normalization rescales the values into a range of [ 0, 1 ]. this might be useful in some cases where all parameters need to have the same positive scale. however, the outliers from the data set are lost. $ $ x _ { changed } = \ frac { x - x _ { min } } { x _ { max } - x _ { min } } $ $ standardization rescales data to have a mean ( $ \ mu $ ) of 0 and standard deviation ( $ \ sigma $ ) of 1 ( unit variance ). $ $ x _ { changed } = \ frac { x - \ mu } { \ sigma } $ $ for most applications standardization is recommended.


you've several options : remove some of the bias. ( a ) by penalizing the likelihood as per @ nick's suggestion. package logistf in r or the firth option in sas's proc logistic implement the method proposed in firth ( 1993 ), " bias reduction of maximum likelihood estimates ", biometrika, 80, 1. ; which removes the first - order bias from maximum likelihood estimates. ( here @ gavin recommends the brglm package, which i'm not familiar with, but i gather it implements a similar approach for non - canonical link functions e. g. probit. ) ( b ) by using median - unbiased estimates in exact conditional logistic regression. package elrm or logistix in r, or the exact statement in sas's proc logistic. exclude cases where the predictor category or value causing separation occurs. these may well be outside your scope ; or worthy of further, focused investigation. ( the r package safebinaryregression is handy for finding them. ) re - cast the model. typically this is something you'd have done beforehand if you'd thought about it, because it's too complex for your sample size. ( a ) remove the predictor from the model. dicey, for the reasons given by @ simon : " you're removing the predictor that best explains the response ". ( b ) by collapsing predictor categories / binning the predictor values. only if this makes sense. ( c ) re - expressing the predictor as two ( or more ) crossed factors without interaction. only if this makes sense. use a bayesian analysis as per @ manoel's suggestion. though it seems unlikely you'd want to just because of separation, worth considering on its other merits. the paper he recommends is gelman et al ( 2008 ), " a weakly informative default prior distribution for logistic & other regression models ", ann. appl. stat., 2, 4 : the default in question is an independent cauchy prior for each coefficient, with a mean of zero & a scale of $ \ frac { 5 } { 2 } $ ; to be used after standardizing all continuous predictors to have a mean of zero & a standard deviation of $ \ frac { 1 } { 2 } $. if you can elucidate strongly informative priors, so much the better. do nothing. ( but calculate confidence intervals based on profile likelihoods


, as the wald estimates of standard error will be badly wrong. ) an often over - looked option. if the purpose of the model is just to describe what you've learnt about the relationships between predictors & response, there's no shame in quoting a confidence interval for an odds ratio of, say, 2. 3 upwards. ( indeed it could seem fishy to quote confidence intervals based on unbiased estimates that exclude the odds ratios best supported by the data. ) problems come when you're trying to predict using point estimates, & the predictor on which separation occurs swamps the others. use a hidden logistic regression model, as described in rousseeuw & christmann ( 2003 ), " robustness against separation and outliers in logistic regression ", computational statistics & data analysis, 43, 3, and implemented in the r package hlr. ( @ user603 suggests this. ) i haven't read the paper, but they say in the abstract " a slightly more general model is proposed under which the observed response is strongly related but not equal to the unobservable true response ", which suggests to me it mightn't be a good idea to use the method unless that sounds plausible. " change a few randomly selected observations from 1 to 0 or 0 to 1 among variables exhibiting complete separation " : @ robertf's comment. this suggestion seems to arise from regarding separation as a problem per se rather than as a symptom of a paucity of information in the data which might lead you to prefer other methods to maximum - likelihood estimation, or to limit inferences to those you can make with reasonable precision — approaches which have their own merits & are not just " fixes " for separation. ( aside from its being unabashedly ad hoc, it's unpalatable to most that analysts asking the same question of the same data, making the same assumptions, should give different answers owing to the result of a coin toss or whatever. )


because it's undecidable whether the program will use the memory again. this means that no algorithm can correctly determine when to call free ( ) in all cases, which means that any compiler that tried to do this would necessarily produce some programs with memory leaks and / or some programs that continued to use memory that had been freed. even if you ensured that your compiler never did the second one and allowed the programmer to insert calls to free ( ) to fix those bugs, knowing when to call free ( ) for that compiler would be even harder than knowing when to call free ( ) when using a compiler that didn't try to help.


there is no tidal bulge. this was one of newton's few mistakes. newton did get the tidal forcing function correct, but the response to that forcing in the oceans : completely wrong. newton's equilibrium theory of the tides with its two tidal bulges is falsified by observation. if this hypothesis was correct, high tide would occur when the moon is at zenith and at nadir. most places on the earth's oceans do have a high tide every 12. 421 hours, but whether those high tides occur at zenith and nadir is sheer luck. in most places, there's a predictable offset from the moon's zenith / nadir and the time of high tide, and that offset is not zero. one of the most confounding places with regard to the tides is newton's back yard. if newton's equilibrium theory was correct, high tide would occur at more or less the same time across the north sea. that is not what is observed. at any time of day, one can always find a place in the north sea that is experiencing high tide, and another that is simultaneously experiencing low tide. why isn't there a bulge? beyond the evidence, there are a number of reasons a tidal bulge cannot exist in the oceans. the tidal bulge cannot exist because the way water waves propagate. if the tidal bulge did exist, it would form a wave with a wavelength of half the earth's circumference. that wavelength is much greater than the depth of the ocean, which means the wave would be a shallow wave. the speed of a shallow wave at some location is approximately $ \ sqrt { gd } $, where $ d $ is the depth of the ocean at that location. this tidal wave could only move at 330 m / s over even the deepest oceanic trench, 205 m / s over the mean depth of 4267 m, and less than that in shallow waters. compare with the 465 m / s rotational velocity at the equator. the shallow tidal wave cannot keep up with the earth's rotation. the tidal bulge cannot exist because the earth isn't completely covered by water. there are two huge north - south barriers to newton's tidal bulge, the americas in the western hemisphere and afro - eurasia in the eastern hemisphere. the tides on the panama's pacific coast are very, very different from the tides just 100 kilometers away on panama's caribbean coast. a third reason the tidal bulge cannot exist is the coriolis effect


. that the earth is rotating at a rate different from the moon's orbital rate means that the coriolis effect would act to sheer the tidal wave apart even if the earth was completely covered by a very deep ocean. what is the right model? what newton got wrong, laplace got right. laplace's dynamic theory of the tides accounts for the problems mentioned above. it explains why it's always high tide somewhere in the north sea ( and patagonia, and the coast of new zealand, and a few other places on the earth where tides are just completely whacko ). the tidal forcing functions combined with oceanic basin depths and outlines results in amphidromic systems. there are points on the surface, " amphidromic points ", that experience no tides, at least with respect to one of the many forcing functions of the tides. the tidal responses rotate about these amphidromic points. there are a large number of frequency responses to the overall tidal forcing functions. the moon is the dominant force with regard to the tides. it helps to look at things from the perspective of the frequency domain. from this perspective, the dominant frequency on most places on the earth is 1 cycle per 12. 421 hours, the m2 tidal frequency. the second largest is the 1 cycle per 12 hours due to the sun, the s2 tidal frequency. since the forcing function is not quite symmetric, there are also 1 cycle per 24. 841 hours responses ( the m1 tidal frequency ), 1 cycle per 24 hours responses ( the s1 tidal frequency ), and a slew of others. each of these has its own amphidromic system. with regard to the north sea, there are three m2 tidal amphidromic points in the neighborhood of the north sea. this nicely explains why the tides are so very goofy in the north sea. images for those who like imagery, here are a few key images. i'm hoping that the owners of these images won't rearrange their websites. the tidal force source : this is what newton did get right. the tidal force is away from the center of the earth when the moon ( or sun ) is at zenith or nadir, inward when the moon ( or sun ) is on the horizon. the vertical component is the driving force behind the response of the earth as a whole to these tidal forces. this question isn't about the earth tides. the question is about the oceanic tides, and there it's the horizontal component that is the


driving force. the global m2 tidal response source : source : link, not archived ] the m2 constituent of the tides is the roughly twice per day response to the tidal forcing function that results from the moon. this is the dominant component of the tides in many parts of the world. the first image shows the m2 amphidromic points, points where there is no m2 component of the tides. even though these points have zero response to this component, these amphidromic points are nonetheless critical in modeling the tidal response. the second image, an animated gif, shows the response over time. the m2 tidal response in the north sea archived source : i mentioned the north sea multiple times in my response. the north atlantic is where 40 % of the m2 tidal dissipation occurs, and the north sea is the hub of this dissipation. energy flow of the semi - diurnal, lunar tidal wave ( m2 ) archived source : ( the above image displays transfer of energy from places where tidal energy is created to places where it is dissipated. this energy transfer explains the weird tides in patagonia, one of the places on the earth where tides are highest and most counterintuitive. those patagonian tides are largely a result of energy transfer from the pacific to the atlantic. it also shows the huge transfer of energy to the north atlantic, which is where 40 % of the m2 tidal dissipation occurs. note that this energy transfer is generally eastward. you can think of this as a representing " net tidal bulge. " or not. i prefer " or not. " extended discussions based on comments (... because we delete comments here ) isn't a tsunami a shallow water wave as well as compared to the ocean basins? i know the wavelength is smaller but it is still a shallow water wave and hence would propagate at the same speed. why don't they suffer from what you mentioned regarding the rotational velocity of the earth. firstly, there's a big difference between a tsunami and the tides. a tsunami is the the result of a non - linear damped harmonic oscillator ( the earth's oceans ) to an impulse ( an earthquake ). the tides are the response to a cyclical driving force. that said, as is the case with any harmonic oscillator, the impulse response is informative of the response to a cyclical driving force. tsunamis are subject to the coriolis effect. the effect is small, but present. the


