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. it was a saturday, so the radiation safety technician on call didn't arrive for half an hour — at which point i was clean, so the detective questions began. i had spent the day sitting on a plastic step stool. the tech looked at it, said that radon's decay products are concentrated by static electricity, and told me that i needed to get a real chair.
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how long would it take for this super - material to convert to the stuff i scribble with? no, despite the fact that james bond said " diamonds are forever ", that is not exactly the case. although bond's statement is a fair approximation of reality it is not a scientifically accurate description of reality. as we will soon see, even though diamond is slightly less stable than graphite ( by ~ 2. 5 kj / mol ), it is kinetically protected by a large activation energy. here is a comparative representation of the structures of diamond and graphite. ( image source : satyanarayana t, rai r. nanotechnology : the future. j interdiscip dentistry 2011 ; 1 : 93 - 100 ) ( image source ) note that diamond is composed of cyclohexane rings and each carbon is bonded to 2 more carbons external to the cyclohexane ring. on the other hand, graphite is comprised of benzene rings and each carbon is bonded to only 1 carbon external to the benzene ring. that means we need to break 6 sigma bonds in diamond and make about 2 pi bonds ( remember it's an extended array of rings, don't double count ) in graphite per 6 - membered ring in order to convert diamond to graphite. a typical aliphatic c – c bond strength is ~ 340 kj / mol and a typical pi bond strength is ~ 260 kj / mol. so to break 6 sigma bonds and make 2 pi bonds would require ~ ( ( 6 * 340 ) - ( 2 * 260 ) ) ~ 1500 kj / mol. if the transition state were exactly midway between diamond and carbon ( with roughly equal bond breaking and bond making ), then we might approximate the activation energy as being half that value or ~ 750 kj / mol. since graphite is a bit more stable than diamond, we can refine our model and realize that the transition state will occur a bit before the mid - point. so our refined model would suggest an activation energy something less than 750 kj / mol. had we attempted to incorporate the effect of aromaticity in graphite our estimate would be even lower. in any case, this is an extremely large activation energy, so, as we anticipated the reaction would be very slow. an estimate ( see p. 171 ) of the activation energy puts the reverse reaction ( graphite to diamond ; but since, as noted above, the energy difference between the two
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is very small the activation energy for the forward reaction is almost the same ) at 367 kj / mol. so at least our rough approximation was in the right ballpark, off by about a factor of 2. however, it appears that the transition state is even further from the midpoint ( closer to starting material ) than we might have guessed. this activation energy tells us that at 25 °c, it would take well over a billion years to convert one cubic centimeter of diamond to graphite. note 04 / 17 / 20 : as mentioned in a comment, the original " estimate " link became defunct and was replaced today with a new estimate link. however the original article and estimate can can still be seen on the wayback machine and it estimates the activation energy to be 538. 45 kj / mol, reasonably close to our estimate.
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in general, all krylov methods essentially seek a polynomial that is small when evaluated on the spectrum of the matrix. in particular, the $ n $ th residual of a krylov method ( with zero initial guess ) can be written in the form $ $ r _ n = p _ n ( a ) b $ $ where $ p _ n $ is some monic polynomial of degree $ n $. if $ a $ is diagonalizable, with $ a = v \ lambda v ^ { - 1 } $, we have \ begin { eqnarray * } \ | r _ n \ | & \ leq & \ | v \ | \ cdot \ | p _ n ( \ lambda ) \ | \ cdot \ | v ^ { - 1 } \ | \ cdot \ | b \ | \ \ & = & \ kappa ( v ) \ cdot \ | p _ n ( \ lambda ) \ | \ cdot \ | b \ |. \ end { eqnarray * } in the event that $ a $ is normal ( e. g., symmetric or unitary ) we know that $ \ kappa ( v ) = 1. $ gmres constructs such a polynomial through arnoldi iteration, while cg constructs the polynomial using a different inner product ( see this answer for details ). similarly, bicg constructs its polynomial through the nonsymmetric lanczos process, while chebyshev iteration uses prior information on the spectrum ( usually estimates of the largest and smallest eigenvalues for symmetric definite matrices ). as a cool example ( motivated by trefethen + bau ), consider a matrix whose spectrum is this : in matlab, i constructed this with : a = rand ( 200, 200 ) ; [ q r ] = qr ( a ) ; a = ( 1 / 2 ) * q + eye ( 200, 200 ) ; if we consider gmres, which constructs polynomials which actually minimize the residual over all monic polynomials of degree $ n $, we can easily predict the residual history by looking at the candidate polynomial $ $ p _ n ( z ) = ( 1 - z ) ^ n $ $ which in our case gives $ $ | p _ n ( z ) | = \ frac { 1 } { 2 ^ n } $ $ for $ z $ in the spectrum of $ a $. now, if we run gmres on a random rhs and compare the residual history with
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this polynomial, they ought to be quite similar ( the candidate polynomial values are smaller than the gmres residual because $ \ | b \ | _ 2 > 1 $ ) :
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it's undecidable because a law book can include arbitrary logic. a silly example censorship law would be " it is illegal to publicize any computer program that does not halt ". the reason results for mtg exist and are interesting is because it has a single fixed set of ( mostly ) unambiguous rules, unlike law which is ever changing, horribly localized and endlessly ambiguous.
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yes, if you can come up with any of the following : deterministic finite automaton ( dfa ), nondeterministic finite automaton ( nfa ), regular expression ( regexp of formal languages ) or regular grammar for some language $ l $, then $ l $ is regular. there are more equivalent models, but the above are the most common. there are also useful properties outside of the " computational " world. $ l $ is also regular if it is finite, you can construct it by performing certain operations on regular languages, and those operations are closed for regular languages, such as intersection, complement, homomorphism, reversal, left - or right - quotient, regular transduction and more, or using myhill – nerode theorem if the number of equivalence classes for $ l $ is finite. in the given example, we have some ( regular ) langage $ l $ as basis and want to say something about a language $ l'$ derived from it. following the first approach - - construct a suitable model for $ l'$ - - we can assume whichever equivalent model for $ l $ we so desire ; it will remain abstract, of course, since $ l $ is unknown. in the second approach, we can use $ l $ directly and apply closure properties to it in order to arrive at a description for $ l'$.
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sure, you can combine prngs like this, if you want, assuming they are seeded independently. however, it will be slower and it probably won't solve the most pressing problems that people have. in practice, if you have a requirement for a very high - quality prng, you use a well - vetted cryptographic - strength prng and you seed it with true entropy. if you do this, your most likely failure mode is not a problem with the prng algorithm itself ; the most likely failure mode is lack of adequate entropy ( or maybe implementation errors ). xor - ing multiple prngs doesn't help with this failure mode. so, if you want a very high - quality prng, there's probably little point in xor - ing them. alternatively, if you want a statistical prng that's good enough for simulation purposes, typically the # 1 concern is either speed ( generate pseudorandom numbers really fast ) or simplicity ( don't want to spend much development time on researching or implementing it ). xor - ing slows down the prng and makes it more complex, so it doesn't address the primary needs in that context, either. as long as you exhibit reasonable care and competence, standard prngs are more than good enough, so there's really no reason why we need anything fancier ( no need for xor - ing ). if you don't have even minimal levels of care or competence, you're probably not going to choose something complex like xor - ing, and the best way to improve things is to focus on more care and competence in the selection of the prng rather than on xor - ing. bottom line : basically, the xor trick doesn't solve the problems people usually actually have when using prngs.
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as suggested, here ’ s an example showing the relevant lines from a description file from a cran / github hosted project that has bioconductor dependencies ( truncated ) : depends : r ( > = 3. 3. 0 ) biocviews : imports : methods, snpstats, dplyr the relevant bit is the empty biocviews : declaration, which allows the bioconductor dependency { snpstats } to be automatically installed.
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i'm not aware of any recent overview articles, but i am actively involved in the development of the pfasst algorithm so can share some thoughts. there are three broad classes of time - parallel techniques that i am aware of : across the method — independent stages of rk or extrapolation integrators can be evaluated in parallel ; see also the ridc ( revisionist integral deferred correction algorithm ) across the problem — waveform relaxation across the time - domain — parareal ; pita ( parallel in time algorithm ) ; and pfasst ( parallel full approximation scheme in space and time ). methods that parallelize across the method usually perform very close to spec but don't scale beyond a handful of ( time ) processors. typically they are relatively easier to implement than other methods and are a good if you have a few extra cores lying around and are looking for predictable and modest speedups. methods that parallelize across the time domain include parareal, pita, pfasst. these methods are all iterative and are comprised of inexpensive ( but inaccurate ) " coarse " propagators and expensive ( but accurate ) " fine " propagators. they achieve parallel efficiency by iteratively evaluating the fine propagator in parallel to improve a serial solution obtained using the coarse propagator. the parareal and pita algorithms suffer from a rather unfortunate upper bound on their parallel efficiency $ e $ : $ e < 1 / k $ where $ k $ is the number of iterations required to obtain convergence throughout the domain. for example, if your parareal implementation required 10 iterations to converge and you are using 100 ( time ) processors, the largest speedup you could hope for would be 10x. the pfasst algorithm relaxes this upper bound by hybridizing the time - parallel iterations with the iterations of the spectral deferred correction time - stepping method and incorporating full approximation scheme corrections to a hierarchy of space / time discretizations. lots of games can be played with all of these methods to try and speed them up, and it seems as though the performance of these across - the - domain techniques depends on what problem you are solving and which techniques are available for speeding up the coarse propagator ( coarsened grids, coarsened operators, coarsened physics etc. ). some references ( see also references listed in the papers ) : this paper demonstrates how various methods can be parallelised across the method : a theoretical comparison of high order explicit runge -
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kutta, extrapolation, and deferred correction methods ; ketcheson and waheed. this paper also shows a nice way of parallelizing across the method, and introduces the ridc algorithm : parallel high - order integrators ; christlieb, macdonald, ong. this paper introduces the pita algorithm : a time - parallel implicit method for accelerating the solution of nonlinear structural dynamics problems ; cortial and farhat. there are lots of papers on parareal ( just google it ). here is a paper on the nievergelt method : a minimal communication approach to parallel time integration ; barker. this paper introduces pfasst : toward an efficient parallel in time method for partial differential equations ; emmett and minion ; this papers describes a neat application of pfasst : a massively space - time parallel n - body solver ; speck, ruprecht, krause, emmett, minion, windel, gibbon. i have written two implementations of pfasst that are available on the'net : pypfasst and libpfasst.
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that molecule is called geosmin. it is mainly produced 1 by actinomycetes such as streptomyces which are filamentous bacteria that live in soil. other organisms also produce geosmin : cyanobacteria certain fungi an amoeba called vanella a liverwort it is an intracellular metabolite and cell damage is the primary reason attributed to its release. however oxidant exposure and transmembrane pressure also causes geosmin release in cyanobacteria. it seems that the release is triggered by some kind of stress. i am not quite sure about their advantage to the host species. 1 or perhaps the most well - studied in
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what's the big deal? when quantum mechanics was being discovered and formalized, in the 1920s and 1930s, our view of physics was deeply rooted in the macroscopic world. we understood that microscopic entities like atoms and molecules existed, and we arrived reasonably quickly at a good understanding of their basic structure, but for a very long time they were very remote objects, whose behaviour was so abstract and disconnected from our everyday experience that it was even kind of pointless to really interrogate it. so, as an example, if you heated up a vial with sodium, then the gas sample in the vial might emit or absorb light at a particular wavelength, and if you worked out the quantum - mechanical maths then you could predict what those wavelengths should be, in terms of quantum jumps between energy levels $ - $ but, could you really say what each individual atom in the gas was doing? how could you be sure that those " quantum jumps " were even real, if you only ever had access to the macroscopic gas sample, and never to any individual atom? moreover, that same quantum - mechanical maths predicts that the dynamics in an atom will be blazingly fast, and indeed many orders of magnitude faster than any experimental techniques available at the time. so, could you really talk about the electrons " moving "? this was aggravated by the fact that the particular choices of quantum - mechanical maths that made sense for this type of experiment talked much more about " orbitals " and " energy levels ", with those mysterious quantum jumps to link them $ - $ so maybe it makes more sense to treat those orbitals and energy levels as the " real " objects, and disregard the notion that there is any movement in the micro - world? however, we live in a very different world now. not only do we have tools like scanning electron microscopy that allow us to observe the atoms that make up a metal surface, we are also now able to hold and control a single atom with delicate electrical " tweezers ", which then allows us to interrogate it directly. and when we look, much to our chagrin, that individual atom is indeed performing the fabled quantum jumps. more generally, since the turn of the millenium the name of the game ( and indeed the routine ) has been the observation and control of individual quantum systems. a similar story holds for the dynamics of microscopic systems, and for our ability to observe them directly. the discoveries of the laser, and then q - switching and mode locking allowed laser pulses to get pretty
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fast, first faster than a microsecond ( $ 10 ^ { - 6 } \ : \ rm s $ ) and then faster than a nanosecond ( $ 10 ^ { - 9 } \ : \ rm s $ ), respectively, and work in the 1970s and 1980s allowed us to create pulses as short as a picosecond ( $ 10 ^ { - 12 } \ : \ rm s $ ) and shorter. if you really push a laser system, using technology known as chirped pulse amplification ( which i wrote about previously here when it won its nobel prize ), you can get down to a few femtoseconds ( $ 10 ^ { - 15 } \ : \ rm s $ ). this is very fast for a pulse of light, and it is actually so fast that the pulse of light is no longer a periodic electric - field oscillation, and instead it lasts only for a few cycles. but it is still not fast enough. why? because atoms are even faster. to understand how fast atoms are, it is enough to do some basic dimensional analysis. the dynamics of the electrons inside an atom are governed by the schrodinger equation, $ $ i \ hbar \ frac { \ partial \ psi } { \ partial t } = - \ frac { \ hbar ^ 2 } { 2m _ e } \ nabla ^ 2 \ psi - \ frac { e ^ 2 } { r } \ psi, $ $ and this has only three core constants involved : the reduced planck constant, $ \ hbar $, the electron's mass, $ m _ e $, and the electron charge $ e $. ( or, if you work in si units, the coulomb constant $ e ^ 2 / 4 \ pi \ epsilon _ 0 $. ) and, as it turns out, those constants can be combined into a unique timescale, known as the atomic unit of time, $ $ t _ \ mathrm { a. u. } = \ frac { \ hbar ^ 3 } { m _ ee ^ 4 } = 24 \ : \ rm as, $ $ which is measured in attoseconds : $ 1 \ : \ rm as = 10 ^ { - 18 } \ : \ rm s $. as a rule of thumb, the dynamics might be somewhat faster, or somewhat slower, depending on the atom and the conditions, but it will generally stick to that rough order
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of magnitude. and that means, in turn, that those dynamics might seem completely out of reach, because the period of oscillation of optical light is still rather slower than this. ( for light of wavelength $ 550 \ : \ rm nm $, the period is of about $ 2 \ : \ rm fs $. ) so that might make you think that a direct observation of something as fast as atomic dynamics must be out of reach. so how do you make an attosecond pulse? this is the real breakthrough that is being rewarded with today's announcement. our workhorse is a process known as high - harmonic generation, which uses a highly nonlinear interaction between a gas and a pulse of laser light to generate sharp bursts of radiation $ - $ the famed attosecond pulses $ - $ which can be much shorter than the period of the pulse that drives the process, and can be as short as a few dozen attoseconds. from an experimental perspective, what you have to do is simply start with a laser pulse with a fairly long wavelength and slow period ( usually in the near - infrared ), shine it into a gas cell, and make sure that the pulse is intense. how intense? very intense. intense enough to directly yank electrons out of the gas atoms and shake them about once they're free. ( and, indeed, intense enough that the pulse will burn out the laser amplifier if you let it, as explained in the thread about chirped pulse amplification. ) this was done in 1987 by a team led by anne l'huillier, and the surprising observation was that the gas emitted harmonics, i. e., additional wavelengths of light at sub - multiples of the original driving wavelength. this was known to occur ( second - harmonic generation is almost as old as the laser itself ), but l'huillier and colleagues discovered that if the driving pulse is intense enough, it can generate all sorts of harmonics at crazy high orders, with a very slow decline in emission as the order increases. ( up until the signal reaches a cutoff and decays exponentially, of course. ) what's going on? the basic physics was worked out by paul corkum ( who was very high in the shortlist for getting the nobel prize if it ever did get awarded to attosecond science ), and it is known as the three - step model. image taken from d. villeneuve, contemp. phys. 59, 47 (
