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CKDMIP meeting: 8 July 2022 at IRS
A short in-person meeting was held at the 2022 International Radiation Symposium to discuss progress and priorities for ways forward in our collaboration.
• Robin Hogan's talk on results of the CKDMIP intercomparison so far is here
• As of July 2022, the entire CKDMIP dataset including the Evaluation-2 spectra (previously withheld) are available from https://dissemination.ecmwf.int/ecpds/home/ckdmip/
• Results from MSTRN, PSLACKD, RRTMGP and PyKdis have also recently been added to the CKDMIP results pages
In terms of priorities for then next steps, we agreed that the results so far raised important questions about how best to generate gas-optics schemes, and that we should address these before, say,
starting to look at cloudy conditions. Specifically:
• We should list the steps involved in creating CKD models, and study in detail the impact of different methodologies in each of these steps. The How CKD Tools Work page has a brief summary of how
our tools work, and could form a basis for this. It would be useful if models who have not yet submitted information for this page could do so.
• What is the best way to spectrally average absorption coefficients to k terms, and what is the impact of different averaging techniques for the various tools?
□ A quick study of the impact on longwave ecCKD models is shown here, implying that the best method amongst those tried is to average the transmission across an equivalent atmospheric layer
over which pressure changes by a factor of 3 (although ecCKD's subsequent optimization step was found to be crucial).
□ A recent paper by Webb, Solovjov and André suggests that the optimal approach is the geometric mean of the absorption coefficients at the bounds of the spectral interval in g space, i.e. k
[opt ]= sqrt(k[1] x k[2]) = exp((ln k[1] + ln k[2])/2). This is similar to logarithmic averaging, but only considers the bounds of the spectral interval.
• Is there a fundamental limit to the accuracy of correlated-k models even with large numbers of k terms? This is implied by Slide 11 or Robin's ecCKD talk, and in some other talks, and is likely
due to the inability to represent the imperfect rank correlation of spectra at different heights. It would be useful if CKDMIP participants could submit models with quite large numbers of k terms
(ideally all using the same "narrow" band structure) so that we can see if this fundamental limit is reached in all models, and whether the limiting accuracy is similar between models. What can
we do about it? SOCRATES has a method to treat imperfect correlation in height - can this be shared and applied by other tools?
• Other crucial aspects are the way spectra are sorted (either separately per pressure/temperature/gas or with a single sorting per gas) and the treatment of gas overlap. Can we test different
approaches using the same tool?
• It would be great to have some more submissions from different tools, ideally using at least the "narrow" band structure.
Other items raised at the meeting were:
• Can we extend the CKDMIP line-by-line database to more profiles? We currently only have 100 base profiles (nominally 50 for training and 50 for evaluation) of temperature, water vapour and
ozone, and this is not enough to fill parameter space in the training. Extending to more profiles is possible but somewhat laborious. There might be a case to use the RFMIP method to select the
profiles according to some criteria such as global coverage/representativity, or covering extreme values of each variable.
• What is the impact of vertical resolution? The LBL datasets typically use 10 points per decade of pressure - is this enough?
• Can we extend the dataset spectrally to the far UV, e.g. the 100-200 nm range? We would only need O2, O3 and Rayleigh, and this would cover a region important for photolysis.
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Next: MINIMUM 3-DIMENSIONAL ASSIGNMENT Up: Weighted Set Problems Previous: Weighted Set Problems   Index
• INSTANCE: Finite set A and a size
• SOLUTION: A partition of A, i.e., a subset
• MEASURE: Number of elements from S on the same side of the partition as
• Bad News: Not approximable within 476].
• Garey and Johnson: Similar to SP12
Viggo Kann
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Balancing Chemical Equations Worksheet Intermediate Level - Equations Worksheets
Balancing Chemical Equations Worksheet Intermediate Level
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These worksheets can be utilized by fifth and eighth grade students. These are great for students who have difficulty calculating percentages. There are three types of problems that you can pick
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you’ve completed a particular level, you can begin to work on solving other types of equations. You can continue to take on the same problems. You could, for instance solve the same problem as an
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Can fans evaluate fielding better than sabermetric statistics?
Can fans evaluate fielding better than sabermetric statistics?
Team defenses differ in how well they turn batted balls into outs. How do you measure the various factors that influence the differences? The fielders obviously have a huge role, but do the pitchers
and parks also have an influence?
Twelve years ago, in a group discussion, Erik Allen, Arvin Hsu, and Tom Tango broke down the variation in batting average on balls in play (BAbip). Their analysis was published in a summary called
"Solving DIPS" (.pdf).
A couple of weeks ago, I independently repeated their analysis -- I had forgotten they had already done it -- and, reassuringly, got roughly the same result. In round numbers, it turns out that:
The SD of team BAbip fielding talent is roughly 30 runs over a season.
There are several competing systems for evaluating which players and teams are best in the field, and by how much. The Fangraphs stats pages list some of those stats, and let you compare.
I looked at those team stats for the 2014 season. Specifically, these three:
1. DRS, from The Fielding Bible -- specifically, the rPM column, runs above average from plays made. (That's the one we want, because it doesn't include outfielder/catcher arms, or double-play
2. The Fan Scouting Report (FSR), which is based on an annual fan survey run by Tom Tango.
3. Ultimate Zone Rating (UZR), a stat originally developed by Mitchel Lichtman, but which, as I understand it, is now public. I used the column "RngR," which is the range portion (again to leave out
arms and other defensive skills).
All three stats are denominated in runs. Here are their team SDs for the 2014 season, rounded:
37 runs -- DRS (rPM column)
23 runs -- Fan Scouting Report (FSR)
29 runs -- UZR (RngR)
30 runs -- team talent
The SD of DRS is much higher than the SD of team talent. Does that mean it's breaching the "speed of light" limit of forecasting, trying to (retrospectively) predict random luck as well as skill?
No, not necessarily. Because DRS isn't actually trying to evaluate talent. It's trying to evaluate what actually happened on the field. That has a wider distribution than just talent, because there's
luck involved.
A team with fielding talent of +30 runs might have actually saved +40 runs last year, just like a player with 30-home-run talent may have actually hit 40.
The thing is, though, that in the second case, we actually KNOW that the player hit 40 homers. For team fielding, we can only ESTIMATE that it saved 40 runs, because we don't have good enough data to
know that the extra runs didn't just result from getting easier balls to field.
In defense, the luck of "made more good plays than average" is all mixed up with "had more easier balls to field than average." The defensive statistics I've seen try their best to figure out which
is which, but they can't, at least not very well.
What they do, basically, is classify every ball in play according to how difficult it was, based on location and trajectory. I found this post from 2003, which shows some of the classifications for
UZR. For instance, a "hard" ground ball to the "56" zone (a specific portion of the field between third and short) gets turned into an out 43.5 percent of the time, and becomes a hit the other 56.5
If it turns out a team had 100 of those balls to field, and converted them to outs at 45 percent instead of 43.5 percent, that's 1.5 extra outs it gets credited for, which is maybe 1.2 runs saved.
The problem with that is: the 43.5 percent is a very imprecise estimate of what the baseline should be. Because, even in the "hard-hit balls to zone 56" category, the opportunities aren't all the
Some of them are hit close to the fielder, and those might be turned into outs 95 percent of the time, even for an average or bad-fielding team. Some are hit with a trajectory and location that makes
them only 8 percent. And, of course, each individual case depends where the fielders are positioned, so the identical ball could be 80 percent in one case and 10 percent in another.
In a "Baseball Guts" thread at Tango's site, data from Sky Andrecheck and BIS suggested that only 20 percent of ground balls, and 10 percent of fly balls, are "in doubt", in the sense that if you
were watching the game, you'd think it could have gone either way. In other words, at least 80% of balls in play are either "easy outs" or "sure hits." ("In doubt" is my phrase, meaning BIPs in which
it wasn't immediately at least 90 percent obvious to the observer whether it would be a hit or an out.)
That means that almost all the differences in talent and performance manifest themselves in just 10 to 20 percent of balls in play.
But, even the best fielding systems have few zones that are less than 20 percent or more than 80 percent. That means that there is still huge variation in difficulty *even accounting for zone*.
So, when a team makes 40 extra plays over a season, it's a combination of:
(a) those 40 plays came from extra performance from the few "in doubt" balls;
(b) those 40 plays came from easier balls overall.
I think (b) is much more a factor than (a), and that you have to regress the +40 to the mean quite a bit to get a true estimate.
Maybe when the zones get good enough to show large differences between teams -- like, say, 20% for a bad fielder and 80% for a good fielder -- well, at that point, you have a system that might work.
But, without that, doesn't it almost have to be the case that most of the difference is just from what kinds of balls you get?
Tango made a very relevant point, indirectly, in a recent post. He asked, "Is it possible that Manny Ramirez never made an above-average play in the outfield?" The consensus answer, which sounds
right to me, was ... it would be very rare to see Manny make a play that an average outfielder wouldn't have made. (Leaving positioning out of the argument for now.)
Suppose BIPs to a certain difficult zone get caught 30% of the time by an average fielder, and Manny catches them 20% of the time. Since ANY outfielder would catch a ball that Manny gets to ... well,
that zone must really be at least TWO zones: a "very easy" zone with a 100% catch rate, and a "harder" zone with an 10% catch rate for an average fielder, and a 0% catch rate for Manny.
In other words, if Manny makes 30% plays in that zone and a Gold Glove outfielder makes 25%, it's almost certain that Manny just got easier balls to catch.
The only way to eliminate that kind of luck is to classify the zones in enough micro detail that you get close to 0% for the worst, or close to 100% for the best.
And that's not what's happening. Which means, there's no way to tell how many runs a defense saved.
And this brings us back to the point I made last month, about figuring out how to split observed runs allowed into observed pitching and observed fielding. There's really no way to do it, because you
can't tell a good fielding play from an average one with the numbers currently available.
Which means: the DRS and UZR numbers in the Fangraphs tables are actually just estimates -- not estimates of talent, but estimates of *what happened in the field*.
There's nothing wrong with that, in principle: but, I don't think it's generally realized that that's what those are, just estimates. They wind up in the same statistical summaries as pitching and
hitting metrics, which themselves are reliable observations.
At baseball-reference, for instance, you can see, on the hitting page, that Robinson Cano hit .302-28-118 (fact), which was worth 31 runs above average (close enough to be called fact).
On his fielding page, you can see that Cano had 323 putouts (fact) and 444 assists (fact), which, by Total Zone Rating, was worth 4 runs below average (uh-oh).
Unlike the other columns, UZR column is an *estimate*. Maybe it really was -4 runs, but it could easily have been -10 runs, or -20 runs, or +6 runs.
To the naked eye, the hitting and fielding numbers both look equally official and reliable, as accurate observations of what happened. But one is based on an observation of what happened, and the
other is based on an estimate of what happened.
OK, that's a bit of an exaggeration, so let me backtrack and explain what I mean.
Cano had 28 home runs, and 444 assists. Those are "facts", in the sense that the error is zero, if the observations are recorded correctly.
Cano's offense was 31 runs above average. I'm saying that's accurate enough to be called a "fact." But admittedly, it is, in fact, an estimate. Even if the Linear Weights formula (or whatever) is
perfectly accurate, the "runs above average" number is after adjusting for park effects (which are imperfect estimates, albeit pretty good ones). Also, the +31 assumes Cano faced league-average
pitching. That, again, is an estimate, but, again, it's a pretty strong one.
For defense, comparatively, the UZR of "-4" is a very, very, weak estimate. It carries an implicit assumption that Cano's "relative difficulty of balls in play" was zero. That's much less reliable
than the estimate that his "relative difficulty of pitchers faced" was zero. If you wanted, you could do the math, and show how much weaker the one estimate is than the other; the difference is huge.
But, here's a thought experiment to make it clear. Suppose Cano faces an the worst pitcher in the league, and hits a home run. In that case, he's at worst 1.3 runs above average for that plate
appearance, instead of our estimate of 1.4. It's a real difference in how we evaluate his performance, but a small one.
On the other hand, suppose Cano faces a grounder in a 50% zone, but one of the easy ones, that almost any fielder would get to. Then, he's maybe +0.01 hits above average, but we're estimating +0.5.
That is a HUGE difference.
It's also completely at odds with our observation of what happens on the field. After an easy ground ball, even the most casual fan would say he observed Cano saving his team 0 runs over what another
player would do. But we write it down as +0.4 runs, which is ... well, it's so big, you have to call it *wrong*. We are not accurately recording what happened on the field.
So, if you take "what happened on the field" in broad, intutive terms, the home run matches: "he did a good thing on the field and created over a run" both to the observer and the statistic. But for
the ground ball, the statistic lies. It says Cano "did a good thing on the field and saved almost half a run," but the observer says Cano "made a routine play."
The batting statistics match what a human would say happened. The fielding stats do not.
How much random error is in those fielding statistics? When UZR gives an SD of 29 runs, how much of that is luck, and how much is talent? If we knew, we could at least regress to the mean. But we
That's because we don't know the idealized actual SD of observed performance, adjusted for the difficulty of the balls in play. It must be somewhere between 47 runs (the SD of observed performance
without adjusting for difficulty), and 30 runs (the SD of talent). But where in between?
In addition: how sure are we that the estimates are even unbiased, in the sense that they're independently just as likely to be too high as too low? If they're truly unbiased, that makes them much
easier to live with -- at the very least, you know they'll get more accurate as you average over multiple seasons. But if they inappropriately adjust for park effects, or pitcher talent, you might
find some teams being consistently overestimated or underestimated. And that could really screw up your evaluations, especially if you're using those fielding estimates to rejig pitching numbers.
For now, the estimates I like best are the ones from Tango's "Fan Scouting Report" (FSR). As I understand it, those are actually estimates of talent, rather than estimates of what happened on the
Team FSR has an SD of 23 runs. That's very reasonable. It's even more conservative than it looks. That 23 includes all the "other than range" stuff -- throwing arm, double plays, and so on. So the
range portion of FSR is probably a bit lower than 23.
We know the true SD of talent is closer to 30, but there's no way for subjective judgments to be that precise. For one thing, the humans that respond to Tango's survey aren't perfect evaluators of
what they see on the field. Second, even if they *were* perfect, a portion of what they're observing is random luck anyway. You have to temper your conclusions for the amount of noise that must be
It might be a little bit apples-to-oranges to compare FSR to the other estimates, because FSR has much more information to work with. The survey respondents don't just use the ball-in-play stats for
a single year -- they consider the individual players' entire careers, ages and trajectories; the opinions of their peers and the press; their personal understanding of how fielding works; and
anything else they deem relevant.
But, that's OK. If your goal is to try to estimate the influence of team fielding, you might as well just use the best estimate you've got.
For my part, I think FSR is the one I trust the most. When it comes to evaluating fielding, I think sabermetrics is still way behind the best subjective evaluations.
Labels: Allen, BABIP, baseball, DSR, fielding, FSR, Hsu, statistics, Tango, UZR
3 Comments:
At Wednesday, August 19, 2015 2:52:00 AM, said...
I'm late to the party, but I have a few relevant comments: One, the "weak estimate" of these fielding metrics actually becomes quite a bit stronger with more data.