reason it is small is because tsunami are, for the most part, short term events relative to the earth's rotation rate. the coriolis effect becomes apparent in the long - term response of the oceans to a tsunami. topography is much more important for a tsunami. the link that follows provides an animation of the 2004 indonesian earthquake tsunami. references for the above : dao, m. h., & tkalich, p. ( 2007 ). tsunami propagation modelling? a sensitivity study. natural hazards and earth system science, 7 ( 6 ), 741 - 754. eze, c. l., uko, d. e., gobo, a. e., sigalo, f. b., & israel - cookey, c. ( 2009 ). mathematical modelling of tsunami propagation. journal of applied sciences and environmental management, 13 ( 3 ). kowalik, z., knight, w., logan, t., & whitmore, p. ( 2005 ). numerical modeling of the global tsunami : indonesian tsunami of 26 december 2004. science of tsunami hazards, 23 ( 1 ), 40 - 56. this is an interesting answer full of cool facts and diagrams, but i think it's a little overstated. newton's explanation wasn't wrong, it was an approximation. he knew it was an approximation - - obviously he was aware that the earth had land as well as water, that tides were of different heights in different places, and so on. i don't think it's a coincidence that the height of the bulge in the equipotential is of very nearly the right size to explain the observed heights of the tides. newton's analysis was a good start. newton certainly did describe the tidal force properly. he didn't have the mathematical tools to do any better than what he did. fourier analysis, proper treatment of non - inertial frames, and fluid dynamics all post - date newton by about a century. besides the issues cited above, newton ignored the horizontal component of the tidal force and only looked at the vertical component. the horizontal component wouldn't be important if the earth was tidally locked to the moon. the dynamical theory of the tides essentially ignores the vertical component and only looks at the horizontal component. this gives a very different picture of the tides. i'm far from alone in saying the tidal bulge doesn't exist. for example, from this lecture, [ archive link ] the page


on dynamic tides [ archive link ] rhetorically asks " but how can water confined to a basin engage in wave motion at all like the “ tidal bulges ” that supposedly sweep around the globe as depicted in equilibrium theory? " and immediately responds ( emphasis mine ) " the answer is – it can ’ t. " in affholder, m., & valiron, f. ( 2001 ). descriptive physical oceanography. crc press, the authors introduce newton's equilibrium tide but then write ( emphasis mine ) " for the tidal wave to move at this enormous speed of 1600 km / h, the ideal ocean depth would have to be 22 km. taking the average depth of the ocean as 3. 9 km, the speed of the tidal elevations can only be 700 km / h. therefore the equilibrium position at any instant required by this theory cannot be established. " oceanographers still teach newton's equilibrium tide theory for a number of reasons. it does give a proper picture of the tidal forcing function. moreover, many students do not understand how many places can have two tides a day. for that matter, most oceanography instructors and textbook authors don't understand! many oceanographers and their texts still hold that the inner bulge is a consequence of gravity but the other bulge is a consequence of a so - called centrifugal force. this drives geophysicists and geodocists absolutely nuts. that's starting to change ; in the last ten years or so, some oceanography texts have finally started teaching that the only force that is needed to explain the tides is gravitation.


log - scale informs on relative changes ( multiplicative ), while linear - scale informs on absolute changes ( additive ). when do you use each? when you care about relative changes, use the log - scale ; when you care about absolute changes, use linear - scale. this is true for distributions, but also for any quantity or changes in quantities. note, i use the word " care " here very specifically and intentionally. without a model or a goal, your question cannot be answered ; the model or goal defines which scale is important. if you're trying to model something, and the mechanism acts via a relative change, log - scale is critical to capturing the behavior seen in your data. but if the underlying model's mechanism is additive, you'll want to use linear - scale. example. stock market. stock a on day 1 : $ \ $ $ 100. on day 2, $ \ $ $ 101. every stock tracking service in the world reports this change in two ways! ( 1 ) + $ \ $ $ 1. ( 2 ) + 1 %. the first is a measure of absolute, additive change ; the second a measure of relative change. illustration of relative change vs absolute : relative change is the same, absolute change is different stock a goes from $ \ $ $ 1 to $ \ $ $ 1. 10. stock b goes from $ \ $ $ 100 to $ \ $ $ 110. stock a gained 10 %, stock b gained 10 % ( relative scale, equal )... but stock a gained 10 cents, while stock b gained $ \ $ $ 10 ( b gained more absolute dollar amount ) if we convert to log space, relative changes appear as absolute changes. stock a goes from $ \ log _ { 10 } ( \ $ 1 ) $ to $ \ log _ { 10 } ( \ $ 1. 10 ) $ = 0 to. 0413 stock b goes from $ \ log _ { 10 } ( \ $ 100 ) $ to $ \ log _ { 10 } ( \ $ 110 ) $ = 2 to 2. 0413 now, taking the absolute difference in log space, we find that both changed by. 0413. both of these measures of change are important, and which one is important to you depends solely on your model of investing. there are two models. ( 1 ) investing a fixed amount of principal, or ( 2 ) investing in a fixed number of shares. model 1 : investing with a fixed amount of principal. say yesterday stock a


cost $ \ $ $ 1 per share, and stock b costs $ \ $ $ 100 a share. today they both went up by one dollar to $ \ $ $ 2 and $ \ $ $ 101 respectively. their absolute change is identical ( $ \ $ $ 1 ), but their relative change is dramatically different ( 100 % for a, 1 % for b ). given that you have a fixed amount of principal to invest, say $ \ $ $ 100, you can only afford 1 share of b or 100 shares of a. if you invested yesterday you'd have $ \ $ $ 200 with a, or $ \ $ $ 101 with b. so here you " care " about the relative gains, specifically because you have a finite amount of principal. model 2 : fixed number of shares. in a different scenario, suppose your bank only lets you buy in blocks of 100 shares, and you've decided to invest in 100 shares of a or b. in the previous case, whether you buy a or b your gains will be the same ( $ \ $ $ 100 - i. e. $ 1 for each share ). now suppose we think of a stock value as a random variable fluctuating over time, and we want to come up with a model that reflects generally how stocks behave. and let's say we want to use this model to maximize profit. we compute a probability distribution whose x - values are in units of'share price ', and y - values in probability of observing a given share price. we do this for stock a, and stock b. if you subscribe to the first scenario, where you have a fixed amount of principal you want to invest, then taking the log of these distributions will be informative. why? what you care about is the shape of the distribution in relative space. whether a stock goes from 1 to 10, or 10 to 100 doesn't matter to you, right? both cases are a 10 - fold relative gain. this appears naturally in a log - scale distribution in that unit gains correspond to fold gains directly. for two stocks whose mean value is different but whose relative change is identically distributed ( they have the same distribution of daily percent changes ), their log distributions will be identical in shape just shifted. conversely, their linear distributions will not be identical in shape, with the higher valued distribution having a higher variance. if you were to look at these same distributions in linear, or absolute space, you would think that higher - valued share prices correspond to greater fluctuations


. for your investing purposes though, where only relative gains matter, this is not necessarily true. example 2. chemical reactions. suppose we have two molecules a and b that undergo a reversible reaction. $ a \ leftrightarrow b $ which is defined by the individual rate constants ( $ k _ { ab } $ ) $ a \ rightarrow b $ ( $ k _ { ba } $ ) $ b \ rightarrow a $ their equilibrium is defined by the relationship : $ k = \ frac { k _ { ab } } { k _ { ba } } = \ frac { [ a ] } { [ b ] } $ two points here. ( 1 ) this is a multiplicative relationship between the concentrations of $ a $ and $ b $. ( 2 ) this relationship isn't arbitrary, but rather arises directly from the fundamental physical - chemical properties that govern molecules bumping into each other and reacting. now suppose we have some distribution of a or b's concentration. the appropriate scale of that distribution is in log - space, because the model of how either concentration changes is defined multiplicatively ( the product of a's concentration with the inverse of b's concentration ). in some alternate universe where $ k ^ * = k _ { ab } - k _ { ba } = [ a ] - [ b ] $, we might look at this concentration distribution in absolute, linear space. that said, if you have a model, be it for stock market prediction or chemical kinetics, you can always interconvert'losslessly'between linear and log space, so long as your range of values is $ ( 0, \ inf ) $. whether you choose to look at the linear or log - scale distribution depends on what you're trying to obtain from the data. edit. an interesting parallel that helped me build intuition is the example of arithmetic means vs geometric means. an arithmetic ( vanilla ) mean computes the average of numbers assuming a hidden model where absolute differences are what matter. example. the arithmetic mean of 1 and 100 is 50. 5. suppose we're talking about concentrations though, where the chemical relationship between concentrations is multiplicative. then the average concentration should really be computed on the log scale. this is called the geometric average. the geometric average of 1 and 100 is 10! in terms of relative differences, this makes sense : 10 / 1 = 10, and 100 / 10 = 10, ie., the relative change


between the average and two values is the same. additively we find the same thing ; 50. 5 - 1 = 49. 5, and 100 - 50. 5 = 49. 5.


there will always be a tradeoff in terms of resource allocation between reproduction and self maintenance. since worker ants forego reproduction to perform other roles ( gathering resources, caring for young etc. ) within the colony, it makes sense that this would favour a longer lifespan. this idea works for most animals ( i. e. higher reproduction = lower lifespan across species in general ) and is well documented ( partridge et al. 1987, gems & riddle 1996 ( pdf link ), westendorp & kirkwood 1998 - to name a few ). however the relationship is reversed in eusocial animals : the queens of ants, some bees and naked mole rats ( which are all eusocial ) tend to be longer lived than their sterile workers ( hartmann & heinze 2003 ). aging patterns within ants and other eusocial organisms are a very popular research topic at the moment, but the mechanisms causing differences between social castes are not fully understood. eusociality is strongly associated with increased lifespan, indeed many studies on the evolution of aging have focussed on eusocial animals which have appeared to overcome aging effects to an extent ( keller & genoud, 1999, buffenstein 2005 ). the naked mole rat is one of very few species of eusocial mammals and the relationship between its lifespan and body mass is very dissimilar to that of other rodents ( ignore the bat points ) : image from buffenstein & pinto ( 2009 ). if a queen has a longer reproductive lifespan then over the course of its life it will create a higher number of offspring. this will cause their increased longevity genes to be more prevalent in the gene pool over time - as long as they are able to reach the limit of their reproductive lifespan and are not killed by externalities ( accident or attack ) in the meantime. therefore this trait ( longer reproductive lifespan ) will be selected for as long as the queen is protected. within eusocial colonies, there is often a protective environment ( buffenstein & jarvis 2002 ) : there is usually a physical structure ( nest or burrow ) ; symbiotic bacteria and / or fungi creating a more hygienic microflora ; and queens are also protected by other castes. this gives queens a lower incidence of death due to accident or attack which ( from the reasoning in the previous paragraph ) supports the selection of a longer reproductive life. this is likely to lead to longer lived workers since they share the same genes. however queens do tend to live much longer than their workers ( o'donnell & jeanne 1995