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2018 ) in essence, the laser can be thought of as a constant force ( and therefore a linear ramp in potential energy ) which slowly oscillates and tilts around the potential well that the atomic electron sits in. at the maximum of field intensity, this is enough to yank the electron away ( though more on this later ), at which point the electron will freely oscillate in the field, gaining energy from the electric field of the light... up until it crashes into the potential well that it just left, at which point it can recombine back with the ion it left behind, and emit its ( now considerable ) kinetic energy as a sharp burst of radiation. the coolest things about this collision are that it is very energetic ( so the burst of radiation has a high photon energy, and therefore very high frequencies ), and that it is very short ( it is over in a flash ), and it is this short duration that means that the pulses of radiation emitted will be extremely short. the other parts of the nobel prize are being awarded for the explicit creation and detection of these sharp bursts of light. one thing that happens quite often is that ( because the driving pulse is long, and has many periods where the three - step model can happen ), the emission is often in the shape of an attosecond - pulse train, sometimes with several dozen sharp bursts following each other in quick succession. pierre agostini was the first to directly observe the duration of the bursts within such a train, using a technique known as rabbitt ( attoscience has since acquired an " animal theme " for our acronyms ), and his group was able to show that they were indeed very short, down to as little as $ 250 \ : \ rm as $. alternatively, you might want to invest some ( considerable ) time and energy into finding a way to " gate " the emission, so that there is only one burst in the train. ( for a fresh - off - the - press review of different ways to " gate " the emission see e. g. this preprint. ) this gating was achieved by ferenc krausz's group, who were able to isolate a single pulse with a duration of $ 650 \ : \ rm as $. of course, the field has continued to innovate, making things more reliable and robust, but also pushing down the shortest duration achievable. if i understand correctly, the current record is $ 43 \ :
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\ rm as $, which is very, very short. ( another cool record is how high you can push the order of nonlinearity in the process, for which, if i understand correctly, a 2012 classic still holds the prize with a minimal order of nonlinearity of 4, 500. ) what can you use these pulses for? we're now down to the most interesting part. say that you have made one of these attosecond pulses. what can you do with it? directly observing the wave oscillations of light for me, the most exciting application from the " classic " experiments in attoscience is a setup known as " attosecond streaking ". the basic idea is to take a short attosecond pulse, and overlap it, inside a gas sample, with a slower pulse of infrared light. the short pulse has enough photon energy to ionize the gas, and we know that this must happen within the duration of the short pulse. after this ionization, the slower infrared pulse has an electric field which oscillates, and this will impact the final energy and momentum of the electron, but the extent of this effect will depend on when the electron is released, so by changing the time delay between the two, we can scan against this electric field. $ \ qquad $ the end result, shown above, is a direct observation of the oscillations of the electric field ( raw data on the left, and reconstructed electric field on the right ), which is a task that was considered somewhere between impossible and unthinkable for many, many decades after we understood that light was a wave ( but only had indirect ways to prove it ). i've discussed this experiment previously here. for more details ( and the source of the figures ), see the landmark publication : direct measurement of light waves. e. goulielmakis et al. science 305, 1267 ( 2004 ) ; author eprint. directly observing electron motion in real time similarly to observing the motion of the electric field of light, we can also observe the motion of electrons inside an atom. i have discussed this in detail in is there oscillating charge in a hydrogen atom?, but the short story is that if you prepare an electron in a quantum superposition of two different energy levels, such as the combination $ $ \ psi = \ psi _ { 1s } + \ psi _ { 2p } $ $ of the hydrogen $ 1s $ and $ 2
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##p $ levels, the charge density in the atom will oscillate over time : mathematica source through import [ " this is not a hypothetical or purely theoretical construct, and we can directly observe it in experiment. the first landmark test, reported in real - time observation of valence electron motion. e. goulielmakis et al. nature 466, 739 ( 2010 ). was able to show a clear oscillation in how much a short pulse was absorbed by an oscillating charge distribution caused by spin - orbit interactions ( where different parts of the oscillations correspond to different orientations of the charge density, and therefore to different absorption profiles ), showing a clear corresponding oscillation in the absorbance : similarly, a much - beloved example is the observation of charge oscillation dynamics in a bio - relevant molecule, phenylalanine, which was reported in ultrafast electron dynamics in phenylalanine initiated by attosecond pulses. f. calegari et al., science 346, 336 ( 2014 ), and where the ionization of the molecule by a ( relatively ) short laser pulse ( in the near - infrared ) is then probed by a ( very ) short attosecond burst. the resulting dynamics inside the molecule are fairly complicated, but they lead to clear oscillations in the signal ( with the graph below showing the overall decay, and the oscillations on top of an exponential background ) at a very short timescale that is only observable thanks to the availability of attosecond pulses. watching quantum interference build up in real time i will do one more direct - timing - of - observation, because i think they're really cool. this one is again about a quantum superposition, but one that happens with a free electron. when you ionize an atom, the electron gets released, and one photon gets absorbed. and, more importantly, the details of the energy states that the electron gets released into will be imprinted into the absorbance spectrum of the light. in particular, it is possible to tune things so that you are ionizing close to a resonance : the electron can either ionize directly, or it can spend some time in a highly - excited autoionizing state ( also explained here and here ) that will fall apart after some time. the end result is that the electron will go into a superposition of both pathways, which will interfere in its spectrum and cause a won
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##ky, nontrivial shape in the absorption spectrum. however, if we have short pulses of radiation, we are able to control how long we let the electron to sit in that autoionizing state, before we come in with a second pulse of light to disrupt it, and kill the interference : and indeed, when we do this, the build - up of the line and the development of the interference features ( and particularly that sharp dip on the right - hand side of the line ) is very clearly seen in experiment : and, just to add some more pretty pictures, here it is all stacked together, on the left - hand figure, and on the right a similar experiment showing very clearly the destructive interference building up over time : for more details, and the sources of the figures, see observing the ultrafast buildup of a fano resonance in the time domain. a. kaldun et al. science 354, 738 ( 2016 ) and attosecond dynamics through a fano resonance : monitoring the birth of a photoelectron. v gruson et al. science 354, 734 ( 2016 ) moreover, it is also possible to use these types of resonances to enhance high - harmonic generation itself, in a process known as resonant hhg. for a nice review written by a colleague ( in a paper i coauthored ) see eur. phys. j d 75, 209 ( 2021 ) ( arxiv : 2101. 09335 ). further reading long as this post is, i have only just scratched the surface. here are some additional places to read more about the field : attosecond science. d. villeneuve, contemp. phys. 59, 47 ( 2018 ) ( author eprint ) attosecond science. p. b. corkum & f. krausz. nature physics 3, 381 ( 2007 ) ( author eprint ) the physics of attosecond light pulses. p. agostini & l. f. dimauro. reports on progress in physics 67, 813 ( 2004 ) ( author eprint ) attosecond electromagnetic pulses : generation, measurement, and application. attosecond metrology and spectroscopy. m. yu. ryabikin et al. * physics - uspekhi 66, 360 ( 2023 ) shining the shortest flashes of light on the secret life of electrons. m. khokhlova, e. pisanty & a.
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zair. advanced photonics 5, 060501 ( 2023 )
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this is not an answer to your question, but an extended comment on the issue that has been raised here in comments by different people, namely : are machine learning " tensors " the same thing as tensors in mathematics? now, according to the cichoki 2014, era of big data processing : a new approach via tensor networks and tensor decompositions, and cichoki et al. 2014, tensor decompositions for signal processing applications, a higher - order tensor can be interpreted as a multiway array, [... ] a tensor can be thought of as a multi - index numerical array, [... ] tensors ( i. e., multi - way arrays ) [... ] so in machine learning / data processing a tensor appears to be simply defined as a multidimensional numerical array. an example of such a 3d tensor would be $ 1000 $ video frames of $ 640 \ times 480 $ size. a usual $ n \ times p $ data matrix is an example of a 2d tensor according to this definition. this is not how tensors are defined in mathematics and physics! a tensor can be defined as a multidimensional array obeying certain transformation laws under the change of coordinates ( see wikipedia or the first sentence in mathworld article ). a better but equivalent definition ( see wikipedia ) says that a tensor on vector space $ v $ is an element of $ v \ otimes \ ldots \ otimes v ^ * $. note that this means that, when represented as multidimensional arrays, tensors are of size $ p \ times p $ or $ p \ times p \ times p $ etc., where $ p $ is the dimensionality of $ v $. all tensors well - known in physics are like that : inertia tensor in mechanics is $ 3 \ times 3 $, electromagnetic tensor in special relativity is $ 4 \ times 4 $, riemann curvature tensor in general relativity is $ 4 \ times 4 \ times 4 \ times 4 $. curvature and electromagnetic tensors are actually tensor fields, which are sections of tensor bundles ( see e. g. here but it gets technical ), but all of that is defined over a vector space $ v $. of course one can construct a tensor product $ v \ otimes w $ of an $ p $ - dimensional $ v $ and $ q $ - dimensional $ w $ but its elements are usually not called " tensors ", as stated e. g. here on
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wikipedia : in principle, one could define a " tensor " simply to be an element of any tensor product. however, the mathematics literature usually reserves the term tensor for an element of a tensor product of a single vector space $ v $ and its dual, as above. one example of a real tensor in statistics would be a covariance matrix. it is $ p \ times p $ and transforms in a particular way when the coordinate system in the $ p $ - dimensional feature space $ v $ is changed. it is a tensor. but a $ n \ times p $ data matrix $ x $ is not. but can we at least think of $ x $ as an element of tensor product $ w \ otimes v $, where $ w $ is $ n $ - dimensional and $ v $ is $ p $ - dimensional? for concreteness, let rows in $ x $ correspond to people ( subjects ) and columns to some measurements ( features ). a change of coordinates in $ v $ corresponds to linear transformation of features, and this is done in statistics all the time ( think of pca ). but a change of coordinates in $ w $ does not seem to correspond to anything meaningful ( and i urge anybody who has a counter - example to let me know in the comments ). so it does not seem that there is anything gained by considering $ x $ as an element of $ w \ otimes v $. and indeed, the common notation is to write $ x \ in \ mathbb r ^ { n \ times p } $, where $ r ^ { n \ times p } $ is a set of all $ n \ times p $ matrices ( which, by the way, are defined as rectangular arrays of numbers, without any assumed transformation properties ). my conclusion is : ( a ) machine learning tensors are not math / physics tensors, and ( b ) it is mostly not useful to see them as elements of tensor products either. instead, they are multidimensional generalizations of matrices. unfortunately, there is no established mathematical term for that, so it seems that this new meaning of " tensor " is now here to stay.
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superoxide, o2− is created by the immune system in phagocytes ( including neutrophils, monocytes, macrophages, dendritic cells, and mast cells ) which use nadph oxidase to produce it from o2 for use against invading microorganisms. however, under normal conditions, the mitochondrial electron transport chain is a major source of o2−, converting up to perhaps 5 % of o2 to superoxide. [ 1 ] as a side note, there are two sides to this coin. while this is a useful tool against microorganisms, the formation of the reactive oxygen species has been incriminated in autoimmune reactions and diabetes ( type 1 ). [ 2 ] [ 1 ] packer l, ed. methods in enzymology, volume 349. san diego, calif : academic press ; 2002 [ 2 ] thayer tc, delano m, et al. ( 2011 ) superoxide production by macrophages and t cells is critical for the induction of autoreactivity and type 1 diabetes, 60 ( 8 ), 2144 - 51.
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intriguing question. first, the best yield would be achieved by selectively producing one enantiomer instead of the other. in this case, white wants d - methamphetamine ( powerful psychoactive drug ), not l - methamphetamine ( vicks vapor inhaler ). reaction processes designed to do this are known as " asymmetric synthesis " reactions, because they favor production of one enantiomer over the other. the pseudoephedrine method for methamphetamine employs one of the more common methods of asymmetric synthesis, called " chiral pool resolution ". as you state, starting with an enantiomerically - pure sample of a chiral reagent ( pseudoephedrine ) as the starting point allows you to preserve the chirality of the finished product, provided the chiral point is not part of any " leaving group " during the reaction. however, again as you show, phenylacetone is achiral, and so the p2p process cannot take advantage of this method. there are other methods of asymmetric synthesis, however none of them seem applicable to the chemistry shown or described on tv either ; none of the reagents or catalysts mentioned would work as chiral catalysts, nor are they bio - or organocatalysts. metal complexes with chiral ligands can be used to selectively catalyze production of one enantiomer, however the aluminum - mercury amalgam is again achiral. i don't remember any mention of using organocatalysis or biocatalysis, but these are possible. the remaining route, then, is chiral resolution ; let the reaction produce the 50 - 50 split, then separate the two enantiomers by some means of reactionary and / or physical chemistry. this seems to be the way it works in the real world. the advantage is that most of the methods are pretty cheap and easy ; the disadvantage is that your maximum possible yield is 50 % ( unless you can then run a racemization reaction on the undesireable half to " reshuffle " the chirality of that half ; then your yield increases by 50 % of the last increase each time you run this step on the undesirable product ). in the case of methamphetamine, this resolution is among the easiest, because methamphetamine forms a " racemic conglomerate " when crystallized. this means,
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for the non - chemists, that each enantiomer molecule prefers to crystallize with others of the same chiral species, so as the solution cools and the solvent is evaporated off, the d - methamphetamine will form one set of homogeneous crystals and the l - methamphetamine will form another set. this means that all white has to do is slow the evaporation of solvent and subsequent cooling of the pan, letting the largest possible crystals form. then, the only remaining trick is identifying which crystals have which enantiomer ( and as these crystals are translucent and " optically active ", observing the polarization pattern of light shone through the crystals will identify which are which ).
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i'll translate an entry in the blog gaussianos ( " gaussians " ) about polya's conjecture, titled : a belief is not a proof. we'll say a number is of even kind if in its prime factorization, an even number of primes appear. for example $ 6 = 2 \ cdot 3 $ is a number of even kind. and we'll say a number is of odd kind if the number of primes in its factorization is odd. for example, $ 18 = 2 · 3 · 3 $ is of odd kind. ( $ 1 $ is considered of even kind ). let $ n $ be any natural number. we'll consider the following numbers : $ e ( n ) = $ number of positive integers less or equal to $ n $ that are of even kind. $ o ( n ) = $ number of positive integers less or equal to $ n $ that are of odd kind. let's consider $ n = 7 $. in this case $ o ( 7 ) = 4 $ ( number 2, 3, 5 and 7 itself ) and $ e ( 7 ) = 3 $ ( 1, 4 and 6 ). so $ o ( 7 ) > e ( 7 ) $. for $ n = 6 $ : $ o ( 6 ) = 3 $ and $ e ( 6 ) = 3 $. thus $ o ( 6 ) = e ( 6 ) $. in 1919 george polya proposed the following result, know as polya's conjecture : for all $ n > 2 $, $ o ( n ) $ is greater than or equal to $ e ( n ) $. polya had checked this for $ n < 1500 $. in the following years this was tested up to $ n = 1000000 $, which is a reason why the conjecture might be thought to be true. but that is wrong. in 1962, lehman found an explicit counterexample : for $ n = 906180359 $, we have $ o ( n ) = e ( n ) – 1 $, so : $ $ o ( 906180359 ) < e ( 906180359 ). $ $ by an exhaustive search, the smallest counterexample is $ n = 906150257 $, found by tanaka in 1980. thus polya's conjecture is false. what do we learn from this? well, it is simple : unfortunately in mathematics we cannot
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trust intuition or what happens for a finite number of cases, no matter how large the number is. until the result is proved for the general case, we have no certainty that it is true.
