a bucket that is a multiple of this factor. therefore, to minimize collisions, it is important to reduce the number of common factors between $ m $ and the elements of $ k $. how can this be achieved? by choosing $ m $ to be a number that has very few factors : a prime number.


good observation! the 3'poly ( a ) tail is actually a very common feature of positive - strand rna viruses, including coronaviruses and picornaviruses. for coronaviruses in particular, we know that the poly ( a ) tail is required for replication, functioning in conjunction with the 3'untranslated region ( utr ) as a cis - acting signal for negative strand synthesis and attachment to the ribosome during translation. mutants lacking the poly ( a ) tail are severely compromised in replication. jeannie spagnolo and brenda hogue report : the 3 ′ poly ( a ) tail plays an important, but as yet undefined role in coronavirus genome replication. to further examine the requirement for the coronavirus poly ( a ) tail, we created truncated poly ( a ) mutant defective interfering ( di ) rnas and observed the effects on replication. bovine coronavirus ( bcv ) and mouse hepatitis coronavirus a59 ( mhv - a59 ) di rnas with tails of 5 or 10 a residues were replicated, albeit at delayed kinetics as compared to di rnas with wild type tail lengths ( > 50 a residues ). a bcv di rna lacking a poly ( a ) tail was unable to replicate ; however, a mhv di lacking a tail did replicate following multiple virus passages. poly ( a ) tail extension / repair was concurrent with robust replication of the tail mutants. binding of the host factor poly ( a ) - binding protein ( pabp ) appeared to correlate with the ability of di rnas to be replicated. poly ( a ) tail mutants that were compromised for replication, or that were unable to replicate at all exhibited less in vitro pabp interaction. the data support the importance of the poly ( a ) tail in coronavirus replication and further delineate the minimal requirements for viral genome propagation. spagnolo j. f., hogue b. g. ( 2001 ) requirement of the poly ( a ) tail in coronavirus genome replication. in : lavi e., weiss s. r., hingley s. t. ( eds ) the nidoviruses. advances in experimental medicine and biology, vol 494. springer, boston, ma yu - hui peng et al. also report that the length of the poly ( a ) tail is regulated during infection : similar to eukaryotic mrna, the positive - strand coronavirus genome of ~ 30 kilobases is 5 ’ - capped and 3


’ - polyadenylated. it has been demonstrated that the length of the coronaviral poly ( a ) tail is not static but regulated during infection ; however, little is known regarding the factors involved in coronaviral polyadenylation and its regulation. here, we show that during infection, the level of coronavirus poly ( a ) tail lengthening depends on the initial length upon infection and that the minimum length to initiate lengthening may lie between 5 and 9 nucleotides. by mutagenesis analysis, it was found that ( i ) the hexamer aguaaa and poly ( a ) tail are two important elements responsible for synthesis of the coronavirus poly ( a ) tail and may function in concert to accomplish polyadenylation and ( ii ) the function of the hexamer aguaaa in coronaviral polyadenylation is position dependent. based on these findings, we propose a process for how the coronaviral poly ( a ) tail is synthesized and undergoes variation. our results provide the first genetic evidence to gain insight into coronaviral polyadenylation. peng y - h, lin c - h, lin c - n, lo c - y, tsai t - l, wu h - y ( 2016 ) characterization of the role of hexamer aguaaa and poly ( a ) tail in coronavirus polyadenylation. plos one 11 ( 10 ) : e0165077 this builds upon prior work by hung - yi wu et al, which showed that the coronaviral 3'poly ( a ) tail is approximately 65 nucleotides in length in both genomic and sgmrnas at peak viral rna synthesis, and also observed that the precise length varied throughout infection. most interestingly, they report : functional analyses of poly ( a ) tail length on specific viral rna species, furthermore, revealed that translation, in vivo, of rnas with the longer poly ( a ) tail was enhanced over those with the shorter poly ( a ). although the mechanisms by which the tail lengths vary is unknown, experimental results together suggest that the length of the poly ( a ) and poly ( u ) tails is regulated. one potential function of regulated poly ( a ) tail length might be that for the coronavirus genome a longer poly ( a ) favors translation. the regulation of coronavirus translation by poly ( a ) tail length resembles that during embryonal development suggesting there may be mechanistic parallels. wu hy, ke ty, liao


wy, chang ny. regulation of coronaviral poly ( a ) tail length during infection. plos one. 2013 ; 8 ( 7 ) : e70548. published 2013 jul 29. doi : 10. 1371 / journal. pone. 0070548 it's also worth pointing out that poly ( a ) tails at the 3'end of rna are not an unusual feature of viruses. eukaryotic mrna almost always contains poly ( a ) tails, which are added post - transcriptionally in a process known as polyadenylation. it should not therefore be surprising that positive - strand rna viruses would have poly ( a ) tails as well. in eukaryotic mrna, the central sequence motif for identifying a polyadenylation region is aauaaa, identified way back in the 1970s, with more recent research confirming its ubiquity. proudfoot 2011 is a nice review article on poly ( a ) signals in eukaryotic mrna.


you are right, the planetary model of the atom does not make sense when one considers the electromagnetic forces involved. the electron in an orbit is accelerating continuously and would thus radiate away its energy and fall into the nucleus. one of the reasons for " inventing " quantum mechanics was exactly this conundrum. the bohr model was proposed to solve this, by stipulating that the orbits were closed and quantized and no energy could be lost while the electron was in orbit, thus creating the stability of the atom necessary to form solids and liquids. it also explained the lines observed in the spectra from excited atoms as transitions between orbits. if you study further into physics you will learn about quantum mechanics and the axioms and postulates that form the equations whose solutions give exact numbers for what was the first guess at a model of the atom. quantum mechanics is accepted as the underlying level of all physical forces at the microscopic level, and sometimes quantum mechanics can be seen macroscopically, as with superconductivity, for example. macroscopic forces, like those due to classical electric and magnetic fields, are limiting cases of the real forces which reign microscopically.


there are basically two major, commercial choices out there : ddt from allinea ( which is what we use at tacc ) and totalview ( as mentioned in the other comment ). they have comparable features, are both actively developed, and are direct competitors. eclipse has their parallel tools platform, which should include mpi and openmp programming support and a parallel debugger.


uranium and thorium in heavy rocks have a decay chain which includes a three - day isotope of radon. if a building has materials with some chemically - insignificant mixture of uranium and thorium, such as concrete or granite, then the radon can diffuse out of the material into the air. this is part of your normal background radiation, unless you have accidentally built a concrete basement with granite countertops and poor air exchange with the outdoors, in which case the radon can accumulate. when radon does decay, the decays leave behind ionized atoms of the heavy metals polonium, lead, and bismuth. these ions neutralize by reacting with the air. here my chemistry is weak, but my assumption is that they are most likely to oxide, and i assume further that the oxide molecules are electrically polarized, like the water molecule ( the stable oxide of hydrogen ) is polarized. polarized or polarizable objects are attracted to strong electric fields, even when the polarized object is electrically neutral. imagine a static electric field around a positive charge. a dipole nearby will feel a torque until its negative end points towards the static positive charge. but because the field gets weaker away from the static charge, there ’ s now more attractive force on the negative end of the dipole than there is repulsive force on the positive end, so the dipole accelerates towards the stronger field. if you used to have a cathode - ray television, you may remember the way the positively - charged screen would attract dust much more than other nearby surfaces. clothes dryers are very effective at making statically charged surfaces. ( dryer sheets help. ) so when radon and its temporary decay products are blown through the dryer, electrically - polarized molecules tend to be attracted to the charged surfaces. the decay chain is isotope half - life decay mode 222 - rn 3. 8 days alpha 218 - po 3. 1 minutes alpha 214 - pb 27 minutes beta 214 - bi 20 minutes beta 214 - po microseconds alpha 210 - pb years irrelevant if your geiger counter is actually detecting radiation, it's almost certainly the half - hour lead and bismuth. constructing a decay curve would make a neat home experiment ( but challenging given what you've told us here ). true story : i was once prevented from leaving a neutron - science facility at los alamos after the seat of my pants set off a radiation alarm on exit. this was odd because the neutron beam had been off for weeks


. it was a saturday, so the radiation safety technician on call didn't arrive for half an hour — at which point i was clean, so the detective questions began. i had spent the day sitting on a plastic step stool. the tech looked at it, said that radon's decay products are concentrated by static electricity, and told me that i needed to get a real chair.