). this is probably due to their reproductive role. all the ants in the colony are investing in reproduction ( whether by physically giving birth to young or by providing it with food and protection ) therefore the normal relationship ( reproduction and lifespan tradeoff ) mentioned at the start is not relevant, but since the queen is the only individual reproducing, it will be the only one for whom lengthened lifespan is beneficial evolutionarily. a mean lifespan of 20 years has been observed in formica exsecta, and a maximum lifespan of 28. 5 years observed in lasius niger. pogonomyrmex owyheei has an observed maximum of 30 years, and mean of 17 years. these ( and many more ) figures were obtained from a review by keller ( 1998 ). other references on this subject include ( svensson & sheldon 1998 ), ( keller & genoud 1997 ) ( pdf link ), ( calabi & porter 1989 ), and ( amdam & omholt 2002 ). references amdam, g. v. & omholt, s. w. ( 2002 ) the regulatory anatomy of honeybee lifespan. journal of theoretical biology, 216, 209 – 228. buffenstein, r. ( 2005 ) the naked mole - rat : a new long - living model for human aging research. the journals of gerontology series a : biological sciences and medical sciences, 60, 1369 – 1377. buffenstein, r. & jarvis, j. u. m. ( 2002 ) the naked mole rat - - a new record for the oldest living rodent. sci. aging knowl. environ., 2002, pe7. buffenstein, r. & pinto, m. ( 2009 ) endocrine function in naturally long - living small mammals. molecular and cellular endocrinology, 299, 101 – 111. calabi, p. & porter, s. d. ( 1989 ) worker longevity in the fire ant solenopsis invicta : ergonomic considerations of correlations between temperature, size and metabolic rates. journal of insect physiology, 35, 643 – 649. hartmann, a. & heinze, j. ( 2003 ) lay eggs, live longer : division of labor and life span in a clonal ant species. evolution, 57, 2424 – 2429. keller, l. ( 1998 ) queen lifespan and colony characteristics in ants and termites. insectes sociaux, 45, 235 – 246. keller, l.


& genoud, m. ( 1997 ) extraordinary lifespans in ants : a test of evolutionary theories of ageing. nature, 389, 958 – 960. keller, l. & genoud, m. ( 1999 ) evolutionary theories of aging. gerontology, 45, 336 – 338. o ’ donnell, s. & jeanne, r. l. ( 1995 ) implications of senescence patterns for the evolution of age polyethism in eusocial insects. behavioral ecology, 6, 269 – 273. svensson, e. & sheldon, b. c. ( 1998 ) the social context of life evolution history. oikos, 83, 466 – 477.


we won't necessarily see any effects. suppose that somebody finds an algorithm that solves 3sat on $ n $ variables in $ 2 ^ { 100 } n $ basic operations. you won't be able to run this algorithm on any instance, since it takes too long. or suppose that she finds an algorithm running in $ n ^ { 100 } $ basic operations. we will only be able to use it on 3sat instances on a single variable, since for more variables it takes too long. on the other hand, suppose that p $ \ neq $ np, and that even the stronger exponential time hypothesis holds. then in general, 3sat should be untractable. yet sat solvers seem to be doing well on certain problems. what's happening here? there are several problems with the p vs. np question : it only concerns the worst case. it is only asymptotic. all polynomial time bounds are the same. these problems cast doubt on its relevance to the real world. now it could happen that some really fast algorithm is found for 3sat, so fast that even symmetric encryption would become breakable. but i consider this highly unlikely. on the other hand, it is perfectly consistent for p to be different from np while factoring being practical ; that would break certain public key encryption schemes. this is a likely situation which would have repercussions, but it is unrelated to the p vs. np question. the p vs. np question might be natural from a mathematical point of view, but its practical relevance is doubtful, in my view. research on the question, on the other hand, might or might not have practical repercussions ; it is not guided by this aspect.


there are different angles this question can be answered : chemical point of view : a full analysis of a totally unknown mixture is painful and extremely costly. it is always helpful to know how many components you are looking for ; what types etc. in this case, it is not enough to analyse the elemental composition or some pure elements, but coke contains a lot of natural products and mixtures like caramel. imagine how you identify ( and correctly describe the production parameters ) caramel in a dilute solution... also, liquids with extremely high concentrations of sugar and different acids are unpleasant for analysis, as you often have to separate these main component, so you can identify only small components. that being said, it is not an impossible task ; there are many methods to identify e. g. natural products based on dna traces. but consider another factor : business point of view : the recipe of coca cola itself worth nothing. most probably any competent soda maker can make a drink that 99 % of the consumers cannot distinguish from coke by the taste ( and could do that decades before ). however they don't do because no one would buy a drink that tastes like coca cola, but made by others. the value is in the brand. people buy a fake rolex if they cannot afford a real. but anyone can afford a coke - why would buy a fake? if you are in beverage business, it is an imperative to make a drink that is somehow different than the other ones!


that is the correct sequence for 2019 - ncov. coronavirus is of course an rna virus and in fact, to my knowledge, every rna virus in genbank is present as cdna ( agct, i. e. thydmine ) and not rna ( agcu, i. e. uracil ). the reason is simple, we never sequence directly from rna because rna is too unstable and easily degraded by rnase. instead the genome is reverse transcribed, either by targeted reverse transcription or random amplification and thus converted to cdna. cdna is stable and is essentially reverse transcribed rna. the cdna is either sequenced directly or further amplified by pcr and then sequenced. hence the sequence we observe is the cdna rather than rna, thus we observe thymine rather than uracil and that is how it is reported.


first of all, i would emphasize that " alignment - free " quantification tools like salmon and kallisto are not reference - free. the basic difference between them and more traditional aligners is that they do not report a specific position ( either in a genome or transcriptome ) to which a read maps. however, their overall purpose is still to quantify the expression levels ( or differences ) of a known set of transcripts ; hence, they require a reference ( which could be arbitrarily defined ). the most important criterion for deciding which approach to use ( and this is true of almost everything in genomics ) is exactly what question you would like to answer. if you are primarily interested in quantifying and comparing expression of mature mrna from known transcripts, then a transcriptome - based alignment may be fastest and best. however, you may miss potentially interesting features outside of those known transcripts, such as new isoforms, non - coding rnas, or information about pre - mrna levels, which can often be gleaned from intronic reads ( see the eisa method ). this paper also has some good considerations about which tools may work best depending on the question you want to answer. finally, another fast and flexible aligner ( which can be used with or without a reference transcriptome ) is star.


someone may have better words to explain this than me, but the big thing you have to remember is voltage is a potential difference. in most cases the " difference " part is a difference between some potential and ground potential. when someone says - 5v, they're saying that you are below ground. you also need to keep in mind that voltage is relative. so like i mentioned before, most people reference to " ground " ; but what is ground? you can say ground is earth ground, but what about the case when you have a battery powered device that has no contact to ground. in this situation we have to treat some arbitrary point as " ground ". usually the negative terminal on the battery is what we consider from this reference. now consider the case that you have 2 batteries in series. if both were 5 volts, then you would say you would have 10 volts total. but the assumption that you get 0 / + 10 is based off of " ground " as being the negative terminal on the battery that isn't touching the other battery and then 10v as being the location of the positive terminal that isn't touching the other battery. in this situation we can make the decision that we want to make the connection between the 2 batteries be our " ground " reference. this would then result in + 5v on one end and - 5v on the other end. here is what i was trying to explain : + 10v + + + + 5v | | | | < battery | | + 5v - - - 0v + + + | | | | < another battery | | 0v - - - - 5v


the forward voltage is the voltage drop across the diode if the voltage at the anode is more positive than the voltage at the cathode ( if you connect + to the anode ). you will be using this value to calculate the power dissipation of the diode and the voltage after the diode. the reverse voltage is the voltage drop across the diode if the voltage at the cathode is more positive than the voltage at the anode ( if you connect + to the cathode ). this is usually much higher than the forward voltage. as with forward voltage, a current will flow if the connected voltage exceeds this value. this is called a " breakdown ". common diodes are usually destroyed but with z and zener diodes this effect is used deliberately.


except for code which does a significant number of floating - point operations on data that are held in cache, most floating - point intensive code is performance limited by memory bandwidth and cache capacity rather than by flops. $ v $ and the products $ av $ and $ bv $ are all vectors of length 2000 ( 16k bytes in double precision ), which will easily fit into a level 1 cache. the matrices $ a $ and $ b $ are 2000 by 2000 or about 32 megabytes in size. your level 3 cache might be large enough to store one of these matrices if you've got a really good processor. computing $ av $ requires reading 32 megabytes ( for $ a $ ) in from memory, reading in 16k bytes ( for $ v $ ) storing intermediate results in the l1 cache and eventually writing 16k bytes out to memory. multiplying $ bv $ takes the same amount of work. adding the two intermediate results to get the final result requires a trivial amount of work. that's a total of roughly 64 megabytes of reads and an insignificant number of writes. computing $ ( a + b ) $ requires reading 32 megabytes ( for a ) plus 32 megabytes ( for b ) from memory and writing 32 megabytes ( for a + b ) out. then you have to do a single matrix - vector multiplication as above which involves reading 32 megabytes from memory ( if you've got a big l3 cache, then perhaps this 32 megabytes is in that l3 cache. ) that's a total of 96 megabytes of reads and 32 megabytes of writes. thus there's twice as much memory traffic involved in computing this as $ ( a + b ) v $ instead of $ av + bv $. note that if you have to do many of these multiplications with different vectors $ v $ but the same $ a $ and $ b $, then it will become more efficient to compute $ a + b $ once and reuse that matrix for the matrix - vector multiplications.