Two, even though hitting and most pitching metrics record exactly what happened, of what use is that? Most of the time we are using these metrics to either estimate true talent or to project
future performance, which is essentially the same thing of course. Given that, even though a single is a single, we care very much about the quality of the single, not just the fact that it was a
single. In fact, we could argue that a MUCH better hitting metric is one that ignores the actual result of a batted ball and uses only the type, location and trajectory of that batted ball. In
fact, there are some hitting and pitching metrics that do this. If we do that, and it is arguably better than using actual results, especially for small samples, now we are in exactly the same
boat as the fielding metrics! So you can't in one breath criticize the fielding metrics because they are only a weak estimate of "what happened" and then in the same breath say that a hitting
metric that estimates "what happened" (based on the characteristics of the batted ball) is better than one in which the actual result of the batted ball is used, which in fact it is!
Basically what I am saying is that the fact that we can only estimate "what happened" in these advanced fielding metrics, at least at the player level, is both a virtue and a curse. But, as can
be proved by looking at it from a hitting perspective, the virtue far outweighs the curse, otherwise would simply use something like simple ZR or even range factor, which does in fact essentially
tell us exactly what happened (a ball was hit near a fielder and he did or did not turn it into an out).
If you wanted to estimate true talent of a hitter, and I told you two things: One, the batter got a single, or two, the batter hit a hard line drive just over the IF (or a medium hit ground ball
to the IF), which one would you prefer? It has to be the second one, and that proves my point that estimating what actually happened on the field is MUCH better than what actually happened, as
long as your estimate of what happened has a decent (not perfect and not necessarily unbiased) amount of information in it.
At Wednesday, August 19, 2015 3:05:00 AM, said...
All that being said, the problem with the current fielding metrics, including my own UZR, is that they do not do the proper regression toward the mean in order to better estimate "what happened"
even in large samples. If they were to do that, then those fielding metrics would be very, very good, even though you call them a "weak estimate" of what happened. However, this problem is a
systematic bias, which means that it is not a huge problem other than the fact that a highly ratef fielder likely did not field as well as these "unregressed" metrics tell is they did and ditto
for a poorly rated fielder. However, since we have to regress these sample data anyway when we estimate true talent and for making projections, we take care of this problem anyway.
For example, say that UZR says that Jeter was a -10 per 150 games in 2012. He probably only fielded at a -8 clip or something like that. If we wanted to estimate his true talent from that one
year only we might regress the -8 to -4 or something like that. OK, since the metric is making a mistake and reporting that -10 is what Jeter actually did in 2012, all we have to do is regress a
little more in order to get that -10 down to -4 rather than more accurate -8. And we do the math to come up with the regressions in order to estimate true talent, we probably come up with the
right regression equation which does in fact take the -10 down to the -4.
So even though the reported "runs saved or cost" is too extreme, we get to the correct place anyway when we estimate true talent.
One interesting thing is that the same bias actually happens with offensive (and pitching) stats. Anytime we report that say, a player, batted .320 in 2012, even though that is a "fact" it is
guaranteed that on average, he will have gotten more than his fair share of "cheap hits." So to some extent, even though he did in fact hit .320, we sort of want to regress that number to report
what actually happened, if we include in the definition of what actually happened, the quality of those hits and whether they were overall lucky or not.
I for one don't ever care what "actually happened anyway" so it doesn't matter to me whether a -10 UZR was really a -8. It also doesn't matter to me whether a player actually hit .320 if he got
more than his fair of cheap hits by luck alone. All I care about is estimates of true talent. I can estimate that from the -10 just as well as I can get that from the -8. I also can get a
player's estimated true talent from the .320 AND I can do even better if I know the quality of the hits and outs withing that .320. In fact, as I said earlier, I do even better with my true
talent BA estimate if I know a lot about those batted balls and I don't even know that the player in question batted .320!
At Thursday, August 20, 2015 10:18:00 AM, Phil Birnbaum said...
MGL (assuming these are MGL),
I generally agree with what you say here. However, we (and you) still do talk about a player's "traditional" statistics (which I refer to as "what happened on the field"). In other words, we DO
say the player hit .320, even when we have trajectory statistics that are better for our purposes.
And there's a legitimate use for "what happened" statistics. Why did the Pirates win? Because "this is what happened on the field" when in Bill Mazeroski's last PA. For those discussions, it
doesn't matter what "true talent" was.
My point is that the .320 for batting is much, much more accurate for "what happened" than the "-10" for fielding, if you want to use those respective numbers to explain why teams won or lost.
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Potter, among many others.
Gilad Gour (University of Calgary)
University of Calgary mathematician Gilad Gour discovered a simple factorization law for multipartite entanglement evolution of a composite quantum system with one subsystem undergoing an arbitrary
physical process. Gour introduced the "entanglement resilience factor", which uniquely determines the multipartite entanglement decay rate. Well known bipartite entanglement evolution emerges as a
special case, and this theory also readily reveals whether a permuted version of a given entangled state can be obtained by local operations.
Aram Harrow (University of Bristol)
Ashley Montanaro (University of Washington)
At the University of Washington, Aram Harrow and collaborator Ashley Montanaro proved the validity of an important, simple, efficient test of whether or not a quantum state is entangled. Entanglement
is a key resource for quantum communication and quantum computation so this test is quite valuable to ascertain the value of a quantum state. One important consequence of this result is that the
tensor optimization problem is not efficiently solvable, even approximately.
Robert Raussendorf (University of British Columbia)
Tzu-Chieh Wei (University of British Columbia)
Ian Affleck (University of British Columbia)
Robert Raussendorf and his colleagues Tzu-Chieh Wei and Ian Affleck at the University of British Columbia showed that Affleck-Kennedy-Lieb-Tasaki states, which are ground states of a simple, highly
symmetric Hamiltonian, can serve as a universal resource for quantum computation by local measurement. Their result, which makes use of percolation theory, opens the possibility that universal
computational resources could be obtained simply by cooling.
Daniel Coombs (University of British Columbia)
Raibatak Das (University of British Columbia)
Christopher W. Cairo (Univesity of Alberta)
Many important biological processes begin when a target molecule binds to a cell surface receptor protein. This event leads to a series of biochemical reactions involving the receptor and signalling
molecules, and ultimately a cellular response. Surface receptors are mobile on the cell surface and their mobility is influenced by their interaction with intracellular proteins. We wanted to
understand the details of these interactions and how they are affected by cellular activation.
Daniel Coombs (University of British Columbia)
Omer Dushek (Oxford University)
Milos Aleksic (Oxford University)
T cells are essential players in the immune response to pathogens such as viruses and bacteria. They can be activated to respond when they recognize molecular signatures of infection (antigens) on
the surface of antigen-presenting-cells of the immune system. The T cell response is highly specific (a particular T cell responds to only the right antigen), sensitive (a T cell will respond to as
few as 1–10 antigens on a single cell) and speedy (antigen binding may induce signalling within just a few seconds).
Daniel Coombs (University of British Columbia)
Jessica M. Conway (University of British Columbia)
While on successful drug treatment, routine testing does not usually detect virus in the blood of an HIV patient. However, more sensitive techniques can detect extremely low levels of virus.
Occasionally, routine blood tests show “viral blips”: short periods of elevated, detectable viral load. In work with postdoctoral fellow Jessica Conway, we explored the hypothesis that residual
low-level viral load can be largely explained by re-activation of cells that were infected before the initiation of treatment, and that viral blips can be viewed as occasional statistical events.
Leah Keshet (University of British Columbia)
Ryan Lukeman (Saint Francis Xavier University)
Research can be challenging and arduous, but sometimes a discovery arises by serendipity. Such was the case with a project carried out by former IGTC PhD student Ryan Lukeman and his co-supervisors
(LEK and Yue Xian Li). Ryan had been writing a PhD thesis on mathematical models for schools and flocks. He derived conditions under which such flocks form perfect structures, with equally-spaced
individuals, all moving coherently. He had already found interesting results and had material ready for a thesis.
|
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1.2 circles
When we have are asked to find the locus of |z-p|=|z-q|, can we immediately say the perpendicular bisector of the line segment joining p and q, or must we plug in z=x+iy and solve to get x=0?
I think it's enough to say that this looks like one of the Apollonius circles in which is row is 1 then a line is given (specifically the perpendicular bisector of the two foci).
|
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Gravitational waves from bodies orbiting the Galactic center black hole and their detectability by LISA
Issue A&A
Volume 627, July 2019
Article Number A92
Number of page(s) 23
Section Astrophysical processes
DOI https://doi.org/10.1051/0004-6361/201935406
Published online 05 July 2019
A&A 627, A92 (2019)
Gravitational waves from bodies orbiting the Galactic center black hole and their detectability by LISA
^1 Laboratoire Univers et Théories, Observatoire de Paris, Université PSL, CNRS, Université Paris Diderot, Sorbonne Paris Cité, 5 Place Jules Janssen, 92190 Meudon, France
e-mail: eric.gourgoulhon@obspm.fr
^2 Laboratoire d’Études Spatiales et d’Instrumentation en Astrophysique, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université Paris Diderot, Sorbonne Paris Cité, 5 Place Jules
Janssen, 92190 Meudon, France
^3 School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
Received: 5 March 2019
Accepted: 29 May 2019
Aims. We present the first fully relativistic study of gravitational radiation from bodies in circular equatorial orbits around the massive black hole at the Galactic center, Sgr A* and we assess the
detectability of various kinds of objects by the gravitational wave detector LISA.
Methods. Our computations are based on the theory of perturbations of the Kerr spacetime and take into account the Roche limit induced by tidal forces in the Kerr metric. The signal-to-noise ratio in
the LISA detector, as well as the time spent in LISA band, are evaluated. We have implemented all the computational tools in an open-source SageMath package, within the Black Hole Perturbation
Toolkit framework.
Results. We find that white dwarfs, neutrons stars, stellar black holes, primordial black holes of mass larger than 10^−4M[⊙], main-sequence stars of mass lower than ∼2.5M[⊙], and brown dwarfs
orbiting Sgr A* are all detectable in one year of LISA data with a signal-to-noise ratio above 10 for at least 10^5 years in the slow inspiral towards either the innermost stable circular orbit
(compact objects) or the Roche limit (main-sequence stars and brown dwarfs). The longest times in-band, of the order of 10^6 years, are achieved for primordial black holes of mass ∼10^−3M[⊙] down to
10^−5M[⊙], depending on the spin of Sgr A*, as well as for brown dwarfs, just followed by white dwarfs and low mass main-sequence stars. The long time in-band of these objects makes Sgr A* a
valuable target for LISA. We also consider bodies on close circular orbits around the massive black hole in the nucleus of the nearby galaxy M 32 and find that, among them, compact objects and brown
dwarfs stay for 10^3–10^4 years in LISA band with a one-year signal-to-noise ratio above ten.
Key words: gravitational waves / black hole physics / Galaxy: center / stars: low-mass / brown dwarfs / stars: black holes
© E. Gourgoulhon et al. 2019
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use,
distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The future space-based Laser Interferometer Space Antenna (LISA; Amaro-Seoane et al. 2017), selected as the L3 mission of ESA, will detect gravitational radiation from various phenomena involving
massive black holes (MBHs), the masses of which range from 10^5 to 10^7M[⊙] (see e.g. Amaro-Seoane 2018; Babak et al. 2017, and references therein). The mass of the MBH Sgr A* at the center of our
galaxy lies within this range (GRAVITY Collaboration 2018a,b):
$M Sgr A ∗ = 4.10 ± 0.03 × 10 6 M ⊙ .$(1)
More precisely, the angular velocity ω[0] on a circular, equatorial orbit at the Boyer–Lindquist radial coordinate r[0] around a Kerr black hole (BH) is given by the formula in Bardeen et al. (1972)
$ω 0 = ( G M ) 1 / 2 r 0 3 / 2 + a ( G M ) 1 / 2 / c ,$(2)
where G is the gravitational constant, c the speed of light, M the BH mass, and a=J/(cM) its reduced spin. Here J is the magnitude of the BH angular momentum (a has the dimension of a length). The
motion of a particle of mass μ≪M on a circular orbit generates some gravitational radiation with a periodic pattern (the dominant mode of which is m=2) and has the frequency f[m=2]=2f[0],
where f[0]≡ω[0]/(2π) is the orbital frequency (details are given in Sect. 2). Combining with Eq. (2), we obtain
$f m = 2 = 1 π ( G M ) 1 / 2 r 0 3 / 2 + a ( G M ) 1 / 2 / c ·$(3)
This frequency is maximal at the (prograde) innermost stable circular orbit (ISCO), which is located at r[0]=6GM/c^2 for a=0 (Schwarzschild BH) and at r[0]=GM/c^2 for a=a[max]≡GM/c^2
(extreme Kerr BH). Equation (3) leads then to
$f m = 2 ISCO , a = 0 = c 3 6 3 / 2 π G M and f m = 2 ISCO , a max = c 3 2 π G M ·$(4)
Substituting the mass of Sgr A* (1) for M, we obtain
$f m = 2 ISCO , a = 0 = 1.1 mHz and f m = 2 ISCO , a max = 7.9 mHz .$(5)
By convenient coincidence, $f m = 2 ISCO , a max$ matches almost exactly the frequency of LISA maximal sensitivity, the latter being 7.86mHz! (see Fig. 1). The spin of Sgr A* is currently not known,
but it is expected to be quite large, due to matter accretion since the birth of the MBH. Actually, the tentative measures of MBH spins in nuclei of other galaxies generally lead to large values of a
. See for example Table 3 of the recent review by Nampalliwar & Bambi (2018), where most entries have a> 0.9GM/c^2.
Fig. 1.
LISA sensitivity curve (Amaro-Seoane et al. 2017) and various gravitational wave frequencies from circular orbits around Sgr A*. The wave frequencies shown above are all for the dominant m=2
mode, except for the dot-dashed and dotted vertical red lines, which correspond to the m=3 and m=4 harmonics of the ISCO of an extreme Kerr BH (a=M). The shaded pink area indicates the
location of the frequencies from the ISCO when a ranges from zero to M. The Roche limits are those discussed in Sect. 5.1.
The adequacy of LISA bandwidth to orbital motions around Sgr A* was first stressed by Freitag (2003a,b), who estimated the gravitational radiation from orbiting stars at the (Newtonian) quadrupole
order. By taking into account the tidal forces exerted by the MBH, he showed that, besides compact objects, low-mass main-sequence stars (mass μ≲0.1M[⊙]) can approach the central MBH sufficiently
close to emit gravitational waves in LISA bandwidth. Via some numerical simulations of the dynamics of the Galactic center stellar cluster, he estimated that there could exist a few such stars
detectable by LISA, whereas the probability of observing a compact object was found to be quite low (Freitag 2003b). This study was refined by Barack & Cutler (2004), who estimated that the
signal-to-noise ratio (S/N) of a μ=0.06M[⊙] main-sequence star observed 10^6yr before plunge is of the order eleven in two years of LISA observations. Moreover, they have shown that the detection
of such an event could lead to the spin measurement of Sgr A* with an accuracy of ∼0.5%. Berry & Gair (2013a) investigated the phenomenon of extreme-mass-ratio burst, which occurs at the periastron
passage of a stellar-mass compact object (mass μ) on a highly eccentric orbit around Sgr A*. These authors have shown that LISA can detect such an event with μ=10M[⊙], provided that the periastron
distance is lower than 65GM/c^2. The event rate of such bursts could be of the order of 1 per year (Berry & Gair 2013b; see Sect. 7.6 of Amaro-Seoane 2018 for some discussion). Linial & Sari (2017)
have computed at the quadrupole order the gravitational wave emission from orbiting main-sequence stars undergoing Roche lobe overflow, treated at the Newtonian level. These authors stressed the
detectability by LISA and have showed the possibility of a reverse chirp signal, the reaction of the accreting system to the angular momentum loss by gravitational radiation being a widening of the
orbit (outspiral) (Dai & Blandford 2013). Recently, Kuhnel et al. (2018) have computed, still at the quadrupole level, the gravitational wave emission from an ensemble of macroscopic dark matter
candidates orbiting Sgr A*, such as primordial BHs, with masses in the range 10^−13−10^3M[⊙].