how long would it take for this super - material to convert to the stuff i scribble with? no, despite the fact that james bond said " diamonds are forever ", that is not exactly the case. although bond's statement is a fair approximation of reality it is not a scientifically accurate description of reality. as we will soon see, even though diamond is slightly less stable than graphite ( by ~ 2. 5 kj / mol ), it is kinetically protected by a large activation energy. here is a comparative representation of the structures of diamond and graphite. ( image source : satyanarayana t, rai r. nanotechnology : the future. j interdiscip dentistry 2011 ; 1 : 93 - 100 ) ( image source ) note that diamond is composed of cyclohexane rings and each carbon is bonded to 2 more carbons external to the cyclohexane ring. on the other hand, graphite is comprised of benzene rings and each carbon is bonded to only 1 carbon external to the benzene ring. that means we need to break 6 sigma bonds in diamond and make about 2 pi bonds ( remember it's an extended array of rings, don't double count ) in graphite per 6 - membered ring in order to convert diamond to graphite. a typical aliphatic c – c bond strength is ~ 340 kj / mol and a typical pi bond strength is ~ 260 kj / mol. so to break 6 sigma bonds and make 2 pi bonds would require ~ ( ( 6 * 340 ) - ( 2 * 260 ) ) ~ 1500 kj / mol. if the transition state were exactly midway between diamond and carbon ( with roughly equal bond breaking and bond making ), then we might approximate the activation energy as being half that value or ~ 750 kj / mol. since graphite is a bit more stable than diamond, we can refine our model and realize that the transition state will occur a bit before the mid - point. so our refined model would suggest an activation energy something less than 750 kj / mol. had we attempted to incorporate the effect of aromaticity in graphite our estimate would be even lower. in any case, this is an extremely large activation energy, so, as we anticipated the reaction would be very slow. an estimate ( see p. 171 ) of the activation energy puts the reverse reaction ( graphite to diamond ; but since, as noted above, the energy difference between the two


is very small the activation energy for the forward reaction is almost the same ) at 367 kj / mol. so at least our rough approximation was in the right ballpark, off by about a factor of 2. however, it appears that the transition state is even further from the midpoint ( closer to starting material ) than we might have guessed. this activation energy tells us that at 25 °c, it would take well over a billion years to convert one cubic centimeter of diamond to graphite. note 04 / 17 / 20 : as mentioned in a comment, the original " estimate " link became defunct and was replaced today with a new estimate link. however the original article and estimate can can still be seen on the wayback machine and it estimates the activation energy to be 538. 45 kj / mol, reasonably close to our estimate.


in general, all krylov methods essentially seek a polynomial that is small when evaluated on the spectrum of the matrix. in particular, the $ n $ th residual of a krylov method ( with zero initial guess ) can be written in the form $ $ r _ n = p _ n ( a ) b $ $ where $ p _ n $ is some monic polynomial of degree $ n $. if $ a $ is diagonalizable, with $ a = v \ lambda v ^ { - 1 } $, we have \ begin { eqnarray * } \ | r _ n \ | & \ leq & \ | v \ | \ cdot \ | p _ n ( \ lambda ) \ | \ cdot \ | v ^ { - 1 } \ | \ cdot \ | b \ | \ \ & = & \ kappa ( v ) \ cdot \ | p _ n ( \ lambda ) \ | \ cdot \ | b \ |. \ end { eqnarray * } in the event that $ a $ is normal ( e. g., symmetric or unitary ) we know that $ \ kappa ( v ) = 1. $ gmres constructs such a polynomial through arnoldi iteration, while cg constructs the polynomial using a different inner product ( see this answer for details ). similarly, bicg constructs its polynomial through the nonsymmetric lanczos process, while chebyshev iteration uses prior information on the spectrum ( usually estimates of the largest and smallest eigenvalues for symmetric definite matrices ). as a cool example ( motivated by trefethen + bau ), consider a matrix whose spectrum is this : in matlab, i constructed this with : a = rand ( 200, 200 ) ; [ q r ] = qr ( a ) ; a = ( 1 / 2 ) * q + eye ( 200, 200 ) ; if we consider gmres, which constructs polynomials which actually minimize the residual over all monic polynomials of degree $ n $, we can easily predict the residual history by looking at the candidate polynomial $ $ p _ n ( z ) = ( 1 - z ) ^ n $ $ which in our case gives $ $ | p _ n ( z ) | = \ frac { 1 } { 2 ^ n } $ $ for $ z $ in the spectrum of $ a $. now, if we run gmres on a random rhs and compare the residual history with


this polynomial, they ought to be quite similar ( the candidate polynomial values are smaller than the gmres residual because $ \ | b \ | _ 2 > 1 $ ) :


it's undecidable because a law book can include arbitrary logic. a silly example censorship law would be " it is illegal to publicize any computer program that does not halt ". the reason results for mtg exist and are interesting is because it has a single fixed set of ( mostly ) unambiguous rules, unlike law which is ever changing, horribly localized and endlessly ambiguous.


yes, if you can come up with any of the following : deterministic finite automaton ( dfa ), nondeterministic finite automaton ( nfa ), regular expression ( regexp of formal languages ) or regular grammar for some language $ l $, then $ l $ is regular. there are more equivalent models, but the above are the most common. there are also useful properties outside of the " computational " world. $ l $ is also regular if it is finite, you can construct it by performing certain operations on regular languages, and those operations are closed for regular languages, such as intersection, complement, homomorphism, reversal, left - or right - quotient, regular transduction and more, or using myhill – nerode theorem if the number of equivalence classes for $ l $ is finite. in the given example, we have some ( regular ) langage $ l $ as basis and want to say something about a language $ l'$ derived from it. following the first approach - - construct a suitable model for $ l'$ - - we can assume whichever equivalent model for $ l $ we so desire ; it will remain abstract, of course, since $ l $ is unknown. in the second approach, we can use $ l $ directly and apply closure properties to it in order to arrive at a description for $ l'$.


sure, you can combine prngs like this, if you want, assuming they are seeded independently. however, it will be slower and it probably won't solve the most pressing problems that people have. in practice, if you have a requirement for a very high - quality prng, you use a well - vetted cryptographic - strength prng and you seed it with true entropy. if you do this, your most likely failure mode is not a problem with the prng algorithm itself ; the most likely failure mode is lack of adequate entropy ( or maybe implementation errors ). xor - ing multiple prngs doesn't help with this failure mode. so, if you want a very high - quality prng, there's probably little point in xor - ing them. alternatively, if you want a statistical prng that's good enough for simulation purposes, typically the # 1 concern is either speed ( generate pseudorandom numbers really fast ) or simplicity ( don't want to spend much development time on researching or implementing it ). xor - ing slows down the prng and makes it more complex, so it doesn't address the primary needs in that context, either. as long as you exhibit reasonable care and competence, standard prngs are more than good enough, so there's really no reason why we need anything fancier ( no need for xor - ing ). if you don't have even minimal levels of care or competence, you're probably not going to choose something complex like xor - ing, and the best way to improve things is to focus on more care and competence in the selection of the prng rather than on xor - ing. bottom line : basically, the xor trick doesn't solve the problems people usually actually have when using prngs.


as suggested, here ’ s an example showing the relevant lines from a description file from a cran / github hosted project that has bioconductor dependencies ( truncated ) : depends : r ( > = 3. 3. 0 ) biocviews : imports : methods, snpstats, dplyr the relevant bit is the empty biocviews : declaration, which allows the bioconductor dependency { snpstats } to be automatically installed.


i'm not aware of any recent overview articles, but i am actively involved in the development of the pfasst algorithm so can share some thoughts. there are three broad classes of time - parallel techniques that i am aware of : across the method — independent stages of rk or extrapolation integrators can be evaluated in parallel ; see also the ridc ( revisionist integral deferred correction algorithm ) across the problem — waveform relaxation across the time - domain — parareal ; pita ( parallel in time algorithm ) ; and pfasst ( parallel full approximation scheme in space and time ). methods that parallelize across the method usually perform very close to spec but don't scale beyond a handful of ( time ) processors. typically they are relatively easier to implement than other methods and are a good if you have a few extra cores lying around and are looking for predictable and modest speedups. methods that parallelize across the time domain include parareal, pita, pfasst. these methods are all iterative and are comprised of inexpensive ( but inaccurate ) " coarse " propagators and expensive ( but accurate ) " fine " propagators. they achieve parallel efficiency by iteratively evaluating the fine propagator in parallel to improve a serial solution obtained using the coarse propagator. the parareal and pita algorithms suffer from a rather unfortunate upper bound on their parallel efficiency $ e $ : $ e < 1 / k $ where $ k $ is the number of iterations required to obtain convergence throughout the domain. for example, if your parareal implementation required 10 iterations to converge and you are using 100 ( time ) processors, the largest speedup you could hope for would be 10x. the pfasst algorithm relaxes this upper bound by hybridizing the time - parallel iterations with the iterations of the spectral deferred correction time - stepping method and incorporating full approximation scheme corrections to a hierarchy of space / time discretizations. lots of games can be played with all of these methods to try and speed them up, and it seems as though the performance of these across - the - domain techniques depends on what problem you are solving and which techniques are available for speeding up the coarse propagator ( coarsened grids, coarsened operators, coarsened physics etc. ). some references ( see also references listed in the papers ) : this paper demonstrates how various methods can be parallelised across the method : a theoretical comparison of high order explicit runge -


kutta, extrapolation, and deferred correction methods ; ketcheson and waheed. this paper also shows a nice way of parallelizing across the method, and introduces the ridc algorithm : parallel high - order integrators ; christlieb, macdonald, ong. this paper introduces the pita algorithm : a time - parallel implicit method for accelerating the solution of nonlinear structural dynamics problems ; cortial and farhat. there are lots of papers on parareal ( just google it ). here is a paper on the nievergelt method : a minimal communication approach to parallel time integration ; barker. this paper introduces pfasst : toward an efficient parallel in time method for partial differential equations ; emmett and minion ; this papers describes a neat application of pfasst : a massively space - time parallel n - body solver ; speck, ruprecht, krause, emmett, minion, windel, gibbon. i have written two implementations of pfasst that are available on the'net : pypfasst and libpfasst.