both ceramic resonators and quartz crystals work on the same principle : the vibrate mechanically when an ac signal is applied to them. quartz crystals are more accurate and temperature stable than ceramic resonators. the resonator or crystal itself has two connections. on the left the crystal, right the ceramic resonator. like you say the oscillator needs extra components, the two capacitors. the active part which makes the oscillator work is an amplifier which supplies the energy to keep the oscillation going. some microcontrollers have a low - frequency oscillator for a 32. 768 khz crystal, which often has the capacitors built - in, so that you only need two connections for the crystal ( left ). most oscillators, however, need the capacitors externally, and then you have thee connections : input from the amplifier, output to the amplifier, and ground for the capacitors. a resonator with three pins has the capacitors integrated. the function of the capacitors : in order to oscillate the closed loop amplifier - crystal must have a total phase shift of 360°. the amplifier is inverting, so that's 180°. together with the capacitors the crystal takes care of the other 180°. edit when you switch a crystal oscillator on it's just an amplifier, you don't get the desired frequency yet. the only thing that's there is a low - level noise over a wide bandwidth. the oscillator will amplify that noise and pass it through the crystal, upon which it enters the oscillator again which amplifies it again and so on. shouldn't that get you just very much noise? no, the crystal's properties are such that it will pass only a very small amount of the noise, around its resonance frequency. all the rest will be attenuated. so in the end it's only that resonance frequency which is left, and then we're oscillating. you can compare it with a trampoline. imagine a bunch of kids jumping on it randomly. the trampoline doesn't move much and the kids have to make a lot of effort to jump just 20cm up. but after some time they will start to synchronize and the trampoline will follow the jumping. the kids will jump higher and higher with less effort. the trampoline


will oscillate at its resonance frequency ( about 1hz ) and it will be hard to jump faster or slower. that's the frequencies that will be filtered out. the kid jumping on the trampoline is the amplifier, she supplies the energy to keep the oscillation going. further reading msp430 32 khz crystal oscillators


you mention that fastqc " fails to find the actual adapter sequences " - i guess you mean in the adapter sequence contamination plot. however, the kmer and sequence content plots are often useful even when the former fails. i've used these in the past - you can sometimes just read off the adapter sequence from the start of the sequence content plot ( or at least see how many bases to trim ).


in the solution of nonlinear hyperbolic pdes, discontinuities ( " shocks " ) appear even when the initial condition is smooth. in the presence of discontinuities, the notion of solution can only be defined in the weak sense. the numerical velocity of a shock depends on the correct rankine - hugoniot conditions being imposed, which in turn depends on numerically satisfying the integral conservation law locally. the lax - wendroff theorem guarantees that a convergent numerical method will converge to a weak solution of the hyperbolic conservation law only if the method is conservative. not only do you need to use a conservative method, in fact you need to use a method that conserves the right quantities. there's a nice example that explains this in leveque's " finite volume methods for hyperbolic problems ", section 11. 12 and section 12. 9. if you discretize burgers'equation $ $ u _ t + 1 / 2 ( u ^ 2 ) _ x = 0 $ $ via the consistent discretization $ $ u ^ { n + 1 } _ i = u ^ n _ i - \ frac { \ delta t } { \ delta x } u ^ n _ i ( u ^ n _ i - u ^ n _ { i - 1 } ) $ $ you will observe that shocks move at the wrong speed, no matter how much you refine the grid. that is, the numerical solution will not converge to the true solution. if you instead use the conservative discretization $ $ u ^ { n + 1 } _ i = u ^ n _ i - \ frac { \ delta t } { 2 \ delta x } ( ( u ^ n _ i ) ^ 2 - ( u ^ n _ { i - 1 } ) ^ 2 ) $ $ based on flux - differencing, shocks will move at the correct speed ( which is the average of the states to the left and the right of the shock, for this equation ). this example is illustrated in this ipython notebook i wrote. for linear hyperbolic pdes, and for other types of pdes which typically have smooth solutions, local conservation is not a necessary ingredient for convergence. however, it may be important for other reasons ( e. g., if the total mass is a quantity of interest ).


a great summary of non - intuitive results in higher dimensions comes from " a few useful things to know about machine learning " by pedro domingos at the university of washington : [ o ] ur intuitions, which come from a three - dimensional world, often do not apply in high - dimensional ones. in high dimensions, most of the mass of a multivariate gaussian distribution is not near the mean, but in an increasingly distant “ shell ” around it ; and most of the volume of a high - dimensional orange is in the skin, not the pulp. if a constant number of examples is distributed uniformly in a high - dimensional hypercube, beyond some dimensionality most examples are closer to a face of the hypercube than to their nearest neighbor. and if we approximate a hypersphere by inscribing it in a hypercube, in high dimensions almost all the volume of the hypercube is outside the hypersphere. this is bad news for machine learning, where shapes of one type are often approximated by shapes of another. the article is also full of many additional pearls of wisdom for machine learning. another application, beyond machine learning, is nearest neighbor search : given an observation of interest, find its nearest neighbors ( in the sense that these are the points with the smallest distance from the query point ). but in high dimensions, a curious phenomenon arises : the ratio between the nearest and farthest points approaches 1, i. e. the points essentially become uniformly distant from each other. this phenomenon can be observed for wide variety of distance metrics, but it is more pronounced for the euclidean metric than, say, manhattan distance metric. the premise of nearest neighbor search is that " closer " points are more relevant than " farther " points, but if all points are essentially uniformly distant from each other, the distinction is meaningless. from charu c. aggarwal, alexander hinneburg, daniel a. keim, " on the surprising behavior of distance metrics in high dimensional space " : it has been argued in [ kevin beyer, jonathan goldstein, raghu ramakrishnan, uri shaft, " when is'nearest neighbor'meaningful? " ] that under certain reasonable assumptions on the data distribution, the ratio of the distances of the nearest and farthest neighbors to a given target in high dimensional space is almost 1 for a wide variety of data distributions and distance functions. in such a case, the nearest neighbor problem becomes ill defined, since the contrast between the distances to diferent data points does


not exist. in such cases, even the concept of proximity may not be meaningful from a qualitative perspective : a problem which is even more fundamental than the performance degradation of high dimensional algorithms.... many high - dimensional indexing structures and algorithms use the [ e ] uclidean distance metric as a natural extension of its traditional use in two - or three - dimensional spatial applications.... in this paper we provide some surprising theoretical and experimental results in analyzing the dependency of the $ l _ k $ norm on the value of $ k $. more specifically, we show that the relative contrasts of the distances to a query point depend heavily on the $ l _ k $ metric used. this provides considerable evidence that the meaningfulness of the $ l _ k $ norm worsens faster within increasing dimensionality for higher values of $ k $. thus, for a given problem with a fixed ( high ) value for the dimensionality $ d $, it may be preferable to use lower values of $ k $. this means that the $ l _ 1 $ distance metric ( manhattan distance metric ) is the most preferable for high dimensional applications, followed by the euclidean metric ( $ l _ 2 $ ).... the authors of the " surprising behavior " paper then propose using $ l _ k $ norms with $ k < 1 $. they produce some results which demonstrate that these " fractional norms " exhibit the property of increasing the contrast between farthest and nearest points. however, later research has concluded against fractional norms. see : " fractional norms and quasinorms do not help to overcome the curse of dimensionality " by mirkes, allohibi, & gorban ( 2020 ). ( thanks to michen00 for the comment and helpful citation. )


try out blosc. it is in many cases faster than memcopy. think about that for a second... wicked. it is super stable, highly - vetted, cross - platform, and performs like a champ.


i was quite amused when a student produced the following when cancelling a fraction : $ $ \ frac { x ^ 2 - y ^ 2 } { x - y } $ $ he began by " cancelling " the $ x $ and the $ y $ on top and bottom, to get : $ $ \ frac { x - y } { - } $ $ and then concluded that " two negatives make a positive ", so the final answer has to be $ x + y $.


first, i think it worthwhile considering'why would internal symmetry be beneficial?'developmental simplicity jumps to mind immediately. you can also consider relationship to external organs ; the stomach and esophagus are lined up with the mouth which is symmetrical about the sagittal plane. or maybe even balance ; the lungs are large organs and if put to one side would likely cause locomotive issues. ( perhaps this is even an interesting topic for another question. ) that said, i feel, at it's core the evolutionary advantage which led to the lack of ubiquitous internal symmetry is space. simply put, there is only so much room inside an organism and every little counts. thus, if there isn't a need for a particular organ to be mirrored about a plane then there is a benefit in putting elsewhere : utilization of space. i think a fantastic example of this is the human digestive tract. the key factor in the shape of the intestines is utilization of space, which directly affects the point at which is connects to the stomach, itself contributing to the asymmetrical shape of the stomach. one could envision other configurations, sure, and nature has. however, this configuration works quite well and the extraordinary use of limited space seems to outweigh all benefits of symmetry. to directly respond to your questions above : question : why do we not have an even number of each organ so it can be placed symmetrically? response : each organ addresses ( or addressed ) a need of the organism. addressing that need with multiple organs working in concert has benefits and consequences, as does addressing the need with a single organ alone. these benefits and consequences are balanced throughout the evolution of an organism. question : if we have a single organ why is it not placed in the middle like the brain or bladder is for instance? response : i feel space. again, there are benefits to symmetry but there are many other factors at play. some of which, it seems, are more important than symmetry at times. question : is there some evolutionary advantage that led to this setup? response : i hope this has been addressed - i don't claim to have'answered'anything, this is a question for discussion. other fuel for discussion : in thinking through this question i found myself able to rationalize why internal symmetry isn't necessary. however, i'd be interested in seeing opinions on why, then, external symmetry is so prevalent.


for logging that allows full reproducibility, i highly recommend the sumatra python package. it nicely links the version control commit number, machine state, and output files to each program run and has a django web interface to interact with the database of run info. the python api makes it very easy to include logging in my scripts.