All the studies mentioned above are based on the quadrupole formula for Newtonian orbits, except that of Berry & Gair (2013a), which is based on the so-called “kludge approximation”. Now, for orbits
close to the ISCO, relativistic effects are expected to be important. In this article, we present the first study of gravitational waves from stellar objects in close orbits around Sgr A* in a fully
relativistic framework: Sgr A* is modeled as a Kerr BH, gravitational waves are computed via the theory of perturbations of the Kerr metric (Teukolsky 1973; Detweiler 1978; Shibata 1994; Kennefick
1998; Hughes 2000; Finn & Thorne 2000; Glampedakis & Kennefick 2002) and tidal effects are evaluated via the theory of Roche potential in the Kerr metric developed by Dai & Blandford (2013).
Moreover, from the obtained waveforms, we carefully evaluate the signal-to-noise ratio in the LISA detector, taking into account the latest LISA sensitivity curve (Robson et al. 2019). There is
another MBH with a mass within the LISA range in the Local Group of galaxies: the 2.5×10^6M[⊙] MBH in the center of the galaxy M 32 (Nguyen et al. 2018). By applying the same techniques, we study
the detectability by LISA of bodies in close circular orbit around it.
The plan of the article is as follows. The method employed to compute the gravitational radiation from a point mass in circular orbit around a Kerr BH is presented in Sect. 2, the open-source code
implementing it being described in Appendix A. The computation of the S/N of the obtained waveforms in the LISA detector is performed in Sect. 3, from which we can estimate the minimal detectable
mass of the orbiting source in terms of the orbital radius. Section 4 investigates the secular evolution of a circular orbit under the reaction to gravitational radiation and provides the frequency
change per year and the inspiral time between two orbits. The potential astrophysical sources are discussed in Sect. 5, taking into account Roche limits for noncompact objects and estimating the
total time spent in LISA band. The case of M 32 is treated in Appendix C. Finally, the main conclusions are drawn in Sect. 6.
2. Gravitational waves from an orbiting point mass
In this section and the remainder of this article, we use geometrized units, for which G=1 and c=1. In addition we systematically use Boyer–Lindquist coordinates (t,r,θ,φ) to describe the Kerr
geometry of a rotating BH of mass M and spin parameter a, with 0⩽a< M. We consider a particle of mass μ≪M on a (stable) prograde circular equatorial orbit of constant coordinate r=r[0].
Hereafter, we call r[0] the orbital radius. The orbital angular velocity ω[0] is given by formula (2). In practice this “particle” can be any object whose extension is negligible with respect to the
orbital radius. In particular, for Sgr A*, it can be an object as large as a solar-type star. Indeed, Sgr A* mass (1) corresponds to a length scale M=6.05×10^6km∼9R[⊙], where R[⊙] is the Sun’s
radius. Moreover, main-sequence stars are centrally condensed objects, so that their “effective” size as gravitational wave generator is smaller that their actual radius. In addition, as we shall see
in Sect. 5.1, their orbital radius must obey r[0]> 34M to avoid tidal disruption (Roche limit), so that R[⊙]/r[0]< 3×10^−3. Hence, regarding Sgr A*, we may safely describe orbiting stars as
point particles.
The gravitational wave emission from a point mass orbiting a Kerr BH has been computed by many groups, starting from the seminal work of Detweiler (1978), which is based on the theory of linear
perturbations of the Kerr metric initiated by Teukolsky (1973). The computations have been extended to eccentric orbits by a number of authors (see e.g. Glampedakis & Kennefick 2002). However, in the
present study, we limit ourselves to circular orbits, mostly for simplicity, but also because some of the scenarii discussed in Sect. 5 lead naturally to low eccentricity orbits; this involves
inspiralling compact objects that result from the tidal disruption of a binary, stars formed in an accretion disk, black holes resulting from the most massive of such stars and a significant
proportion (∼1/4) of the population of brown dwarfs that might be in LISA band.
In Sect. 2.1, we recall the gravitational waveform obtained from perturbation analysis of the Kerr metric. It requires the numerical computation of many mode amplitudes. This is quite technical and
we describe the technique we use to perform the computation in Sect. 2.2. We discuss the limiting case of distant orbits in Sect. 2.3 and evaluate the Fourier spectrum of the waveform in Sect. 2.4,
where we present some specific waveforms.
2.1. Gravitational waveform
The gravitational waves generated by the orbital motion of the particle are conveniently encoded in the linear combination h[+]−ih[×] of the two polarization states h[+] and h[×]. A standard result
from the theory of linear perturbations of the Kerr BH (Teukolsky 1973; Detweiler 1978; Shibata 1994; Kennefick 1998; Hughes 2000; Finn & Thorne 2000; Glampedakis & Kennefick 2002) yielded the
asymptotic waveform as
$h + − i h × = 2 μ r ∑ ℓ = 2 + ∞ ∑ m = − ℓ m ≠ 0 ℓ Z ℓ m ∞ ( r 0 ) ( m ω 0 ) 2 − 2 S ℓ m a m ω 0 ( θ , φ ) e − i m ( ω 0 ( t − r ∗ ) + φ 0 ) ,$(6)
where (h[+],h[×]) are evaluated at the spacetime event of Boyer–Lindquist coordinates (t,r,θ,φ) and r[*] is the so-called “tortoise coordinate”, defined as
$r ∗ ≡ r + 2 M r + r + − r − ln ( r − r + 2 M ) − 2 M r − r + − r − ln ( r − r − 2 M ) ,$(7)
where $r ± ≡ M ± M 2 − a 2$ denote the coordinate locations of the outer (+) and inner (−) event horizons. The phase φ[0] in Eq. (6) can always be absorbed into a shift of the origin of t. The
spin-weighted spheroidal harmonics $−2 S lm am ω 0 (θ,φ)$ encode the dependency of the waveform with respect to the polar angles (θ,φ) of the observer. For each harmonic (ℓ,m), they depend on the
(dimensionless) product aω[0] of the Kerr spin parameter and the orbital angular velocity, and they reduce to the more familiar spin-weighted spherical harmonics [−2]Y[ℓm](θ,φ) when a=0. The
coefficients $Z lm ∞ ( r 0 )$ encode the amplitude and phase of each mode. They depend on M and a and are computed by solving the radial component of the Teukolsky equation (Teukolsky 1973); they
satisfy $Z ℓ , − m ∞ = ( − 1 ) ℓ Z ℓ m ∞ ∗$, where the star denotes the complex conjugation.
Given the distance r=8.12±0.03kpc to Sgr A* (GRAVITY Collaboration 2018a), the prefactor μ/r in formula (6) takes the following numerical value:
$μ r = 5.89 × 10 − 18 ( μ 1 M ⊙ ) ( 8.12 kpc r ) ·$(8)
2.2. Mode amplitudes
The factor $| Z lm ∞ ( r 0 )|/ (m ω 0 ) 2$ sets the amplitude of the mode (ℓ,m) of h[+] and h[×] according to Eq. (6). The complex amplitudes, $Z lm ∞$, are computed by solving the Teukolsky equation
(Teukolsky 1973), where, typically, the secondary is modeled as a structureless point mass. Generally, the Teukolsky equation is solved in either the time or frequency domain. Time domain
calculations are computationally expensive but well suited to modeling a source moving along an arbitrary trajectory. Frequency domain calculations have the advantage that the Teukolsky equation is
completely separable in this domain and this reduces the problem from solving partial to ordinary differential equations. This leads to very efficient calculations so long as the Fourier spectrum of
the source is sufficiently narrow. Over short timescales^1 the trajectory of a small body with μ≪M orbiting a MBH is well approximated by a bound geodesic of the background spacetime. Motion along
a bound geodesic is periodic (or bi-periodic Schmidt 2002) and so the spectrum of the source is discrete. This allows the Teukolsky equation to be solved efficiently in the frequency domain, at least
for orbits with up to a moderate eccentricity (for large eccentricities the Fourier spectrum broadens to a point where time domain calculations can be more efficient Barton et al. 2008). Frequency
domain calculations have been carried out for circular (Detweiler 1978), spherical (Hughes 2000), eccentric equatorial (Glampedakis & Kennefick 2002) and generic orbits (Drasco & Hughes 2006; Fujita
et al. 2009; van de Meent 2018) and we follow this approach in this work.
In the frequency domain the Teukolsky equation separates into spin-weighted spheroidal harmonics and frequency modes. The former can be computed via eigenvalue (Hughes 2000) or continuous fraction
methods (Leaver 1985). The main task is then finding solutions to the Teukolsky radial equation. Typically, this is a two step process whereby one first finds the homogeneous solutions and then
computes the inhomogeneous solutions via the method of variation of parameters. Finding the homogeneous solutions is usually done by either numerical integration or via an expansion of the solution
in a series of special functions (Sasaki & Tagoshi 2003). In this work we make use of both methods as a cross check. Direct numerical integration of the Teukolsky equation is numerically unstable but
this can be overcome by transforming the equation to a different form (Sasaki & Nakamura 1982a,b). Our implementation is based off the code developed for Gralla et al. (2015). For the series method
our code is based off of codes used in Kavanagh et al. (2016), Buss & Casals (2018). Both of these codes, as well as code to compute spin-weighted spheroidal harmonics, are now publicly available as
part of the Black Hole Perturbation Toolkit^2.
The final step is to compute the inhomogeneous radial solutions. In this work we consider circular, equatorial orbits. With a point particle source, this reduces the application of variation of
parameters to junction conditions at the particle’s radius (Detweiler 1978). The asymptotic complex amplitudes, $Z lm ∞$, can then be computed by evaluating the radial solution in the limit r→∞.
The mode amplitudes are plotted in Fig. 2 as functions of the orbital radius r[0] for 2⩽ℓ⩽5, 1⩽m⩽ℓ and some selected values of the MBH spin parameter a. Each curve starts at the value of r[0]
corresponding to the prograde ISCO for the considered a.
Fig. 2.
Amplitude factor $| Z lm ∞ ( r 0 )|/ (m ω 0 ) 2$ for the harmonic (ℓ,m) of the gravitational wave emitted by an orbiting point mass (cf. Eq. (6)), in terms of the orbital radius r[0]. Each panel
corresponds to a given value of the MBH spin: a=0 (Schwarzschild BH), a=0.5M, a=0.9M and a=0.98M. A given color corresponds to a fixed value of ℓ and the line style indicates the value
of m: solid: m=ℓ, dashed: m=ℓ−1, dot-dashed: m=ℓ−2, dotted: 0< m⩽ℓ−3.
2.3. Waveform for distant orbits (r[0]≫M)
When the orbital radius obeys r[0]≫M, we see from Fig. 2 that the modes (ℓ,m)=(2,±2) dominate the waveform (cf. the solid red curves in the four panels of Fig. 2). Moreover, for r[0]≫M, the
effects of the MBH spin become negligible. This is also apparent on Fig. 2: the value of $| Z lm ∞ ( r 0 )|/ (m ω 0 ) 2$ for (ℓ,m)=(2,2) and r[0]=50M appears to be independent of a, being equal
to roughly 7×10^−2 in all the four panels. The value of $Z 2,±2 ∞ ( r 0 )$ at the lowest order in M/r[0] is given by e.g. Eq. (5.6) of Poisson (1993a), and reads^3
$Z 2 , ± 2 ∞ ( r 0 ) = 16 π 5 M 2 r 0 4 [ 1 + O ( M r 0 ) ] ·$(9)
The dependency with respect to a would appear only at the relative order (M/r[0])^3/2 (see Eq. (24) of Poisson 1993b) and can safely be ignored, as already guessed from Fig. 2. Besides, for r[0]≫M,
Eq. (2) reduces to the standard Newtonian expression:
Combining with Eq. (9), we see that the amplitude factor in the waveform (6) is
$Z 2 , ± 2 ∞ ( r 0 ) ( 2 ω 0 ) 2 ≃ 4 π 5 M r 0 ·$(11)
Besides, when r[0]≫M, Eq. (2) leads to Mω[0]≪1 and therefore to aω[0]≪1 since |a|⩽M. Accordingly the spheroidal harmonics $−2 S lm am ω 0 (θ,φ)$ in Eq. (6) can be approximated by the spherical
harmonics [−2]Y[ℓm](θ,φ). For (ℓ,m)=(2,±2), the latter are
$− 2 Y 2 , ± 2 ( θ , φ ) = 1 8 5 π ( 1 ± cos θ ) 2 e ± 2 i φ .$(12)
Keeping only the terms (ℓ,m)=(2,±2) in the summations involved in Eq. (6) and substituting expression (11) for the amplitude factor and expression (12) for $−2 S 2,±2 2a ω 0 (θ,φ) ≃ −2 Y 2,±2 (θ,φ)$
, we get
$h + − i h × = μ r M r 0 [ ( 1 − cos θ ) 2 e 2 i ψ + ( 1 + cos θ ) 2 e − 2 i ψ ] ,$(13)
$ψ ≡ ω 0 ( t − r ∗ ) + φ 0 − φ .$(14)
Expanding (13) leads immediately to
$h + (t,r,θ,φ)=2 μ r M r 0 (1+ cos 2 θ)cos[ 2 ω 0 (t− r * )+2( φ 0 −φ) ],$(15a)
$h × (t,r,θ,φ)=4 μ r M r 0 cosθsin[ 2 ω 0 (t− r * )+2( φ 0 −φ) ].$(15b)
As expected for r[0]≫M, we recognize the waveform obtained from the standard quadrupole formula applied to a point mass μ on a Newtonian circular orbit around a mass M≫μ (compare with e.g. Eqs.
(3.13) and (3.14) of Blanchet 2001).