that molecule is called geosmin. it is mainly produced 1 by actinomycetes such as streptomyces which are filamentous bacteria that live in soil. other organisms also produce geosmin : cyanobacteria certain fungi an amoeba called vanella a liverwort it is an intracellular metabolite and cell damage is the primary reason attributed to its release. however oxidant exposure and transmembrane pressure also causes geosmin release in cyanobacteria. it seems that the release is triggered by some kind of stress. i am not quite sure about their advantage to the host species. 1 or perhaps the most well - studied in


what's the big deal? when quantum mechanics was being discovered and formalized, in the 1920s and 1930s, our view of physics was deeply rooted in the macroscopic world. we understood that microscopic entities like atoms and molecules existed, and we arrived reasonably quickly at a good understanding of their basic structure, but for a very long time they were very remote objects, whose behaviour was so abstract and disconnected from our everyday experience that it was even kind of pointless to really interrogate it. so, as an example, if you heated up a vial with sodium, then the gas sample in the vial might emit or absorb light at a particular wavelength, and if you worked out the quantum - mechanical maths then you could predict what those wavelengths should be, in terms of quantum jumps between energy levels $ - $ but, could you really say what each individual atom in the gas was doing? how could you be sure that those " quantum jumps " were even real, if you only ever had access to the macroscopic gas sample, and never to any individual atom? moreover, that same quantum - mechanical maths predicts that the dynamics in an atom will be blazingly fast, and indeed many orders of magnitude faster than any experimental techniques available at the time. so, could you really talk about the electrons " moving "? this was aggravated by the fact that the particular choices of quantum - mechanical maths that made sense for this type of experiment talked much more about " orbitals " and " energy levels ", with those mysterious quantum jumps to link them $ - $ so maybe it makes more sense to treat those orbitals and energy levels as the " real " objects, and disregard the notion that there is any movement in the micro - world? however, we live in a very different world now. not only do we have tools like scanning electron microscopy that allow us to observe the atoms that make up a metal surface, we are also now able to hold and control a single atom with delicate electrical " tweezers ", which then allows us to interrogate it directly. and when we look, much to our chagrin, that individual atom is indeed performing the fabled quantum jumps. more generally, since the turn of the millenium the name of the game ( and indeed the routine ) has been the observation and control of individual quantum systems. a similar story holds for the dynamics of microscopic systems, and for our ability to observe them directly. the discoveries of the laser, and then q - switching and mode locking allowed laser pulses to get pretty


fast, first faster than a microsecond ( $ 10 ^ { - 6 } \ : \ rm s $ ) and then faster than a nanosecond ( $ 10 ^ { - 9 } \ : \ rm s $ ), respectively, and work in the 1970s and 1980s allowed us to create pulses as short as a picosecond ( $ 10 ^ { - 12 } \ : \ rm s $ ) and shorter. if you really push a laser system, using technology known as chirped pulse amplification ( which i wrote about previously here when it won its nobel prize ), you can get down to a few femtoseconds ( $ 10 ^ { - 15 } \ : \ rm s $ ). this is very fast for a pulse of light, and it is actually so fast that the pulse of light is no longer a periodic electric - field oscillation, and instead it lasts only for a few cycles. but it is still not fast enough. why? because atoms are even faster. to understand how fast atoms are, it is enough to do some basic dimensional analysis. the dynamics of the electrons inside an atom are governed by the schrodinger equation, $ $ i \ hbar \ frac { \ partial \ psi } { \ partial t } = - \ frac { \ hbar ^ 2 } { 2m _ e } \ nabla ^ 2 \ psi - \ frac { e ^ 2 } { r } \ psi, $ $ and this has only three core constants involved : the reduced planck constant, $ \ hbar $, the electron's mass, $ m _ e $, and the electron charge $ e $. ( or, if you work in si units, the coulomb constant $ e ^ 2 / 4 \ pi \ epsilon _ 0 $. ) and, as it turns out, those constants can be combined into a unique timescale, known as the atomic unit of time, $ $ t _ \ mathrm { a. u. } = \ frac { \ hbar ^ 3 } { m _ ee ^ 4 } = 24 \ : \ rm as, $ $ which is measured in attoseconds : $ 1 \ : \ rm as = 10 ^ { - 18 } \ : \ rm s $. as a rule of thumb, the dynamics might be somewhat faster, or somewhat slower, depending on the atom and the conditions, but it will generally stick to that rough order


of magnitude. and that means, in turn, that those dynamics might seem completely out of reach, because the period of oscillation of optical light is still rather slower than this. ( for light of wavelength $ 550 \ : \ rm nm $, the period is of about $ 2 \ : \ rm fs $. ) so that might make you think that a direct observation of something as fast as atomic dynamics must be out of reach. so how do you make an attosecond pulse? this is the real breakthrough that is being rewarded with today's announcement. our workhorse is a process known as high - harmonic generation, which uses a highly nonlinear interaction between a gas and a pulse of laser light to generate sharp bursts of radiation $ - $ the famed attosecond pulses $ - $ which can be much shorter than the period of the pulse that drives the process, and can be as short as a few dozen attoseconds. from an experimental perspective, what you have to do is simply start with a laser pulse with a fairly long wavelength and slow period ( usually in the near - infrared ), shine it into a gas cell, and make sure that the pulse is intense. how intense? very intense. intense enough to directly yank electrons out of the gas atoms and shake them about once they're free. ( and, indeed, intense enough that the pulse will burn out the laser amplifier if you let it, as explained in the thread about chirped pulse amplification. ) this was done in 1987 by a team led by anne l'huillier, and the surprising observation was that the gas emitted harmonics, i. e., additional wavelengths of light at sub - multiples of the original driving wavelength. this was known to occur ( second - harmonic generation is almost as old as the laser itself ), but l'huillier and colleagues discovered that if the driving pulse is intense enough, it can generate all sorts of harmonics at crazy high orders, with a very slow decline in emission as the order increases. ( up until the signal reaches a cutoff and decays exponentially, of course. ) what's going on? the basic physics was worked out by paul corkum ( who was very high in the shortlist for getting the nobel prize if it ever did get awarded to attosecond science ), and it is known as the three - step model. image taken from d. villeneuve, contemp. phys. 59, 47 (


2018 ) in essence, the laser can be thought of as a constant force ( and therefore a linear ramp in potential energy ) which slowly oscillates and tilts around the potential well that the atomic electron sits in. at the maximum of field intensity, this is enough to yank the electron away ( though more on this later ), at which point the electron will freely oscillate in the field, gaining energy from the electric field of the light... up until it crashes into the potential well that it just left, at which point it can recombine back with the ion it left behind, and emit its ( now considerable ) kinetic energy as a sharp burst of radiation. the coolest things about this collision are that it is very energetic ( so the burst of radiation has a high photon energy, and therefore very high frequencies ), and that it is very short ( it is over in a flash ), and it is this short duration that means that the pulses of radiation emitted will be extremely short. the other parts of the nobel prize are being awarded for the explicit creation and detection of these sharp bursts of light. one thing that happens quite often is that ( because the driving pulse is long, and has many periods where the three - step model can happen ), the emission is often in the shape of an attosecond - pulse train, sometimes with several dozen sharp bursts following each other in quick succession. pierre agostini was the first to directly observe the duration of the bursts within such a train, using a technique known as rabbitt ( attoscience has since acquired an " animal theme " for our acronyms ), and his group was able to show that they were indeed very short, down to as little as $ 250 \ : \ rm as $. alternatively, you might want to invest some ( considerable ) time and energy into finding a way to " gate " the emission, so that there is only one burst in the train. ( for a fresh - off - the - press review of different ways to " gate " the emission see e. g. this preprint. ) this gating was achieved by ferenc krausz's group, who were able to isolate a single pulse with a duration of $ 650 \ : \ rm as $. of course, the field has continued to innovate, making things more reliable and robust, but also pushing down the shortest duration achievable. if i understand correctly, the current record is $ 43 \ :


\ rm as $, which is very, very short. ( another cool record is how high you can push the order of nonlinearity in the process, for which, if i understand correctly, a 2012 classic still holds the prize with a minimal order of nonlinearity of 4, 500. ) what can you use these pulses for? we're now down to the most interesting part. say that you have made one of these attosecond pulses. what can you do with it? directly observing the wave oscillations of light for me, the most exciting application from the " classic " experiments in attoscience is a setup known as " attosecond streaking ". the basic idea is to take a short attosecond pulse, and overlap it, inside a gas sample, with a slower pulse of infrared light. the short pulse has enough photon energy to ionize the gas, and we know that this must happen within the duration of the short pulse. after this ionization, the slower infrared pulse has an electric field which oscillates, and this will impact the final energy and momentum of the electron, but the extent of this effect will depend on when the electron is released, so by changing the time delay between the two, we can scan against this electric field. $ \ qquad $ the end result, shown above, is a direct observation of the oscillations of the electric field ( raw data on the left, and reconstructed electric field on the right ), which is a task that was considered somewhere between impossible and unthinkable for many, many decades after we understood that light was a wave ( but only had indirect ways to prove it ). i've discussed this experiment previously here. for more details ( and the source of the figures ), see the landmark publication : direct measurement of light waves. e. goulielmakis et al. science 305, 1267 ( 2004 ) ; author eprint. directly observing electron motion in real time similarly to observing the motion of the electric field of light, we can also observe the motion of electrons inside an atom. i have discussed this in detail in is there oscillating charge in a hydrogen atom?, but the short story is that if you prepare an electron in a quantum superposition of two different energy levels, such as the combination $ $ \ psi = \ psi _ { 1s } + \ psi _ { 2p } $ $ of the hydrogen $ 1s $ and $ 2