this answer is general to processors and peripherals, and has an sram specific comment at the end, which is probably pertinent to your specific ram and cpu. output pins can be driven in three different modes : open drain - a transistor connects to low and nothing else open drain, with pull - up - a transistor connects to low, and a resistor connects to high push - pull - a transistor connects to high, and a transistor connects to low ( only one is operated at a time ) input pins can be a gate input with a : pull - up - a resistor connected to high pull - down - a resistor connected to low pull - up and pull - down - both a resistor connected to high and a resistor connected to low ( only useful in rare cases ). there is also a schmitt triggered input mode where the input pin is pulled with a weak pull - up to an initial state. when left alone it persists in its state, but may be pulled to a new state with minimal effort. open drain is useful when multiple gates or pins are connected together with an ( external or internal ) pull - up. if all the pin are high, they are all open circuits and the pull - up drives the pins high. if any pin is low they all go low as they tied together. this configuration effectively forms an and gate. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ note added november 2019 - 7 + years on : the configuration of combining multiple open collector / drain outputs has traditionally been referred to as a " wired or " configuration. calling it an or ( even traditionally ) does not make it one. if you use negative logic ( which traditionally may have been the case ) things will be different, but in the following i'll stick to positive logic convention which is what is used as of right unless specifically stated. the above comment about forming an'and'gate has been queried a number of times over the years - and it has been suggested that the result is'really'an'or'gate. it's complex. the simple picture'is that if several open collector outputs are connected together then if any one of the open collector transistors is turned on then the common output will be low. for the common output to be high all outputs must be off. if you consider combining 3 outputs - for the result to be high all


3 would need to have been high individually. 111 - > 1. that's an'and '. if you consider each of the output stages as an inverter then for each one to have a high output it's input must be low. so to get a combined high output you need three 000 - > 1. that's a'nor '. some have suggested that this is an or - any of xyz with at least 1 of these is a 1 - > 1. i can't really " force " that idea onto the situation. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ when driving an sram you probably want to drive either the data lines or the address lines high or low as solidly and rapidly as possible so that active up and down drive is needed, so push - pull is indicated. in some cases with multiple rams you may want to do something clever and combine lines, where another mode may be more suitable. with sram with data inputs from the sram if the ram ic is always asserting data then a pin with no pull - up is probably ok as the ram always sets the level and this minimises load. if the ram data lines are sometimes open circuit or tristate you will need the input pins to be able to set their own valid state. in very high speed communications you may want to use a pull - up and a pull - down so the parallel effective resistance is the terminating resistance, and the bus idle voltage is set by the two resistors, but this is somewhat specialist.


humans do not produce vitamin c due to a mutation in the gulo ( gulonolactone oxidase ) gene, which results in the inability to synthesize the protein. normal gulo is an enzyme that catalyses the reaction of d - glucuronolactone with oxygen to l - xylo - hex - 3 - gulonolactone. this then spontaneously forms ascorbic acid ( vitamin c ). however without the gulo enzyme, no vitamin c is produced. this has not been selected against in natural selection as we are able to consume more than enough vitamin c from our diet. it is also suggested that organisms without a functional gulo gene have a method of " recycling " the vitamin c that they obtain from their diets using red blood cells ( see montel - hagen et al. 2008 ). a 2008 published study ( li et al. 2008 ) claimed to have successfully re - instated the ability to produce vitamin c in mice. simply as trivia : other than humans ; guinea pigs, bats and dry - nosed primates have lost their ability to produce vitamin c in the same way. references li, y., shi, c. - x., mossman, k. l., rosenfeld, j., boo, y. c. & schellhorn, h. e. ( 2008 ) restoration of vitamin c synthesis in transgenic gulo - / - mice by helper - dependent adenovirus - based expression of gulonolactone oxidase. human gene therapy. [ online ] 19 ( 12 ), 1349 – 1358. available from : doi : 10. 1089 / hgt. 2008. 106 [ accessed : 31 december 2011 ]. montel - hagen, a., kinet, s., manel, n., mongellaz, c., prohaska, r., battini, j. - l., delaunay, j., sitbon, m. & taylor, n. ( 2008 ) erythrocyte glut1 triggers dehydroascorbic acid uptake in mammals unable to synthesize vitamin c. cell. [ online ] 132 ( 6 ), 1039 – 1048. available from : doi : 10. 1016 / j. cell. 2008. 01. 042 [ accessed : 31 december 2011 ].


when people say that kohn - sham orbitals bear no physical meaning, they mean it in the sense that nobody has proved mathematically that they mean anything. however, it has been empirically observed that many times, kohn - sham orbitals often do look very much like hartree - fock orbitals, which do have accepted physical interpretations in molecular orbital theory. in fact, the reference in the op lends evidence to precisely this latter viewpoint. to say that orbitals are " good " or " bad " is not really that meaningful in the first place. a basic fact that can be found in any electronic structure textbook is that in theories that use determinantal wavefunctions such as hartree - fock theory or kohn - sham dft, the occupied orbitals form an invariant subspace in that any ( unitary ) rotation can be applied to the collection of occupied orbitals while leaving the overall density matrix unchanged. since any observable you would care to construct is a functional of the density matrix in scf theories, this means that individuals orbitals themselves aren't physical observables, and therefore interpretations of any orbitals should always be undertaken with caution. even the premise of this question is not quite true. the energies of kohn - sham orbitals are known to correspond to ionization energies and electron affinities of the true electronic system due to janak's theorem, which is the dft analogue of koopmans'theorem. it would be exceedingly strange if the eigenvalues were meaningful while their corresponding eigenvectors were completely meaningless.


this problem reminds me of tension field theory and related problems in studying the shape of inflated inextensible membranes ( like helium balloons ). what follows is far from a solution, but some initial thoughts about the problem. first, since you're allowing creasing and folding, by nash - kuiper it's enough to consider short immersions $ $ \ phi : p \ subset \ mathbb { r } ^ 2 \ to \ mathbb { r } ^ 3, \ qquad \ | d \ phi ^ td \ phi \ | _ 2 \ leq 1 $ $ of the piece of paper $ p $ into $ \ mathbb { r } ^ 3 $, the intuition being that you can always " hide " area by adding wrinkling / corrugation, but cannot " create " area. it follows that we can assume, without loss of generality, that $ \ phi $ sends the paper boundary $ \ partial p $ to a curve $ \ gamma $ in the plane. we can thus partition your problem into two pieces : ( i ) given a fixed curve $ \ gamma $, what is the volume of the volume - maximizing surface $ m _ { \ gamma } $ with $ \ phi ( \ partial p ) = \ gamma $? ( ii ) can we characterize $ \ gamma $ for which $ m _ { \ gamma } $ has maximum volume? let's consider the case where $ \ gamma $ is given. we can partition $ m _ { \ gamma } $ into 1 ) regions of pure tension, where $ d \ phi ^ td \ phi = i $ ; in these regions $ m _ { \ gamma } $ is, by definition, developable ; 2 ) regions where one direction is in tension and one in compression, $ \ | d \ phi ^ td \ phi \ | _ 2 = 1 $ but $ \ det d \ phi ^ td \ phi < 1 $. we need not consider $ \ | d \ phi ^ td \ phi \ | _ 2 < 1 $ as in such regions of pure compression, one could increase the volume while keeping $ \ phi $ a short map. let us look at the regions of type ( 2 ). we can trace on these regions a family of curves $ \ tau $ along which $ \ phi $ is an isometry. since $ m _ { \ gamma } $ maximizes volume, we can imagine the situation physically as follows : pressure inside $ m _ { \ gamma }


$ pushes against the surface, and is exactly balanced by stress along inextensible fibers $ \ tau $. in other words, for some stress $ \ sigma $ constant along each $ \ tau $, at all points $ \ tau ( s ) $ along $ \ tau $ we have $ $ \ hat { n } = \ sigma \ tau'' ( s ) $ $ where $ \ hat { n } $ the surface normal ; it follows that ( 1 ) the $ \ tau $ follow geodesics on $ m _ { \ gamma } $, ( 2 ) each $ \ tau $ has constant curvature. the only thing i can say about problem ( ii ) is that for the optimal $ \ gamma $, the surface $ m _ \ gamma $ must meet the plane at a right angle. but there are many locally - optimal solutions that are not globally optimal ( for example, consider a half - cylinder ( type 1 region ) with two quarter - spherical caps ( type 2 region ) ; it has volume $ \ approx 1. 236 $ liters, less than joriki's solution ). i got curious so i implemented a quick - and - dirty tension field simulation that optimizes for $ \ gamma $ and $ m _ { \ gamma } $. source code is here ( needs the header - only eigen and libigl libraries ) : here is a rendering of the numerical solution, from above and below ( the volume is roughly 1. 56 liters ). edit 2 : a sketch of the orientation of $ \ tau $ on the surface :


i found many plausible claims that fingerprints increase friction. however, the following article claims, at least under their experimental conditions, that fingerprints actually decrease friction with smooth surfaces by reducing contact area. fingerprints are unlikely to increase the friction of primate fingerpads. it is generally assumed that fingerprints improve the grip of primates, but the efficiency of their ridging will depend on the type of frictional behaviour the skin exhibits. ridges would be effective at increasing friction for hard materials, but in a rubbery material they would reduce friction because they would reduce contact area. in this study we investigated the frictional performance of human fingertips on dry acrylic glass using a modified universal mechanical testing machine, measuring friction at a range of normal loads while also measuring the contact area. tests were carried out on different fingers, fingers at different angles and against different widths of acrylic sheet to separate the effects of normal force and contact area. the results showed that fingertips behaved more like rubbers than hard solids ; their coefficients of friction fell at higher normal forces and friction was higher when fingers were held flatter against wider sheets and hence when contact area was greater. the shear stress was greater at higher pressures, suggesting the presence of a biofilm between the skin and the surface. fingerprints reduced contact area by a factor of one - third compared with flat skin, however, which would have reduced the friction ; this casts severe doubt on their supposed frictional function. that said, the author does later discuss their potential role in gripping of rough or wet surfaces : so why do we have fingerprints? one possibility is that they increase friction on rougher surfaces compared with flat skin, because the ridges project into the depressions of such surfaces and provide a higher contact area. experiments on materials of contrasting known roughness are needed to test this possibility. a second possibility is that they facilitate runoff of water like the tread of a car tyre or grooves in the feet of tree frogs ( federle et al., 2006 ), so that they improve grip on wet surfaces. though there is evidence that friction falls on fingers coated with high levels of moisture ( andre et al., 2008 ) it is possible that it falls less quickly on fingertips than on flatter skin. once more, suitable experiments could test this idea. there seems to be more consensus on the idea that fingerprints are useful for tactile sensation. the following are just some articles which discuss this. effect of fingerprints orientation on skin vibrations during tactile exploration of textured surfaces. in humans, the tactile perception of