2.4. Fourier series expansion
Observed at a fixed location (r,θ,φ), the waveform (h[+],h[×]) as given by Eq. (6) is a periodic function of t, or equivalently of the retarded time u≡t−r[*], the period being nothing but the
orbital period of the particle: T[0]=2π/ω[0]. It can therefore be expanded in Fourier series. Noticing that the φ-dependency of the spheroidal harmonic $−2 S lm am ω 0 (θ,φ)$ is simply e^imφ, we
may rewrite Eq. (6) as an explicit Fourier series expansion^4:
$h + , × = μ r ∑ m = 1 + ∞ [ A m + , × ( θ ) cos ( m ψ ) + B m + , × ( θ ) sin ( m ψ ) ] ,$(16)
where ψ is given by Eq. (14) and $A m + (θ)$, $A m × (θ)$, $B m + (θ)$ and $B m × (θ)$ are real-valued functions of θ, involving M, a and r[0]:
$A m + ( θ ) = 2 ( m ω 0 ) 2 ∑ ℓ = 2 + ∞ Re ( Z ℓ m ∞ ( r 0 ) ) × [ ( − 1 ) ℓ − 2 S ℓ , − m − a m ω 0 ( θ , 0 ) + − 2 S ℓ m a m ω 0 ( θ , 0 ) ] ,$(17a)
$B m + ( θ ) = 2 ( m ω 0 ) 2 ∑ ℓ = 2 + ∞ Im ( Z ℓ m ∞ ( r 0 ) ) × [ ( − 1 ) ℓ − 2 S ℓ , − m − a m ω 0 ( θ , 0 ) + − 2 S ℓ m a m ω 0 ( θ , 0 ) ] ,$(17b)
$A m × ( θ ) = 2 ( m ω 0 ) 2 ∑ ℓ = 2 + ∞ Im ( Z ℓ m ∞ ( r 0 ) ) × [ ( − 1 ) ℓ − 2 S ℓ , − m − a m ω 0 ( θ , 0 ) − − 2 S ℓ m a m ω 0 ( θ , 0 ) ] ,$(17c)
$B m × ( θ ) = 2 ( m ω 0 ) 2 ∑ ℓ = 2 + ∞ Re ( Z ℓ m ∞ ( r 0 ) ) × [ ( − 1 ) ℓ + 1 − 2 S ℓ , − m − a m ω 0 ( θ , 0 ) + − 2 S ℓ m a m ω 0 ( θ , 0 ) ] .$(17d)
We then define the spectrum of the gravitational wave at a fixed value of θ as the two series (one per polarization mode):
$H m + , × ( θ ) ≡ ( A m + , × ( θ ) ) 2 + ( B m + , × ( θ ) ) 2 , 1 ⩽ m < + ∞ .$(18)
We have developed an open-source SageMath package, kerrgeodesic_gw (cf. Appendix A), implementing the above formulas, and more generally all the computations presented in this article, like the S/N
and Roche limit ones to be discussed below. The spectrum, as well as the corresponding waveform, computed via kerrgeodesic_gw, are depicted in Figs. 3 and 4 for a=0 and a=0.98M respectively. In
each figure, φ=φ[0] and three values of θ are selected: θ=0 (orbit seen face-on), π/4 and π/2 (orbit seen edge-on).
Fig. 3.
Waveform (left column) and Fourier spectrum (right column) of gravitational radiation from a point mass orbiting on the ISCO of a Schwarzschild BH (a=0). All amplitudes are rescaled by r/μ, where
r is the Boyer–Lindquist radial coordinate of the observer and μ the mass of the orbiting point. Three values of the colatitude θ of the observer are considered: θ=0 (first row), θ=π/4 (second
row) and θ=π/2 (third row).
Fig. 4.
Same as Fig. 3 but for a point mass orbiting on the prograde ISCO of a Kerr BH with a=0.98M.
We notice that for θ=0, only the Fourier mode m=2 is present and that h[+] and h[×] have identical amplitudes and are in quadrature. This behavior is identical to that given by the large radius
(quadrupole-formula) approximation (15). For θ> 0, all modes with m≥1 are populated, whereas the approximation () contains only m=2. For θ=π/2, h[×] vanishes identically and the relative
amplitude of the modes m ≠ 2 with respect to the mode m=2 is the largest one, reaching ∼75% for m=3 and ∼50% for m=4 when a=0.98M.
Some tests of our computations, in particular comparisons with previous results by Poisson (1993a; a=0) and Detweiler (1978; a=0.5M and a=0.9M) are presented in Appendix A.
3. Signal-to-noise ratio in the LISA detector
The results in Sect. 2 are valid for any BH. We now specialize them to Sgr A* and evaluate the S/N in the LISA detector, as a function of the mass μ of the orbiting object, the orbital radius r[0]
and the spin parameter a of Sgr A*.
3.1. Computation
Assuming that its noise is stationary and Gaussian, a given detector is characterized by its one-sided noise power spectral density (PSD) S[n](f). For a gravitational wave search based on the matched
filtering technique, the S/N ρ is given by the following formula (see e.g. Jaranowski & Królak 2012; Moore et al. 2015):
$ρ 2 = 4 ∫ 0 + ∞ | h ∼ ( f ) | 2 S n ( f ) d f ,$(19)
where $h ∼ ( f )$ is the Fourier transform of the imprint h(t) of the gravitational wave on the detector,
$h ∼ ( f ) = ∫ − ∞ + ∞ h ( t ) e − 2 π i f t d t ,$(20)
h(t) being a linear combination of the two polarization modes h[+] and h[×] at the detector location:
$h ( t ) = F + ( Θ , Φ , Ψ ) h + ( t , r , θ , φ ) + F × ( Θ , Φ , Ψ ) h × ( t , r , θ , φ ) .$(21)
In the above expression, (t,r,θ,φ) are the Boyer–Lindquist coordinates of the detector (“Sgr A* frame”), while F[+] and F[×] are the detector beam-pattern coefficients (or response functions),
which depend on the direction (Θ,Φ) of the source with respect to the detector’s frame and on the polarization angle Ψ, the latter being the angle between the direction of constant azimuth Φ and the
principal direction “+” in the wavefront plane (i.e. the axis of the h[+] mode or equivalently the direction of the semi-major axis of the orbit viewed as an ellipse in the detector’s sky) (
Apostolatos et al. 1994). For a detector like LISA, where, for high enough frequencies, the gravitational wavelength can be comparable or smaller than the arm length (2.5Gm), the response functions
F[+] and F[×] depend a priori on the gravitational wave frequency f, in addition to (Θ,Φ,Ψ) (Robson et al. 2019). However for the gravitational waves considered here, a reasonable upper bound of
the frequency is that of the harmonic m=4 (say) of waves from the prograde ISCO of an extreme Kerr BH (see Fig. 4). From the value given by Eq. (5), this is f[max]=2×7.9≃15.8mHz, the
multiplication by 2 taking into account the transition from m=2 to m=4. This value being lower than LISA’s transfer frequency f[*]=19.1mHz (Robson et al. 2019), we may consider that F[+] and F
[×] do not depend on f (see Fig. 2 in Robson et al. 2019). They are given in terms of (Θ,Φ,Ψ) by Eq. (3.12) of Cutler (1998) (with the prefactor $3 / 2$ appearing in Eq. (3.11) included in them).
Generally, the function S[n](f) considered in the LISA literature, and in particular to present the LISA sensitivity curve, is not the true noise PSD of the instrument, P[n](f) say, but rather P[n](f
)/R(f), where R(f) is the average over the sky (angles (Θ,Φ)) and over the polarization (angle Ψ) of the square of the response functions F[+] and F[×], so that Eq. (19) yields directly the sky and
polarization average S/N by substituting $| h ∼ + ( f ) | 2 + | h ∼ × ( f ) | 2$ for $| h ∼ ( f ) | 2$ (see Robson et al. 2019 for details). With Sgr A* as a target, the direction angles (Θ,Φ) are
of course known and, for a short observation time (1 day say), they are approximately constant. However, on longer observation times, theses angles varies due to the motion of LISA spacecrafts on
their orbits around the Sun. Moreover, the polarization angle Ψ is not known at all, since it depends on the orientation of the orbital plane around the MBH, which is assumed to be the equatorial
plane, the latter being currently unknown. For these reasons, we consider the standard sky and polarization average sensitivity of LISA, S[n](f)=P[n](f)/R(f), as given e.g. by Eq. (13) of Robson et
al. (2019), and define the effective (S/N)ρ by
$ρ 2 = 4 ∫ 0 + ∞ | h ∼ + ( f ) | 2 + | h ∼ × ( f ) | 2 S n ( f ) d f ,$(22)
where $h ∼ + ( f )$ and $h ∼ × ( f )$ are the Fourier transforms of the two gravitational wave signals h[+](t) and h[×](t), as given by Eq. (6) or Eq. (16), over some observation time T:
$h ∼ + , × ( f ) = ∫ − T / 2 T / 2 h + , × ( t ) e − 2 π i f t d t .$(23)
As shown in Appendix B, plugging the expressions (16) for h[+](t) and h[×](t) into Eqs. (22) and (23) leads to the following S/N value:
$ρ = μ r T ( ∑ m = 1 + ∞ ( H m + ( θ ) ) 2 + ( H m × ( θ ) ) 2 S n ( m f 0 ) ) 1 / 2 for f 0 T ≫ 1 ,$(24)
where the coefficients $H m + (θ)$ and $H m × (θ)$ are defined by Eq. (18) and f[0]=ω[0]/(2π) is the orbital frequency, ω[0] being the function of M, a and r[0] given by Eq. (2).
The effective S/N resulting from Eq. (24) is shown in Figs. 5 and 6. We use the value (8) for μ/r and the analytic model of Robson et al. (2019; their Eq. (13)) for LISA sky and polarization average
sensitivity S[n](f). We notice that for a given value of the orbital radius r[0] and a given MBH spin a, the S/N is maximum for the inclination angle θ=0 and minimal for θ=π/2, the ratio between
the two values varying from ∼2 for a=0 to ∼3 for a=0.98M. This behavior was already present in the waveform amplitudes displayed in Figs. 3 and 4.
Fig. 5.
Effective (direction and polarization averaged) S/N in LISA for a 1-day observation of an object of mass μ orbiting Sgr A*, as a function of the orbital radius r[0] and for selected values of the
Sgr A*’s spin parameter a and well as selected values of the inclination angle θ. Each curve starts at the ISCO radius of the corresponding value of a.
Fig. 6.
Same as Fig. 5, except for r[0] ranging up to 50M. For r[0]> 15M, only the a=0 curves are plotted, since the MBH spin plays a negligible role at large distance.
Another feature apparent from Figs. 5 and 6 is that at fixed orbital radius r[0], the S/N is a decaying function of a. This results from the fact that the orbital frequency f[0] is a decaying
function of a (cf. Eq. (2)), which both reduces the gravitational wave amplitude and displaces the wave frequency to less favorable parts of LISA’s sensitivity curve.
At the ISCO, the S/N for θ=0 is
$ρ ISCO = α 10 5 ( μ 1 M ⊙ ) ( T 1 d ) 1 / 2 ,$(25)
with the coefficient α given in Table 1. It should be noted that if the observation time is one year, then the factor (T/1d)^1/2 is $365.25 ≃ 19.1$.
Table 1.
Coefficient α in formula (25) for the S/N from the ISCO.
3.2. Minimal detectable mass
As clear from Eq. (24), the S/N ρ is proportional to the mass μ of the orbiting body and to the square root of the observing time T. It is then easy to evaluate the minimal mass μ[min] that can be
detected by analyzing one year of LISA data, setting the detection threshold to
where S/N[1yr] stands for the value of ρ for T=1yr. The result is shown in Fig. 7. If one does not take into account any Roche limit, it is worth noticing that the minimal detectable mass is
quite small: μ[min]≃3×10^−5M[⊙] at the ISCO of a Schwarzschild BH (a=0), down to μ[min]≃2×10^−6M[⊙] (the Earth mass) at the ISCO of a rapidly rotating Kerr BH (a⩾0.90M).
Fig. 7.
Minimal detectable mass with a S/N larger than 10 in one year of LISA observation, as a function of the orbital radius r[0]. The various Roche limits are those considered in Sect. 5.1. As in Fig. 6
, for r[0]> 15M, only the a=0 curves are shown, the MBH spin playing a negligible role at large distance.
4. Radiated energy and orbital decay
In the above sections, we have assumed that the orbits are exactly circular, i.e. we have neglected the reaction to gravitational radiation. We now take it into account and discuss the resulting
secular evolution of the orbits.
4.1. Total radiated power
The total power (luminosity) emitted via gravitational radiation is given by (Detweiler 1978):
$L= lim r → + ∞ r 2 16π ∮ S r | h ˙ + −i h ˙ × | 2 sinθdθdφ,$(27)
where 𝒮[r] is the sphere of constant value of r and an overdot stands for the partial derivative with respect to the time coordinate t, i.e. $h ˙ + , × ≡ ∂ h + , × / ∂ t$. Substituting the waveform (
6) into this expression leads to
$L= lim r→+∞ μ 2 4π ∮ S r | ∑ l=2 +∞ ∑ m=−l m = 0 l Z lm ∞ ( r 0 ) m ω 0 −2 S lm am ω 0 (θ,φ) e −im( ω 0 (t− r * )+ φ 0 ) | 2 ×sinθdθdφ.$(28)
Thanks to the orthonormality property of the spin-weighted spheroidal harmonics,
$∮ S 2 − 2 S ℓ m a m ω 0 ( θ , φ ) − 2 S ℓ ′ m ′ a m ′ ω 0 ( θ , φ ) ∗ sin θ d θ d φ = δ ℓ ℓ ′ δ m m ′ ,$(29)
the above expression simplifies to
$L = ( μ M ) 2 L ∼ ( r 0 M ) with L ∼ ( r 0 M ) ≡ M 2 4 π ∑ ℓ = 2 ∞ ∑ m = − ℓ m ≠ 0 ℓ | Z ℓ m ∞ ( r 0 ) | 2 ( m ω 0 ) 2 ·$(30)
It should be noted that $L ∼$ is a dimensionless function of x≡r[0]/M, the dimension of $Z lm ∞$ being an inverse squared length (see e.g. Eq. (9)) and ω[0] being the function of r[0]/M given by
Eq. (2). Moreover, the function $L ∼ ( x )$ depends only on the parameter a/M of the MBH.
As a check of Eq. (30), let us consider the limit of large orbital radius: r[0]≫M. As discussed in Sect. 2.3, only the terms (ℓ,m)=(2,±2) are pertinent in this case, with $Z 2,±2 ∞ ( r 0 )$ given
by Eq. (9) and ω[0] related to r[0] by Eq. (10). Equation (30) reduces then to
$L ≃ 32 5 ( μ M ) 2 ( M r 0 ) 5 ( r 0 ≫ M ) ⟺ L ∼ ( x ) ≃ 32 5 x 5 ( x ≫ 1 ) .$(31)
We recognize the standard result from the quadrupole formula at Newtonian order (Landau & Lifshitz 1971; see also the lowest order of formula (314) in the review by Blanchet 2014).
The total emitted power L (actually the function $L ∼ ( r 0 / M )$) is depicted in Fig. 8. A test of our computations is provided by the comparison with Figs. 6 and 7 of Detweiler (1978)’s study. At
the naked eye, the agreement is quite good, in particular for the values of L at the ISCO’s. Moreover, for large values of r[0] all curves converge towards the curve of the quadrupole formula (31)
(dotted curve), as they should. However, as the inset of Fig. 8 reveals, the relative deviation from the quadrupole formula is still ∼5% for orbital radii as large as r[0]∼50M. This is not
negligibly small and justifies the fully relativistic approach that we have adopted.
Fig. 8.
Gravitational wave luminosity L for an object of mass μ in circular equatorial orbit around a Kerr BH of mass M and spin parameter a, as a function of the orbital radius r[0]. Each curve starts at
the prograde ISCO radius of the corresponding value of a. The dotted curve corresponds to the quadrupole approximation as given by Eq. (31). The inset shows the relative difference with respect to
the quadrupole formula (31) up to r[0]=50M.