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a rolling hash function for dna sequences called nthash has recently been published in bioinformatics and the authors dealt with reverse complements : using this table, we can easily compute the hash value for the reverse - complement ( as well as the canonical form ) of a sequence efficiently, without actually reverse - complementing the input sequence, as follows :... edit ( by @ user172818 ) : i will add more details about how nthash works. the notations used in its paper are somewhat uncommon. the source code is more informative. let's first define rotation functions for 64 - bit integers : rol ( x, k ) : = x < < k | x > > ( 64 - k ) ror ( x, k ) : = x > > k | x < < ( 64 - k ) we then define a hash function h ( ) for each base. in the implementation, the authors are using : h ( a ) = 0x3c8bfbb395c60474 h ( c ) = 0x3193c18562a02b4c h ( g ) = 0x20323ed082572324 h ( t ) = 0x295549f54be24456 h ( n ) = 0 the rolling hash function of a forward k - mer s [ i, i + k - 1 ] is : f ( s [ i, i + k - 1 ] ) : = rol ( h ( s [ i ] ), k - 1 ) ^ rol ( h ( s [ i + 1 ] ), k - 2 ) ^... ^ h ( s [ i + k - 1 ] ) where ^ is the xor operator. the hash function of its reverse complement is : r ( s [ i, i + k - 1 ] ) : = f ( ~ s [ i, i + k - 1 ] ) = rol ( h ( ~ s [ i + k - 1 ] ), k - 1 ) ^ rol ( h ( ~ s [ i + k - 2 ] ), k - 2 ) ^... ^ h ( ~ s [ i ] ) where ~ gives the reverse complement of a dna sequence. knowing f ( s [ i, i + k - 1 ] ) and r ( s [ i, i + k - 1 ] ), we can compute their values for the next k - mer : f (
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s [ i + 1, i + k ] ) = rol ( f ( s [ i, i + k - 1 ] ), 1 ) ^ rol ( h ( s [ i ] ), k ) ^ h ( s [ i + k ] ) r ( s [ i + 1, i + k ] ) = ror ( r ( s [ i, i + k - 1 ] ), 1 ) ^ ror ( h ( ~ s [ i ] ), 1 ) ^ rol ( h ( ~ s [ i + k ] ), k - 1 ) in other words, for the forward kmer for each additional base, xor the following three values together : a single left rotation of the previous hash, f ( s [ i, i + k - 1 ] ) a $ k $ - times left rotation of the base hash of s [ i ] the base hash of s [ i + k ] similarly for the reverse kmer, xor the following three values together : a single right rotation of the previous reverse hash, r ( s [ i, i + k - 1 ] ) a single right rotation of the base hash of the reverse complement of s [ i ] a $ k - 1 $ - times left rotation of the base hash of the reverse complement of s [ i + k ] this works because rol, ror and ^ can all be switched in order. finally, for a k - mer s, the hash function considering both strands is the smaller between f ( s ) and r ( s ) : h ( s ) = min ( f ( s ), r ( s ) ) this is a linear algorithm regardless of the k - mer length. it only uses simple arithmetic operations, so should be fairly fast. i have briefly tested its randomness. it seems comparable to murmur. nthash is probably the best algorithm so far if you want to hash an arbitrarily long k - mer into 64 bits.
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as willie wong observed, including an expression of the form $ \ displaystyle \ frac { | \ alpha | } { \ alpha } $ is a way of ensuring that $ \ alpha > 0 $. ( as $ \ sqrt { | \ alpha | / \ alpha } $ is $ 1 $ if $ \ alpha > 0 $ and non - real if $ \ alpha < 0 $. ) the ellipse $ \ displaystyle \ left ( \ frac { x } { 7 } \ right ) ^ { 2 } + \ left ( \ frac { y } { 3 } \ right ) ^ { 2 } - 1 = 0 $ looks like this : so the curve $ \ left ( \ frac { x } { 7 } \ right ) ^ { 2 } \ sqrt { \ frac { \ left | \ left | x \ right | - 3 \ right | } { \ left | x \ right | - 3 } } + \ left ( \ frac { y } { 3 } \ right ) ^ { 2 } \ sqrt { \ frac { \ left | y + 3 \ frac { \ sqrt { 33 } } { 7 } \ right | } { y + 3 \ frac { \ sqrt { 33 } } { 7 } } } - 1 = 0 $ is the above ellipse, in the region where $ | x | > 3 $ and $ y > - 3 \ sqrt { 33 } / 7 $ : that's the first factor. the second factor is quite ingeniously done. the curve $ \ left | \ frac { x } { 2 } \ right | \ ; - \ ; \ frac { \ left ( 3 \ sqrt { 33 } - 7 \ right ) } { 112 } x ^ { 2 } \ ; - \ ; 3 \ ; + \ ; \ sqrt { 1 - \ left ( \ left | \ left | x \ right | - 2 \ right | - 1 \ right ) ^ { 2 } } - y = 0 $ looks like : this is got by adding $ y = \ left | \ frac { x } { 2 } \ right | - \ frac { \ left ( 3 \ sqrt { 33 } - 7 \ right ) } { 112 } x ^ { 2 } - 3 $, a parabola on the positive - x side, reflected : and $ y = \ sqrt { 1 - \ left ( \ left
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| \ left | x \ right | - 2 \ right | - 1 \ right ) ^ { 2 } } $, the upper halves of the four circles $ \ left ( \ left | \ left | x \ right | - 2 \ right | - 1 \ right ) ^ 2 + y ^ 2 = 1 $ : the third factor $ 9 \ sqrt { \ frac { \ left ( \ left | \ left ( 1 - \ left | x \ right | \ right ) \ left ( \ left | x \ right | -. 75 \ right ) \ right | \ right ) } { \ left ( 1 - \ left | x \ right | \ right ) \ left ( \ left | x \ right | -. 75 \ right ) } } \ ; - \ ; 8 \ left | x \ right | \ ; - \ ; y \ ; = \ ; 0 $ is just the pair of lines y = 9 - 8 | x | : truncated to the region $ 0. 75 < | x | < 1 $. similarly, the fourth factor $ 3 \ left | x \ right | \ ; + \ ;. 75 \ sqrt { \ left ( \ frac { \ left | \ left (. 75 - \ left | x \ right | \ right ) \ left ( \ left | x \ right | -. 5 \ right ) \ right | } { \ left (. 75 - \ left | x \ right | \ right ) \ left ( \ left | x \ right | -. 5 \ right ) } \ right ) } \ ; - \ ; y \ ; = \ ; 0 $ is the pair of lines $ y = 3 | x | + 0. 75 $ : truncated to the region $ 0. 5 < | x | < 0. 75 $. the fifth factor $ 2. 25 \ sqrt { \ frac { \ left | \ left (. 5 - x \ right ) \ left ( x +. 5 \ right ) \ right | } { \ left (. 5 - x \ right ) \ left ( x +. 5 \ right ) } } \ ; - \ ; y \ ; = \ ; 0 $ is the line $ y = 2. 25 $ truncated to $ - 0. 5 < x < 0. 5 $. finally, $ \ frac { 6 \ sqrt { 10 } } { 7 } \ ; + \ ; \ left ( 1. 5 \ ; - \ ;. 5 \
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left | x \ right | \ right ) \ ; - \ ; \ frac { \ left ( 6 \ sqrt { 10 } \ right ) } { 14 } \ sqrt { 4 - \ left ( \ left | x \ right | - 1 \ right ) ^ { 2 } } \ ; - \ ; y \ ; = \ ; 0 $ looks like : so the sixth factor $ \ frac { 6 \ sqrt { 10 } } { 7 } \ ; + \ ; \ left ( 1. 5 \ ; - \ ;. 5 \ left | x \ right | \ right ) \ sqrt { \ frac { \ left | \ left | x \ right | - 1 \ right | } { \ left | x \ right | - 1 } } \ ; - \ ; \ frac { \ left ( 6 \ sqrt { 10 } \ right ) } { 14 } \ sqrt { 4 - \ left ( \ left | x \ right | - 1 \ right ) ^ { 2 } } \ ; - \ ; y \ ; = \ ; 0 $ looks like as a product of factors is $ 0 $ iff any one of them is $ 0 $, multiplying these six factors puts the curves together, giving : ( the software, grapher. app, chokes a bit on the third factor, and entirely on the fourth )
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look carefully, it's ( distorted ) tetrahedral - - four groups at nearly symmetrically positions in 3d space { * }. so the hybridization is $ sp ^ 3 $. as you can see, the shape is distorted, but it's tetrahedral. technically, the banana bonds can be said to be made up of orbitals similar to $ sp ^ 3 $ but not exactly ( like two $ sp ^ { 3. 1 } $ and two $ sp ^ { 2. 9 } $ orbitals - - since hybridization is just addition of wavefunctions, we can always change the coefficients to give proper geometry ). i'm not too sure of this though. $ \ ce { b } $ has an $ 2s ^ 22p ^ 1 $ valence shell, so three covalent bonds gives it an incomplete octet. $ \ ce { bh3 } $ has an empty $ 2p $ orbital. this orbital overlaps the existing $ \ ce { b - h } $ $ \ sigma $ bond cloud ( in a nearby $ \ ce { bh3 } $ ), and forms a 3c2e bond. it seems that there are a lot more compounds with 3c2e geometry. i'd completely forgotten that there were entire homologous series'under'boranes'which all have 3c2e bonds ( though not the same structure ) and there are indium and gallium compounds as well. still group iiia, though these are metals. i guess they, like $ \ ce { al } $, still form covalent bonds. so the basic reason for this happening is due to an incomplete octet wanting to fill itself. note that " banana " is not necessarily only for 3c2e bonds. any bent bond is called a " banana " bond. regarding similar structures, $ \ ce { becl2 } $ and $ \ ce { alcl3 } $ come to mind, but both of them have the structure via dative ( coordinate ) bonds. additionally, $ \ ce { becl2 } $ is planar. sneaks off and checks wikipedia. wikipedia says $ \ ce { al2 ( ch3 ) 6 } $ is similar in structure and bond type. i guess we have less such compounds because there are comparatively few elements ( $ \ ce { b } $ group pretty much ) with $ \ leq3 $ valence electrons which form covalent bonds ( criteria for the empty orbital )
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. additionally, $ \ ce { al } $ is an iffy case - - it like both covalent and ionic bonds. also, for this geometry ( either by banana bonds or by dative bonds ), i suppose the relative sizes matter as well - - since $ \ ce { bcl3 } $ is a monomer even though $ \ ce { cl } $ has a lone pair and can form a dative bond. * maybe you're used to the view of tetrahedral structure with an atom at the top? mentally tilt the boron atom till a hydrogen is up top. you should realize that this is tetrahedral as well.
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the global alliance for genomics and health has been working on the issue of representing sequencing data and metadata for storage and sharing for quite some time, though with mixed results. they do offer a model and api for storing ngs data in their github repository, but it can be a bit of a pain to get a high - level view. i am not sure if any better representation of this exists elsewhere. i can say from personal experience ( having built over a dozen genomic databases ), there is no ideal data model and storage best practices. genomic data comes in many shapes and sizes, and your needs are going to vary from every other organization, so what works for one bioinformatics group won't necessarily work for you. the best thing to do is design and implement a model that will cover all of the data types in your workflow and downstream analyses you might do with the data and metadata.