fine textures is mediated by skin vibrations when scanning the surface with the fingertip. these vibrations are encoded by specific mechanoreceptors, pacinian corpuscules ( pcs ), located about 2 mm below the skin surface. in a recent article, we performed experiments using a biomimetic sensor which suggest that fingerprints ( epidermal ridges ) may play an important role in shaping the subcutaneous stress vibrations in a way which facilitates their processing by the pc channel. here we further test this hypothesis by directly recording the modulations of the fingerpad / substrate friction force induced by scanning an actual fingertip across a textured surface. when the fingerprints are oriented perpendicular to the scanning direction, the spectrum of these modulations shows a pronounced maximum around the frequency v / λ, where v is the scanning velocity and λ the fingerprints period. this simple biomechanical result confirms the relevance of our previous finding for human touch. the role of fingerprints in the coding of tactile information probed with a biomimetic sensor. in humans, the tactile perception of fine textures ( spatial scale < 200 micrometers ) is mediated by skin vibrations generated as the finger scans the surface. to establish the relationship between texture characteristics and subcutaneous vibrations, a biomimetic tactile sensor has been designed whose dimensions match those of the fingertip. when the sensor surface is patterned with parallel ridges mimicking the fingerprints, the spectrum of vibrations elicited by randomly textured substrates is dominated by one frequency set by the ratio of the scanning speed to the interridge distance. for human touch, this frequency falls within the optimal range of sensitivity of pacinian afferents, which mediate the coding of fine textures. thus, fingerprints may perform spectral selection and amplification of tactile information that facilitate its processing by specific mechanoreceptors. this paper also asserts a reason for the elliptical nature of fingerprints : in humans, fingerprints are organized in elliptical twirls so that each region of the fingertip ( and thus each pc ) can be ascribed with an optimal scanning orientation.


the relation of " minimum " to " phase " in a minimum phase system or filter can be seen if you plot the unwrapped phase against frequency. you can use a pole zero diagram of the system response to help do a incremental graphical plot of the frequency response and phase angle. this method helps in doing a phase plot without phase wrapping discontinuities. put all the zeros inside the unit circle ( or in left half plane in the continuous - time case ), where all the poles have to be as well for system stability. add up the angles from all the poles, and the negative of the angles from all the zeros, to calculate total phase to a point on the unit circle, as that frequency response reference point moves around the unit circle. plot phase vs. frequency. now compare this plot with a similar plot for a pole - zero diagram with any of the zeros swapped outside the unit circle ( non - minimum phase ). the overall average slope of the line with all the zeros inside will be lower than the average slope of any other line representing the same lti system response ( e. g. with a zero reflected outside the unit circle ). this is because the " wind ups " in phase angle are all mostly cancelled by the " wind downs " in phase angle only when both the poles and zeros are on the same side of the unit circle line. otherwise, for each zero outside, there will be an extra " wind up " of increasing phase angle that will remain mostly uncancelled as the plot reference point " winds " around the unit circle from 0 to pi. (... or up the vertical axis in the continuous - time case. ) this arrangement, all the zeros inside the unit circle, thus corresponds to the minimum total increase in phase, which corresponds to minimum average total phase delay, which corresponds to maximum compactness in time, for any given ( stable ) set of poles and zeros with the exact same frequency magnitude response. thus the relationship between " minimum " and " phase " for this particular arrangement of poles and zeros. also see my old word picture with strange crank handles in the ancient usenet comp. dsp archives :


john kruschke released a book in mid 2011 called doing bayesian data analysis : a tutorial with r and bugs. ( a second edition was released in nov 2014 : doing bayesian data analysis, second edition : a tutorial with r, jags, and stan. ) it is truly introductory. if you want to walk from frequentist stats into bayes though, especially with multilevel modelling, i recommend gelman and hill. john kruschke also has a website for the book that has all the examples in the book in bugs and jags. his blog on bayesian statistics also links in with the book.


stereo imaging given the large field of view you need in relation to the accuracy you want, and how close you want to be, i think that stereo imaging may be a challenging, so you need to somehow amplify the differences you are trying to measure. structured lighting if you are essentially trying to measure the profile of an object, have you considered a single high resolution camera and structured lighting? thanks to looptechnology for this image, used without permission, but hopefully attribution will be enough. note, the shallower the grazing angle, the the greater accuracy you can measure, but the lower the supported depth of field would be, so for your application you would need to optimise for your needs or make your system adjustable ( one laser angle for 0 - 500um, another for 500 - 1500um and so on ). in this case though, you would probably have to calibrate each time you changed laser position. incidentally, a very cheap way to try this out would be to pick up a pair of laser scissors which include a basic line laser led. finally, you can remove the vibration problem by sampling multiple times, rejecting outliers and then averaging. a better solution though would be to mount the whole test apparatus on a block of granite. this worked well for laser micro - machining tools i've worked with in the past, which require micron level position and depth of focus accuracy, even when located in factories. some back of the envelope calculations. lets assume an incident angle of 10 degree from horizontal, and a camera with a 640x480 resolution and a field of view of 87 x 65mm. if we place the beam so that it is right at the bottom of the portrait frame with no sample, and then place the sample with the beam crossing it, this should give us a maximum height of around 15mm and thus an uncorrected resolution of around 24um for each pixel the line walks up the screen. with this setup, a 0. 1mm variation should be visible as a 4 pixel variation in position. similarly, if we use an incident angle of 2 degrees from horizontal then this should give us a maximum height of around 3mm ( tan ( 2deg ) * 87mm ) and thus an uncorrected resolution of around 4. 7um per pixel, for a much more noticeable 20 pixel jump. this would probably require a much more accurate line laser however. note, if the camera is close enough then the you may need to do a second trig calculation


, using the camera height, to determine the the true position of the line relative to the base line. also note that if you don't need absolute accuracy, and local repeatability is enough ( say you are profiling the flatness of a sample to ensure it is within given tolerances ) then just being able to see the relative position of the laser line might be enough.


qualimap will do this for you. go to qualimap. bioinfo. cipf. es run qualimap ( default params are fine ) on each bam file open up the html output, and you can read off the % identity ( they measure the opposite, i. e. mismatch rate, but 100 % - mismatch rate is % identity of course ), indel rate, etc. one thing to watch out for ( you don't mention it in your question, but just in case ) is that you cannot directly compare q scores - these are a bit of a mess and calculated very differently in each piece of software. unsolicited suggestion : you might also try ngm - lr for mapping minion data. we've found it beats the others for our data ( though we map to a distant reference ).


there are good books on this such as gelman and hill. what follows is essentially a summary of their perspective. first of all, you should not get too caught up in the terminology. in statistics, jargon should never be used as a substitute for a mathematical understanding of the models themselves. that is especially true for random and mixed effects models. " mixed " just means the model has both fixed and random effects, so let's focus on the difference between fixed and random. random versus fixed effects let's say you have a model with a categorical predictor, which divides your observations into groups according to the category values. * the model coefficients, or " effects ", associated to that predictor can be either fixed or random. the most important practical difference between the two is this : random effects are estimated with partial pooling, while fixed effects are not. partial pooling means that, if you have few data points in a group, the group's effect estimate will be based partially on the more abundant data from other groups. this can be a nice compromise between estimating an effect by completely pooling all groups, which masks group - level variation, and estimating an effect for all groups completely separately, which could give poor estimates for low - sample groups. random effects are simply the extension of the partial pooling technique as a general - purpose statistical model. this enables principled application of the idea to a wide variety of situations, including multiple predictors, mixed continuous and categorical variables, and complex correlation structures. ( but with great power comes great responsibility : the complexity of modeling and inference is substantially increased, and can give rise to subtle biases that require considerable sophistication to avoid. ) to motivate the random effects model, ask yourself : why would you partial pool? probably because you think the little subgroups are part of some bigger group with a common mean effect. the subgroup means can deviate a bit from the big group mean, but not by an arbitrary amount. to formalize that idea, we posit that the deviations follow a distribution, typically gaussian. that's where the " random " in random effects comes in : we're assuming the deviations of subgroups from a parent follow the distribution of a random variable. once you have this idea in mind, the mixed - effects model equations follow naturally. unfortunately, users of mixed effect models often have false preconceptions about what random effects are and how they differ from fixed effects. people hear " random


" and think it means something very special about the system being modeled, like fixed effects have to be used when something is " fixed " while random effects have to be used when something is " randomly sampled ". but there's nothing particularly random about assuming that model coefficients come from a distribution ; it's just a soft constraint, similar to the $ \ ell _ 2 $ penalty applied to model coefficients in ridge regression. there are many situations when you might or might not want to use random effects, and they don't necessarily have much to do with the distinction between " fixed " and " random " quantities. unfortunately, the concept confusion caused by these terms has led to a profusion of conflicting definitions. of the five definitions at this link, only # 4 is completely correct in the general case, but it's also completely uninformative. you have to read entire papers and books ( or failing that, this post ) to understand what that definition implies in practical work. example let's look at a case where random effects modeling might be useful. suppose you want to estimate average us household income by zip code. you have a large dataset containing observations of households'incomes and zip codes. some zip codes are well represented in the dataset, but others have only a couple households. for your initial model you would most likely take the mean income in each zip. this will work well when you have lots of data for a zip, but the estimates for your poorly sampled zips will suffer from high variance. you can mitigate this by using a shrinkage estimator ( aka partial pooling ), which will push extreme values towards the mean income across all zip codes. but how much shrinkage / pooling should you do for a particular zip? intuitively, it should depend on the following : how many observations you have in that zip how many observations you have overall the individual - level mean and variance of household income across all zip codes the group - level variance in mean household income across all zip codes if you model zip code as a random effect, the mean income estimate in all zip codes will be subjected to a statistically well - founded shrinkage, taking into account all the factors above. the best part is that random and mixed effects models automatically handle ( 4 ), the variability estimation, for all random effects in the model. this is harder than it seems at first glance : you could try the variance of the sample mean for each zip, but this will be biased high, because some of the variance


between estimates for different zips is just sampling variance. in a random effects model, the inference process accounts for sampling variance and shrinks the variance estimate accordingly. having accounted for ( 1 ) - ( 4 ), a random / mixed effects model is able to determine the appropriate shrinkage for low - sample groups. it can also handle much more complicated models with many different predictors. relationship to hierarchical bayesian modeling if this sounds like hierarchical bayesian modeling to you, you're right - it is a close relative but not identical. mixed effects models are hierarchical in that they posit distributions for latent, unobserved parameters, but they are typically not fully bayesian because the top - level hyperparameters will not be given proper priors. for example, in the above example we would most likely treat the mean income in a given zip as a sample from a normal distribution, with unknown mean and sigma to be estimated by the mixed - effects fitting process. however, a ( non - bayesian ) mixed effects model will typically not have a prior on the unknown mean and sigma, so it's not fully bayesian. that said, with a decent - sized data set, the standard mixed effects model and the fully bayesian variant will often give very similar results. * while many treatments of this topic focus on a narrow definition of " group ", the concept is in fact very flexible : it is just a set of observations that share a common property. a group could be composed of multiple observations of a single person, or multiple people in a school, or multiple schools in a district, or multiple varieties of a single kind of fruit, or multiple kinds of vegetable from the same harvest, or multiple harvests of the same kind of vegetable, etc. any categorical variable can be used as a grouping variable.