##p $ levels, the charge density in the atom will oscillate over time : mathematica source through import [ " this is not a hypothetical or purely theoretical construct, and we can directly observe it in experiment. the first landmark test, reported in real - time observation of valence electron motion. e. goulielmakis et al. nature 466, 739 ( 2010 ). was able to show a clear oscillation in how much a short pulse was absorbed by an oscillating charge distribution caused by spin - orbit interactions ( where different parts of the oscillations correspond to different orientations of the charge density, and therefore to different absorption profiles ), showing a clear corresponding oscillation in the absorbance : similarly, a much - beloved example is the observation of charge oscillation dynamics in a bio - relevant molecule, phenylalanine, which was reported in ultrafast electron dynamics in phenylalanine initiated by attosecond pulses. f. calegari et al., science 346, 336 ( 2014 ), and where the ionization of the molecule by a ( relatively ) short laser pulse ( in the near - infrared ) is then probed by a ( very ) short attosecond burst. the resulting dynamics inside the molecule are fairly complicated, but they lead to clear oscillations in the signal ( with the graph below showing the overall decay, and the oscillations on top of an exponential background ) at a very short timescale that is only observable thanks to the availability of attosecond pulses. watching quantum interference build up in real time i will do one more direct - timing - of - observation, because i think they're really cool. this one is again about a quantum superposition, but one that happens with a free electron. when you ionize an atom, the electron gets released, and one photon gets absorbed. and, more importantly, the details of the energy states that the electron gets released into will be imprinted into the absorbance spectrum of the light. in particular, it is possible to tune things so that you are ionizing close to a resonance : the electron can either ionize directly, or it can spend some time in a highly - excited autoionizing state ( also explained here and here ) that will fall apart after some time. the end result is that the electron will go into a superposition of both pathways, which will interfere in its spectrum and cause a won


##ky, nontrivial shape in the absorption spectrum. however, if we have short pulses of radiation, we are able to control how long we let the electron to sit in that autoionizing state, before we come in with a second pulse of light to disrupt it, and kill the interference : and indeed, when we do this, the build - up of the line and the development of the interference features ( and particularly that sharp dip on the right - hand side of the line ) is very clearly seen in experiment : and, just to add some more pretty pictures, here it is all stacked together, on the left - hand figure, and on the right a similar experiment showing very clearly the destructive interference building up over time : for more details, and the sources of the figures, see observing the ultrafast buildup of a fano resonance in the time domain. a. kaldun et al. science 354, 738 ( 2016 ) and attosecond dynamics through a fano resonance : monitoring the birth of a photoelectron. v gruson et al. science 354, 734 ( 2016 ) moreover, it is also possible to use these types of resonances to enhance high - harmonic generation itself, in a process known as resonant hhg. for a nice review written by a colleague ( in a paper i coauthored ) see eur. phys. j d 75, 209 ( 2021 ) ( arxiv : 2101. 09335 ). further reading long as this post is, i have only just scratched the surface. here are some additional places to read more about the field : attosecond science. d. villeneuve, contemp. phys. 59, 47 ( 2018 ) ( author eprint ) attosecond science. p. b. corkum & f. krausz. nature physics 3, 381 ( 2007 ) ( author eprint ) the physics of attosecond light pulses. p. agostini & l. f. dimauro. reports on progress in physics 67, 813 ( 2004 ) ( author eprint ) attosecond electromagnetic pulses : generation, measurement, and application. attosecond metrology and spectroscopy. m. yu. ryabikin et al. * physics - uspekhi 66, 360 ( 2023 ) shining the shortest flashes of light on the secret life of electrons. m. khokhlova, e. pisanty & a.


zair. advanced photonics 5, 060501 ( 2023 )


this is not an answer to your question, but an extended comment on the issue that has been raised here in comments by different people, namely : are machine learning " tensors " the same thing as tensors in mathematics? now, according to the cichoki 2014, era of big data processing : a new approach via tensor networks and tensor decompositions, and cichoki et al. 2014, tensor decompositions for signal processing applications, a higher - order tensor can be interpreted as a multiway array, [... ] a tensor can be thought of as a multi - index numerical array, [... ] tensors ( i. e., multi - way arrays ) [... ] so in machine learning / data processing a tensor appears to be simply defined as a multidimensional numerical array. an example of such a 3d tensor would be $ 1000 $ video frames of $ 640 \ times 480 $ size. a usual $ n \ times p $ data matrix is an example of a 2d tensor according to this definition. this is not how tensors are defined in mathematics and physics! a tensor can be defined as a multidimensional array obeying certain transformation laws under the change of coordinates ( see wikipedia or the first sentence in mathworld article ). a better but equivalent definition ( see wikipedia ) says that a tensor on vector space $ v $ is an element of $ v \ otimes \ ldots \ otimes v ^ * $. note that this means that, when represented as multidimensional arrays, tensors are of size $ p \ times p $ or $ p \ times p \ times p $ etc., where $ p $ is the dimensionality of $ v $. all tensors well - known in physics are like that : inertia tensor in mechanics is $ 3 \ times 3 $, electromagnetic tensor in special relativity is $ 4 \ times 4 $, riemann curvature tensor in general relativity is $ 4 \ times 4 \ times 4 \ times 4 $. curvature and electromagnetic tensors are actually tensor fields, which are sections of tensor bundles ( see e. g. here but it gets technical ), but all of that is defined over a vector space $ v $. of course one can construct a tensor product $ v \ otimes w $ of an $ p $ - dimensional $ v $ and $ q $ - dimensional $ w $ but its elements are usually not called " tensors ", as stated e. g. here on


wikipedia : in principle, one could define a " tensor " simply to be an element of any tensor product. however, the mathematics literature usually reserves the term tensor for an element of a tensor product of a single vector space $ v $ and its dual, as above. one example of a real tensor in statistics would be a covariance matrix. it is $ p \ times p $ and transforms in a particular way when the coordinate system in the $ p $ - dimensional feature space $ v $ is changed. it is a tensor. but a $ n \ times p $ data matrix $ x $ is not. but can we at least think of $ x $ as an element of tensor product $ w \ otimes v $, where $ w $ is $ n $ - dimensional and $ v $ is $ p $ - dimensional? for concreteness, let rows in $ x $ correspond to people ( subjects ) and columns to some measurements ( features ). a change of coordinates in $ v $ corresponds to linear transformation of features, and this is done in statistics all the time ( think of pca ). but a change of coordinates in $ w $ does not seem to correspond to anything meaningful ( and i urge anybody who has a counter - example to let me know in the comments ). so it does not seem that there is anything gained by considering $ x $ as an element of $ w \ otimes v $. and indeed, the common notation is to write $ x \ in \ mathbb r ^ { n \ times p } $, where $ r ^ { n \ times p } $ is a set of all $ n \ times p $ matrices ( which, by the way, are defined as rectangular arrays of numbers, without any assumed transformation properties ). my conclusion is : ( a ) machine learning tensors are not math / physics tensors, and ( b ) it is mostly not useful to see them as elements of tensor products either. instead, they are multidimensional generalizations of matrices. unfortunately, there is no established mathematical term for that, so it seems that this new meaning of " tensor " is now here to stay.


superoxide, o2− is created by the immune system in phagocytes ( including neutrophils, monocytes, macrophages, dendritic cells, and mast cells ) which use nadph oxidase to produce it from o2 for use against invading microorganisms. however, under normal conditions, the mitochondrial electron transport chain is a major source of o2−, converting up to perhaps 5 % of o2 to superoxide. [ 1 ] as a side note, there are two sides to this coin. while this is a useful tool against microorganisms, the formation of the reactive oxygen species has been incriminated in autoimmune reactions and diabetes ( type 1 ). [ 2 ] [ 1 ] packer l, ed. methods in enzymology, volume 349. san diego, calif : academic press ; 2002 [ 2 ] thayer tc, delano m, et al. ( 2011 ) superoxide production by macrophages and t cells is critical for the induction of autoreactivity and type 1 diabetes, 60 ( 8 ), 2144 - 51.


intriguing question. first, the best yield would be achieved by selectively producing one enantiomer instead of the other. in this case, white wants d - methamphetamine ( powerful psychoactive drug ), not l - methamphetamine ( vicks vapor inhaler ). reaction processes designed to do this are known as " asymmetric synthesis " reactions, because they favor production of one enantiomer over the other. the pseudoephedrine method for methamphetamine employs one of the more common methods of asymmetric synthesis, called " chiral pool resolution ". as you state, starting with an enantiomerically - pure sample of a chiral reagent ( pseudoephedrine ) as the starting point allows you to preserve the chirality of the finished product, provided the chiral point is not part of any " leaving group " during the reaction. however, again as you show, phenylacetone is achiral, and so the p2p process cannot take advantage of this method. there are other methods of asymmetric synthesis, however none of them seem applicable to the chemistry shown or described on tv either ; none of the reagents or catalysts mentioned would work as chiral catalysts, nor are they bio - or organocatalysts. metal complexes with chiral ligands can be used to selectively catalyze production of one enantiomer, however the aluminum - mercury amalgam is again achiral. i don't remember any mention of using organocatalysis or biocatalysis, but these are possible. the remaining route, then, is chiral resolution ; let the reaction produce the 50 - 50 split, then separate the two enantiomers by some means of reactionary and / or physical chemistry. this seems to be the way it works in the real world. the advantage is that most of the methods are pretty cheap and easy ; the disadvantage is that your maximum possible yield is 50 % ( unless you can then run a racemization reaction on the undesireable half to " reshuffle " the chirality of that half ; then your yield increases by 50 % of the last increase each time you run this step on the undesirable product ). in the case of methamphetamine, this resolution is among the easiest, because methamphetamine forms a " racemic conglomerate " when crystallized. this means,


for the non - chemists, that each enantiomer molecule prefers to crystallize with others of the same chiral species, so as the solution cools and the solvent is evaporated off, the d - methamphetamine will form one set of homogeneous crystals and the l - methamphetamine will form another set. this means that all white has to do is slow the evaporation of solvent and subsequent cooling of the pan, letting the largest possible crystals form. then, the only remaining trick is identifying which crystals have which enantiomer ( and as these crystals are translucent and " optically active ", observing the polarization pattern of light shone through the crystals will identify which are which ).