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what follows is taken ( mostly ) from more extensive discussions in the following sci. math posts : [ 23 january 2000 ] [ 6 november 2006 ] [ 20 december 2006 ] note : the term interval is restricted to nondegenerate intervals ( i. e. intervals containing more than one point ). the continuity set of a derivative on an open interval $ j $ is dense in $ j. $ in fact, the continuity set has cardinality $ c $ in every subinterval of $ j. $ on the other hand, the discontinuity set $ d $ of a derivative can have the following properties : $ d $ can be dense in $ \ mathbb r $. $ d $ can have cardinality $ c $ in every interval. $ d $ can have positive measure. ( hence, the function can fail to be riemann integrable. ) $ d $ can have positive measure in every interval. $ d $ can have full measure in every interval ( i. e. measure zero complement ). $ d $ can have a hausdorff dimension zero complement. $ d $ can have an $ h $ - hausdorff measure zero complement for any specified hausdorff measure function $ h. $ more precisely, a subset $ d $ of $ \ mathbb r $ can be the discontinuity set for some derivative if and only if $ d $ is an $ f _ { \ sigma } $ first category ( i. e. an $ f _ { \ sigma } $ meager ) subset of $ \ mathbb r. $ this characterization of the discontinuity set of a derivative can be found in the following references : benedetto [ 1 ] ( chapter 1. 3. 2, proposition, 1. 10, p. 30 ) ; bruckner [ 2 ] ( chapter 3, section 2, theorem 2. 1, p. 34 ) ; bruckner / leonard [ 3 ] ( theorem at bottom of p. 27 ) ; goffman [ 5 ] ( chapter 9, exercise 2. 3, p. 120 states the result ) ; klippert / williams [ 7 ]. regarding this characterization of the discontinuity set of a derivative, bruckner and leonard [ 3 ] ( bottom of p. 27 ) wrote the following in 1966 : although we imagine that this theorem is known, we have been unable to find a reference. i have found the result stated in goffman's 1953 text [ 5 ], but
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nowhere else prior to 1966 ( including goffman's ph. d. dissertation ). interestingly, in a certain sense most derivatives have the property that $ d $ is large in all of the ways listed above ( # 1 through # 7 ). in 1977 cliff weil [ 8 ] published a proof that, in the space of derivatives with the sup norm, all but a first category set of such functions are discontinuous almost everywhere ( in the sense of lebesgue measure ). when weil's result is paired with the fact that derivatives ( being baire $ 1 $ functions ) are continuous almost everywhere in the sense of baire category, we get the following : ( a ) every derivative is continuous at the baire - typical point. ( b ) the baire - typical derivative is not continuous at the lebesgue - typical point. note that weil's result is stronger than simply saying that the baire - typical derivative fails to be riemann integrable ( i. e. $ d $ has positive lebesgue measure ), or even stronger than saying that the baire - typical derivative fails to be riemann integrable on every interval. note also that, for each of these baire - typical derivatives, $ \ { d, \ ; { \ mathbb r } - d \ } $ gives a partition of $ \ mathbb r $ into a first category set and a lebesgue measure zero set. in 1984 bruckner / petruska [ 4 ] ( theorem 2. 4 ) strengthened weil's result by proving the following : given any finite borel measure $ \ mu, $ the baire - typical derivative is such that the set $ d $ is the complement of a set that has $ \ mu $ - measure zero. in 1993 kirchheim [ 5 ] strengthened weil's result by proving the following : given any hausdorff measure function $ h, $ the baire - typical derivative is such that the set $ d $ is the complement of a set that has hausdorff $ h $ - measure zero. [ 1 ] john j. benedetto, real variable and integration with historical notes, mathematische leitfaden. stuttgart : b. g. teubne, 1976, 278 pages. [ mr 58 # 28328 ; zbl 336. 26001 ] [ 2 ] andrew m. bruckner, differentiation of real functions, 2nd edition, crm monograph series #
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5, american mathematical society, 1994, xii + 195 pages. [ the first edition was published in 1978 as springer - verlag's lecture notes in mathematics # 659. the second edition is essentially unchanged from the first edition with the exception of a new chapter on recent developments ( 23 pages ) and 94 additional bibliographic items. ] [ mr 94m : 26001 ; zbl 796. 26001 ] [ 3 ] andrew m. bruckner and john l. leonard, derivatives, american mathematical monthly 73 # 4 ( april 1966 ) [ part ii : papers in analysis, herbert ellsworth slaught memorial papers # 11 ], 24 - 56. [ mr 33 # 5797 ; zbl 138. 27805 ] [ 4 ] andrew m. bruckner and gyorgy petruska, some typical results on bounded baire $ 1 $ functions, acta mathematica hungarica 43 ( 1984 ), 325 - 333. [ mr 85h : 26004 ; zbl 542. 26004 ] [ 5 ] casper goffman, real functions, prindle, weber & schmidt, 1953 / 1967, x + 261 pages. [ mr 14, 855e ; zbl 53. 22502 ] [ 6 ] bernd kirchheim, some further typical results on bounded baire one functions, acta mathematica hungarica 62 ( 1993 ), 119 - 129. [ 94k : 26008 ; zbl 786. 26002 ] [ 7 ] john clayton klippert and geoffrey williams, on the existence of a derivative continuous on a $ g _ { \ delta } $, international journal of mathematical education in science and technology 35 ( 2004 ), 91 - 99. [ 8 ] clifford weil, the space of bounded derivatives, real analysis exchange 3 ( 1977 - 78 ), 38 - 41. [ zbl 377. 26005 ]
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five point summary yes, the idea is to give a quick summary of the distribution. it should be roughly symmetrical about mean, the median should be close to 0, the 1q and 3q values should ideally be roughly similar values. coefficients and $ \ hat { \ beta _ i } s $ each coefficient in the model is a gaussian ( normal ) random variable. the $ \ hat { \ beta _ i } $ is the estimate of the mean of the distribution of that random variable, and the standard error is the square root of the variance of that distribution. it is a measure of the uncertainty in the estimate of the $ \ hat { \ beta _ i } $. you can look at how these are computed ( well the mathematical formulae used ) on wikipedia. note that any self - respecting stats programme will not use the standard mathematical equations to compute the $ \ hat { \ beta _ i } $ because doing them on a computer can lead to a large loss of precision in the computations. $ t $ - statistics the $ t $ statistics are the estimates ( $ \ hat { \ beta _ i } $ ) divided by their standard errors ( $ \ hat { \ sigma _ i } $ ), e. g. $ t _ i = \ frac { \ hat { \ beta _ i } } { \ hat { \ sigma _ i } } $. assuming you have the same model in object modas your q : > mod < - lm ( sepal. width ~ petal. width, data = iris ) then the $ t $ values r reports are computed as : > tstats < - coef ( mod ) / sqrt ( diag ( vcov ( mod ) ) ) ( intercept ) petal. width 53. 277950 - 4. 786461 where coef ( mod ) are the $ \ hat { \ beta _ i } $, and sqrt ( diag ( vcov ( mod ) ) ) gives the square roots of the diagonal elements of the covariance matrix of the model parameters, which are the standard errors of the parameters ( $ \ hat { \ sigma _ i } $ ). the p - value is the probability of achieving a $ | t | $ as large as or larger than the observed absolute t value if the null hypothesis ( $ h _ 0 $ ) was true, where $ h _ 0 $ is $ \ beta _ i = 0 $. they are computed as ( using tstats from above
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) : > 2 * pt ( abs ( tstats ), df = df. residual ( mod ), lower. tail = false ) ( intercept ) petal. width 1. 835999e - 98 4. 073229e - 06 so we compute the upper tail probability of achieving the $ t $ values we did from a $ t $ distribution with degrees of freedom equal to the residual degrees of freedom of the model. this represents the probability of achieving a $ t $ value greater than the absolute values of the observed $ t $ s. it is multiplied by 2, because of course $ t $ can be large in the negative direction too. residual standard error the residual standard error is an estimate of the parameter $ \ sigma $. the assumption in ordinary least squares is that the residuals are individually described by a gaussian ( normal ) distribution with mean 0 and standard deviation $ \ sigma $. the $ \ sigma $ relates to the constant variance assumption ; each residual has the same variance and that variance is equal to $ \ sigma ^ 2 $. adjusted $ r ^ 2 $ adjusted $ r ^ 2 $ is computed as : $ $ 1 - ( 1 - r ^ 2 ) \ frac { n - 1 } { n - p - 1 } $ $ the adjusted $ r ^ 2 $ is the same thing as $ r ^ 2 $, but adjusted for the complexity ( i. e. the number of parameters ) of the model. given a model with a single parameter, with a certain $ r ^ 2 $, if we add another parameter to this model, the $ r ^ 2 $ of the new model has to increase, even if the added parameter has no statistical power. the adjusted $ r ^ 2 $ accounts for this by including the number of parameters in the model. $ f $ - statistic the $ f $ is the ratio of two variances ( $ ssr / sse $ ), the variance explained by the parameters in the model ( sum of squares of regression, ssr ) and the residual or unexplained variance ( sum of squares of error, sse ). you can see this better if we get the anova table for the model via anova ( ) : > anova ( mod ) analysis of variance table response : sepal. width df sum sq mean sq f value pr ( > f ) petal. width 1 3. 7945 3. 7945 22. 91 4. 073e - 06 * * *
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residuals 148 24. 5124 0. 1656 - - - signif. codes : 0 ‘ * * * ’ 0. 001 ‘ * * ’ 0. 01 ‘ * ’ 0. 05 ‘. ’ 0. 1 ‘ ’ 1 the $ f $ s are the same in the anova output and the summary ( mod ) output. the mean sq column contains the two variances and $ 3. 7945 / 0. 1656 = 22. 91 $. we can compute the probability of achieving an $ f $ that large under the null hypothesis of no effect, from an $ f $ - distribution with 1 and 148 degrees of freedom. this is what is reported in the final column of the anova table. in the simple case of a single, continuous predictor ( as per your example ), $ f = t _ { \ mathrm { petal. width } } ^ 2 $, which is why the p - values are the same. this equivalence only holds in this simple case.
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theorem ( false ) : one can arbitrarily rearrange the terms in a convergent series without changing its value.
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it's difficult to get this to go massively quicker i think - as with this question working with large gzipped fastq files is mostly io - bound. we could instead focus on making sure we are getting the right answer. people deride them too often, but this is where a well - written parser is worth it's weight in gold. heng li gives us this fastq parser in c. i downloaded the example tarball and modified the example code ( excuse my c... ) : # include < zlib. h > # include < stdio. h > # include " kseq. h " kseq _ init ( gzfile, gzread ) int main ( int argc, char * argv [ ] ) { gzfile fp ; kseq _ t * seq ; int l ; if ( argc = = 1 ) { fprintf ( stderr, " usage : % s < in. seq > \ n ", argv [ 0 ] ) ; return 1 ; } fp = gzopen ( argv [ 1 ], " r " ) ; seq = kseq _ init ( fp ) ; int seqcount = 0 ; long seqlen = 0 ; while ( ( l = kseq _ read ( seq ) ) > = 0 ) { seqcount = seqcount + 1 ; seqlen = seqlen + ( long ) strlen ( seq - > seq. s ) ; } kseq _ destroy ( seq ) ; gzclose ( fp ) ; printf ( " number of sequences : % d \ n ", seqcount ) ; printf ( " number of bases in sequences : % ld \ n ", seqlen ) ; return 0 ; } then make and kseq _ test foo. fastq. gz. for my example file ( ~ 35m reads of ~ 75bp ) this took : real 0m49. 670s user 0m49. 364s sys 0m0. 304s compared with your example : real 0m43. 616s user 1m35. 060s sys 0m5. 240s konrad's solution ( in my hands ) : real 0m39. 682s user 1m11. 900s sys 0m
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##5. 112s ( by the way, just zcat - ing the data file to / dev / null ) : real 0m38. 736s user 0m38. 356s sys 0m0. 308s so, i get pretty close in speed, but am likely to be more standards compliant. also this solution gives you more flexibility with what you can do with the data. and my horrible c can almost certainly be optimised. same test, with kseq. h from github, as suggested in the comments : my machine is under different load this morning, so i've retested. wall clock times : op : 0m44. 813s konrad : 0m40. 061s zcat > / dev / null : 0m34. 508s kseq. h ( github ) : 0m32. 909s so most recent version of kseq. h is faster than simply zcat - ing the file ( consistently in my tests... ).
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the name " ring " is derived from hilbert's term " zahlring " ( number ring ), introduced in his zahlbericht for certain rings of algebraic integers. as for why hilbert chose the name " ring ", i recall reading speculations that it may have to do with cyclical ( ring - shaped ) behavior of powers of algebraic integers. namely, if $ \ : \ alpha \ : $ is an algebraic integer of degree $ \ rm \ : n \ : $ then $ \ : \ alpha ^ n \ : $ is a $ \ rm \ : \ mathbb z $ - linear combination of lower powers of $ \ rm \ : \ alpha \ :, \ : $ thus so too are all higher powers of $ \ rm \ : \ alpha \ :. \ : $ hence all powers cycle back onto $ \ rm \ : 1, \ : \ alpha, \ :, \ ldots, \ alpha ^ { n - 1 } \ :, \ : $ i. e. $ \ rm \ : \ mathbb z [ \ alpha ] \ : $ is a finitely generated $ \ : \ mathbb z $ - module. possibly also the motivation for the name had to do more specifically with rings of cyclotomic integers. however, as plausible as that may seem, i don't recall the existence of any historical documents that provide solid evidence in support of such speculations. beware that one has to be very careful when reading such older literature. some authors mistakenly read modern notions into terms which have no such denotation in their original usage. to provide some context i recommend reading lemmermeyer and schappacher's introduction to the english edition of hilbert ’ s zahlbericht. below is a pertinent excerpt. below is an excerpt from leo corry's modern algebra and the rise of mathematical structures, p. 149. below are a couple typical examples of said speculative etymology of the term " ring " via the " circling back " nature of integral dependence, from harvey cohn's advanced number theory, p. 49. $ \ quad $ the designation of the letter $ \ mathfrak d $ for the integral domain has some historical importance going back to gauss's work on quadratic forms. gauss $ \ left ( 1800 \ right ) $ noted that for certain quadratic forms $ ax ^ 2 + bxy + cy ^ 2 $ the discriminant need not be square - free, although $ a
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$, $ b $, $ c $ are relatively prime. for example, $ x ^ 2 - 45y ^ 2 $ has $ d = 4 \ cdot45 $. the $ 4 $ was ignored for the reason that $ 4 | d $ necessarily by virtue of gauss's requirement that $ b $ be even, but the factor of $ 3 ^ 2 $ in $ d $ caused gauss to refer to the form as one of " order $ 3 $. " eventually, the forms corresponding to a value of $ d $ were called an " order " ( ordnung ). dedekind retained this word for what is here called an " integral domain. " $ \ quad $ the term " ring " is a contraction of " zahlring " introduced by hilbert $ \ left ( 1892 \ right ) $ to denote ( in our present context ) the ring generated by the rational integers and a quadratic integer $ \ eta $ defined by $ $ \ eta ^ 2 + b \ eta + c = 0. $ $ it would seem that module $ \ left [ 1, \ eta \ right ] $ is called a zahlring because $ \ eta ^ 2 $ equals $ - b \ eta - c $ " circling directly back " to an element of $ \ left [ 1, \ eta \ right ] $. this word has been maintained today. incidentally, every zahlring is an integral domain and the converse is true for quadratic fields. and from rotman's advanced modern algebra, p. 81.
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ok, my bad. the error is in the last equation : \ begin { align } kl ( p, q ) & = - \ int p ( x ) \ log q ( x ) dx + \ int p ( x ) \ log p ( x ) dx \ \ \ \ & = \ frac { 1 } { 2 } \ log ( 2 \ pi \ sigma _ 2 ^ 2 ) + \ frac { \ sigma _ 1 ^ 2 + ( \ mu _ 1 - \ mu _ 2 ) ^ 2 } { 2 \ sigma _ 2 ^ 2 } - \ frac { 1 } { 2 } ( 1 + \ log 2 \ pi \ sigma _ 1 ^ 2 ) \ \ \ \ & = \ log \ frac { \ sigma _ 2 } { \ sigma _ 1 } + \ frac { \ sigma _ 1 ^ 2 + ( \ mu _ 1 - \ mu _ 2 ) ^ 2 } { 2 \ sigma _ 2 ^ 2 } - \ frac { 1 } { 2 } \ end { align } note the missing $ - \ frac { 1 } { 2 } $. the last line becomes zero when $ \ mu _ 1 = \ mu _ 2 $ and $ \ sigma _ 1 = \ sigma _ 2 $.
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in practice, most people stick to relatively low orders, usually first or second order. this view is often challenged by more theoretical researchers that believe in more accurate answers. the rate of convergence for simple smooth problems is well documented, for example see bill mitchell's comparison of hp adaptivity. while for theoretical works it is nice to see what the convergence rate are, for more application oriented among us this concern is balanced with constitutive laws, necessary precision, and code complexity. it doesn't make much since in many porous media problems that solve over a highly discontinuous media to have high order methods, the numerical error will dominate the discretization errors. the same concern applies for problems that include a large number of degrees of freedom. since low - order implicit methods have a smaller bandwidth and often a better conditioning, the high order method becomes too costly to solve. finally the code complexity of switching orders and types of polynomials is usually too much for the graduate students running the application codes.
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there are some cases of bio - metallic materials, as hinted at by the comments. but these are relatively small amount of metal. it's not that there is a lack of metal available. iron in particular is the fourth most common element in the earth's crust. most soil that has a reddish color has iron in it. there are several reasons you don't see iron exoskeletons on animals all the time. firstly, metallic iron ( in chemistry terms, fully reduced, oxidation state 0 ) has a high energetic cost to create. iron is the second most common metal after aluminum on the earth's crust but it's almost entirely present in oxidized states - that's to say : as rust. most biological iron functions in the + 2 / + 3 oxidation state, which is more similar to rust than metal. cytochromes and haemoglobin are examples of how iron is more valuable as a chemically active biological agent than a structural agent, using oxidized iron ions as they do. aluminium, the most common metal on earth, has relatively little biological activity - one might assume because its redox costs are even higher than iron. as to why reduced biometal doesn't show up very often, inability of biological systems to deposit reduced ( metallic ) metals is not one of them. there are cases of admittedly small pieces of reduced metal being produced by biological systems. the magnetosomes in magnetotactic bacteria are mentioned, but there are also cases of reduced gold being accumulated by microorganisms. bone and shell are examples of biomineralization where the proteins depositing the calcium carbonate or other minerals in the material are structured by the proteins to be stronger than they would be as a simple crystal. most of the examples here have very little or no metal, but rather minerals like the chrysomallon squamiferum cited by @ navyguymarko and @ loki'sbane here. the iron sulfide looks metallic but it is a mineral, akin to a bone. while iron skeletons might seem to be an advantage, they are electrochemically unstable - oxygen and water will tend to oxidize ( rust ) them quickly and the organism would have to spend a lot of energy keeping it in working form. electrical conductivity sounds useful, but the nervous system favors exquisite levels of control over bulk current flow, even in cases like electric eels, whose current is produced by gradients from acetylcholine. what's more,
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biological materials actually perform as well as or better than metal when they need to. spider silk has a greater tensile strength than steel ( along the direction of the thread ). mollusk shells are models for tank armor - they are remarkably resistant to puncture and breakage. bone is durable for most purposes and flexible in addition. the time it would take for metallized structures to evolve biologically are likely too long. by the time the metalized version of an organ or skeleton got started, the bones, shells and fibers we know probably have a big lead and selective advantage.