i'll try to summarize my experiences obtained in the course of developing viennacl, where we have cuda and opencl backends with mostly 1 : 1 translations of a lot of compute kernels. from your question i'll also assume that we are mostly taking about gpus here. performance portability. first of all, there is no such thing as performance - portable kernels in the sense that you write a kernel once and it will run efficiently on every hardware. not in opencl, where it is more apparent due to the broader range of hardware supported, but also not in cuda. in cuda it is less apparent because of the smaller range of hardware supported, but even here we have to distinguish at least three hardware architectures ( pre - fermi, fermi, kepler ) already. these performance fluctuations can easily result in a 20 percent performance variation depending on how you orchestrate threads and which work group sizes you choose, even if the kernel is as simple as a buffer copy. it's probably also worth mentioning that on pre - fermi and fermi gpus it was possible to write fast matrix - matrix multiplication kernels directly in cuda, while for the latest kepler gpus it seems that one has to go down to the ptx pseudo - assembly language in order to get close to cublas'performance. thus, even a vendor - controlled language such as cuda appears to have issues to keep the pace with hardware developments. moreover, all cuda code gets compiled statically when you run nvcc, which somewhat requires a balancing act via the - arch flag, while opencl kernels get compiled at run - time from the just - in - time compiler, so you can in principle tailor kernels down to the very specifics of a particular compute device. the latter is, however, quite involved and usually only becomes a very attractive option as your code matures and as your experience accumulates. the price to pay is the o ( 1 ) time required for just - in - time compilation, which can be an issue in certain situations. opencl 2. 0 has some great improvements to address this. debugging and profiling. the cuda debugging and profiling tools are the best available for gpgpu. amd's tools are not bad either, but they do not include gems like cuda - gdb or cuda - memcheck. also, still today nvidia provides the most robust drivers and sdks for gp


##gpu, system freezes due to buggy kernels are really the exception, not the rule, both with opencl and cuda. for reasons i probably do not need to explain here, nvidia no longer offers debugging and profiling for opencl with cuda 5. 0 and above. accessibility and convenience. it is a lot easier to get the first cuda codes up and running, particularly since cuda code integrates rather nicely with host code. ( i'll discuss the price to pay later. ) there are plenty of tutorials out there on the web as well as optimization guides and some libraries. with opencl you have to go through quite a bit of initialization code and write your kernels in strings, so you only find compilation errors during execution when feeding the sources to the jit - compiler. thus, it takes longer to go through one code / compile / debug cycle with opencl, so your productivity is usually lower during this initial development stage. software library aspects. while the previous items were in favor of cuda, the integration into other software is a big plus for opencl. you can use opencl by just linking with the shared opencl library and that's it, while with cuda you are required to have the whole cuda toolchain available. even worse, you need to use the correct host compilers for nvcc to work. if you ever tried to use e. g. cuda 4. 2 with gcc 4. 6 or newer, you'll have a hard time getting things to work. generally, if you happen to have any compiler in use which is newer than the cuda sdk, troubles are likely to occur. integration into build systems like cmake is another source of headache ( you can also find ample of evidence on e. g. the petsc mailinglists ). this may not be an issue on your own machine where you have full control, but as soon as you distribute your code you will run into situations where users are somewhat restricted in their software stack. in other words, with cuda you are no longer free to choose your favourite host compiler, but nvidia dictates which compilers you are allowed to use. other aspects. cuda is a little closer to hardware ( e. g. warps ), but my experience with linear algebra is that you rarely get a significant benefit from it. there are a few more software libraries out there for cuda, but more


and more libraries use multiple compute backends. viennacl, vexcl, or paralution all support opencl and cuda backends in the meanwhile, a similar trend can be seen with libraries in other areas. gpgpu is not a silver bullet. gpgpu has been shown to provide good performance for structured operations and compute - limited tasks. however, for algorithms with a non - negligible share of sequential processing, gpgpu cannot magically overcome amdahl's law. in such situations you are better off using a good cpu implementation of the best algorithm available rather than trying to throw a parallel, but less suitable algorithm at your problem. also, pci - express is a serious bottleneck, so you need to check in advance whether the savings from gpus can compensate the overhead of moving data back and forth. my recommendation. please consider cuda and opencl rather than cuda or opencl. there is no need to unnecessarily restrict yourself to one platform, but instead take the best out of both worlds. what works well for me is to set up an initial implementation in cuda, debug it, profile it, and then port it over to opencl by simple string substitutions. ( you may even parametrize your opencl kernel string generation routines such that you have some flexibility in tuning to the target hardware. ) this porting effort will usually consume less than 10 percent of your time, but gives you the ability to run on other hardware as well. you may be surprised about how well non - nvidia hardware can perform in certain situations. most of all, consider the reuse of functionality in libraries to the largest extent possible. while a quick & dirty reimplementation of some functionality often works acceptable for single - threaded execution on a cpu, it will often give you poor performance on massively parallel hardware. ideally you can even offload everything to libraries and don't ever have to care about whether they use cuda, opencl, or both internally. personally i would never dare to write vendor - locked code for something i want to rely on in several years from now, but this ideological aspect is should go into a separate discussion.


biology is rarely black or white, all or nothing. protective immunity is generally not an on / off switch, where from the moment you're vaccinated you're infinitely resistant for the rest of your life. you shouldn't expect that, having received a smallpox vaccine, you could have billions of smallpox viruses squirted directly into your lungs and shrug it off without noticing. given that ( fairly obvious ) fact, you should immediately think of scenarios where vaccinated people are still at risk of disease following exposure to unvaccinated people. what about older people who were vaccinated 20 years ago, 50 years ago? what about people whose immune systems are slightly weakened through lack of sleep or obesity or stress? any of these vaccinated people might well be protected against a brief encounter, but not against, say, being in an airplane seat for 18 hours beside an infected child shedding huge amounts of virus, or caring for their sick child. it's all sliders, not switches. you can have a slight loss of immunity ( 4 hours sleep last night ) and be protected against everything except a large exposure ( your baby got infected and won't rest unless you hold him for 8 hours ). you can have a moderate loss of immunity ( you were vaccinated twenty years ago ) and be protected against most exposures, but you're sitting next to someone on the subway for an hour. you may have a significant loss of immunity ( you're a frail 80 - year - old ) and still be protected against a moderate exposure, but your grandchild is visiting for a week.


water, as simple as it might appear, has quite a few extraordinary things to offer. most does not seem to be as it appears. before diving deeper, a few cautionary words about hybridisation. hybridisation is an often misconceived concept. it only is a mathematical interpretation, which explains a certain bonding situation ( in an intuitive fashion ). in a molecule the equilibrium geometry will result from various factors, such as steric and electronic interactions, and furthermore interactions with the surroundings like a solvent or external field. the geometric arrangement will not be formed because a molecule is hybridised in a certain way, it is the other way around, i. e. a result of the geometry or more precise and interpretation of the wave function for the given molecular arrangement. in molecular orbital theory linear combinations of all available ( atomic ) orbitals will form molecular orbitals ( mo ). these are spread over the whole molecule, or delocalised, and in a quantum chemical interpretation they are called canonical orbitals. such a solution ( approximation ) of the wave function can be unitary transformed form localised molecular orbitals ( lmo ). the solution ( the energy ) does not change due to this transformation. these can then be used to interpret a bonding situation in a simpler theory. each lmo can be expressed as a linear combination of the atomic orbitals, hence it is possible to determine the coefficients of the atomic orbitals and describe these also as hybrid orbitals. it is absolutely wrong to assume that there are only three types of spx hybrid orbitals. therefore it is very well possible, that there are multiple different types of orbitals involved in bonding for a certain atom. for more on this, read about bent's rule on the network. [ 1 ] let's look at water, wikipedia is so kind to provide us with a schematic drawing : the bonding angle is quite close to the ideal tetrahedral angle, so one would assume, that the involved orbitals are sp3 hybridised. there is also a connection between bond angle and hybridisation, called coulson's theorem, which lets you approximate hybridisation. [ 2 ] in this case the orbitals involved in the bonds would be sp4 hybridised. ( close enough. ) let us also consider the symmetry of the molecule. the point group of water is c2v. because there are mirror planes, in the canonical bonding picture π - type orbitals [ 3 ] are necessary. we have an orbital with appropriate symmetry, which is


the p - orbital sticking out of the bonding plane. this interpretation is not only valid it is one that comes as the solution of the schrodinger equation. [ 4 ] that leaves for the other orbital a hybridisation of sp ( 2 / 3 ). if we make the reasonable assumption, that the oxygen hydrogen bonds are sp3 hybridised, and the out - of - plane lone pair is a p orbital, then the maths is a bit easier and the in - plane lone pair is sp hybridised. [ 5 ] a calculation on the mo6 / def2 - qzvpp level of theory gives us the following canonical molecular orbitals : ( orbital symmetries : $ 2 \ mathrm { a } _ 1 $, $ 1 \ mathrm { b } _ 2 $, $ 3 \ mathrm { a } _ 1 $, $ 1 \ mathrm { b } _ 1 $ ) [ 6, 7 ] since the interpretation with hybrid orbitals is equivalent, i used the natural bond orbital theory to interpret the results. this method transforms the canonical orbitals into localised orbitals for easier interpretation. here is an excerpt of the output ( core orbital and polarisation functions omitted ) giving us the calculated hybridisations : ( occupancy ) bond orbital / coefficients / hybrids - - - - - - - - - - - - - - - - - - lewis - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 2. ( 1. 99797 ) lp ( 1 ) o 1 s ( 53. 05 % ) p 0. 88 ( 46. 76 % ) d 0. 00 ( 0. 19 % ) 3. ( 1. 99770 ) lp ( 2 ) o 1 s ( 0. 00 % ) p 1. 00 ( 99. 69 % ) d 0. 00 ( 0. 28 % ) 4. ( 1. 99953 ) bd ( 1 ) o 1 - h 2 ( 73. 49 % ) 0. 8573 * o 1 s ( 23. 41 % ) p 3. 26 ( 76. 25 % ) d 0. 01 ( 0. 31 % ) ( 26. 51 % ) 0. 5149 * h 2 s ( 99. 65 % ) p 0. 00 ( 0. 32