4.2. Secular evolution of the orbit
For a particle moving along any geodesic in Kerr spacetime, in particular along a circular orbit, the conserved energy is E≡−p[a]ξ^a, where p[a] is the particle’s 4-momentum 1-form and ξ^a the
Killing vector associated with the pseudo-stationarity of Kerr spacetime (ξ=∂/∂t in Boyer–Lindquist coordinates). Far from the MBH, E coincides with the particle’s energy as an inertial observer at
rest with respect to the MBH would measure. For a circular orbit of radius r[0] in the equatorial plane of a Kerr BH of mass M and spin parameter a, the expression of E is (Bardeen et al. 1972)
$E = μ 1 − 2 M / r 0 + a M 1 / 2 / r 0 3 / 2 ( 1 − 3 M / r 0 + 2 a M 1 / 2 / r 0 3 / 2 ) 1 / 2 ,$(32)
where μ≡(−p[a]p^a)^1/2 is the particle’s rest mass.
Due to the reaction to gravitational radiation, the particle’s worldline is actually not a true timelike geodesic of Kerr spacetime, but is slowly inspiralling towards the central MBH. In particular,
E is not truly constant. Its secular evolution is governed by the balance law (Finn & Thorne 2000; Barack & Pound 2019; Isoyama et al. 2019)
where Ė ≡ dE/dt, L is the gravitational wave luminosity evaluated in Sect. 4.1 and L[H] is the power radiated down to the event horizon of the MBH. It turns out that in practice, L[H] is quite small
compared to L. From Table VII of Finn & Thorne (2000), we notice that for a=0, one has always |L[H]/Ė| < 4 × 10^−3 and for a=0.99M, one has |L[H]/Ė| < 9.5 × 10^−2, with |L[H]/Ė| < 10^−2 as soon
as r[0]> 7.3M. In the following, we will neglect the term L[H] in our numerical evaluations of Ė.
From Eq. (32), we have
$E ˙ = μ M 2 r 0 2 1 − 6 M / r 0 + 8 a M 1 / 2 / r 0 3 / 2 − 3 a 2 / r 0 2 ( 1 − 3 M / r 0 + 2 a M 1 / 2 / r 0 3 / 2 ) 3 / 2 r ˙ 0 .$(34)
In view of Eq. (2), the secular evolution of the orbital frequency f[0]=ω[0]/(2π) is related to ṙ[0] by
$f ˙ 0 f 0 = − 3 2 1 1 + a M 1 / 2 / r 0 3 / 2 r ˙ 0 r 0 ·$(35)
By combining successively Eqs. (34), (33) and (30), we get
$f ˙ 0 f 0 = 3 μ M 2 [ L ∼ ( r 0 M ) + L ∼ H ( r 0 M ) ] r 0 / M 1 + a M 1 / 2 / r 0 3 / 2 × ( 1 − 3 M / r 0 + 2 a M 1 / 2 / r 0 3 / 2 ) 3 / 2 1 − 6 M / r 0 + 8 a M 1 / 2 / r 0 3 / 2 − 3 a 2 / r 0 2
where we have introduced the rescaled horizon flux function $L ∼ H$, such that
$L H = ( μ M ) 2 L ∼ H ( r 0 M ) ·$(37)
This relative change in orbital frequency is depicted in Fig. 9, with a y-axis scaled to the mass (1) of Sgr A* for M and to μ=1M[⊙]. One can note that ḟ[0] diverges at the ISCO. This is due to
the fact that E is minimal at the ISCO, so that dE/dr[0]=0 there. At this point, a loss of energy cannot be compensated by a slight decrease of the orbit.
Fig. 9.
Relative change in orbital frequency ḟ[0]/f[0] induced by the reaction to gravitational radiation for an object of mass μ in circular equatorial orbit around Kerr BH of mass M equal to that of Sgr
A* as a function of the orbital radius r[0] (Eq. (36)). Each curve has a vertical asymptote at the ISCO radius for the corresponding value of a. The dotted curve corresponds to the quadrupole
approximation and quasi-Newtonian orbits. The inset shows the curves extended upto r[0]=50M.
Another representation of the orbital frequency evolution, via the adiabaticity parameter $ε ≡ f ˙ 0 / f 0 2$, is shown in Fig. 10. The adiabaticity parameter ε is a dimensionless quantity, the
smallness of which guarantees the validity of approximating the inspiral trajectory by a succession of circular orbits of slowly shrinking radii. As we can see on Fig. 10, ε< 10^−4 except very near
the ISCO, where ḟ[0] diverges.
Fig. 10.
Adiabaticity parameter $f ˙ 0 / f 0 2$ as a function of the orbital radius r[0]. The dotted curve corresponds to the quadrupole approximation and quasi-Newtonian orbits.
4.3. Inspiral time
By combining Eqs. (34), (33), (37) and (30), we get an expression for $r ˙ 0 − 1 = d t / d r 0$ as a function of r[0]. Once integrated, this leads to the time required for the orbit to shrink from an
initial radius r[0] to a given radius r[1]< r[0]:
$T ins ( r 0 , r 1 ) = M 2 2 μ ∫ r 1 / M r 0 / M 1 − 6 / x + 8 a ¯ / x 3 / 2 − 3 a ¯ 2 / x 2 ( 1 − 3 / x + 2 a ¯ / x 3 / 2 ) 3 / 2 × d x x 2 ( L ∼ ( x ) + L ∼ H ( x ) ) ,$(38)
where $a ¯ ≡ a / M = J / M 2$ is the dimensionless Kerr parameter. We shall call T[ins](r[0],r[1]) the inspiral time from r[0] to r[1]. For an object whose evolution is only driven by the reaction
to gravitation radiation (e.g. a compact object, cf. Sect. 5.3), we define then the life time from the orbit r[0] as
$T life ( r 0 ) ≡ T ins ( r 0 , r ISCO ) .$(39)
Indeed, once the ISCO is reached, the plunge into the MBH is very fast, so that T[life](r[0]) is very close to the actual life time outside the MBH, starting from the orbit of radius r[0].
The life time is depicted in Fig. 11, which is drawn for M=M[SgrA^*]. It appears from Fig. 11 that the life time near the ISCO is pretty short; for instance, for a=0 and a solar-mass object, it
is only 34 days at r[0]=6.1M. Far from the ISCO, it is much larger and reaches ∼3×10^5yr at r[0]=50M (still for μ=1M[⊙]). The dotted curve in Fig. 11 corresponds to the value obtained for
Newtonian orbits and the quadrupole formula (31): T[life]=5/256(M^2/μ)(r[0]/M)^4 (Peters 1964), a value which can be recovered by taking the limit x→+∞ in Eqs. (38) and (39) and using expression
(31) for $L ∼ ( x )$, as well as $L ∼ H ( x ) = 0$. For M=M[SgrA^*], the quadrupole formula becomes
$T life quad ≃ 4.2 × 10 4 ( 1 M ⊙ μ ) ( r 0 30 M ) 4 yr .$(40)
Fig. 11.
Life time of a (compact) object of mass μ in circular equatorial orbit around a Kerr BH with a mass M equal to that of Sgr A* as a function of the orbital radius r[0] (Eq. (39)). The inset shows
the curves extended up to r[0]=50M.
The relative difference between the exact formula (39) and the quadrupole approximation (40) is plotted in Fig. 12. Not surprisingly, the difference is very large in the strong field region, reaching
100% close to the ISCO. For r[0]=20M, it is still ∼10%. Even for r[0]=50M, it is as large as 3 to 5% for a⩾0.5M and 0.1% for a=0.
Fig. 12.
Relative difference between the life time given by Eq. (39) and the value given the quadrupole formula, Eq. (40), as a function of the orbital radius r[0].
5. Potential sources
Having established the signal properties and detectability by LISA, let us now discuss astrophysical candidates for the orbiting object. A preliminary required for the discussion is the evaluation of
the tidal effects exerted by Sgr A* on the orbiting body, since this can make the innermost orbit to be significantly larger than the ISCO. We thus start by investigating the tidal limits in Sect.
5.1. Then, in Sect. 5.2, we review the scenarios which might lead to the presence of stellar objects in circular orbits close to Sgr A*. The various categories of sources are then discussed in the
remaining subsections: compact objects (Sect. 5.3), main-sequence stars (5.4), brown dwarfs (Sect. 5.5), accretion flow (5.6), dark matter (Sect. 5.7) and artificial sources (Sect. 5.8). As it will
appear in the discussion, not all these sources are on the same footing regarding the probability of detection by LISA.
5.1. Tidal radius and Roche radius
In Sects. 2–4, we have considered an idealized point mass. When the orbiting object has some extension, a natural question is whether the object integrity can be maintained in presence of the tidal
forces exerted by the central MBH. This leads to the concept of tidal radiusr[T], defined as the minimal orbital radius for which the tidal forces cannot disrupt the orbiting body. In other words,
the considered object cannot move on an orbit with r[0]< r[T]. The tidal radius is given by the formula
where M is the mass of the MBH, ρ the mean density of the orbiting object and α is a coefficient of order 1, the value of which depends on the object internal structure and rotational state. From the
naive argument of equating the self-gravity and the tidal force at the surface of a spherical Newtonian body, one gets α=(3/(2π))^1/3=0.78. If one further assumes that the object is corotating,
i.e. is in synchronous rotation with respect to the orbital motion, then one gets α=(9/(4π))^1/3=0.89. Hills (1975) uses α=(6/π)^1/3=1.24, while Rees (1988) uses α=(3/(4π))^1/3=0.62. For
a Newtonian incompressible fluid ellipsoid in synchronous rotation, α=1.51 (Chandrasekhar 1969). This result has been generalized by Fishbone (1973) to incompressible fluid ellipsoids in the Kerr
metric: α increases then from 1.51 for r≫M to 1.60 (resp. 1.56) for r=10M and a=0 (resp. a=0.99M) (cf. Fig. 5 of Fishbone 1973, which displays 1/(πα^3)). Taking into account the
compressibility decreases α: α=1.34 for a polytrope of index n=1.5 (Ishii et al. 2005).
For a stellar type object on a circular orbit, a more relevant quantity is the Roche radius, which marks the onset of tidal stripping near the surface of the star, leading to some steady accretion to
the MBH (Roche lobe overflow) without the total disruption of the star (Dai et al. 2013; Dai & Blandford 2013). For centrally condensed bodies, like main-sequence stars, the Roche radius is given by
the condition that the stellar material fills the Roche lobe. In the Kerr metric, the volume V[R] of the Roche lobe generated by a mass μ on a circular orbit of radius r[0] has been evaluated by Dai
& Blandford (2013), yielding to the approximate formula^5V[R]≃μM^2𝒱[R], with
$V R ≡ ( r 0 M ) 3 [ 0.683 1 + χ 2.78 + 0.456 1 + χ 4.09 − 0.683 1 + χ 2.78 r 0 r ISCO + F ( a , χ ) ( r 0 r ISCO ( a ) − 1 ) ] ,$(42)
where r[ISCO] is the radius of the prograde ISCO, χ≡Ω/ω[0] is the ratio between the angular velocity Ω of the star (assumed to be a rigid rotator) with respect to some inertial frame to the orbital
angular velocity ω[0] and F(a,χ) is the function defined by
$F ( a , χ ) ≡ − 23.3 + 13.9 2.8 + χ + ( 23.8 − 14.8 2.8 + χ ) ( 1 − a ) 0.02 + ( 0.9 − 0.4 2.6 + χ ) ( 1 − a ) − 0.16 .$(43)
It should be noted that χ=1 for a corotating star. The Roche limit is reached when the actual volume of the star equals the volume of the Roche lobe. If ρ stands for the mean mass density of the
star, this corresponds to the condition μ=ρV[R], or equivalently
Solving this equation for r[0] leads to the orbital radius r[R] at the Roche limit, i.e. the Roche radius. The mass μ has disappeared from Eq. (44), so that r[R] depends only on the mean density ρ
and the rotational parameter χ. For r[0]≫r[ISCO], we can neglect the second term in the square brackets in Eq. (42) and obtain an explicit expression:
$r R ≃ 1.14 ( 1 + χ 2.78 ) 1 / 3 ( M ρ ) 1 / 3 for r R ≫ M .$(45)
This equation has the same shape as the tidal radius formula (41). Using Sgr A* value (1) for M, we may rewrite the above formula as
$r R M ≃ 33.8 ( 1 + χ 2.78 ) 1 / 3 ( ρ ⊙ ρ ) 1 / 3 for r R ≫ M ,$(46)
where ρ[⊙]≡1.41×10^3kg⋅m^−3 is the mean density of the Sun.
The numerical resolution of Eq. (44) for r[R] has been implemented in the kerrgeodesic_gw package (cf. Appendix A) and the results are shown in Fig. 13 and Table 2. The straight line behavior in the
left part of Fig. 13 corresponds to the power law r[R]∝ρ^−1/3 in the asymptotic formula (46). In Table 2, the characteristics of the red dwarf star are taken from Fig. 1 of Chabrier et al. (2007) –
it corresponds to a main-sequence star of spectral type M4V. The brown dwarf model of Table 2 is the model of minimal radius along the 5 Gyr isochrone in Fig. 1 of Chabrier et al. (2009). This brown
dwarf is close to the hydrogen burning limit and to the maximum mean mass density ρ among brown dwarfs and main-sequence stars. We note from Table 2 that it has a Roche radius very close to the
Schwarzschild ISCO. We note as well that r[R]< M for a white dwarf. This means that such a star is never tidally disrupted above Sgr A*’s event horizon. A fortiori, neutron stars share the same
Fig. 13.
Roche radius r[R] as a function of the mean density ρ of the star (in solar units), for two values of the MBH spin a and two rotational states of the star: irrotational (χ=0) and corotating (χ=
1). The blue (resp. red) dotted horizontal line marks the ISCO radius for a=0 (resp. a=0.98M).
Table 2.
Roche radius r[R] for different types of objects orbiting Sgr A*.
5.2. Presence of stellar objects in the vicinity of Sgr A*
The Galactic center is undoubtably a very crowded region. For instance, it is estimated that there are ∼2×10^4 stellar BHs in the central parsec, a tenth of which are located within 0.1pc of Sgr
A* (Freitag et al. 2006). The recent detection of a dozen of X-ray binaries in the central parsec (Hailey et al. 2018) supports these theoretical predictions. The two-body relaxation in the central
cluster causes some mass segregation: massive stars lose energy to lighter ones and drift to the center (Hopman & Alexander 2005; Freitag et al. 2006). Accordingly BHs are expected to dominate the
mass density within 0.2pc. However, they do not dominate the number density, main-sequence stars being more numerous than BHs (Freitag et al. 2006; Amaro-Seoane 2018). The number of stars or stellar
BHs very close to Sgr A* (i.e. located at r< 100M) is expected to be quite small though. Indeed the central parsec region is very extended in terms of Sgr A*’s length scale: 1pc=5.1×10^6M,
where M is Sgr A*’s mass. At the moment, the closest known stellar object orbiting Sgr A* is the star S2, the periastron of which is located at r[p]=120au≃3×10^3M (GRAVITY Collaboration 2018a
The most discussed process for populating the vicinity of the central MBH is the extreme mass ratio inspiral (EMRI) of a (compact) star or stellar BH (Amaro-Seoane et al. 2007; Amaro-Seoane 2018). In
the standard scenario (see e.g. Amaro-Seoane 2018 for a review), the inspiralling object originates from the two-body scattering by other stars in the Galactic center cluster. It keeps a very high
eccentricity until the final plunge in the MBH, despite the circularization effect of gravitational radiation (Hopman & Alexander 2005). Such an EMRI is thus not an eligible source for the process
considered in the present article, which is limited to circular orbits.