i'll translate an entry in the blog gaussianos ( " gaussians " ) about polya's conjecture, titled : a belief is not a proof. we'll say a number is of even kind if in its prime factorization, an even number of primes appear. for example $ 6 = 2 \ cdot 3 $ is a number of even kind. and we'll say a number is of odd kind if the number of primes in its factorization is odd. for example, $ 18 = 2 · 3 · 3 $ is of odd kind. ( $ 1 $ is considered of even kind ). let $ n $ be any natural number. we'll consider the following numbers : $ e ( n ) = $ number of positive integers less or equal to $ n $ that are of even kind. $ o ( n ) = $ number of positive integers less or equal to $ n $ that are of odd kind. let's consider $ n = 7 $. in this case $ o ( 7 ) = 4 $ ( number 2, 3, 5 and 7 itself ) and $ e ( 7 ) = 3 $ ( 1, 4 and 6 ). so $ o ( 7 ) > e ( 7 ) $. for $ n = 6 $ : $ o ( 6 ) = 3 $ and $ e ( 6 ) = 3 $. thus $ o ( 6 ) = e ( 6 ) $. in 1919 george polya proposed the following result, know as polya's conjecture : for all $ n > 2 $, $ o ( n ) $ is greater than or equal to $ e ( n ) $. polya had checked this for $ n < 1500 $. in the following years this was tested up to $ n = 1000000 $, which is a reason why the conjecture might be thought to be true. but that is wrong. in 1962, lehman found an explicit counterexample : for $ n = 906180359 $, we have $ o ( n ) = e ( n ) – 1 $, so : $ $ o ( 906180359 ) < e ( 906180359 ). $ $ by an exhaustive search, the smallest counterexample is $ n = 906150257 $, found by tanaka in 1980. thus polya's conjecture is false. what do we learn from this? well, it is simple : unfortunately in mathematics we cannot


trust intuition or what happens for a finite number of cases, no matter how large the number is. until the result is proved for the general case, we have no certainty that it is true.


a rolling hash function for dna sequences called nthash has recently been published in bioinformatics and the authors dealt with reverse complements : using this table, we can easily compute the hash value for the reverse - complement ( as well as the canonical form ) of a sequence efficiently, without actually reverse - complementing the input sequence, as follows :... edit ( by @ user172818 ) : i will add more details about how nthash works. the notations used in its paper are somewhat uncommon. the source code is more informative. let's first define rotation functions for 64 - bit integers : rol ( x, k ) : = x < < k | x > > ( 64 - k ) ror ( x, k ) : = x > > k | x < < ( 64 - k ) we then define a hash function h ( ) for each base. in the implementation, the authors are using : h ( a ) = 0x3c8bfbb395c60474 h ( c ) = 0x3193c18562a02b4c h ( g ) = 0x20323ed082572324 h ( t ) = 0x295549f54be24456 h ( n ) = 0 the rolling hash function of a forward k - mer s [ i, i + k - 1 ] is : f ( s [ i, i + k - 1 ] ) : = rol ( h ( s [ i ] ), k - 1 ) ^ rol ( h ( s [ i + 1 ] ), k - 2 ) ^... ^ h ( s [ i + k - 1 ] ) where ^ is the xor operator. the hash function of its reverse complement is : r ( s [ i, i + k - 1 ] ) : = f ( ~ s [ i, i + k - 1 ] ) = rol ( h ( ~ s [ i + k - 1 ] ), k - 1 ) ^ rol ( h ( ~ s [ i + k - 2 ] ), k - 2 ) ^... ^ h ( ~ s [ i ] ) where ~ gives the reverse complement of a dna sequence. knowing f ( s [ i, i + k - 1 ] ) and r ( s [ i, i + k - 1 ] ), we can compute their values for the next k - mer : f (


s [ i + 1, i + k ] ) = rol ( f ( s [ i, i + k - 1 ] ), 1 ) ^ rol ( h ( s [ i ] ), k ) ^ h ( s [ i + k ] ) r ( s [ i + 1, i + k ] ) = ror ( r ( s [ i, i + k - 1 ] ), 1 ) ^ ror ( h ( ~ s [ i ] ), 1 ) ^ rol ( h ( ~ s [ i + k ] ), k - 1 ) in other words, for the forward kmer for each additional base, xor the following three values together : a single left rotation of the previous hash, f ( s [ i, i + k - 1 ] ) a $ k $ - times left rotation of the base hash of s [ i ] the base hash of s [ i + k ] similarly for the reverse kmer, xor the following three values together : a single right rotation of the previous reverse hash, r ( s [ i, i + k - 1 ] ) a single right rotation of the base hash of the reverse complement of s [ i ] a $ k - 1 $ - times left rotation of the base hash of the reverse complement of s [ i + k ] this works because rol, ror and ^ can all be switched in order. finally, for a k - mer s, the hash function considering both strands is the smaller between f ( s ) and r ( s ) : h ( s ) = min ( f ( s ), r ( s ) ) this is a linear algorithm regardless of the k - mer length. it only uses simple arithmetic operations, so should be fairly fast. i have briefly tested its randomness. it seems comparable to murmur. nthash is probably the best algorithm so far if you want to hash an arbitrarily long k - mer into 64 bits.


as willie wong observed, including an expression of the form $ \ displaystyle \ frac { | \ alpha | } { \ alpha } $ is a way of ensuring that $ \ alpha > 0 $. ( as $ \ sqrt { | \ alpha | / \ alpha } $ is $ 1 $ if $ \ alpha > 0 $ and non - real if $ \ alpha < 0 $. ) the ellipse $ \ displaystyle \ left ( \ frac { x } { 7 } \ right ) ^ { 2 } + \ left ( \ frac { y } { 3 } \ right ) ^ { 2 } - 1 = 0 $ looks like this : so the curve $ \ left ( \ frac { x } { 7 } \ right ) ^ { 2 } \ sqrt { \ frac { \ left | \ left | x \ right | - 3 \ right | } { \ left | x \ right | - 3 } } + \ left ( \ frac { y } { 3 } \ right ) ^ { 2 } \ sqrt { \ frac { \ left | y + 3 \ frac { \ sqrt { 33 } } { 7 } \ right | } { y + 3 \ frac { \ sqrt { 33 } } { 7 } } } - 1 = 0 $ is the above ellipse, in the region where $ | x | > 3 $ and $ y > - 3 \ sqrt { 33 } / 7 $ : that's the first factor. the second factor is quite ingeniously done. the curve $ \ left | \ frac { x } { 2 } \ right | \ ; - \ ; \ frac { \ left ( 3 \ sqrt { 33 } - 7 \ right ) } { 112 } x ^ { 2 } \ ; - \ ; 3 \ ; + \ ; \ sqrt { 1 - \ left ( \ left | \ left | x \ right | - 2 \ right | - 1 \ right ) ^ { 2 } } - y = 0 $ looks like : this is got by adding $ y = \ left | \ frac { x } { 2 } \ right | - \ frac { \ left ( 3 \ sqrt { 33 } - 7 \ right ) } { 112 } x ^ { 2 } - 3 $, a parabola on the positive - x side, reflected : and $ y = \ sqrt { 1 - \ left ( \ left


| \ left | x \ right | - 2 \ right | - 1 \ right ) ^ { 2 } } $, the upper halves of the four circles $ \ left ( \ left | \ left | x \ right | - 2 \ right | - 1 \ right ) ^ 2 + y ^ 2 = 1 $ : the third factor $ 9 \ sqrt { \ frac { \ left ( \ left | \ left ( 1 - \ left | x \ right | \ right ) \ left ( \ left | x \ right | -. 75 \ right ) \ right | \ right ) } { \ left ( 1 - \ left | x \ right | \ right ) \ left ( \ left | x \ right | -. 75 \ right ) } } \ ; - \ ; 8 \ left | x \ right | \ ; - \ ; y \ ; = \ ; 0 $ is just the pair of lines y = 9 - 8 | x | : truncated to the region $ 0. 75 < | x | < 1 $. similarly, the fourth factor $ 3 \ left | x \ right | \ ; + \ ;. 75 \ sqrt { \ left ( \ frac { \ left | \ left (. 75 - \ left | x \ right | \ right ) \ left ( \ left | x \ right | -. 5 \ right ) \ right | } { \ left (. 75 - \ left | x \ right | \ right ) \ left ( \ left | x \ right | -. 5 \ right ) } \ right ) } \ ; - \ ; y \ ; = \ ; 0 $ is the pair of lines $ y = 3 | x | + 0. 75 $ : truncated to the region $ 0. 5 < | x | < 0. 75 $. the fifth factor $ 2. 25 \ sqrt { \ frac { \ left | \ left (. 5 - x \ right ) \ left ( x +. 5 \ right ) \ right | } { \ left (. 5 - x \ right ) \ left ( x +. 5 \ right ) } } \ ; - \ ; y \ ; = \ ; 0 $ is the line $ y = 2. 25 $ truncated to $ - 0. 5 < x < 0. 5 $. finally, $ \ frac { 6 \ sqrt { 10 } } { 7 } \ ; + \ ; \ left ( 1. 5 \ ; - \ ;. 5 \