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in a svm you are searching for two things : a hyperplane with the largest minimum margin, and a hyperplane that correctly separates as many instances as possible. the problem is that you will not always be able to get both things. the c parameter determines how great your desire is for the latter. i have drawn a small example below to illustrate this. to the left you have a low c which gives you a pretty large minimum margin ( purple ). however, this requires that we neglect the blue circle outlier that we have failed to classify correct. on the right you have a high c. now you will not neglect the outlier and thus end up with a much smaller margin. so which of these classifiers are the best? that depends on what the future data you will predict looks like, and most often you don't know that of course. if the future data looks like this : then the classifier learned using a large c value is best. on the other hand, if the future data looks like this : then the classifier learned using a low c value is best. depending on your data set, changing c may or may not produce a different hyperplane. if it does produce a different hyperplane, that does not imply that your classifier will output different classes for the particular data you have used it to classify. weka is a good tool for visualizing data and playing around with different settings for an svm. it may help you get a better idea of how your data look and why changing the c value does not change the classification error. in general, having few training instances and many attributes make it easier to make a linear separation of the data. also that fact that you are evaluating on your training data and not new unseen data makes separation easier. what kind of data are you trying to learn a model from? how much data? can we see it?
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it is not possible to produce white light without an efficient blue led, either using rgb leds or a blue led + yellow phosphor. the breakthrough was the invention of the high - brightness gallium - nitride blue led by shuji nakamura at nichia in the early 1990s. it still took a while to get the overall efficiency up to the level of fluorescent bulbs, and it's only in the last decade that leds finally came out on top.
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there is a very different mechanism for generation ( and detection ) of ultraviolet, visible and infrared light vs radio waves. for the first, it is possible to generate it using chemical reactions ( that is, chemiluminescence, bioluminescence ) with a typical energy of order of 2 ev ( electronovolts ). also, it is easy to detect with similar means - coupling to a bond ( e. g. using opsins ). for much longer electromagnetic waves, and much lower energies per photon, such mechanism does not work. there are two reasons : typical energy levels for molecules ( but it can be worked around ), thermal noise has energies ( 0. 025 ev ) which are higher than radio wave photon energies ( < 0. 001 ev ) ( it rules out both controlled creation and detection using molecules ). in other words - radiation which is less energetic than thermal radiation ( far infrared ) is not suitable for communication using molecular mechanisms, as thermal noise jams transmission ( making the sender firing at random and making the receiver being blind by noise way stronger than the signal ). however, one can both transmit, and detect it, using wires. in principle it is possible ; however, without good conductors ( like metals, not - salt solutions ) it is not an easy task ( not impossible though ).
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simulators designed specifically for oxford nanopore : nanosim nanosim - h silico readsim deepsimulator general long read simulators : loresim loresim 2 fastqsim longislnd for an exhaustive list of existing read simulators, see page 15 of my thesis, novel computational techniques for mapping and classifying next - generation sequencing data.
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yes, you can. and you do not even need to leave the earth to do it. you are always viewing things in the past, just as you are always hearing things in the past. if you see someone do something, who is 30 meters away, you are seeing what happened $ ( 30 \ ; \ mathrm { m } ) / ( 3 \ times10 ^ 8 \ ; \ mathrm { m } / \ mathrm { s } ) = 0. 1 \ ; \ mu \ mathrm { s } $ in the past. if you had a mirror on the moon ( about 238k miles away ), you could see about 2. 5 seconds into earth's past. if that mirror was on pluto, you could see about 13. 4 hours into earth's past. if you are relying on hearing, you hear an event at 30 m away about 0. 1 s after it occurs. that is why runners often watch the starting pistol at an event, because they can see a more recent picture of the past than they can hear. to more directly answer the intent of your question : yes, if you could magically be transported 27 lightyears away, or had a mirror strategically placed 13. 5 lightyears away, you could see yourself being born.
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in general terms, there are the $ o ( n ^ 2 ) $ sorting algorithms, such as insertion sort, bubble sort, and selection sort, which you should typically use only in special circumstances ; quicksort, which is worst - case $ o ( n ^ 2 ) $ but quite often $ o ( n \ log n ) $ with good constants and properties and which can be used as a general - purpose sorting procedure ; the $ o ( n \ log n ) $ algorithms, like merge - sort and heap - sort, which are also good general - purpose sorting algorithms ; and the $ o ( n ) $, or linear, sorting algorithms for lists of integers, such as radix, bucket and counting sorts, which may be suitable depending on the nature of the integers in your lists. if the elements in your list are such that all you know about them is the total order relationship between them, then optimal sorting algorithms will have complexity $ \ omega ( n \ log n ) $. this is a fairly cool result and one for which you should be able to easily find details online. the linear sorting algorithms exploit further information about the structure of elements to be sorted, rather than just the total order relationship among elements. even more generally, optimality of a sorting algorithm depends intimately upon the assumptions you can make about the kind of lists you're going to be sorting ( as well as the machine model on which the algorithm will run, which can make even otherwise poor sorting algorithms the best choice ; consider bubble sort on machines with a tape for storage ). the stronger your assumptions, the more corners your algorithm can cut. under very weak assumptions about how efficiently you can determine " sortedness " of a list, the optimal worst - case complexity can even be $ \ omega ( n! ) $. this answer deals only with complexities. actual running times of implementations of algorithms will depend on a large number of factors which are hard to account for in a single answer.
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the " stuff " sticks to itself better than it sticks to the cookie. now if you pull the cookies apart, you create a region of local stress, and one of the two interfaces will begin to unstick. at that point, you get something called " stress concentration " at the tip of the crack ( red arrow ) - where the tensile force concentrates : to get the stuff to start separating at a different part of the cookie, you need to tear the stuffing ( which is quite good at sticking to itself ) and initiate a delamination at a new point ( where there is no stress concentration ). those two things together explain your observation. cookie picture credit ( also explanation about manufacturing process introducing a bias ) update a plausible explanation was given in this article describing work by cannarella et al : nabisco won ’ t divulge its oreo secrets, but in 2010, newman ’ s own — which makes a very similar “ newman - o ” — let the discovery channel into its factory to see how their version of cookies are made. the key aspect for twist - off purposes : a pump applies the cream onto one wafer, which is then sent along the line until a robotic arm places a second wafer on top of the cream shortly after. the cream always adheres better to one of these wafers — and all of the cookies in a single box end up oriented in the same direction. which side is the stronger wafer - to - cream interface? “ we think we know, ” says spechler. the key is that fluids flow better at high temperatures. so the hot cream flows easily over the first wafer, filling in the tiny cracks of the cookie and sticking to it like hot glue, whereas the cooler cream just kind of sits on the edges of those crevices.
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you are right that the two algorithms of dijkstra ( shortest paths from a single start node ) and prim ( minimal weight spanning tree starting from a given node ) have a very similar structure. they are both greedy ( take the best edge from the present point of view ) and build a tree spanning the graph. the value they minimize however is different. dijkstra selects as next edge the one that leads out from the tree to a node not yet chosen closest to the starting node. ( then with this choice, distances are recalculated. ) prim choses as edge the shortest one leading out of the tree constructed so far. so, both algorithms chose a " minimal edge ". the main difference is the value chosen to be minimal. for dijkstra it is the length of the complete path from start node to the candidate node, for prim it is just the weight of that single edge. to see the difference you should try to construct a few examples to see what happens, that is really instructive. the simplest example that shows different behaviour is a triangle $ x, y, z $ with edges $ \ { x, y \ } $ and $ \ { x, z \ } $ of length 2, while $ \ { y, z \ } $ has length 1. starting in $ x $ dijkstra will choose $ \ { x, y \ } $ and $ \ { x, z \ } $ ( giving two paths of length 2 ) while prim chooses $ \ { x, y \ } $ and $ \ { y, z \ } $ ( giving spanning tree of weight 3 ). as for kruskal, that is slightly different. it solves the minimal spanning tree, but during execution it chooses edge that may not form a tree, they just avoid cycles. so the partial solutions may be disconnected. in the end you get a tree.
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the power spectral density describes the density of power in a stationary random process $ x ( t ) $ per unit of frequency. by the wiener - khinchin theorem, it can be calculated as follows for a wide - sense stationary random process : $ $ s _ { xx } ( f ) = \ int _ { - \ infty } ^ { \ infty } r _ { xx } ( \ tau ) e ^ { - j2 \ pi f \ tau } d \ tau $ $ where $ r _ { xx } ( \ tau ) $ is the autocorrelation function of the process $ x ( t ) $ : $ $ r _ { xx } ( \ tau ) = \ mathbb { e } \ left ( x ( t ) x ( t - \ tau ) \ right ) $ $ this is only valid for a wide - sense stationary process because its autocorrelation function is only a function of the time lag $ \ tau $ and not the absolute time $ t $ ; stated differently, this means that its second - order statistics don't change as a function of time. with that said, if you have a sufficiently - detailed and accurate statistical model for your signal, then you can calculate its power spectral density using the relationship above. as an example, this can be used to calculate the power spectral density of communications signals, given the statistics of the information symbols carried by the signal and any pulse shaping employed during transmission. in most practical situations, this level of information is not available, however, and one must resort to estimating a given signal's power spectral density. one very straightforward approach is to take the squared magnitude of its fourier transform ( or, perhaps, the squared magnitude of several short - time fourier transforms and average them ) as the estimate of the psd. however, assuming that the signal you're observing contains some stochastic component ( which is often the case ), this is again just an estimate of what the true underlying psd is based upon a single realization ( i. e. a single observation ) of the random process. whether the power spectrum that you calculate bears any meaningful resemblance to the actual psd of the process is situation - dependent. as this previous post notes, there are many methods for psd estimation ; which is most suitable depends upon the character of the random process, any a priori information that you might have, and what features of the signal you're most interested in.
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it does. you would find the average percentage of the atmosphere that is argon is very slightly higher at the floor of valleys. however, bear in mind first of all it wouldn't be anywhere near a complete stratification - - a layer of pure argon, then another of pure n2, and so on. a mixture of nearly ideal gases doesn't do that, at least at equilibrium, because it would eliminate the considerable entropy of mixing. ( it can happen in liquids because liquids have strong intermolecular forces that normally favor separation and oppose the entropy of mixing. ) another way to think about it is that since the atoms and molecules in gases don't ( much ) interact, there's nothing stopping an individual argon atom going slightly faster than nearby nitrogen and oxygen molecules from bouncing up higher than they do. what you would get in a theoretical ideal ( uniform gravitational field, complete stillness - - no wind - - and uniform temperature ) would be an exponential fall of pressure with altitude, and the exponential for heavier gases would be steeper than for lighter gases. that would result in enrichment of the heavier gases at lower altitudes. a little work starting from the boltzmann distribution of gravitational potential energies of each type of atom and molecule would get you an ideal estimate of the argon excess as a function of altitude. in practice the lower atmosphere has so much mixing due to wind and big thermal gradients that i doubt you could even measure the mild excess of argon and other heavy gases. there is one fascinating short - term exception, which bears directly on your question. sometimes volcanoes will belch out a considerable quantity of co2, which is significantly denser than air, and this co2 can accumulate briefly in a thick layer at the bottom of a valley or over a lake, if there isn't much wind. it can persist for some hours, perhaps days, before it diffuses away and is mixed with the rest of the atmosphere. then indeed the valley bottom becomes an invisible death trap for humans and animals : walk into the valley, or be unable to exit fast enough when it happens, and you will suffocate for no reason you can see. the most famous example of this is the lake nyos disaster in 1986 which killed thousands of humans and animals. i think the government now has mixing devices installed in that lake to prevent any future sudden release of co2.
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here's a video of physicist richard feynman discussing this question. imagine a blue dot and a red dot. they are in front of you, and the blue dot is on the right. behind them is a mirror, and you can see their image in the mirror. the image of the blue dot is still on the right in the mirror. what's different is that in the mirror, there's also a reflection of you. from that reflection's point of view, the blue dot is on the left. what the mirror really does is flip the order of things in the direction perpendicular to its surface. going on a line from behind you to in front of you, the order in real space is your back your front dots mirror the order in the image space is mirror dots your front your back although left and right are not reversed, the blue dot, which in reality is lined up with your right eye, is lined up with your left eye in the image. the key is that you are roughly left / right symmetric. the eye the blue dot is lined up with is still your right eye, even in the image. imagine instead that two - face was looking in the mirror. ( this is a fictional character whose left and right side of his face look different. his image on wikipedia looks like this : ) if two - face looked in the mirror, he would instantly see that it was not himself looking back! if he had an identical twin and looked right at the identical twin, the " normal " sides of their face would be opposite each other. two - face's good side is the right. when he looked at his twin, the twin's good side would be to the original two - face's left. instead, the mirror two - face's good side is also to the right. here is an illustration : so two - face would not be confused by the dots. if the blue dot is lined up with two - face's good side, it is still lined up with his good side in the mirror. here it is with the dots : two - face would recognize that left and right haven't been flipped so much as forward and backward, creating a different version of himself that cannot be rotated around to fit on top the original.
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you can think of the dct as a compression step. typically with mfccs, you will take the dct and then keep only the first few coefficients. this is basically the same reason that the dct is used in jpeg compression. dcts are chosen because their boundary conditions work better on these types of signals. let's contrast the dct with the fourier transform. the fourier transform is made up of sinusoids that have an integer number of cycles. this means, all of the fourier basis functions start and end at the same value - - they do not do a good job of representing signals that start and end at different values. remember that the fourier transform assumes a periodic extension : if you imagine your signal on a sheet of paper, the fourier transform wants to roll that sheet into a cylinder so that the left and right sides meet. think of a spectrum that is shaped roughly like a line with negative slope ( which is pretty typical ). the fourier transform will have to use a lot of different coefficients to fit this shape. on the other hand, the dct has cosines with half - integer numbers of cycles. there is, for example, a dct basis function that looks vaguely like that line with negative slope. it does not assume a period extension ( instead, an even extension ), so it will do a better job of fitting that shape. so, let's put this together. once you've computed the mel - frequency spectrum, you have a representation of the spectrum that is sensitive in a way similar to how human hearing works. some aspects of this shape are more relevant than others. usually, the larger more overarching spectral shape is more important than the noisy fine details in the spectrum. you can imagine drawing a smooth line to follow the spectral shape, and that the smooth line you draw might tell you just about as much about the signal. when you take the dct and discard the higher coefficients, you are taking this spectral shape, and only keeping the parts that are more important for representing this smooth shape. if you used the fourier transform, it wouldn't do such a good job of keeping the important information in the low coefficients. if you think about feeding the mfccs as features to a machine learning algorithm, these lower - order coefficients will make good features, since they represent some simple aspects of the spectral shape, while the higher - order coefficients that you discard are more noise - like and are not important to train on. additionally, training on the mel spectrum
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magnitudes themselves would probably not be as good because the particular amplitude at different frequencies are less important than the general shape of the spectrum.
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here is my favourite " wow " proof. theorem there exist two positive irrational numbers $ s, t $ such that $ s ^ t $ is rational. proof if $ \ sqrt2 ^ \ sqrt 2 $ is rational, we may take $ s = t = \ sqrt 2 $. if $ \ sqrt 2 ^ \ sqrt 2 $ is irrational, we may take $ s = \ sqrt 2 ^ \ sqrt 2 $ and $ t = \ sqrt 2 $ since $ ( \ sqrt 2 ^ \ sqrt 2 ) ^ \ sqrt 2 = ( \ sqrt 2 ) ^ 2 = 2 $.