Another kind of EMRI results from the tidal separation of a binary by the MBH (Miller et al. 2005). In such a process, a member of the binary is ejected at high speed while the other one is captured
by the MBH and inspirals towards it, on an initially low eccentricity orbit. Gravitational radiation is then efficient in circularizing the orbit, making it almost circular when it enters LISA band.
Such an EMRI is thus fully relevant to the study presented here. The rate of formation of these zero-eccentricity EMRIs is very low, being comparable to those of high-eccentricities EMRIs (Miller et
al. 2005), which is probably below 10^−6yr^−1 (Amaro-Seoane 2018; Hopman & Alexander 2005). However, as discussed in Sect. 5.3, due to their long life time (> 10^5yr) in the LISA band, the
probability of detection of these EMRIs is not negligibly small.
Another process discussed in the literature and leading to objects on almost circular orbits is the formation of stars in an accretion disk surrounding the MBH (see e.g. Collin & Zahn 1999, 2008;
Nayakshin et al. 2007, and references therein). Actually, it was particularly surprising to find in the inner parsec of the Galaxy a population of massive (few 10M[⊙]) young stars, that were formed
≈6 Myr ago (Genzel et al. 2010). Indeed, forming stars in the extreme environment of a MBH is not obvious because of the strong tidal forces that would break typical molecular clouds. A few scenarios
were proposed to account for this young stellar population; see Mapelli & Gualandris (2016) for a recent dedicated review. Among these, in situ formation might take place in a geometrically thin
Keplerian (circularly orbiting) accretion disk surrounding the MBH (Collin & Zahn 1999, 2008; Nayakshin et al. 2007). Such an accretion disk is not presently detected, and would have existed in past
periods of AGN activity at the Galactic center (Ponti et al. 2013, 2014).
Stellar formation in a disk is supported by the fact that the massive young stellar population proper motion was found to be consistent with rotational motion in a disk (Paumard et al. 2006). It is
interesting to note that the on-sky orientation of this stellar disk is similar to the orientation of the orbital plane of a recently detected flare of Sgr A* (GRAVITY Collaboration 2018b). However,
such a scenario suffers from the fact that the young stars observed have a median eccentricity of 0.36±0.06 (Bartko et al. 2009), while formation in a Keplerian disk leads to circular orbits. On
the other side, the recently detected X-ray binaries (Hailey et al. 2018) mentioned above are most probably quiescent BH binaries. These BHs are likely to have formed in situ in a disk (Generozov et
al. 2018), giving more support to the scenario discussed here.
A population of stellar-mass BHs will form after the death of the most massive stars born in the accretion disk. These would be good candidates for the scenario discussed here, provided the initially
circular orbit is maintained after supernova explosion. The recent study of Bortolas et al. (2017) shows that BHs formed from the supernova explosion of one of the members of a massive binary keep
their initial orbit without noticeable kink from the supernova explosion. Given that a large fraction (tens of percent) of the Galactic center massive young stars are likely to be binaries (Sana et
al. 2011), this shows that circular-orbiting BHs are likely to exist within the framework of the Keplerian in-situ star formation model. This scenario was already advocated by Levin (2007), which
considers the fragmentation of a self-gravitating thin accretion disk that forms massive stars, leading to the formation of BHs that inspiral in towards Sgr A*, following quasi-circular orbits, in a
typical time of ≈10 Myr.
5.3. Compact objects
As discussed in Sect. 5.1, compact objects – BHs, neutron stars and white dwarfs – do not suffer any tidal disruption above the event horizon of Sgr A*. Their evolution around Sgr A* is thus entirely
given by the reaction to gravitational radiation with the timescale shown by Fig. 11.
Let us define the entry in LISA band as the moment in the slow inspiral when S/N[1yr] reaches 10, which is the threshold we adopt for a positive detection (Eq. (26)). The orbital radius at the entry
in LISA band in plotted in Fig. 14 as a function of the mass μ of the inspiralling object. It is denoted by r[0,max] since it is the maximum radius at which the detection is possible. Some selected
values are displayed in Table 3. The mass of the primordial BH has been chosen arbitrarily to be the mass of Jupiter (10^−3M[⊙]), as a representative of a low mass compact object.
Fig. 14.
Maximum orbital radius r[0,max] for a S/N=10 detection by LISA in one year of data, as a function of the mass μ of the object orbiting around Sgr A*.
Table 3.
Orbital radius r[0,max] at the entry in LISA band (S/N[1yr]⑾10), the corresponding gravitational wave frequency f[m=2](r[0,max]) and the time spent in LISA band until the ISCO, T[in-band], for
various compact objects orbiting Sgr A*.
For a compact object, the time T[in-band] spent in LISA band is nothing but the inspiral time from r[0,max] to the ISCO:
$T in-band ≡ T ins ( r 0 , max , r ISCO ) = T life ( r 0 , max ) ,$(47)
where T[ins] is given by Eq. (38) and T[life] by Eq. (39). The time in LISA band is depicted in Fig. 15 and some selected values are given in Table 3. The trends in Fig. 15 can be understood by
noticing that, at fixed initial radius, the inspiral time is a decreasing function of μ (as μ^−1, cf. Eq. (38)), while it is an increasing function of the initial radius (as $r 0 4$ at large
distance, cf. Eq. (40)), the latter being larger for larger values of μ, since r[0] marks the point where S/N[1yr]=10, the S/N being an increasing function of μ (cf. Eq. (24)). The behavior of the
T[in-band] curves in Fig. 15 results from the balance between these two competing effects. The maximum is reached for masses around 10^−3M[⊙] for a=0 (maxT[in-band]∼9×10^5yr) and around 10^
−5 M[⊙] for a=0.98M (maxT[in-band]∼2×10^6yr), which correspond to hypothetical primordial BHs.
Fig. 15.
Time elapsed between the entry in LISA band (S/N[1yr] reaching 10) and the ISCO for a compact object inspiralling around Sgr A*, as a function of the object’s mass μ.
The key feature of Fig. 15 and Table 3 is that the values of T[in-band] are very large, of the order of 10^5yr, except for very small values of μ (below 10^−4M[⊙]). This contrasts with the time in
LISA band for extragalactic EMRIs, which is only 1–10^2yr. This is of course due to the much larger S/N resulting from the proximity of the Galactic center. This large time scale counter-balances
the low event rate for the capture of a compact object by Sgr A* via the processes discussed in Sect. 5.2: even if only a single compact object is driven to the close vicinity of Sgr A* every 10^
6yr, the fact that it remains there in the LISA band for ∼10^5yr makes the probability of detection of order 0.1. Given the large uncertainty on the capture event rate, one can be reasonably
One may stress as well that white dwarfs, which are generally not considered as extragalactic EMRI sources for LISA because of their low mass, have a larger value of T[in-band] than BHs (cf. Table 3
). Given that they are probably more numerous than BHs in the Galactic center, despite mass segregation (cf. the discussion in Sect. 5.2 and Freitag 2003b), they appear to be good candidates for a
detection by LISA.
5.4. Main-sequence stars
As discussed in Sect. 5.1 (see Table 2), main-sequence stars orbiting Sgr A* have a Roche limit above the ISCO. Away from the Roche limit, the evolution of a star on a quasi-circular orbit is driven
by the loss of energy and angular angular momentum via gravitational radiation, as for the compact objects discussed above. The orbit thus shrinks until the Roche limit is reached. At this point, the
star starts to loose mass through the Lagrange point L[1] (Hameury et al. 1994; Dai & Blandford 2013; standard accretion onto the MBH by Roche lobe overflow) and possibly through the outer Lagrange
point L[2] as well for stars of mass μ≳1M[⊙] (Linial & Sari 2017). In any case, the mass loss is stable and proceeds on a secular time scale (with respect to the orbital period). The net effect on
the orbit is an increase of its radius (Hameury et al. 1994; Dai & Blandford 2013; Linial & Sari 2017), at least for masses μ< 5.6M[⊙] (Linial & Sari 2017). Accordingly, instead of an EMRI, one
may speak about an extreme mass ratio outspiral (EMRO; Dai & Blandford 2013), or a reverse chirp gravitational wave signal (Linial & Sari 2017; Jaranowski & Krolak 1992) when describing the evolution
of such systems after they have reached the Roche limit.
For stars, let us denote by $T in-band ins$ the inspiral time from the entry in LISA band (r[0]=r[0,max], cf. Fig. 14) to the Roche limit (r[0]=r[R], cf. Table 2). $T in-band ins$ is a lower
bound for the total time spent in LISA band, the latter being $T in-band ins$ augmented by the mass-loss time at the Roche limit, which can be quite large, of the order of 10^5yr (Dai & Blandford
2013). The values of $T in-band ins$ are given in Table 4 for three typical main-sequence stars: a Sun-like one, a red dwarf (μ=0.2M[⊙], same as in Table 2) and a main-sequence star of mass μ=
2.4M[⊙], which corresponds to a spectral type A0V. Let us mention that current observational data cannot rule out the presence of such a rather luminous star in the vicinity of Sgr A*: GRAVITY
observations (GRAVITY Collaboration 2017) have set the upper luminosity threshold to a B8V star, which is a main-sequence star of mass μ=3.8M[⊙].
Table 4.
Inspiral time to the Roche limit in LISA band for a μ=0.062M[⊙] brown dwarf and different types of main-sequence stars.
$T in-band ins$ appears to be very large, of the order of 10^5yr, except for the 2.4M[⊙]-star for the inclination angle θ=π/2, which has r[R]> r[0,max], i.e. it is not detectable by LISA. As
already argued for compact objects, this large value of the time spent in LISA enhances the detection probability.
Regarding main-sequence stars, we note that the recently claimed detection of a 149min periodicity in the X-ray flares from Sgr A* (Leibowitz 2018) has been interpreted as being caused by a μ=
0.18M[⊙] star orbiting at r[0]=13.2M, where it is filling its Roche lobe (Leibowitz 2018). If such a star exists, we can read from Fig. 6 that LISA can detect it with a S/N equal to 76 (resp.
35) for θ=0 (resp. θ=π/2) in a single day of data.
5.5. Brown dwarfs
Brown dwarfs are less massive than main-sequence stars, their mass range being ∼10^−2 to ∼0.08M[⊙] (Chabrier & Baraffe 2000; Chabrier et al. 2009). Accordingly, they enter later (i.e. at smaller
orbital radii) in the LISA band. However, they are more dense than main-sequence stars, so that their Roche limit is closer to the MBH, as already noticed in Sect. 5.1: the μ=0.062M[⊙] brown dwarf
of Table 2 has a Roche radius of order 7M, i.e. quite close to the Schwarzschild ISCO. In this region the S/N is quite high, despite the low value of μ: for μ=0.062M[⊙] and θ=0, S/N[1yr]=
7.4×10^3 (resp. S/N[1yr]=5.4×10^3) at the Roche limit with χ=0 (resp. χ=1). For θ=π/2, these numbers become S/N[1yr]=3.7×10^3 (χ=0) and S/N[1yr]=2.6×10^3 (χ=1). Moreover,
brown dwarfs stay longer in this region than compact objets since the inspiral time is inversely proportional to the mass μ of the orbiting object (cf. Eq. (42)). As we can see from the values in
Table 4, the inspiral time in LISA band of brown dwarfs is even larger than that of main-sequence stars: $T in-band ins ∼ 5 × 10 5 yr$ for θ=0 and $T in-band ins ∼ 3 × 10 5 yr$ for θ=π/2. These
large values tends to make brown dwarfs good candidates for detection by LISA. To conclude, one should know the capture rate of brown dwarfs by Sgr A*. It is highly uncertain but estimates have been
provided very recently by Amaro-Seoane (2019), which lead to a detection probability of one, with ∼20 brown dwarfs in LISA band at any moment, among which ∼5 have almost circular orbits.
5.6. Inner accretion flow
Sgr A*’s accretion flow is known for generating particularly low-luminosity radiation, orders of magnitude below the Eddington limit, and orders of magnitude below what could be available from the
gas supply at a Bondi radius (Falcke & Markoff 2013). This means that accretion models should be very inefficient in converting viscously dissipated energy into radiation. This energy will rather be
stored in the disk as heat, so that Sgr A* accretion flow must be part of the hot accretion flow family (Yuan & Narayan 2014). Such systems are made of a geometrically thick, optically thin, hot
(i.e. close to the virial temperature) accretion flow, probably accompanied by outflows. A plethora of studies have been devoted to modeling the hot flow of Sgr A*, see Falcke & Markoff (2000),
Vincent et al. (2015), Broderick et al. (2016), Ressler et al. (2017), Davelaar et al. (2018), among many others, and references therein.
There is reasonable agreement between these different authors regarding the typical number density and geometry of the geometrically thick hot flow in the close vicinity of Sgr A*. The electron
maximum number density is of order 10^8cm^−3 (to within one order of magnitude), and the density maximum is located at a Boyer–Lindquist radius of around 10M (to within a factor of a few). It is
thus straightforward to give a very rough estimate of the mass of the flow, which is of the order of ≈5×10^−11M[⊙] (where we consider a constant-density torus with circular cross section of radius
4M, such that its inner radius is at the Schwarzschild ISCO). This extremely small total mass of Sgr A*’s accretion flow makes it impossible to detect gravitational waves from orbiting
inhomogeneities. Figure 6 shows that the LISA S/N would be vanishingly small, assuming for instance an inhomogeneity of 10% of the total mass.
5.7. Dark matter
The dark matter (DM) density profile in the inner regions of galaxies is subject to debate. There is a controversy between observations and cold-dark-matter simulations regarding the value of the DM
density power-law slope in the inner kpc, observations advocating a cored profile ρ(r)∝r^0, while simulations predict ρ(r)∝r^−1 (de Blok 2010). The parsec-scale profile is even less well known.
Gondolo & Silk (1999) have proposed a model of the interaction of the central MBH with the surrounding DM distribution for the Milky Way. According to these authors, the presence of the MBH should
lead to an even more spiky inner profile, with a scaling of ρ(r)∝r^−2.3. Such a dark matter spike can be constrained by high-angular resolution observation at the Galactic center (Lacroix 2018).
Figure 1 of Lacroix (2018) shows the enclosed DM mass at the Galactic center as a function of radius, for various DM models: either nonannihilating DM, or selfannihilating DM (with particle mass
equal to 1 TeV) for various cross sections. Weakly-interacting DM (⟨σv⟩ < 10^−30cm^3s^−1) leads to an enclosed mass higher than 10^−4M[⊙] in the inner 10M. Figure 6 shows that this leads to S/N
[1yr]> 0.2, assuming that 10% inhomogeneities would appear in the DM distribution and orbit circularly around the MBH around 10M. For nonannihilating DM, the S/N values can be as high as S/N
[1yr]∼10^4. This makes a DM spike an interesting candidate for a potential gravitational wave source at the Galactic center, to be studied in details in a forthcoming article (Le Tiec et al., in
5.8. Artificial sources
The MBH Sgr A* is indubitably a unique object in our Galaxy. If^6 an advanced civilization exists, or has existed, in the Galaxy, it would seem unlikely that it has not shown any interest in Sgr A*.