left | x \ right | \ right ) \ ; - \ ; \ frac { \ left ( 6 \ sqrt { 10 } \ right ) } { 14 } \ sqrt { 4 - \ left ( \ left | x \ right | - 1 \ right ) ^ { 2 } } \ ; - \ ; y \ ; = \ ; 0 $ looks like : so the sixth factor $ \ frac { 6 \ sqrt { 10 } } { 7 } \ ; + \ ; \ left ( 1. 5 \ ; - \ ;. 5 \ left | x \ right | \ right ) \ sqrt { \ frac { \ left | \ left | x \ right | - 1 \ right | } { \ left | x \ right | - 1 } } \ ; - \ ; \ frac { \ left ( 6 \ sqrt { 10 } \ right ) } { 14 } \ sqrt { 4 - \ left ( \ left | x \ right | - 1 \ right ) ^ { 2 } } \ ; - \ ; y \ ; = \ ; 0 $ looks like as a product of factors is $ 0 $ iff any one of them is $ 0 $, multiplying these six factors puts the curves together, giving : ( the software, grapher. app, chokes a bit on the third factor, and entirely on the fourth )


look carefully, it's ( distorted ) tetrahedral - - four groups at nearly symmetrically positions in 3d space { * }. so the hybridization is $ sp ^ 3 $. as you can see, the shape is distorted, but it's tetrahedral. technically, the banana bonds can be said to be made up of orbitals similar to $ sp ^ 3 $ but not exactly ( like two $ sp ^ { 3. 1 } $ and two $ sp ^ { 2. 9 } $ orbitals - - since hybridization is just addition of wavefunctions, we can always change the coefficients to give proper geometry ). i'm not too sure of this though. $ \ ce { b } $ has an $ 2s ^ 22p ^ 1 $ valence shell, so three covalent bonds gives it an incomplete octet. $ \ ce { bh3 } $ has an empty $ 2p $ orbital. this orbital overlaps the existing $ \ ce { b - h } $ $ \ sigma $ bond cloud ( in a nearby $ \ ce { bh3 } $ ), and forms a 3c2e bond. it seems that there are a lot more compounds with 3c2e geometry. i'd completely forgotten that there were entire homologous series'under'boranes'which all have 3c2e bonds ( though not the same structure ) and there are indium and gallium compounds as well. still group iiia, though these are metals. i guess they, like $ \ ce { al } $, still form covalent bonds. so the basic reason for this happening is due to an incomplete octet wanting to fill itself. note that " banana " is not necessarily only for 3c2e bonds. any bent bond is called a " banana " bond. regarding similar structures, $ \ ce { becl2 } $ and $ \ ce { alcl3 } $ come to mind, but both of them have the structure via dative ( coordinate ) bonds. additionally, $ \ ce { becl2 } $ is planar. sneaks off and checks wikipedia. wikipedia says $ \ ce { al2 ( ch3 ) 6 } $ is similar in structure and bond type. i guess we have less such compounds because there are comparatively few elements ( $ \ ce { b } $ group pretty much ) with $ \ leq3 $ valence electrons which form covalent bonds ( criteria for the empty orbital )


. additionally, $ \ ce { al } $ is an iffy case - - it like both covalent and ionic bonds. also, for this geometry ( either by banana bonds or by dative bonds ), i suppose the relative sizes matter as well - - since $ \ ce { bcl3 } $ is a monomer even though $ \ ce { cl } $ has a lone pair and can form a dative bond. * maybe you're used to the view of tetrahedral structure with an atom at the top? mentally tilt the boron atom till a hydrogen is up top. you should realize that this is tetrahedral as well.


the global alliance for genomics and health has been working on the issue of representing sequencing data and metadata for storage and sharing for quite some time, though with mixed results. they do offer a model and api for storing ngs data in their github repository, but it can be a bit of a pain to get a high - level view. i am not sure if any better representation of this exists elsewhere. i can say from personal experience ( having built over a dozen genomic databases ), there is no ideal data model and storage best practices. genomic data comes in many shapes and sizes, and your needs are going to vary from every other organization, so what works for one bioinformatics group won't necessarily work for you. the best thing to do is design and implement a model that will cover all of the data types in your workflow and downstream analyses you might do with the data and metadata.


what follows is taken ( mostly ) from more extensive discussions in the following sci. math posts : [ 23 january 2000 ] [ 6 november 2006 ] [ 20 december 2006 ] note : the term interval is restricted to nondegenerate intervals ( i. e. intervals containing more than one point ). the continuity set of a derivative on an open interval $ j $ is dense in $ j. $ in fact, the continuity set has cardinality $ c $ in every subinterval of $ j. $ on the other hand, the discontinuity set $ d $ of a derivative can have the following properties : $ d $ can be dense in $ \ mathbb r $. $ d $ can have cardinality $ c $ in every interval. $ d $ can have positive measure. ( hence, the function can fail to be riemann integrable. ) $ d $ can have positive measure in every interval. $ d $ can have full measure in every interval ( i. e. measure zero complement ). $ d $ can have a hausdorff dimension zero complement. $ d $ can have an $ h $ - hausdorff measure zero complement for any specified hausdorff measure function $ h. $ more precisely, a subset $ d $ of $ \ mathbb r $ can be the discontinuity set for some derivative if and only if $ d $ is an $ f _ { \ sigma } $ first category ( i. e. an $ f _ { \ sigma } $ meager ) subset of $ \ mathbb r. $ this characterization of the discontinuity set of a derivative can be found in the following references : benedetto [ 1 ] ( chapter 1. 3. 2, proposition, 1. 10, p. 30 ) ; bruckner [ 2 ] ( chapter 3, section 2, theorem 2. 1, p. 34 ) ; bruckner / leonard [ 3 ] ( theorem at bottom of p. 27 ) ; goffman [ 5 ] ( chapter 9, exercise 2. 3, p. 120 states the result ) ; klippert / williams [ 7 ]. regarding this characterization of the discontinuity set of a derivative, bruckner and leonard [ 3 ] ( bottom of p. 27 ) wrote the following in 1966 : although we imagine that this theorem is known, we have been unable to find a reference. i have found the result stated in goffman's 1953 text [ 5 ], but


nowhere else prior to 1966 ( including goffman's ph. d. dissertation ). interestingly, in a certain sense most derivatives have the property that $ d $ is large in all of the ways listed above ( # 1 through # 7 ). in 1977 cliff weil [ 8 ] published a proof that, in the space of derivatives with the sup norm, all but a first category set of such functions are discontinuous almost everywhere ( in the sense of lebesgue measure ). when weil's result is paired with the fact that derivatives ( being baire $ 1 $ functions ) are continuous almost everywhere in the sense of baire category, we get the following : ( a ) every derivative is continuous at the baire - typical point. ( b ) the baire - typical derivative is not continuous at the lebesgue - typical point. note that weil's result is stronger than simply saying that the baire - typical derivative fails to be riemann integrable ( i. e. $ d $ has positive lebesgue measure ), or even stronger than saying that the baire - typical derivative fails to be riemann integrable on every interval. note also that, for each of these baire - typical derivatives, $ \ { d, \ ; { \ mathbb r } - d \ } $ gives a partition of $ \ mathbb r $ into a first category set and a lebesgue measure zero set. in 1984 bruckner / petruska [ 4 ] ( theorem 2. 4 ) strengthened weil's result by proving the following : given any finite borel measure $ \ mu, $ the baire - typical derivative is such that the set $ d $ is the complement of a set that has $ \ mu $ - measure zero. in 1993 kirchheim [ 5 ] strengthened weil's result by proving the following : given any hausdorff measure function $ h, $ the baire - typical derivative is such that the set $ d $ is the complement of a set that has hausdorff $ h $ - measure zero. [ 1 ] john j. benedetto, real variable and integration with historical notes, mathematische leitfaden. stuttgart : b. g. teubne, 1976, 278 pages. [ mr 58 # 28328 ; zbl 336. 26001 ] [ 2 ] andrew m. bruckner, differentiation of real functions, 2nd edition, crm monograph series #


5, american mathematical society, 1994, xii + 195 pages. [ the first edition was published in 1978 as springer - verlag's lecture notes in mathematics # 659. the second edition is essentially unchanged from the first edition with the exception of a new chapter on recent developments ( 23 pages ) and 94 additional bibliographic items. ] [ mr 94m : 26001 ; zbl 796. 26001 ] [ 3 ] andrew m. bruckner and john l. leonard, derivatives, american mathematical monthly 73 # 4 ( april 1966 ) [ part ii : papers in analysis, herbert ellsworth slaught memorial papers # 11 ], 24 - 56. [ mr 33 # 5797 ; zbl 138. 27805 ] [ 4 ] andrew m. bruckner and gyorgy petruska, some typical results on bounded baire $ 1 $ functions, acta mathematica hungarica 43 ( 1984 ), 325 - 333. [ mr 85h : 26004 ; zbl 542. 26004 ] [ 5 ] casper goffman, real functions, prindle, weber & schmidt, 1953 / 1967, x + 261 pages. [ mr 14, 855e ; zbl 53. 22502 ] [ 6 ] bernd kirchheim, some further typical results on bounded baire one functions, acta mathematica hungarica 62 ( 1993 ), 119 - 129. [ 94k : 26008 ; zbl 786. 26002 ] [ 7 ] john clayton klippert and geoffrey williams, on the existence of a derivative continuous on a $ g _ { \ delta } $, international journal of mathematical education in science and technology 35 ( 2004 ), 91 - 99. [ 8 ] clifford weil, the space of bounded derivatives, real analysis exchange 3 ( 1977 - 78 ), 38 - 41. [ zbl 377. 26005 ]