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edit : i now think that this list is long enough that i shall be maintaining it over time - - updating it whenever i use a new book / learn a new subject. while every suggestion below should be taken with a grain of salt - - i will say that i spend a huge amount of time sifting through books to find the ones that conform best to my ( and hopefully your! ) learning style. here is my two cents ( for whatever that's worth ). i tried to include all the topics i could imagine you could want to know at this point. i hope i picked the right level of difficult. feel absolutely free to ask my specific opinion about any book. basic analysis : rudin - - apostol measure theory : royden ( only if you get the newest fourth edition ) - - folland general algebra : d & f - - rotman - - lang - - grillet finite group theory : isaacs - - kurzweil general group theory : robinson - - rotman ring theory : t. y. lam - - times two commutative algebra : eisenbud - - a & m - - reid homological algebra : weibel - - rotman - - vermani category theory : mac lane - - adamek et. al - - berrick et. al - - awodey - - mitchell linear algebra : roman - - hoffman and kunze - - golan field theory : morandi - - roman complex analysis : ahlfors - - cartan - - freitag riemann surfaces : varolin ( great first read, can be a little sloppy though ) - - freitag ( overall great book for a second course in complex analysis! ) - - forster ( a little more old school, and with a slightly more algebraic bend then a differential geometric one ) - - donaldson scv : gunning et. al - - ebeling point - set topology : munkres - - steen et. al - - kelley differential topology : pollack et. al - - milnor - - lee algebraic topology : bredon - - may - - bott and tu ( great, great book ) - - rotman - - massey - - tom dieck differential geometry : do carmo - - spivak - - jost - - lee representation theory of finite groups : serre - - steinberg - - liebeck - - isaacs general representation theory : fulton and harris - - humphreys - - hall representation theory of compact groups :
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tom dieck et. al - - sepanski ( linear ) algebraic groups : springer - - humphreys " elementary " number theory : niven et. al - - ireland et. al algebraic number theory : ash - - lorenzini - - neukirch - - marcus - - washington fourier analysis - - katznelson modular forms : diamond and shurman - - stein local fields : lorenz and levy - - read chapters 23, 24, 25. this is by far my favorite quick reference, as well as " learning text " for the basics of local fields one needs to break into other topics ( e. g. class field theory ). serre - - this is the classic book. it is definitely low on the readability side, especially notationally. it also has a tendency to consider things in more generality than is needed at a first go. this isn't bad, but is not good if you're trying to " brush up " or quickly learn local fields for another subject. fesenko et. al - - a balance between 1. and 2. definitely more readable than 2., but more comprehensive than 1. if you are wondering whether or not so - and - so needs henselian, this is the place i'd check. iwasawa - - a great place to learn the bare - bones of what one might need to learn class field theory. i am referencing, in particular, the first three chapters. if you are dead - set on just learning what you need to, this is a pretty good reference, but if you're likely to wonder about why so - and - so theorem is true, or get a broader understanding of the basics of local fields, i recommend 1. class field theory : lorenz and levy - - read chapters 28 - 32, second only to iwasawa, but with a different flavor ( cohomological vs. formal group laws ) tate and artin - - the classic book. a little less readable then any of the alternatives here. childress - - focused mostly on the global theory opposed to the local. actually deduces local at the end as a result of global. thus, very old school. iwasawa ( read the rest of it! ) milne - - where i first started learning it. very good, but definitely roughly hewn. a lot of details are left out, and he sometimes forgets to tell you where you are going. metric groups : markley algebraic geometry : reid - -
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shafarevich - - hartshorne - - griffiths and harris - - mumford
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ok, this question appears to have generated some controversy. on the one hand is the answer by niels nielsen ( currently accepted ), which implies that the orange color is from sodium. on the other hand is the answer by stessenj, which implies that the orange is normal black body radiation from the soot. plus there are lots of commentators arguing about rightness or wrongness of the sodium answer. the only good way to settle the matter is an experiment. i did it, with some modifications. first, instead of gas stove i used a jet lighter ( zl - 3 zengaz ). second, instead of humidifier i used a simple barber water spray. the third necessary component is a diffraction grating, a cheap one i had bought on aliexpress. i inserted it into colorless safety goggles to avoid necessity for a third hand. when i lit the lighter i saw a set of images in the first diffraction order : violet, blue, green, yellow and some blurred dim red. so far consistent with the spectrum of blue flame given on wikipedia. then i sprayed water in the air, simultaneously moving the lighter trying to find the place where the flame will change color. as the flame got orange jets instead of initial blue, i noticed orange image of the flame appear between red and yellow images in the diffraction grating. below is a photo i could take with the grating attached to a photo camera's lens, having mounted the camera on a tripod and holding the lighter and spray in both hands while 10s exposure was in progress ( sorry for bad quality ). notice the yellow / orange ( colors are not calibrated ) tall spike at the rhs : that is the part only present in the orange flame. ( the jet indeed became visibly taller when it changed its color to orange. ) from this follows that the orange color indeed comes from sodium, otherwise the orange flame's image would be much wider and spread into multiple colors like the flame from a candle or a non - jet lighter. the readers are welcome to replicate this experiment. edit ok, i've managed to measure some spectra using my amadeus spectrometer with custom driver. i used 15 s integration time with the flame about 3 - 5 cm from the sma905 connector on the spectrometer body. below the two spectra are superimposed, with the blue curve corresponding to the blue flame, and the orange one corresponds to the flame with some orange. i've filtered the data
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with 5 - point moving average before plotting. the spectrometer has lower sensitivity near uv and ir, so disregard the noise there. ( click the image for a larger version. ) what's worth noting is that not only the sodium 590 nm line is present in the orange flame, but also two potassium lines – 766 nm and 770 nm. edit2 just tried the same with a humidifier instead of the spray. the result with filtered tap water is the same : orange flame with sodium peak. with distilled water, although the experiment with the spray still resulted in orange flame ( basically the same as with tap water ), with the humidifier i got no orange at all. anyway, in no one case was i able to make the lighter emit continuous spectrum. whenever i got orange flame, it always appeared to be sodium d doublet, not continuous spectrum.
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as an extension to moyner's answer, the on - chip sqrt is usually an rsqrt, i. e. a reciprocal square root that computes $ a \ rightarrow 1 / \ sqrt { a } $. so if in your code you're only going to use $ 1 / r $ ( if you're doing molecular dynamics, you are ), you can compute r = rsqrt ( r2 ) directly and save yourself the division. the reason why rsqrt is computed instead of sqrt is that its newton iteration has no divisions, only additions and multiplications. as a side - note, divisions are also computed iteratively and are almost just as slow as rsqrt in hardware. if you're looking for efficiency, you're better off trying to remove superfluous divisions. some more modern architectures such as ibm's power architectures do not provide rsqrt per - se, but an estimate accurate to a few bits, e. g. frsqrte. when a user calls rsqrt, this generates an estimate and then one or two ( as many as required ) iterations of newton's or goldschmidt's algorithm using regular multiplications and additions. the advantage of this approach is that the iteration steps may be pipelined and interleaved with other instructions without blocking the fpu ( for a very nice overview of this concept, albeit on older architectures, see rolf strebel's phd thesis ). for interaction potentials, the sqrt operation can be avoided entirely by using a polynomial interpolant of the potential function, but my own work ( implemented in mdcore ) in this area show that, at least on x86 - type architectures, the sqrt instruction is fast enough. update since this answer seems to be getting quite a bit of attention, i would also like to address the second part of your question, i. e. is it really worth it to try to improve / eliminate basic operations such as sqrt? in the context of molecular dynamics simulations, or any particle - based simulation with cutoff - limited interactions, there is a lot to be gained from better algorithms for neighbour finding. if you're using cell lists, or anything similar, to find neighbours or create a verlet list, you will be computing a large number of spurious pairwise distances. in the naive case, only 16 % of particle pairs inspected will actually be within the cutoff distance of
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each other. although no interaction is computed for such pairs, accessing the particle data and computing the spurious pairwise distance carries a large cost. my own work in this area ( here, here, and here ), as well as that of others ( e. g. here ), show how these spurious computations can be avoided. these neighbour - finding algorithms even out - perform verlet lists, as described here. the point i want to emphasize is that although there may be some improvements to gain from better knowing / exploiting the underlying hardware architecture, there are also potentially larger gains to be had in re - thinking the higher - level algorithms.
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edit : i am rewriting the answer in response to updates to the original question. tl ; dr : use cram background 1 : quality binning and fastq compression in the old days, base callers outputted base quality at full resolution – you could see quality from q2 to q40 in full range. as a result, quality strings were like semi - random strings and very difficult to compress. later people gradually realized that keeping base quality in low resolution wouldn't affect downstream analysis. the illumina basecaller started to output quality in 8 distinct values and later changed that to 4 bins. this change greatly simplified quality string and made them compressed better. for example, in old days, a 30x bam would take ~ 100 gb. with quality binning, it would only take ~ 60 gb. background 2 : gatk base quality recalibration in early 2010s, illumina base quality was not calibrated well. gatk people introduced bqsr to correct that and observed noticeable improvement in snp accuracy. nonetheless, with improved illumina base caller, their base quality became more accurate. meanwhile, the world moved to 30x deep sequencing. the depth overwhelms slight inaccuracy in quality. i would say around 2015, bqsr was already unnecessary for data produced at the time. does it hurt to apply bqsr? yes. first, bqsr introduces subtle biases towards the reference and towards known snps. second, bqsr distorts the data. at least for some datasets, i observed that snp accuracy dropped with variant quality after bqsr ; i didn't observe this with raw quality. third, bqsr is slow. fourth, for new sequencers producing data at higher quality, bqsr is likely to decrease data quality. last, related to the question, bqsr added another semi - random quality string and made compression even harder. nowadays, running bqsr is a waste of resource for worse results. the official gatk best practice no longer uses bqsr according to their wdl file. cram these days a 30x human cram only takes ~ 15 gb ( see this file ). this is a huge contrast to ~ 100 gb bam in early 2010s. op only saw ~ 10 % saving probably due to a ) bqsr and / or b ) old data with full quality resolution. on encoding / decoding speed, cram was much
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slower than bam. not any more. the latest htslib implementation of cram is faster than bam on encoding and only slightly slower on decoding. the poor performance of igv on cram could be that the java cram decoder is not as optimized. it is true that cram is not as widely supported as bam. however, all the other alternatives are much worse. petagene said they had igv - pg ( pdf ) for their format. that is not official igv and i couldn't find more recent update beyond the 2019 press release. i don't see other viable options. note that the common practice is to keep all raw reads, mapped or not, in bam or cram such that you can get raw reads back later. bam / cram additionally keeps metadata like read group, sample name, run information etc and is actually more popular than fastq in large sequencing centers. also note that you don't need to sort cram by coordinate. unsorted cram is only a little larger than sorted cram. cram and its competitors the core cram developer, james bonfield, is one of the most knowledgeable researchers on compression ( and one of the best c programmers ) in this field. he has done a lot of compression evaluation over the years. the conclusion is that on a fair benchmark, cram is comparable to the best tools so far in terms of compression ratio. petagene could compress better in the plot @ terdon showed mostly because it has a special treatment of the oq tag generated by gatk bqsr. it is a typical trick marketing people use to make their methods look better. with bqsr phased out, this plot is no longer relevant. on commercial software in general, i welcome commercial tools and think they are invaluable to users. i also have huge respect to dragen developers. however, on fastq storage, i would strongly recommend against closed - source compressors. if those tools go under, you may lose your data. not worth it.
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since general relativity is a local theory just like any good classical field theory, the earth will respond to the local curvature which can change only once the information about the disappearance of the sun has been communicated to the earth's position ( through the propagation of gravitational waves ). so yes, the earth would continue to orbit what should've been the position of the sun for 8 minutes before flying off tangentially. but i should add that such a disappearance of mass is unphysical anyway since you can't have mass - energy just poofing away or even disappearing and instantaneously appearing somewhere else. ( in the second case, mass - energy would be conserved only in the frame of reference in which the disappearance and appearance are simultaneous - this is all a consequence of gr being a classical field theory ). a more realistic situation would be some mass configuration shifting its shape non - spherically in which case the orbits of satellites would be perturbed but only once there has been enough time for gravitational waves to reach the satellite.