On the contrary, it would seem natural that such a civilization has put some material in close orbit around Sgr A*, for instance to extract energy from it via the Penrose process. Whatever the reason
for which the advanced civilization acted so (it could be for purposes that we humans simply cannot imagine), the orbital motion of this material necessarily emits gravitational waves and if the mass
is large enough, these waves could be detected by LISA. Given the S/N values obtained in Sect. 3 and assuming that Sgr A* is a fast rotator, an object of mass as low^7 as the Earth mass orbiting
close to the ISCO is detectable by LISA. This scenario is discussed further by Abramowicz et al. (2019), who consider a long lasting Jupiter-mass orbiter, left as a “messenger” by an advanced
civilization, which possibly disappeared billions of years ago.
6. Discussion and conclusions
We have conducted a fully relativistic study of gravitational radiation from bodies on circular orbits in the equatorial plane of the 4.1×10^6M[⊙] MBH at the Galactic center, Sgr A*. We have
performed detailed computations of the S/N in the LISA detector, taking into account all the harmonics in the signal, whereas previous studies (Freitag 2003b; Dai & Blandford 2013; Linial & Sari 2017
; Kuhnel et al. 2018) were limited to the Newtonian quadrupole approximation, which yields only the m=2 harmonic for circular orbits. The Roche limits have been evaluated in a relativistic
framework as well, being based on the computation of the Roche volume in the Kerr metric (Dai & Blandford 2013). This is specially important for brown dwarfs, since their Roche limit occurs in the
strong field region.
Setting the detection threshold to S/N[1yr]=10, we have found that LISA has the capability to detect orbiting masses close to Sgr A*’s ISCO as small as ten Earth masses or even one Earth mass if
Sgr A* is a fast rotator (a≳0.9M). Given the strong tidal forces at the ISCO, these small bodies have to be compact objects, i.e. small BHs. Planets and main-sequence stars have a Roche limit
quite far from the ISCO: r[R]∼34M for a solar-type star (or Jupiter-type planet) and r[R]∼13M for a 0.2M[⊙] star. However, even at these distances, main-sequence stars are still detectable by
LISA, the entry in LISA band (defined by S/N[1yr]=10) being achieved for r[0,max]∼47M for a solar-type star and at r[0,max]∼35M for a 0.2M[⊙] main-sequence star, assuming an inclination
angle θ=0. Because they are more dense, brown dwarfs have a Roche limit pretty close to the ISCO, the minimal Roche radius being r[R]∼7M, which is achieved for a 0.062M[⊙] brown dwarf. For such
an object, the entry in LISA band occurs at r[0,max]∼28M.
Beside the S/N at a given orbit, a key parameter is the total time spent in LISA band, i.e. the time T[in-band] during which the source has S/N[1yr]⩾10. We have found that, once they have entered
LISA band from the low frequency side, all the considered objects, be they compact objects, main-sequence stars or brown dwarfs, spend more than 10^5yr in LISA band^8. The minimal time in-band
occurs for high-mass BHs (μ∼30M[⊙]), for which T[in-band]∼1×10^5yr (assuming θ=0) and the maximal one, of the order of one million years, is achieved for a Jupiter-mass BH (μ∼10^−3M[⊙])
if Sgr A* is a slow rotator (a/M≪1): T[in-band]∼9×10^5yr, or for a μ∼10^−5M[⊙] BH if Sgr A* is a rapid rotator (a/M≳0.9): T[in-band]∼2×10^6yr. These small BH masses regard primordial
BHs. Among stars and stellar BHs, the maximum time spent in LISA band is achieved for brown dwarfs: T[in-band]⩾5×10^5yr, just followed by low-mass main-sequence stars (red dwarfs) and white
dwarfs, for which T[in-band]⩾3×10^5yr. These large values of T[in-band] contrast with those for extragalactic EMRIs, which are typically of the order of 1–10^2yr. This is of course due to the
much larger S/N resulting from the proximity of Sgr A*, which allows one to catch compact objects at much larger orbital radii, where the orbital decay is not too fast, and to catch main-sequence
stars above their Roche limit.
To predict some LISA detection rate from T[in-band], one shall know the rate at which the considered objects are brought to close circular orbits around Sgr A* (“capture” rate). While we have briefly
described some scenarios proposed in the literature in Sect. 5.2, it is not the purpose of this work to make precise estimates. Having those is probably very difficult, given the involved
uncertainties, both on the observational ground (strong absorption in the direction of the Galactic center) and the theoretical one (dynamics of the tens of thousands of stars and BHs in the central
parsec). Some optimistic scenarios mentioned in Sect. 5.2 predict a capture rate of the order of 10^−6yr^−1 for BHs. For T[in-band]∼10^5yr, this would result in a detection probability of 0.1 by
LISA. For white dwarfs, low mass main-sequence stars and brown dwarfs, the capture rate could possibly be higher (Freitag 2003b), leading to a significant detection probability by LISA, especially
for brown dwarfs. Instead of making any concrete prediction, we prefer an “agnostic” approach, stating that Sgr A* is definitely a target worth of attention for LISA, which may reveal various bodies
orbiting around it.
Let us point out that Amaro-Seoane (2019) has recently performed a study of gravitational radiation from main-sequence stars and brown dwarfs orbiting Sgr A*. He finds results similar to ours
regarding the S/N in LISA. Also, he derives the event rate for the Galactic center taking into account the relativistic loss-cone and eccentric orbits, which are more typical in an astrophysical
context. The high event rate that he has obtained makes brown dwarfs promising candidates for LISA.
In Appendix C, we have considered bodies in close circular orbit around the 2.5×10^6M[⊙] MBH in the center of the nearby galaxy M 32. We find that main-sequence stars with μ⩾0.2M[⊙] are not
detectable by LISA in this case, while compact objects and brown dwarfs are still detectable, with a lower probability: the time they are spending in LISA band with S/N[1yr]⩾10 is 10^3 to 10^4
years, that is two orders of magnitude lower than for Sgr A*.
A natural extension of the work presented here is towards noncircular orbits. Gravitational waves from a compact body on eccentric, equatorial (Glampedakis & Kennefick 2002), spherical (Hughes 2000),
and generic bound (Drasco & Hughes 2006) geodesics have been studied before. The application of these results to Sgr A* including the calculation of the orbital decay for generic orbits, exploration
of the inspiral parameter space, and the analysis of the tidal and Roche radii remains to be completed. Another extension would be to study the gravitational emission from a (stochastic) ensemble of
small masses, such as brown dwarfs, in the case they are numerous around Sgr A*, or from dark matter clumps as mentioned in Sect. 5.7 (Le Tiec et al., in prep.).
As discussed further in Sect. 4, an orbiting body’s true worldline spirals inwards due to gravitational radiation reaction. A geodesic that is tangent to the worldline at an instance will dephase
from the inspiraling worldline on a timescale ∼Mϵ^−1/2 where ϵ≡μ/M is the mass ratio. By approximating the radiation reaction force at each instance by that computed along a tangent geodesic one
can compute a worldline that dephases from the true inspiral over the radiation reaction timescale of ∼Mϵ^−1 (Hinderer & Flanagan 2008).
Our values of $Z lm ∞ ( r 0 )$ have a sign opposite to those of Poisson (1993a) due to a different choice of metric signature, namely (+,−,−,−) in Poisson (1993a) vs. (−,+,+,+) here,
and hence a different sign of (h[+],h[×]).
Equation (B.7a) immediately follows from the well known identity $∫ − ∞ ∞ sinc ( π x ) d x = 1$.
Andromeda Galaxy itself harbors a MBH in its nucleus, but it has M∼10^8M[⊙] (Bender et al. 2005), which is too massive for LISA band. Beyond the Local Group, nearby galaxies with a MBH in the LISA
range have been considered by Berry & Gair (2013c) in their study of extreme mass ratio burts (cf. Sect. 1).
We are grateful to Antoine Petiteau for having provided us with the LISA noise power spectral density curve and to Pau Amaro-Seoane, Michał Bejger, Christopher Berry, Gilles Chabrier, Suzy
Collin-Zahn, and Thibaut Paumard for fruitful discussions. NW gratefully acknowledges support from a Royal Society – Science Foundation Ireland University Research Fellowship.
Appendix A: The kerrgeodesic_gw package
We have developed the open-source package kerrgeodesic_gw for the Python-based free mathematics software system SageMath^9. This package implements all the computations presented in this article. The
installation of kerrgeodesic_gw is very easy, since it relies on the standard pip mechanism for Python packages. One only needs to run
sage -pip install kerrgeodesic_gw
to download and install the package in any working SageMath environment. The sources of the package are available at the following git repository, as part of the Black Hole Perturbation Toolkit^10:
The reference manual of kerrgeodesic_gw includes many examples and is online at
Various Jupyter notebooks making use of kerrgeodesic_gw are publicly available on the cloud platform CoCalc, including those used to generate all the figures presented in the current article:
Other notebooks regard tests of the package, like the comparison with the 1.5 PN waveforms obtained by Poisson (1993a) for a=0 and with the fully relativistic waveforms obtained by Detweiler (1978)
for a=0.5M and a=0.9M:
Appendix B: Computation of the S/N integral
In order to evaluate the S/N integral (22), we need to compute the Fourier transforms $h ∼ + ( f )$ and $h ∼ × ( f )$ over the observation time T via Eq. (23). Let us focus first on h[+](t) and
rewrite its Fourier series (16) as
$h + ( t ) = μ r ∑ m = 1 + ∞ H m + ( θ ) cos ( 2 π m f 0 t + χ m ) ,$(B.1)
where the amplitude $H m + (θ)$ is defined by Eq. (18) and the phase angle χ[m] is defined by (cf. Eqs. (16) and (14))
$χ m = m ( φ 0 − φ − 2 π f 0 r ∗ ) + Φ m ,$(B.2)
$cos Φ m = A m + ( θ ) H m + ( θ ) and sin Φ m = − B m + ( θ ) H m + ( θ ) .$(B.3)
The Fourier transform (23) is then
$h ∼ + ( f ) = μ r ∑ m = 1 + ∞ H m + ( θ ) ∫ − T / 2 T / 2 cos ( 2 π m f 0 t + χ m ) e − 2 π i f t d t = μ 2 r ∑ m = 1 + ∞ H m + ( θ ) ∫ − T / 2 T / 2 [ e 2 π i m f 0 t + i χ m − 2 π i f t + e − 2 π
i m f 0 t − i χ m − 2 π i f t ] d t = μ 2 r ∑ m = 1 + ∞ H m + ( θ ) [ e i χ m ∫ − T / 2 T / 2 e 2 π i ( m f 0 − f ) t d t + e − i χ m ∫ − T / 2 T / 2 e − 2 π i ( m f 0 + f ) t d t ] = μ 2 r ∑ m = 1 +
∞ H m + ( θ ) [ e i χ m 2 i sin ( π ( m f 0 − f ) T ) 2 π i ( m f 0 − f ) + e − i χ m − 2 i sin ( π ( m f 0 + f ) T ) − 2 π i ( m f 0 + f ) ] = μ 2 r T ∑ m = 1 + ∞ H m + ( θ ) [ e i χ m sinc ( π ( f
− m f 0 ) T ) + e − i χ m sinc ( π ( f + m f 0 ) T ) ] ,$(B.4)
where sinc stands for the cardinal sine function: sinc(x)≡sinx/x. The square of the modulus of $h ∼ + ( f )$, which appears in the S/N formula (22), is then
$| h ∼ + ( f ) | 2 = h ∼ + ( f ) h ∼ + ( f ) ∗ = ( μ 2 r T ) 2 ( ∑ m = 1 + ∞ H m + ( θ ) [ e i χ m sinc ( π ( f − m f 0 ) T ) + e − i χ m sinc ( π ( f + m f 0 ) T ) ] ) × ( ∑ n = 1 + ∞ H n + ( θ ) [
e − i χ n sinc ( π ( f − n f 0 ) T ) + e i χ n sinc ( π ( f + n f 0 ) T ) ] ) = ( μ r ) 2 T 4 ∑ m = 1 + ∞ ∑ n = 1 + ∞ H m + ( θ ) H n + ( θ ) × [ e i ( χ m − χ n ) T sinc ( π ( f − m f 0 ) T ) sinc (
π ( f − n f 0 ) T ) + e i ( χ m + χ n ) T sinc ( π ( f − m f 0 ) T ) sinc ( π ( f + n f 0 ) T ) + e − i ( χ m + χ n ) T sinc ( π ( f + m f 0 ) T ) sinc ( π ( f − n f 0 ) T ) + e i ( χ n − χ m ) T
sinc ( π ( f + m f 0 ) T ) sinc ( π ( f + n f 0 ) T ) ] = ( μ r ) 2 T 4 ∑ m = 1 + ∞ ∑ n = 1 + ∞ H m + ( θ ) H n + ( θ ) × [ e i ( χ m − χ n ) Δ T , m f 0 ( f ) sinc ( π ( f − n f 0 ) T ) + e i ( χ m
+ χ n ) Δ T , m f 0 ( f ) sinc ( π ( f + n f 0 ) T ) + e − i ( χ m + χ n ) Δ T , − m f 0 ( f ) sinc ( π ( f − n f 0 ) T ) + e i ( χ n − χ m ) Δ T , − m f 0 ( f ) sinc ( π ( f + n f 0 ) T ) ] ,$(B.5)
where the functions Δ[T,f[*]](f) are defined for any pair of real parameters (T,f[*]) by
$Δ T , f ∗ ( f ) ≡ T sinc ( π ( f − f ∗ ) T ) .$(B.6)
For each value of f[*], the Δ[T,f[*]] constitute a family of nascent delta functions, i.e. they obey^11
$∫ − ∞ + ∞ Δ T , f ∗ ( f ) d f = 1$(B.7a)
$∀δf>0, lim T→+∞ ∫ ℝ∖( f * −δf, f * +δf) Δ T, f * (f)df=0.$(B.7b)
These two properties imply that, for any integrable function F,
$lim T→+∞ ∫ −∞ ∞ F (f) Δ T, f * (f)df=F( f * ).$(B.8)
In other words, when T→+∞, Δ[T,f[*]] tends to the Dirac delta distribution centered on f[*]. Considering successively the four terms that appear in Eq. (B.5) and gathering them two by two by means
of ±, we have then
$∫ 0 + ∞ Δ T , m f 0 ( f ) sinc ( π ( f ± n f 0 ) T ) S n ( f ) d f ≃ sinc ( π ( m ± n ) f 0 T ) S n ( m f 0 ) when T → + ∞ ,$(B.9a)
$∫ 0 + ∞ Δ T , − m f 0 ( f ) sinc ( π ( f ± n f 0 ) T ) S n ( f ) d f → 0 when T → + ∞ .$(B.9b)
It should be noted that (B.9b) readily follows from property (B.7b) since −mf[0]< 0. Regarding Eq. (B.9a), we note that
$lim T→+∞ sinc ( π ( m − n ) f 0 T ) = { 1 if n = m 0 if n ≠ m and lim T→+∞ sinc ( π ( m + n ) f 0 T ) = 0 ,$(B.10)
the last property resulting from m + n ≠ 0 for m⩾1 and n⩾1. In view of Eqs. (B.5) and (B.9a)–(B.10), we see that, when T→+∞, the only contribution to the S/N integral (22) arises from the first
term in Eq. (B.5) with moreover n=m, which implies e^i(χ[m]−χ[n])=1. Hence we have
$∫ 0 + ∞ | h ∼ + ( f ) | 2 S n ( f ) d f ≃ ( μ r ) 2 T 4 ∑ m = 1 + ∞ H m + ( θ ) 2 S n ( m f 0 ) for T → + ∞ .$(B.11)
The limit T→+∞, which arises from Eqs. (B.9a) and (B.10), can be translated by mf[0]T≫1 for all m, i.e. by f[0]T≫1. Obviously, we get a similar formula for the contribution of $| h ∼ × ( f ) |
2$ to the S/N, so that Eq. (22) becomes
$ρ 2 = 4 ∫ 0 + ∞ | h ∼ + ( f ) | 2 + | h ∼ × ( f ) | 2 S n ( f ) d f ≃ ( μ r ) 2 T ∑ m = 1 + ∞ H m + ( θ ) 2 + H m × ( θ ) 2 S n ( m f 0 ) for f 0 T ≫ 1 ,$(B.12)
hence the S/N value (24).