% ) d 0. 00 ( 0. 02 % ) 5. ( 1. 99955 ) bd ( 1 ) o 1 - h 3 ( 73. 48 % ) 0. 8572 * o 1 s ( 23. 41 % ) p 3. 26 ( 76. 27 % ) d 0. 01 ( 0. 30 % ) ( 26. 52 % ) 0. 5150 * h 3 s ( 99. 65 % ) p 0. 00 ( 0. 32 % ) d 0. 00 ( 0. 02 % ) - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - as we can see, that pretty much matches the assumption of sp3 oxygen hydrogen bonds, a p lone pair, and a sp lone pair. does that mean that the lone pairs are non - equivalent? well, that is at least one interpretation. and we only deduced all that from a gas phase point of view. when we go towards condensed phase, things will certainly change. hydrogen bonds will break the symmetry, dynamics will play an important role and in the end, both will probably behave quite similarly or even identical. now let's get to the juicy part : second, if so, does this have any significance in actual physical systems ( i. e. is it a measurable phenomenon ), and what is the approximate energy difference between the pairs of electrons? well the first part is a bit tricky to answer, because that is dependent on a lot more conditions. but the part in parentheses is easy. it is measurable with photoelectron spectroscopy. there is a nice orbital scheme correlated to the orbital ionisation potential on the homepage of michael k. denk for water. [ 8 ] unfortunately i cannot find license information, or a reference to reproduce, hence i am hesitant to post it here. however, i found a nice little publication on the photoelectron spectroscopy of water in the bonding region. [ 9 ] i'll quote some relevant data from the article. $ \ ce { h2o } $ is a non - linear, triatomic molecule consisting of an oxygen atom covalently bonded to two hydrogen atoms. the ground state of the $ \


ce { h2o } $ molecule is classified as belonging to the $ c _ \ mathrm { 2v } $ point group and so the electronic states of water are described using the irreducible representations $ \ mathrm { a } _ 1 $, $ \ mathrm { a } _ 2 $, $ \ mathrm { b } _ 1 $, $ \ mathrm { b } _ 2 $. the electronic configuration of the ground state of the $ \ ce { h2o } $ molecule is described by five doubly occupied molecular orbitals : $ $ \ begin { align } \ underbrace { ( 1 \ mathrm { a } _ 1 ) ^ 2 } _ { \ text { core } } & & \ underbrace { ( 2 \ mathrm { a } _ 1 ) ^ 2 } _ { \ text { inner - valence orbital } } & & \ underbrace { ( 1 \ mathrm { b } _ 2 ) ^ 2 ( 3 \ mathrm { a } _ 1 ) ^ 2 ( 1 \ mathrm { b } _ 1 ) ^ 2 } _ { \ text { outer - valence orbital } } & & \ mathrm { x ~ ^ 1a _ 1 } \ end { align } $ $ [.. ] in addition to the three band systems observed in hei pes of $ \ ce { h2o } $, a fourth band system in the tpe spectrum close to 32 ev is also observed. as indicated in fig. 1, these band systems correspond to the removal of a valence electron from each of the molecular orbitals $ ( 1 \ mathrm { b } _ 1 ) ^ { - 1 } $, $ ( 3 \ mathrm { a } _ 1 ) ^ { - 1 } $, $ ( 1 \ mathrm { b } _ 2 ) ^ { - 1 } $ and $ ( 2 \ mathrm { a } _ 1 ) ^ { - 1 } $ of $ \ ce { h2o } $. as you can see, it fits quite nicely with the calculated data. from the image i would say that the difference between $ ( 1 \ mathrm { b } _ 1 ) ^ { - 1 } $ and $ ( 3 \ mathrm { a } _ 1 ) ^ { - 1 } $ is about 1 - 2 ev. tl ; dr as you see your hunch paid off quite well. photoelectron spectroscopy of water in the


gas phase confirms that the lone pairs are non - equivalent. conclusions for condensed phases might be different, but that is a story for another day. notes and references what is bent's rule? utility of bent's rule - what can bent's rule explain that other qualitative considerations cannot? formal theory of bent's rule, derivation of coulson's theorem ( wikipedia ). worked example for cyclo propane by ron. a π orbital has one nodal plane collinear with the bonding axis, it is asymmetric with respect to this plane. a bit more explanation in my question what would follow in the series sigma, pi and delta bonds? with in the approximation that molecular orbitals are a linear combination of atomic orbitals ( mo = lcao ). the terminology we use for hybridisation actually is just an abbreviation : $ $ \ mathrm { sp } ^ { x } = \ mathrm { s } ^ { \ frac { 1 } { x + 1 } } \ mathrm { p } ^ { \ frac { x } { x + 1 } } $ $ in theory $ x $ can have any value ; since it is just a unitary transformation the representation does not change, hence \ begin { align } 1 \ times \ mathrm { s }, 3 \ times \ mathrm { p } & \ leadsto 4 \ times \ mathrm { sp } ^ 3 \ \ & \ leadsto 3 \ times \ mathrm { sp } ^ 2, 1 \ times \ mathrm { p } \ \ & \ leadsto 2 \ times \ mathrm { sp }, 2 \ times \ mathrm { p } \ \ & \ leadsto 2 \ times \ mathrm { sp } ^ 3, 1 \ times \ mathrm { sp }, 1 \ times \ mathrm { p } \ \ & \ leadsto \ text { etc. pp. } \ \ & \ leadsto 2 \ times \ mathrm { sp } ^ 4, 1 \ times \ mathrm { p }, 1 \ times \ mathrm { sp } ^ { ( 2 / 3 ) } \ end { align } there are virtually infinite possibilities of combination. this and the next footnote address a couple of points that were raised in a comment by davephd. while i already extensively answered that there, i want to include a few more clarifying points here. ( if i do it right, the comments become obsolete. ) what is the reason


for concluding 2 lone pairs versus 1 or 3? for example mulliken has in table v the b1 orbital being a definite lone pair ( no h population ) but the two a1 orbitals both have about 0. 3e population on h. would it be wrong to say only one of the pes energy levels corresponds to a lone pair, and the other 3 has some significant population on hydrogen? are mulliken's calculations still valid? – davephd the article dave refers to is r. s. mulliken, j. chem. phys. 1955, 23, 1833., which introduces mulliken population analysis. in this paper mulliken analyses wave functions on the scf - lcao - mo level of theory. this is essentially hartree fock with a minimal basis set. ( i will address this in the next footnote. ) we have to understand that this was state - of - the - art computational chemistry back then. what we take for granted nowadays, calculating the same thing in a few seconds, was revolutionary back then. today we have a lot fancier methods. i used density functional theory with a very large basis set. the main difference between these approaches is that the level i use recovers a lot more of electron correlation than the method of mulliken. however, if you look closely at the results it is quite impressive how well these early approximations perform. on the m06 / def2 - qzvpp level of theory the geometry of the molecule is optimised to have an oxygen hydrogen distance of 95. 61 pm and a bond angle of 105. 003°. this is quite close to the experimental results. the contribution to the orbitals are given as follows. i include the orbital energies ( oe ), too. the contributions of the atomic orbitals are given to 1. 00 being the total for each molecular orbital. because the basis set has polarisation functions the missing parts are attributed to this. the threshold for printing is 3 %. ( i also rearranged the gaussian output for better readability. ) atomic contributions to molecular orbitals : 2 : 2a1 oe = - 1. 039 is o1 - s = 0. 81 o1 - p = 0. 03 h2 - s = 0. 07 h3 - s = 0. 07 3 : 1b2 oe = - 0. 547 is o1 - p = 0. 63 h2 - s = 0. 18 h3


- s = 0. 18 4 : 3a1 oe = - 0. 406 is o1 - s = 0. 12 o1 - p = 0. 74 h2 - s = 0. 06 h3 - s = 0. 06 5 : 1b1 oe = - 0. 332 is o1 - p = 0. 95 we can see that there is indeed some contribution by the hydrogens to the in - plane lone pair of oxygen. on the other hand we see that there is only one orbital where there is a large contribution by hydrogen. one could here easily come up with the theory of one or three lone pairs of oxygen, depending on your own point of view. mulliken's analysis is based on the canonical orbitals, which are delocalised, so we will never have a pure lone pair orbital. when we refer to orbitals as being of a certain type, then we imply that this is the largest contribution. often we also use visual aides like pictures of these orbitals to decide if they are of bonding or anti - bonding nature, or if their contribution is on the bonding axis. all these analyses are highly biased by your point of view. there is no right or wrong when it comes to separation schemes. there is no hard evidence for any of these obtainable. these are mathematical interpretations that do in the best case help us understand bonding better. thus deciding whether water has one, two or three ( or even four ) lone pairs is somewhat playing with numbers until something seems to fit. bonding is too difficult to transform it in easy pictures. ( that's why i am not an advocate for cautiously using lewis structures. ) the nbo analysis is another separation scheme. one that aims to transform the obtained canonical orbitals into a lewis like picture for a better understanding. this transformation does not change the wave function and in this way is as equally a representation as other approaches. what you loose by this approach are the orbital energies, since you break the symmetry of the wave function, but this is going much too far to explain. in a nutshell, the localisation scheme aims to transform the delocalised orbitals into orbitals that correspond to bonds. from a quite general point of view, mulliken's calculations ( he actually only interpreted the results of others ) and conclusion hold up to a certain point. nowadays we know that his population analysis has severe problems, but within the minimal basis they still produce justifiable results. the popularity of this method comes mainly