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pca computes eigenvectors of the covariance matrix ( " principal axes " ) and sorts them by their eigenvalues ( amount of explained variance ). the centered data can then be projected onto these principal axes to yield principal components ( " scores " ). for the purposes of dimensionality reduction, one can keep only a subset of principal components and discard the rest. ( see here for a layman's introduction to pca. ) let $ \ mathbf x _ \ text { raw } $ be the $ n \ times p $ data matrix with $ n $ rows ( data points ) and $ p $ columns ( variables, or features ). after subtracting the mean vector $ \ boldsymbol \ mu $ from each row, we get the centered data matrix $ \ mathbf x $. let $ \ mathbf v $ be the $ p \ times k $ matrix of some $ k $ eigenvectors that we want to use ; these would most often be the $ k $ eigenvectors with the largest eigenvalues. then the $ n \ times k $ matrix of pca projections ( " scores " ) will be simply given by $ \ mathbf z = \ mathbf { xv } $. this is illustrated on the figure below : the first subplot shows some centered data ( the same data that i use in my animations in the linked thread ) and its projections on the first principal axis. the second subplot shows only the values of this projection ; the dimensionality has been reduced from two to one : in order to be able to reconstruct the original two variables from this one principal component, we can map it back to $ p $ dimensions with $ \ mathbf v ^ \ top $. indeed, the values of each pc should be placed on the same vector as was used for projection ; compare subplots 1 and 3. the result is then given by $ \ hat { \ mathbf x } = \ mathbf { zv } ^ \ top = \ mathbf { xvv } ^ \ top $. i am displaying it on the third subplot above. to get the final reconstruction $ \ hat { \ mathbf x } _ \ text { raw } $, we need to add the mean vector $ \ boldsymbol \ mu $ to that : $ $ \ boxed { \ text { pca reconstruction } = \ text { pc scores } \ cdot \ text {
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eigenvectors } ^ \ top + \ text { mean } } $ $ note that one can go directly from the first subplot to the third one by multiplying $ \ mathbf x $ with the $ \ mathbf { vv } ^ \ top $ matrix ; it is called a projection matrix. if all $ p $ eigenvectors are used, then $ \ mathbf { vv } ^ \ top $ is the identity matrix ( no dimensionality reduction is performed, hence " reconstruction " is perfect ). if only a subset of eigenvectors is used, it is not identity. this works for an arbitrary point $ \ mathbf z $ in the pc space ; it can be mapped to the original space via $ \ hat { \ mathbf x } = \ mathbf { zv } ^ \ top $. discarding ( removing ) leading pcs sometimes one wants to discard ( to remove ) one or few of the leading pcs and to keep the rest, instead of keeping the leading pcs and discarding the rest ( as above ). in this case all the formulas stay exactly the same, but $ \ mathbf v $ should consist of all principal axes except for the ones one wants to discard. in other words, $ \ mathbf v $ should always include all pcs that one wants to keep. caveat about pca on correlation when pca is done on correlation matrix ( and not on covariance matrix ), the raw data $ \ mathbf x _ \ mathrm { raw } $ is not only centered by subtracting $ \ boldsymbol \ mu $ but also scaled by dividing each column by its standard deviation $ \ sigma _ i $. in this case, to reconstruct the original data, one needs to back - scale the columns of $ \ hat { \ mathbf x } $ with $ \ sigma _ i $ and only then to add back the mean vector $ \ boldsymbol \ mu $. image processing example this topic often comes up in the context of image processing. consider lenna - - one of the standard images in image processing literature ( follow the links to find where it comes from ). below on the left, i display the grayscale variant of this $ 512 \ times 512 $ image ( file available here ). we can treat this grayscale image as a $ 512 \ times 512 $ data matrix $ \ mathbf x _ \ text { raw } $. i perform pc
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##a on it and compute $ \ hat { \ mathbf x } _ \ text { raw } $ using the first 50 principal components. the result is displayed on the right. reverting svd pca is very closely related to singular value decomposition ( svd ), see relationship between svd and pca. how to use svd to perform pca? for more details. if a $ n \ times p $ matrix $ \ mathbf x $ is svd - ed as $ \ mathbf x = \ mathbf { usv } ^ \ top $ and one selects a $ k $ - dimensional vector $ \ mathbf z $ that represents the point in the " reduced " $ u $ - space of $ k $ dimensions, then to map it back to $ p $ dimensions one needs to multiply it with $ \ mathbf s ^ \ phantom \ top _ { 1 : k, 1 : k } \ mathbf v ^ \ top _ { :, 1 : k } $. examples in r, matlab, python, and stata i will conduct pca on the fisher iris data and then reconstruct it using the first two principal components. i am doing pca on the covariance matrix, not on the correlation matrix, i. e. i am not scaling the variables here. but i still have to add the mean back. some packages, like stata, take care of that through the standard syntax. thanks to @ stask and @ kodiologist for their help with the code. we will check the reconstruction of the first datapoint, which is : 5. 1 3. 5 1. 4 0. 2 matlab load fisheriris x = meas ; mu = mean ( x ) ; [ eigenvectors, scores ] = pca ( x ) ; ncomp = 2 ; xhat = scores ( :, 1 : ncomp ) * eigenvectors ( :, 1 : ncomp )'; xhat = bsxfun ( @ plus, xhat, mu ) ; xhat ( 1, : ) output : 5. 083 3. 5174 1. 4032 0. 21353 r x = iris [, 1 : 4 ] mu = colmeans ( x ) xpca = prcomp ( x ) ncomp = 2 xhat = xpca $ x [, 1 : ncomp ] % * % t ( xpca $ rotation [, 1 : ncomp ] ) xhat
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= scale ( xhat, center = - mu, scale = false ) xhat [ 1, ] output : sepal. length sepal. width petal. length petal. width 5. 0830390 3. 5174139 1. 4032137 0. 2135317 for worked out r example of pca reconstruction of images see also this answer. python import numpy as np import sklearn. datasets, sklearn. decomposition x = sklearn. datasets. load _ iris ( ). data mu = np. mean ( x, axis = 0 ) pca = sklearn. decomposition. pca ( ) pca. fit ( x ) ncomp = 2 xhat = np. dot ( pca. transform ( x ) [ :, : ncomp ], pca. components _ [ : ncomp, : ] ) xhat + = mu print ( xhat [ 0, ] ) output : [ 5. 08718247 3. 51315614 1. 4020428 0. 21105556 ] note that this differs slightly from the results in other languages. that is because python's version of the iris dataset contains mistakes. stata webuse iris, clear pca sep * pet *, components ( 2 ) covariance predict _ seplen _ sepwid _ petlen _ petwid, fit list in 1 iris seplen sepwid petlen petwid _ seplen _ sepwid _ petlen _ petwid setosa 5. 1 3. 5 1. 4 0. 2 5. 083039 3. 517414 1. 403214. 2135317
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i prefer to treat software tools and computers in a similar fashion to laboratory equipment, and in some sense biology in general. biologists are used to unexpected things happening in their experiments, and it's not uncommon for a new discovery to change the way that people look at something. things break down, cells die off quicker on a wednesday afternoon, results are inconsistent, and that third reviewer keeps on about doing that thing that's been done a hundred times before without anything surprising happening ( just not this time ). it's a good idea to record as much as can be thought of that might influence an experiment, and for software that includes any input data or command line options, and especially software version numbers. in this sense, a discovered software bug can be treated as a new discovery of how the world works. if the discovery is made public, and other people consider that it's important enough, then some people might revisit old research to see if it changes things. of course, the nice thing about software is that bugs can be reported back to the creators of programs, and possibly fixed, resulting in an improved version of the software at a later date. if the bug itself doesn't spark interest and the program gets fixed anyway, people unknowingly use newer versions, and there might be a bit more confusion and discussion about why results don't match similar studies carried out before the software change. if you want a bit of an idea of the biological equivalent of a major software bug, have a look at the barcode index switching issue, or the cell line contamination issue.
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you have some very strong color and geometry cues you can leverage. i would try the following : extract the green channel & apply watershed type algorithm on it, followed by connected components. subsequently compute component statistics ( area & bounding box ) for each component. retain only the components with area ~ = bounding box size. this will be true only for rectangular objects and will eliminate forests / wooded areas etc. isolate the white channel ( r = g = b ) and apply hough transform on the output. this will give you the lines. combine 1 & 2 to get your tennis courts.
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the problem with equispaced points is that the interpolation error polynomial, i. e. $ $ f ( x ) - p _ n ( x ) = \ frac { f ^ { ( n + 1 ) } ( \ xi ) } { ( n + 1 )! } \ prod _ { i = 0 } ^ n ( x - x _ i ), \ quad \ xi \ in [ x _ 0, x _ n ] $ $ behaves differently for different sets of nodes $ x _ i $. in the case of equispaced points, this polynomial blows up at the edges. if you use gauss - legendre points, the error polynomial is significantly better behaved, i. e. it doesn't blow up at the edges. if you use chebyshev nodes, this polynomial equioscillates and the interpolation error is minimal.
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no - one mentioned the inverse hyperbolic sine transformation. so for completeness i'm adding it here. this is an alternative to the box - cox transformations and is defined by \ begin { equation } f ( y, \ theta ) = \ text { sinh } ^ { - 1 } ( \ theta y ) / \ theta = \ log [ \ theta y + ( \ theta ^ 2y ^ 2 + 1 ) ^ { 1 / 2 } ] / \ theta, \ end { equation } where $ \ theta > 0 $. for any value of $ \ theta $, zero maps to zero. there is also a two parameter version allowing a shift, just as with the two - parameter bc transformation. burbidge, magee and robb ( 1988 ) discuss the ihs transformation including estimation of $ \ theta $. the ihs transformation works with data defined on the whole real line including negative values and zeros. for large values of $ y $ it behaves like a log transformation, regardless of the value of $ \ theta $ ( except 0 ). the limiting case as $ \ theta \ rightarrow0 $ gives $ f ( y, \ theta ) \ rightarrow y $. it looks to me like the ihs transformation should be a lot better known than it is.
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without a transformer the live wire is live relative to ground. if you are at " ground " potential then touching the live wire makes you part of the return path. { this image taken from an excellent discussion here with a transformer the output voltage is not referenced to ground - see diagram ( a ) below. there is no " return path " so you could ( stupidly ) safely touch the " live " conductor and ground and not received a shock. from the electricians guide i say " stupidly " as, while this arrangement is safer it is not safe unconditionally. this is because, if there is leakage or hard connection from the other side of the transformer to ground then there may still be a return path - as shown in ( b ) above. in the diagram the return path is shown as either capacitive or direct. if the coupling is capacitive then you may feel a " tickle " or somewhat mild " bite " from the live conductor. if the other conductor is grounded then you are back to the original transformlerless situation. ( capacitive coupling may occur when an appliance body is connected to a conductor but there is no direct connection from body to ground. the body to ground proximity forms a capacitor. ) so a transformer makes things safer by providing isolation relative to ground. murphy / circumstance will work to defeat this isoation. this is why, ideally, an isolating transformer should be used to protect only one item of equipment at a time. with one item a fault in the equipment will propbably not produce a dangerous situation. the transformer has done its job. but with n items of equipment - if one has a fault from neutral to case or is wired wrongly this may defeat the transformer such that a second faulty device may then present a hazard to the user. in figure ( b ) above, the first faulty device provides the link at bottom and the second provides the link at top. similarly :
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the probabilistic way : this is $ p [ n _ n \ leqslant n ] $ where $ n _ n $ is a random variable with poisson distribution of parameter $ n $. hence each $ n _ n $ is distributed like $ x _ 1 + \ cdots + x _ n $ where the random variables $ ( x _ k ) $ are independent and identically distributed with poisson distribution of parameter $ 1 $. by the central limit theorem, $ y _ n = \ frac1 { \ sqrt { n } } ( x _ 1 + \ cdots + x _ n - n ) $ converges in distribution to a standard normal random variable $ z $, in particular, $ p [ y _ n \ leqslant 0 ] \ to p [ z \ leqslant0 ] $. finally, $ p [ z \ leqslant0 ] = \ frac12 $ and $ [ n _ n \ leqslant n ] = [ y _ n \ leqslant 0 ] $ hence $ p [ n _ n \ leqslant n ] \ to \ frac12 $, qed. the analytical way, completing your try : hence, i know that what i need to do is to find $ \ lim \ limits _ { n \ to \ infty } i _ n $, where $ $ i _ n = \ frac { e ^ { - n } } { n! } \ int _ { 0 } ^ n ( n - t ) ^ ne ^ tdt. $ $ to begin with, let $ u ( t ) = ( 1 - t ) e ^ t $, then $ i _ n = \ dfrac { e ^ { - n } n ^ n } { n! } nj _ n $ with $ $ j _ n = \ int _ { 0 } ^ 1 u ( t ) ^ n \ mathrm dt. $ $ now, $ u ( t ) \ leqslant \ mathrm e ^ { - t ^ 2 / 2 } $ hence $ $ j _ n \ leqslant \ int _ 0 ^ 1 \ mathrm e ^ { - nt ^ 2 / 2 } \ mathrm dt \ leqslant \ int _ 0 ^ \ infty \ mathrm e ^ { - nt ^ 2 / 2 } \ mathrm dt = \ sqrt { \ frac { \ pi } {
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2n } }. $ $ likewise, the function $ t \ mapsto u ( t ) \ mathrm e ^ { t ^ 2 / 2 } $ is decreasing on $ t \ geqslant0 $ hence $ u ( t ) \ geqslant c _ n \ mathrm e ^ { - t ^ 2 / 2 } $ on $ t \ leqslant1 / n ^ { 1 / 4 } $, with $ c _ n = u ( 1 / n ^ { 1 / 4 } ) \ mathrm e ^ { - 1 / ( 2 \ sqrt { n } ) } $, hence $ $ j _ n \ geqslant c _ n \ int _ 0 ^ { 1 / n ^ { 1 / 4 } } \ mathrm e ^ { - nt ^ 2 / 2 } \ mathrm dt = \ frac { c _ n } { \ sqrt { n } } \ int _ 0 ^ { n ^ { 1 / 4 } } \ mathrm e ^ { - t ^ 2 / 2 } \ mathrm dt = \ frac { c _ n } { \ sqrt { n } } \ sqrt { \ frac { \ pi } { 2 } } ( 1 + o ( 1 ) ). $ $ since $ c _ n \ to1 $, all this proves that $ \ sqrt { n } j _ n \ to \ sqrt { \ frac \ pi2 } $. stirling formula shows that the prefactor $ \ frac { e ^ { - n } n ^ n } { n! } $ is equivalent to $ \ frac1 { \ sqrt { 2 \ pi n } } $. regrouping everything, one sees that $ i _ n \ sim \ frac1 { \ sqrt { 2 \ pi n } } n \ sqrt { \ frac \ pi { 2n } } = \ frac12 $. moral : the probabilistic way is shorter, easier, more illuminating, and more fun. caveat : my advice in these matters is, clearly, horribly biased.
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it's a rather rough algorithm, but i'd use the following procedure for a crude estimate : if, as you say, the purported $ f ( x ) $ that represents your $ ( x _ i, y _ i ) $ is already almost linear as $ x $ increases, what i'd do is to take differences $ \ dfrac { y _ { i + 1 } - y _ i } { x _ { i + 1 } - x _ i } $, and then use an extrapolation algorithm like the shanks transformation to estimate the limit of the differences. the result is hopefully a good estimate of this asymptotic slope. what follows is a mathematica demonstration. the wynn $ \ epsilon $ algorithm is a convenient implementation of the shanks transformation, and it is built in as the ( hidden ) function sequencelimit [ ]. we try out the procedure on the function $ $ \ frac4 { x ^ 2 + 3 } + 2 x + e ^ { - 4 x } + 3 $ $ xdata = randomreal [ { 20, 40 }, 25 ] ; ydata = table [ ( 3 + 13 * e ^ ( 4 * x ) + 6 * e ^ ( 4 * x ) * x + x ^ 2 + 3 * e ^ ( 4 * x ) * x ^ 2 + 2 * e ^ ( 4 * x ) * x ^ 3 ) / ( e ^ ( 4 * x ) * ( 3 + x ^ 2 ) ), { x, xdata } ] ; sequencelimit [ differences [ ydata ] / differences [ xdata ], method - > { " wynnepsilon ", degree - > 2 } ] 1. 999998 i might as well show off how simple the algorithm is : wynnepsilon [ seq _? vectorq ] : = module [ { n = length [ seq ], ep, res, v, w }, res = { } ; do [ ep [ k ] = seq [ [ k ] ] ; w = 0 ; do [ v = w ; w = ep [ j ] ; ep [ j ] = v + ( if [ abs [ ep [ j + 1 ] - w ] > 10 ^ - ( precision [ w ] ), ep [ j + 1 ] - w, 10 ^ - ( precision [ w ] ) ] ) ^ - 1 ;, { j, k - 1, 1, -
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1 } ] ; res = { res, ep [ if [ oddq [ k ], 1, 2 ] ] } ;, { k, n } ] ; flatten [ res ] ] last [ wynnepsilon [ differences [ ydata ] / differences [ xdata ] ] ] 1. 99966 this implementation is adapted from weniger's paper.
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all models are wrong, but some are useful. ( george e. p. box ) reference : box & draper ( 1987 ), empirical model - building and response surfaces, wiley, p. 424. also : g. e. p. box ( 1979 ), " robustness in the strategy of scientific model building " in robustness in statistics ( launer & wilkinson eds. ), p. 202.