five point summary yes, the idea is to give a quick summary of the distribution. it should be roughly symmetrical about mean, the median should be close to 0, the 1q and 3q values should ideally be roughly similar values. coefficients and $ \ hat { \ beta _ i } s $ each coefficient in the model is a gaussian ( normal ) random variable. the $ \ hat { \ beta _ i } $ is the estimate of the mean of the distribution of that random variable, and the standard error is the square root of the variance of that distribution. it is a measure of the uncertainty in the estimate of the $ \ hat { \ beta _ i } $. you can look at how these are computed ( well the mathematical formulae used ) on wikipedia. note that any self - respecting stats programme will not use the standard mathematical equations to compute the $ \ hat { \ beta _ i } $ because doing them on a computer can lead to a large loss of precision in the computations. $ t $ - statistics the $ t $ statistics are the estimates ( $ \ hat { \ beta _ i } $ ) divided by their standard errors ( $ \ hat { \ sigma _ i } $ ), e. g. $ t _ i = \ frac { \ hat { \ beta _ i } } { \ hat { \ sigma _ i } } $. assuming you have the same model in object modas your q : > mod < - lm ( sepal. width ~ petal. width, data = iris ) then the $ t $ values r reports are computed as : > tstats < - coef ( mod ) / sqrt ( diag ( vcov ( mod ) ) ) ( intercept ) petal. width 53. 277950 - 4. 786461 where coef ( mod ) are the $ \ hat { \ beta _ i } $, and sqrt ( diag ( vcov ( mod ) ) ) gives the square roots of the diagonal elements of the covariance matrix of the model parameters, which are the standard errors of the parameters ( $ \ hat { \ sigma _ i } $ ). the p - value is the probability of achieving a $ | t | $ as large as or larger than the observed absolute t value if the null hypothesis ( $ h _ 0 $ ) was true, where $ h _ 0 $ is $ \ beta _ i = 0 $. they are computed as ( using tstats from above


) : > 2 * pt ( abs ( tstats ), df = df. residual ( mod ), lower. tail = false ) ( intercept ) petal. width 1. 835999e - 98 4. 073229e - 06 so we compute the upper tail probability of achieving the $ t $ values we did from a $ t $ distribution with degrees of freedom equal to the residual degrees of freedom of the model. this represents the probability of achieving a $ t $ value greater than the absolute values of the observed $ t $ s. it is multiplied by 2, because of course $ t $ can be large in the negative direction too. residual standard error the residual standard error is an estimate of the parameter $ \ sigma $. the assumption in ordinary least squares is that the residuals are individually described by a gaussian ( normal ) distribution with mean 0 and standard deviation $ \ sigma $. the $ \ sigma $ relates to the constant variance assumption ; each residual has the same variance and that variance is equal to $ \ sigma ^ 2 $. adjusted $ r ^ 2 $ adjusted $ r ^ 2 $ is computed as : $ $ 1 - ( 1 - r ^ 2 ) \ frac { n - 1 } { n - p - 1 } $ $ the adjusted $ r ^ 2 $ is the same thing as $ r ^ 2 $, but adjusted for the complexity ( i. e. the number of parameters ) of the model. given a model with a single parameter, with a certain $ r ^ 2 $, if we add another parameter to this model, the $ r ^ 2 $ of the new model has to increase, even if the added parameter has no statistical power. the adjusted $ r ^ 2 $ accounts for this by including the number of parameters in the model. $ f $ - statistic the $ f $ is the ratio of two variances ( $ ssr / sse $ ), the variance explained by the parameters in the model ( sum of squares of regression, ssr ) and the residual or unexplained variance ( sum of squares of error, sse ). you can see this better if we get the anova table for the model via anova ( ) : > anova ( mod ) analysis of variance table response : sepal. width df sum sq mean sq f value pr ( > f ) petal. width 1 3. 7945 3. 7945 22. 91 4. 073e - 06 * * *


residuals 148 24. 5124 0. 1656 - - - signif. codes : 0 ‘ * * * ’ 0. 001 ‘ * * ’ 0. 01 ‘ * ’ 0. 05 ‘. ’ 0. 1 ‘ ’ 1 the $ f $ s are the same in the anova output and the summary ( mod ) output. the mean sq column contains the two variances and $ 3. 7945 / 0. 1656 = 22. 91 $. we can compute the probability of achieving an $ f $ that large under the null hypothesis of no effect, from an $ f $ - distribution with 1 and 148 degrees of freedom. this is what is reported in the final column of the anova table. in the simple case of a single, continuous predictor ( as per your example ), $ f = t _ { \ mathrm { petal. width } } ^ 2 $, which is why the p - values are the same. this equivalence only holds in this simple case.


theorem ( false ) : one can arbitrarily rearrange the terms in a convergent series without changing its value.


it's difficult to get this to go massively quicker i think - as with this question working with large gzipped fastq files is mostly io - bound. we could instead focus on making sure we are getting the right answer. people deride them too often, but this is where a well - written parser is worth it's weight in gold. heng li gives us this fastq parser in c. i downloaded the example tarball and modified the example code ( excuse my c... ) : # include < zlib. h > # include < stdio. h > # include " kseq. h " kseq _ init ( gzfile, gzread ) int main ( int argc, char * argv [ ] ) { gzfile fp ; kseq _ t * seq ; int l ; if ( argc = = 1 ) { fprintf ( stderr, " usage : % s < in. seq > \ n ", argv [ 0 ] ) ; return 1 ; } fp = gzopen ( argv [ 1 ], " r " ) ; seq = kseq _ init ( fp ) ; int seqcount = 0 ; long seqlen = 0 ; while ( ( l = kseq _ read ( seq ) ) > = 0 ) { seqcount = seqcount + 1 ; seqlen = seqlen + ( long ) strlen ( seq - > seq. s ) ; } kseq _ destroy ( seq ) ; gzclose ( fp ) ; printf ( " number of sequences : % d \ n ", seqcount ) ; printf ( " number of bases in sequences : % ld \ n ", seqlen ) ; return 0 ; } then make and kseq _ test foo. fastq. gz. for my example file ( ~ 35m reads of ~ 75bp ) this took : real 0m49. 670s user 0m49. 364s sys 0m0. 304s compared with your example : real 0m43. 616s user 1m35. 060s sys 0m5. 240s konrad's solution ( in my hands ) : real 0m39. 682s user 1m11. 900s sys 0m


##5. 112s ( by the way, just zcat - ing the data file to / dev / null ) : real 0m38. 736s user 0m38. 356s sys 0m0. 308s so, i get pretty close in speed, but am likely to be more standards compliant. also this solution gives you more flexibility with what you can do with the data. and my horrible c can almost certainly be optimised. same test, with kseq. h from github, as suggested in the comments : my machine is under different load this morning, so i've retested. wall clock times : op : 0m44. 813s konrad : 0m40. 061s zcat > / dev / null : 0m34. 508s kseq. h ( github ) : 0m32. 909s so most recent version of kseq. h is faster than simply zcat - ing the file ( consistently in my tests... ).


the name " ring " is derived from hilbert's term " zahlring " ( number ring ), introduced in his zahlbericht for certain rings of algebraic integers. as for why hilbert chose the name " ring ", i recall reading speculations that it may have to do with cyclical ( ring - shaped ) behavior of powers of algebraic integers. namely, if $ \ : \ alpha \ : $ is an algebraic integer of degree $ \ rm \ : n \ : $ then $ \ : \ alpha ^ n \ : $ is a $ \ rm \ : \ mathbb z $ - linear combination of lower powers of $ \ rm \ : \ alpha \ :, \ : $ thus so too are all higher powers of $ \ rm \ : \ alpha \ :. \ : $ hence all powers cycle back onto $ \ rm \ : 1, \ : \ alpha, \ :, \ ldots, \ alpha ^ { n - 1 } \ :, \ : $ i. e. $ \ rm \ : \ mathbb z [ \ alpha ] \ : $ is a finitely generated $ \ : \ mathbb z $ - module. possibly also the motivation for the name had to do more specifically with rings of cyclotomic integers. however, as plausible as that may seem, i don't recall the existence of any historical documents that provide solid evidence in support of such speculations. beware that one has to be very careful when reading such older literature. some authors mistakenly read modern notions into terms which have no such denotation in their original usage. to provide some context i recommend reading lemmermeyer and schappacher's introduction to the english edition of hilbert ’ s zahlbericht. below is a pertinent excerpt. below is an excerpt from leo corry's modern algebra and the rise of mathematical structures, p. 149. below are a couple typical examples of said speculative etymology of the term " ring " via the " circling back " nature of integral dependence, from harvey cohn's advanced number theory, p. 49. $ \ quad $ the designation of the letter $ \ mathfrak d $ for the integral domain has some historical importance going back to gauss's work on quadratic forms. gauss $ \ left ( 1800 \ right ) $ noted that for certain quadratic forms $ ax ^ 2 + bxy + cy ^ 2 $ the discriminant need not be square - free, although $ a


$, $ b $, $ c $ are relatively prime. for example, $ x ^ 2 - 45y ^ 2 $ has $ d = 4 \ cdot45 $. the $ 4 $ was ignored for the reason that $ 4 | d $ necessarily by virtue of gauss's requirement that $ b $ be even, but the factor of $ 3 ^ 2 $ in $ d $ caused gauss to refer to the form as one of " order $ 3 $. " eventually, the forms corresponding to a value of $ d $ were called an " order " ( ordnung ). dedekind retained this word for what is here called an " integral domain. " $ \ quad $ the term " ring " is a contraction of " zahlring " introduced by hilbert $ \ left ( 1892 \ right ) $ to denote ( in our present context ) the ring generated by the rational integers and a quadratic integer $ \ eta $ defined by $ $ \ eta ^ 2 + b \ eta + c = 0. $ $ it would seem that module $ \ left [ 1, \ eta \ right ] $ is called a zahlring because $ \ eta ^ 2 $ equals $ - b \ eta - c $ " circling directly back " to an element of $ \ left [ 1, \ eta \ right ] $. this word has been maintained today. incidentally, every zahlring is an integral domain and the converse is true for quadratic fields. and from rotman's advanced modern algebra, p. 81.