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the exact mechanism is unclear. here are some possible causes : rapid collapsing of cavities inside the joint [ 1 ] ; rapid ligament stretching [ 1 ] ; breaking of intra - articular adhesions [ 1 ] ; escaping gases from synovial fluid [ 2 ] ; movements of joints, tendons and ligaments [ 2 ] ; mechanic interaction between rough surfaces [ 2 ], mostly in pathological situations like arthritis ( and it is called crepitus [ 3 ] ). there are no known bad effects of joint cracking [ 1, 4 ]. there are no long term sequelae of these noises, and they do not lead to future problems. there is no basis for the admonition to not crack your knuckles because it can lead to arthritis. there are no supplements or exercises to prevent these noises [ 4 ]. and no good effects either : knuckle " cracking " has not been shown to be harmful or beneficial. more specifically, knuckle cracking does not cause arthritis [ 5 ]. references : wikipedia contributors, " cracking joints, " wikipedia, the free encyclopedia, ( accessed july 22, 2014 ). the library of congress. everyday mysteries. what causes the noise when you crack a joint? available from ( accessed 22. 07. 2014 ) wikipedia contributors, " crepitus, " wikipedia, the free encyclopedia, ( accessed july 22, 2014 ). johns hopkins sports medicine patient guide to joint cracking & popping. available from ( accessed 22. 07. 2014 ) webmd, llc. will joint cracking cause osteoarthritis? available from ( accessed 22. 07. 2014 )
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( this is a fairly long answer, there is a summary at the end ) you are not wrong in your understanding of what nested and crossed random effects are in the scenario that you describe. however, your definition of crossed random effects is a little narrow. a more general definition of crossed random effects is simply : not nested. we will look at this at the end of this answer, but the bulk of the answer will focus on the scenario you presented, of classrooms within schools. first note that : nesting is a property of the data, or rather the experimental design, not the model. also, nested data can be encoded in at least 2 different ways, and this is at the heart of the issue you found. the dataset in your example is rather large, so i will use another schools example from the internet to explain the issues. but first, consider the following over - simplified example : here we have classes nested in schools, which is a familiar scenario. the important point here is that, between each school, the classes have the same identifier, even though they are distinct if they are nested. class1 appears in school1, school2 and school3. however if the data are nested then class1 in school1 is not the same unit of measurement as class1 in school2 and school3. if they were the same, then we would have this situation : which means that every class belongs to every school. the former is a nested design, and the latter is a crossed design ( some might also call it multiple membership. edit : for a discussion of the differences between multiple membership and crossed random effects, see here ), and we would formulate these in lme4 using : ( 1 | school / class ) or equivalently ( 1 | school ) + ( 1 | class : school ) and ( 1 | school ) + ( 1 | class ) respectively. due to the ambiguity of whether there is nesting or crossing of random effects, it is very important to specify the model correctly as these models will produce different results, as we shall show below. moreover, it is not possible to know, just by inspecting the data, whether we have nested or crossed random effects. this can only be determined with knowledge of the data and the experimental design. but first let us consider a case where the class variable is coded uniquely across schools : there is no longer any ambiguity concerning nesting or crossing. the nesting is explicit. let us now see this with an example in r, where we
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have 6 schools ( labelled i - vi ) and 4 classes within each school ( labelled a to d ) : > dt < - read. table ( " header = true, sep = ", ", na. strings = " na ", dec = ". ", strip. white = true ) > # update 1 : data was previously publicly available from > # > # but the link is now broken. > # update 2 : the link is broken again. a new link is used. the previous link was : > xtabs ( ~ school + class, dt ) class school a b c d i 50 50 50 50 ii 50 50 50 50 iii 50 50 50 50 iv 50 50 50 50 v 50 50 50 50 vi 50 50 50 50 we can see from this cross tabulation that every class id appears in every school, which satisfies your definition of crossed random effects ( in this case we have fully, as opposed to partially, crossed random effects, because every class occurs in every school ). so this is the same situation that we had in the first figure above. however, if the data are really nested and not crossed, then we need to explicitly tell lme4 : > m0 < - lmer ( extro ~ open + agree + social + ( 1 | school / class ), data = dt ) > summary ( m0 ) random effects : groups name variance std. dev. class : school ( intercept ) 8. 2043 2. 8643 school ( intercept ) 93. 8421 9. 6872 residual 0. 9684 0. 9841 number of obs : 1200, groups : class : school, 24 ; school, 6 fixed effects : estimate std. error t value ( intercept ) 60. 2378227 4. 0117909 15. 015 open 0. 0061065 0. 0049636 1. 230 agree - 0. 0076659 0. 0056986 - 1. 345 social 0. 0005404 0. 0018524 0. 292 > m1 < - lmer ( extro ~ open + agree + social + ( 1 | school ) + ( 1 | class ), data = dt ) summary ( m1 ) random effects : groups name variance std. dev. school ( intercept ) 95. 887 9. 792 class ( intercept ) 5. 790 2. 406 residual 2. 787 1. 669 number of obs : 1200, groups : school
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, 6 ; class, 4 fixed effects : estimate std. error t value ( intercept ) 60. 198841 4. 212974 14. 289 open 0. 010834 0. 008349 1. 298 agree - 0. 005420 0. 009605 - 0. 564 social - 0. 001762 0. 003107 - 0. 567 as expected, the results differ because m0 is a nested model while m1 is a crossed model. now, if we introduce a new variable for the class identifier : > dt $ classid < - paste ( dt $ school, dt $ class, sep = ". " ) > xtabs ( ~ school + classid, dt ) classid school i. a i. b i. c i. d ii. a ii. b ii. c ii. d iii. a iii. b iii. c iii. d iv. a iv. b i 50 50 50 50 0 0 0 0 0 0 0 0 0 0 ii 0 0 0 0 50 50 50 50 0 0 0 0 0 0 iii 0 0 0 0 0 0 0 0 50 50 50 50 0 0 iv 0 0 0 0 0 0 0 0 0 0 0 0 50 50 v 0 0 0 0 0 0 0 0 0 0 0 0 0 0 vi 0 0 0 0 0 0 0 0 0 0 0 0 0 0 classid school iv. c iv. d v. a v. b v. c v. d vi. a vi. b vi. c vi. d i 0 0 0 0 0 0 0 0 0 0 ii 0 0 0 0 0 0 0 0 0 0 iii 0 0 0 0 0 0 0 0 0 0 iv 50 50 0 0 0 0 0 0 0 0 v 0 0 50 50 50 50 0 0 0 0 vi 0 0 0 0 0 0 50 50 50 50 the cross tabulation shows that each level of class occurs only in one level of school, as per your definition of nesting. this is also the case with your data, however it is difficult to show that with your data because it is very sparse. both model formulations will now produce the same output ( that of the nested model m0 above ) : > m2 < - lmer ( extro ~ open + agree + social + ( 1 | school / classid ), data = dt ) > summary ( m2 ) random effects : groups name variance std. dev. classid : school
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( intercept ) 8. 2043 2. 8643 school ( intercept ) 93. 8419 9. 6872 residual 0. 9684 0. 9841 number of obs : 1200, groups : classid : school, 24 ; school, 6 fixed effects : estimate std. error t value ( intercept ) 60. 2378227 4. 0117882 15. 015 open 0. 0061065 0. 0049636 1. 230 agree - 0. 0076659 0. 0056986 - 1. 345 social 0. 0005404 0. 0018524 0. 292 > m3 < - lmer ( extro ~ open + agree + social + ( 1 | school ) + ( 1 | classid ), data = dt ) > summary ( m3 ) random effects : groups name variance std. dev. classid ( intercept ) 8. 2043 2. 8643 school ( intercept ) 93. 8419 9. 6872 residual 0. 9684 0. 9841 number of obs : 1200, groups : classid, 24 ; school, 6 fixed effects : estimate std. error t value ( intercept ) 60. 2378227 4. 0117882 15. 015 open 0. 0061065 0. 0049636 1. 230 agree - 0. 0076659 0. 0056986 - 1. 345 social 0. 0005404 0. 0018524 0. 292 it is worth noting that crossed random effects do not have to occur within the same factor - in the above the crossing was completely within school. however, this does not have to be the case, and very often it is not. for example, sticking with a school scenario, if instead of classes within schools we have pupils within schools, and we were also interested in the doctors that the pupils were registered with, then we would also have nesting of pupils within doctors. there is no nesting of schools within doctors, or vice versa, so this is also an example of crossed random effects, and we say that schools and doctors are crossed. a similar scenario where crossed random effects occur is when individual observations are nested within two factors simultaneously, which commonly occurs with so - called repeated measures subject - item data. typically each subject is measured / tested multiple times with / on different items and these same items are measured / tested by different subjects. thus, observations are clustered within subjects and within items, but
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items are not nested within subjects or vice - versa. again, we say that subjects and items are crossed. summary : tl ; dr the difference between crossed and nested random effects is that nested random effects occur when one factor ( grouping variable ) appears only within a particular level of another factor ( grouping variable ). this is specified in lme4 with : ( 1 | group1 / group2 ) where group2 is nested within group1. crossed random effects are simply : not nested. this can occur with three or more grouping variables ( factors ) where one factor is separately nested in both of the others, or with two or more factors where individual observations are nested separately within the two factors. these are specified in lme4 with : ( 1 | group1 ) + ( 1 | group2 )
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there is considerable overlap among these, but some distinctions can be made. of necessity, i will have to over - simplify some things or give short - shrift to others, but i will do my best to give some sense of these areas. firstly, artificial intelligence is fairly distinct from the rest. ai is the study of how to create intelligent agents. in practice, it is how to program a computer to behave and perform a task as an intelligent agent ( say, a person ) would. this does not have to involve learning or induction at all, it can just be a way to'build a better mousetrap '. for example, ai applications have included programs to monitor and control ongoing processes ( e. g., increase aspect a if it seems too low ). notice that ai can include darn - near anything that a machine does, so long as it doesn't do it'stupidly '. in practice, however, most tasks that require intelligence require an ability to induce new knowledge from experiences. thus, a large area within ai is machine learning. a computer program is said to learn some task from experience if its performance at the task improves with experience, according to some performance measure. machine learning involves the study of algorithms that can extract information automatically ( i. e., without on - line human guidance ). it is certainly the case that some of these procedures include ideas derived directly from, or inspired by, classical statistics, but they don't have to be. similarly to ai, machine learning is very broad and can include almost everything, so long as there is some inductive component to it. an example of a machine learning algorithm might be a kalman filter. data mining is an area that has taken much of its inspiration and techniques from machine learning ( and some, also, from statistics ), but is put to different ends. data mining is carried out by a person, in a specific situation, on a particular data set, with a goal in mind. typically, this person wants to leverage the power of the various pattern recognition techniques that have been developed in machine learning. quite often, the data set is massive, complicated, and / or may have special problems ( such as there are more variables than observations ). usually, the goal is either to discover / generate some preliminary insights in an area where there really was little knowledge beforehand, or to be able to predict future observations accurately. moreover, data mining procedures could be either'unsupervised'( we don't know
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the answer - - discovery ) or'supervised'( we know the answer - - prediction ). note that the goal is generally not to develop a more sophisticated understanding of the underlying data generating process. common data mining techniques would include cluster analyses, classification and regression trees, and neural networks. i suppose i needn't say much to explain what statistics is on this site, but perhaps i can say a few things. classical statistics ( here i mean both frequentist and bayesian ) is a sub - topic within mathematics. i think of it as largely the intersection of what we know about probability and what we know about optimization. although mathematical statistics can be studied as simply a platonic object of inquiry, it is mostly understood as more practical and applied in character than other, more rarefied areas of mathematics. as such ( and notably in contrast to data mining above ), it is mostly employed towards better understanding some particular data generating process. thus, it usually starts with a formally specified model, and from this are derived procedures to accurately extract that model from noisy instances ( i. e., estimation - - by optimizing some loss function ) and to be able to distinguish it from other possibilities ( i. e., inferences based on known properties of sampling distributions ). the prototypical statistical technique is regression.
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" touch not the cat, bot a glove " dttah / acnr / ianal / ymmv * equipment : high impedance voltmeter / oscilloscope with hv probe. high voltage low capacitance capacitors ( 1 10 100 1000 pf ) x 2 of each. pretest - charge capacitors to some semi known high voltage and measure with voltmeter to determine measurement ability. for purrfect results there should be minimal paws between first and second iterations of 2. 3. 4. select cap - say 100 pf. discharge cap ( short ) connect one end of cap to ground - one end of cap to cat..... ( how " to cat " is achieved is left as an exercise for the reader. ).... ( cap and cat are now at same purrtential ) disconnect cap from cat measure vcap repeat 2. 3. 4. compare readings. repeat with higher and lower caps. aim is range where v1 / v2 is usefully high - say about 2 : 1. processing. when cap connects to cat cap is charged. cat and cap share charge in proportion to capacitances. overall voltage drops to reflect increase in system capacitance from addin cap to ccat. if vcat before and after transfer was known you could calculate ccat. but vcat'a bit hard'to determine. repeating process gives a second point and 2 simultaneous equations can be solved to give ccat. if ccap < < ccat the delta v is small and results are ill conditioned. if ccap > > ccat the delta v is large and results are ill conditioned. if ccap ~ ~ ~ = ccat the porridge is just right and the bed is just right. if ccap = ccat then voltage will halve on second reading. v = vcat _ original / 2 otherwise ratio change is related to inverse proportion to capacitances. v2 = v1 x ccat / ( ccat + ccap ) or say v1 / v2 = 0. 75 ccat = 3 x ccap. e & oe.... dttah...... don't try this at home acnr........ all care, no responsibility ianal....... i am not a lawyer ymmv....... your mileage will vary e
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& oe........ errors & omissions excepted.
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it's an interesting question and one that has been asked before. npr did a story in 2013 on this topic, but their question was a bit more focused than just " why are so many black people good runners? " the observation that led to their story wasn't just that black people in general were over - represented among long - distance running medalists, but that kenyans in particular were over - represented. digging deeper, the story's investigators found that the best runners in kenya also tended to come from the same tribal group : the kalenjin. i'm not going to repeat all the details in that story ( which i encourage you to read ), but the working answer that the investigators came up with is that there are both genetic traits and certain cultural practices that contribute to this tribe's success on the track. unfortunately, from the point of view of someone who wants a concise answer, it is very difficult to separate and quantify the exact contributions that each genetic and cultural modification makes to the runners'successes. pubmed also has a number of peer - reviewed papers detailing the kalenjin running phenomenon, but i could only find two with free full - access and neither had the promising title of " analysis of the kenyan distance - running phenomenon, " for which you have to pay. insert annoyed frowning face here. i did a quick search of some kenyan gold medalist runners in the 2016 olympics and sure enough, several ( though certainly not all ) are kalenjin. i'm less sure about the ethiopian runners, since most research that i found online seems to focus on the kenyans, but i'd feel safe hypothesizing that something similar can explain their dominance at the podium. so, the short answer to your question is that it's not just " black people " who dominate the world of competitive long - distance running, but that very specific subsets of people ( who, as it turns out, are black ) do display a competitive advantage and that both genetics and culture account for much of this advantage.
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why do we age is a classical question in evolutionary biology. there are several things to consider when we think of how genes that cause disease, aging, and death to evolve. one explanation for the evolution of aging is the mutation accumulation ( ma ) hypothesis. this hypothesis by p. medawar states that mutations causing late life deleterious ( damaging ) effects can build up in the genome more than diseases that cause early life disease. this is because selection on late acting mutations is weaker. mutations that cause early life disease will more severely reduce the fitness of its carrier than late acting mutations. for example, if we said in an imaginary species that all individuals cease to reproduce at 40 years old and a mutation arises that causes a fatal disease at 50 years old then selection can not remove it from the population - carriers will have as many children as those who do not have the gene. under the mutation accumulation hypothesis it is then possible for mutations to drift through the population. another hypothesis which could contribute to aging is the antagonistic pleiotropy ( ap ) hypothesis of g. c. williams. pleiotropy is when genes have more than one effect, such genes tend to cause correlations between traits, height and arm length probably have many of the same genes affecting them, otherwise there would be no correlation between arm length and height ( though environment and linkage can also cause these patterns )... back to ap as an explanation for aging, if a gene improves fitness early in life, but causes late life disease it can spread through the population via selection. the favourable early effect spreads well because of selection and, just as with ma, selection can not " see " the late acting disease. under both ma and ap the key point is that selection is less efficient at removing late acting deleterious mutations, and they may spread more rapidly thanks to beneficial early life effects. also if there is extrinsic mortality ( predation etc. ) then the effect of selection is also weakened on alleles that affect late life. the same late - life reduction in the efficacy of selection also slows the rate at which alleles increasing lifespan spread. a third consideration is the disposable - soma model, a description by t. kirkwood of life - history trade - offs which might explain why aging and earlier death could be favoured. the idea is that individuals have a limited amount of resources available to them - perhaps because of environmental constraints or ability to acquire / allocate the resources. if we then assume that individuals have
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