ok, my bad. the error is in the last equation : \ begin { align } kl ( p, q ) & = - \ int p ( x ) \ log q ( x ) dx + \ int p ( x ) \ log p ( x ) dx \ \ \ \ & = \ frac { 1 } { 2 } \ log ( 2 \ pi \ sigma _ 2 ^ 2 ) + \ frac { \ sigma _ 1 ^ 2 + ( \ mu _ 1 - \ mu _ 2 ) ^ 2 } { 2 \ sigma _ 2 ^ 2 } - \ frac { 1 } { 2 } ( 1 + \ log 2 \ pi \ sigma _ 1 ^ 2 ) \ \ \ \ & = \ log \ frac { \ sigma _ 2 } { \ sigma _ 1 } + \ frac { \ sigma _ 1 ^ 2 + ( \ mu _ 1 - \ mu _ 2 ) ^ 2 } { 2 \ sigma _ 2 ^ 2 } - \ frac { 1 } { 2 } \ end { align } note the missing $ - \ frac { 1 } { 2 } $. the last line becomes zero when $ \ mu _ 1 = \ mu _ 2 $ and $ \ sigma _ 1 = \ sigma _ 2 $.


in practice, most people stick to relatively low orders, usually first or second order. this view is often challenged by more theoretical researchers that believe in more accurate answers. the rate of convergence for simple smooth problems is well documented, for example see bill mitchell's comparison of hp adaptivity. while for theoretical works it is nice to see what the convergence rate are, for more application oriented among us this concern is balanced with constitutive laws, necessary precision, and code complexity. it doesn't make much since in many porous media problems that solve over a highly discontinuous media to have high order methods, the numerical error will dominate the discretization errors. the same concern applies for problems that include a large number of degrees of freedom. since low - order implicit methods have a smaller bandwidth and often a better conditioning, the high order method becomes too costly to solve. finally the code complexity of switching orders and types of polynomials is usually too much for the graduate students running the application codes.


there are some cases of bio - metallic materials, as hinted at by the comments. but these are relatively small amount of metal. it's not that there is a lack of metal available. iron in particular is the fourth most common element in the earth's crust. most soil that has a reddish color has iron in it. there are several reasons you don't see iron exoskeletons on animals all the time. firstly, metallic iron ( in chemistry terms, fully reduced, oxidation state 0 ) has a high energetic cost to create. iron is the second most common metal after aluminum on the earth's crust but it's almost entirely present in oxidized states - that's to say : as rust. most biological iron functions in the + 2 / + 3 oxidation state, which is more similar to rust than metal. cytochromes and haemoglobin are examples of how iron is more valuable as a chemically active biological agent than a structural agent, using oxidized iron ions as they do. aluminium, the most common metal on earth, has relatively little biological activity - one might assume because its redox costs are even higher than iron. as to why reduced biometal doesn't show up very often, inability of biological systems to deposit reduced ( metallic ) metals is not one of them. there are cases of admittedly small pieces of reduced metal being produced by biological systems. the magnetosomes in magnetotactic bacteria are mentioned, but there are also cases of reduced gold being accumulated by microorganisms. bone and shell are examples of biomineralization where the proteins depositing the calcium carbonate or other minerals in the material are structured by the proteins to be stronger than they would be as a simple crystal. most of the examples here have very little or no metal, but rather minerals like the chrysomallon squamiferum cited by @ navyguymarko and @ loki'sbane here. the iron sulfide looks metallic but it is a mineral, akin to a bone. while iron skeletons might seem to be an advantage, they are electrochemically unstable - oxygen and water will tend to oxidize ( rust ) them quickly and the organism would have to spend a lot of energy keeping it in working form. electrical conductivity sounds useful, but the nervous system favors exquisite levels of control over bulk current flow, even in cases like electric eels, whose current is produced by gradients from acetylcholine. what's more,


biological materials actually perform as well as or better than metal when they need to. spider silk has a greater tensile strength than steel ( along the direction of the thread ). mollusk shells are models for tank armor - they are remarkably resistant to puncture and breakage. bone is durable for most purposes and flexible in addition. the time it would take for metallized structures to evolve biologically are likely too long. by the time the metalized version of an organ or skeleton got started, the bones, shells and fibers we know probably have a big lead and selective advantage.


in a svm you are searching for two things : a hyperplane with the largest minimum margin, and a hyperplane that correctly separates as many instances as possible. the problem is that you will not always be able to get both things. the c parameter determines how great your desire is for the latter. i have drawn a small example below to illustrate this. to the left you have a low c which gives you a pretty large minimum margin ( purple ). however, this requires that we neglect the blue circle outlier that we have failed to classify correct. on the right you have a high c. now you will not neglect the outlier and thus end up with a much smaller margin. so which of these classifiers are the best? that depends on what the future data you will predict looks like, and most often you don't know that of course. if the future data looks like this : then the classifier learned using a large c value is best. on the other hand, if the future data looks like this : then the classifier learned using a low c value is best. depending on your data set, changing c may or may not produce a different hyperplane. if it does produce a different hyperplane, that does not imply that your classifier will output different classes for the particular data you have used it to classify. weka is a good tool for visualizing data and playing around with different settings for an svm. it may help you get a better idea of how your data look and why changing the c value does not change the classification error. in general, having few training instances and many attributes make it easier to make a linear separation of the data. also that fact that you are evaluating on your training data and not new unseen data makes separation easier. what kind of data are you trying to learn a model from? how much data? can we see it?


it is not possible to produce white light without an efficient blue led, either using rgb leds or a blue led + yellow phosphor. the breakthrough was the invention of the high - brightness gallium - nitride blue led by shuji nakamura at nichia in the early 1990s. it still took a while to get the overall efficiency up to the level of fluorescent bulbs, and it's only in the last decade that leds finally came out on top.


there is a very different mechanism for generation ( and detection ) of ultraviolet, visible and infrared light vs radio waves. for the first, it is possible to generate it using chemical reactions ( that is, chemiluminescence, bioluminescence ) with a typical energy of order of 2 ev ( electronovolts ). also, it is easy to detect with similar means - coupling to a bond ( e. g. using opsins ). for much longer electromagnetic waves, and much lower energies per photon, such mechanism does not work. there are two reasons : typical energy levels for molecules ( but it can be worked around ), thermal noise has energies ( 0. 025 ev ) which are higher than radio wave photon energies ( < 0. 001 ev ) ( it rules out both controlled creation and detection using molecules ). in other words - radiation which is less energetic than thermal radiation ( far infrared ) is not suitable for communication using molecular mechanisms, as thermal noise jams transmission ( making the sender firing at random and making the receiver being blind by noise way stronger than the signal ). however, one can both transmit, and detect it, using wires. in principle it is possible ; however, without good conductors ( like metals, not - salt solutions ) it is not an easy task ( not impossible though ).


simulators designed specifically for oxford nanopore : nanosim nanosim - h silico readsim deepsimulator general long read simulators : loresim loresim 2 fastqsim longislnd for an exhaustive list of existing read simulators, see page 15 of my thesis, novel computational techniques for mapping and classifying next - generation sequencing data.


yes, you can. and you do not even need to leave the earth to do it. you are always viewing things in the past, just as you are always hearing things in the past. if you see someone do something, who is 30 meters away, you are seeing what happened $ ( 30 \ ; \ mathrm { m } ) / ( 3 \ times10 ^ 8 \ ; \ mathrm { m } / \ mathrm { s } ) = 0. 1 \ ; \ mu \ mathrm { s } $ in the past. if you had a mirror on the moon ( about 238k miles away ), you could see about 2. 5 seconds into earth's past. if that mirror was on pluto, you could see about 13. 4 hours into earth's past. if you are relying on hearing, you hear an event at 30 m away about 0. 1 s after it occurs. that is why runners often watch the starting pistol at an event, because they can see a more recent picture of the past than they can hear. to more directly answer the intent of your question : yes, if you could magically be transported 27 lightyears away, or had a mirror strategically placed 13. 5 lightyears away, you could see yourself being born.


in general terms, there are the $ o ( n ^ 2 ) $ sorting algorithms, such as insertion sort, bubble sort, and selection sort, which you should typically use only in special circumstances ; quicksort, which is worst - case $ o ( n ^ 2 ) $ but quite often $ o ( n \ log n ) $ with good constants and properties and which can be used as a general - purpose sorting procedure ; the $ o ( n \ log n ) $ algorithms, like merge - sort and heap - sort, which are also good general - purpose sorting algorithms ; and the $ o ( n ) $, or linear, sorting algorithms for lists of integers, such as radix, bucket and counting sorts, which may be suitable depending on the nature of the integers in your lists. if the elements in your list are such that all you know about them is the total order relationship between them, then optimal sorting algorithms will have complexity $ \ omega ( n \ log n ) $. this is a fairly cool result and one for which you should be able to easily find details online. the linear sorting algorithms exploit further information about the structure of elements to be sorted, rather than just the total order relationship among elements. even more generally, optimality of a sorting algorithm depends intimately upon the assumptions you can make about the kind of lists you're going to be sorting ( as well as the machine model on which the algorithm will run, which can make even otherwise poor sorting algorithms the best choice ; consider bubble sort on machines with a tape for storage ). the stronger your assumptions, the more corners your algorithm can cut. under very weak assumptions about how efficiently you can determine " sortedness " of a list, the optimal worst - case complexity can even be $ \ omega ( n! ) $. this answer deals only with complexities. actual running times of implementations of algorithms will depend on a large number of factors which are hard to account for in a single answer.