Appendix C: Case of M 32
Apart from Sgr A*, the only MBH in the Local Group of galaxies whose mass fits LISA band is the one in the center of M 32 – the compact elliptical galaxy satellite of the Andromeda Galaxy M31^12. Its
mass is $M = 2 . 5 − 1.0 + 0.6 × 10 6 M ⊙$ (Nguyen et al. 2018). The distance to the Earth is r≃790kpc (Nguyen et al. 2018), i.e. roughly a hundred time farther than Sgr A*.
The LISA S/N for objects on circular equatorial orbits around M 32 MBH is depicted as a function of the orbital radius in Fig. C.1. The minimal mass μ[min] detectable with S/N[1yr]≥10 at a given
orbital radius is shown in Fig. C.2. We note that the minimal detectable mass is ∼2×10^−3M[⊙] (close to the ISCO) if M 32 MBH is a slow rotator, down to ∼2×10^−4M[⊙] in the case of a fast
rotator. The Roche limits for the various kinds of stars considered in Sect. 5.1, reevaluated to take into account M 32 MBH mass M, have been drawn in Fig. C.2. It appears then clearly that a
solar-type star in circular orbit around M 32 MBH cannot be detected by LISA and that a 0.2M[⊙] red dwarf can be marginally detected, while there is no issue in detecting a brown dwarf at its Roche
Fig. C.1.
Effective (direction and polarization averaged) signal-to-noise in LISA for a T=1yr observation of an object of mass μ=1M[⊙] orbiting M 32 MBH, as a function of the orbital radius r[0] (in
units of M, the mass of M 32 MBH), and for selected values of the MBH spin parameter a as well as selected values of the inclination angle θ. Each curve starts at the ISCO radius of the
corresponding value of a. It should be noted that this figure is scaled for T=1yr, while the equivalent figure for Sgr A* (Fig. 6) is scaled for T=1d.
Fig. C.2.
Minimal detectable mass with S/N[1yr]⩾10 in LISA observations of M 32 center, as a function of the orbital radius r[0]. The various Roche limits are those considered in Sect. 5.1.
Regarding the detection probability, the important parameter is the time T[in-band] spent in LISA band, i.e. the time elapsed between the orbit at which the object starts to be detectable by LISA
(cf. Fig. C.3) and either the ISCO (for a compact object, cf. Fig. C.4 and Table C.1) or the Roche limit (brown dwarfs and red dwarfs, cf. Table C.2). From Fig. C.4, the largest values of T[in-band]
are T[in-band]∼1.×10^4yr (resp. T[in-band]∼2×10^4yr) for a=0 (resp. a=0.98M) and are achieved for μ∼0.1M[⊙] (resp. μ∼10^−3M[⊙]), which corresponds to hypothetical primordial BHs.
We note that for a 0.5M[⊙] white dwarf, T[in-band]∼1×10^4yr. For stellar mass BHs, T[in-band] is of the order of a few 10^3yr.
Fig. C.3.
Maximum orbital radius r[0,max] for a S/N[1yr]=10 detection by LISA, as a function of the mass μ of the object orbiting around M 32 MBH.
Fig. C.4.
Time elapsed between the entry in LISA band (S/N[1yr]⩾10) and the ISCO for a compact object inspiralling around M 32 MBH, as a function of the object’s mass μ.
Table C.1.
Orbital radius r[0,max] at the entry in LISA band (S/N[1yr] reaching 10), the corresponding gravitational wave frequency f[m=2](r[0,max]) and the time spent in LISA band until the ISCO, T
[in-band], for various compact objects orbiting M 32 MBH.
For the 0.2M[⊙] red dwarf, we conclude from Table C.2 that it can be detected by LISA only if the inclination angle θ is small and if it is not corotating (|χ|≪1). One has then $T in-band > T
in-band ins ∼ 2 × 10 3 yr$.
Regarding the 0.062M[⊙] brown dwarf, we read in Table C.2 that $T in-band > T in-band ins ∼ 1 × 10 4 yr$ for low inclinations and ∼3×10^3yr for large inclinations.
Table C.2.
Inspiral time to the Roche limit in LISA band (S/N[1yr]⩾10) for the brown dwarf and red dwarf models considered in Sect. 5.1, when orbiting M 32 MBH.
All Tables
Table 1.
Coefficient α in formula (25) for the S/N from the ISCO.
Table 2.
Roche radius r[R] for different types of objects orbiting Sgr A*.
Table 3.
Orbital radius r[0,max] at the entry in LISA band (S/N[1yr]⑾10), the corresponding gravitational wave frequency f[m=2](r[0,max]) and the time spent in LISA band until the ISCO, T[in-band], for
various compact objects orbiting Sgr A*.
Table 4.
Inspiral time to the Roche limit in LISA band for a μ=0.062M[⊙] brown dwarf and different types of main-sequence stars.
Table C.1.
Orbital radius r[0,max] at the entry in LISA band (S/N[1yr] reaching 10), the corresponding gravitational wave frequency f[m=2](r[0,max]) and the time spent in LISA band until the ISCO, T
[in-band], for various compact objects orbiting M 32 MBH.
Table C.2.
Inspiral time to the Roche limit in LISA band (S/N[1yr]⩾10) for the brown dwarf and red dwarf models considered in Sect. 5.1, when orbiting M 32 MBH.
All Figures
Fig. 1.
LISA sensitivity curve (Amaro-Seoane et al. 2017) and various gravitational wave frequencies from circular orbits around Sgr A*. The wave frequencies shown above are all for the dominant m=2
mode, except for the dot-dashed and dotted vertical red lines, which correspond to the m=3 and m=4 harmonics of the ISCO of an extreme Kerr BH (a=M). The shaded pink area indicates the
location of the frequencies from the ISCO when a ranges from zero to M. The Roche limits are those discussed in Sect. 5.1.
In the text
Fig. 2.
Amplitude factor $| Z lm ∞ ( r 0 )|/ (m ω 0 ) 2$ for the harmonic (ℓ,m) of the gravitational wave emitted by an orbiting point mass (cf. Eq. (6)), in terms of the orbital radius r[0]. Each panel
corresponds to a given value of the MBH spin: a=0 (Schwarzschild BH), a=0.5M, a=0.9M and a=0.98M. A given color corresponds to a fixed value of ℓ and the line style indicates the value
of m: solid: m=ℓ, dashed: m=ℓ−1, dot-dashed: m=ℓ−2, dotted: 0< m⩽ℓ−3.
In the text
Fig. 3.
Waveform (left column) and Fourier spectrum (right column) of gravitational radiation from a point mass orbiting on the ISCO of a Schwarzschild BH (a=0). All amplitudes are rescaled by r/μ, where
r is the Boyer–Lindquist radial coordinate of the observer and μ the mass of the orbiting point. Three values of the colatitude θ of the observer are considered: θ=0 (first row), θ=π/4 (second
row) and θ=π/2 (third row).
In the text
Fig. 4.
Same as Fig. 3 but for a point mass orbiting on the prograde ISCO of a Kerr BH with a=0.98M.
In the text
Fig. 5.
Effective (direction and polarization averaged) S/N in LISA for a 1-day observation of an object of mass μ orbiting Sgr A*, as a function of the orbital radius r[0] and for selected values of the
Sgr A*’s spin parameter a and well as selected values of the inclination angle θ. Each curve starts at the ISCO radius of the corresponding value of a.
In the text
Fig. 6.
Same as Fig. 5, except for r[0] ranging up to 50M. For r[0]> 15M, only the a=0 curves are plotted, since the MBH spin plays a negligible role at large distance.
In the text
Fig. 7.
Minimal detectable mass with a S/N larger than 10 in one year of LISA observation, as a function of the orbital radius r[0]. The various Roche limits are those considered in Sect. 5.1. As in Fig. 6
, for r[0]> 15M, only the a=0 curves are shown, the MBH spin playing a negligible role at large distance.
In the text
Fig. 8.
Gravitational wave luminosity L for an object of mass μ in circular equatorial orbit around a Kerr BH of mass M and spin parameter a, as a function of the orbital radius r[0]. Each curve starts at
the prograde ISCO radius of the corresponding value of a. The dotted curve corresponds to the quadrupole approximation as given by Eq. (31). The inset shows the relative difference with respect to
the quadrupole formula (31) up to r[0]=50M.
In the text
Fig. 9.
Relative change in orbital frequency ḟ[0]/f[0] induced by the reaction to gravitational radiation for an object of mass μ in circular equatorial orbit around Kerr BH of mass M equal to that of Sgr
A* as a function of the orbital radius r[0] (Eq. (36)). Each curve has a vertical asymptote at the ISCO radius for the corresponding value of a. The dotted curve corresponds to the quadrupole
approximation and quasi-Newtonian orbits. The inset shows the curves extended upto r[0]=50M.
In the text
Fig. 10.
Adiabaticity parameter $f ˙ 0 / f 0 2$ as a function of the orbital radius r[0]. The dotted curve corresponds to the quadrupole approximation and quasi-Newtonian orbits.
In the text
Fig. 11.
Life time of a (compact) object of mass μ in circular equatorial orbit around a Kerr BH with a mass M equal to that of Sgr A* as a function of the orbital radius r[0] (Eq. (39)). The inset shows
the curves extended up to r[0]=50M.
In the text
Fig. 12.
Relative difference between the life time given by Eq. (39) and the value given the quadrupole formula, Eq. (40), as a function of the orbital radius r[0].
In the text
Fig. 13.
Roche radius r[R] as a function of the mean density ρ of the star (in solar units), for two values of the MBH spin a and two rotational states of the star: irrotational (χ=0) and corotating (χ=
1). The blue (resp. red) dotted horizontal line marks the ISCO radius for a=0 (resp. a=0.98M).
In the text
Fig. 14.
Maximum orbital radius r[0,max] for a S/N=10 detection by LISA in one year of data, as a function of the mass μ of the object orbiting around Sgr A*.
In the text
Fig. 15.
Time elapsed between the entry in LISA band (S/N[1yr] reaching 10) and the ISCO for a compact object inspiralling around Sgr A*, as a function of the object’s mass μ.
In the text
Fig. C.1.
Effective (direction and polarization averaged) signal-to-noise in LISA for a T=1yr observation of an object of mass μ=1M[⊙] orbiting M 32 MBH, as a function of the orbital radius r[0] (in
units of M, the mass of M 32 MBH), and for selected values of the MBH spin parameter a as well as selected values of the inclination angle θ. Each curve starts at the ISCO radius of the
corresponding value of a. It should be noted that this figure is scaled for T=1yr, while the equivalent figure for Sgr A* (Fig. 6) is scaled for T=1d.
In the text
Fig. C.2.
Minimal detectable mass with S/N[1yr]⩾10 in LISA observations of M 32 center, as a function of the orbital radius r[0]. The various Roche limits are those considered in Sect. 5.1.
In the text
Fig. C.3.
Maximum orbital radius r[0,max] for a S/N[1yr]=10 detection by LISA, as a function of the mass μ of the object orbiting around M 32 MBH.
In the text
Fig. C.4.
Time elapsed between the entry in LISA band (S/N[1yr]⩾10) and the ISCO for a compact object inspiralling around M 32 MBH, as a function of the object’s mass μ.
In the text
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singapore math 7a
Because of his low score, you move him down a level and have him take the 3B test. The new series, Dimensions Math, should be a great “next step” for those completing Primary Mathematics. Singapore
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If you want to change selection, open document below and click on "Move attachment"
Isomorphism - Wikipedia, the free encyclopedia
and exponential 1.2 Integers modulo 6 1.3 Relation-preserving isomorphism 2 Isomorphism vs. bijective morphism 3 Applications 4 Relation with equality 5 See also 6 Notes 7 References 8 Further
reading 9 External links Examples[edit] <span>
Logarithm and exponential[edit] Let be the multiplicative group of positive real numbers, and let be the additive group of real numbers. The logarithm function satisfies for all , so it is a group
homomorphism. The exponential function satisfies for all , so it too is a homomorphism. The identities and show that and are inverses of each other. Since is a homomorphism that has an inverse that
is also a homomorphism, is an isomorphism of groups. Because is an isomorphism, it translates multiplication of positive real numbers into addition of real numbers. This facility makes it possible to
multiply real numbers using a ruler and a table of logarithms, or using a slide rule with a logarithmic scale. Integers modulo 6[edit] Consider the group , the integers from 0 to 5 with addition
modulo 6. Also consider the group , the ordered pairs where the x coordinates can be 0 or 1, and the
status not read reprioritisations
last reprioritisation on suggested re-reading day
started reading on finished reading on
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Interest Coverage Ratio Calculator - Find Formula, Check Example, Calculate & more
Interest Coverage Ratio Calculator – Find Formula, Check Example, Calculate & more
Last Updated Date: Nov 17, 2022
This article has all the necessary information about the Interest Coverage Ratio Calculator, including the base number which is considered appropriate for lending.
Lenders want assurance for their money, they want to know the borrower and how capable they are in paying timely interest of borrowed funds. Based on this, the future borrowing criteria is judged,
and the lender or the creditor lends money based on the assurance.
Interest coverage ratio can be taken up as the assurance which lender evaluate, off their borrowers, for a guarantee of interest payment.
Interest Coverage Ratio Calculator
EBIT (Rs.) *
Interest Expense (Rs.)*
Gross Profit Margin (%)
Interest Coverage Ratio = EBIT / Interest Expense
Interest Coverage Ratio Calculator Details
Factors, when put together, will helps find the formula, and we need:
• Earning before interest and taxes (EBIT)
• Interest expenses
A couple of factors need to be figured out in order to calculate the product. The factors are EBIT an interest expenses.
Earning before interest and tax is generally the income which is made by the company, before it pays off the interest and taxes associated with it. EBIT is found in the income statement of a company.
Interest expense is the non-operating expense which is borne by a company and is deducted from the earnings, which as well can be found in the income statement.
Check out more Financial Calculators here –
Inventory Turnover Ratio Calculator Payback Period Calculator Retention Ratio Calculator
Net Profit Margin Calculator Profitability Index Calculator Return On Assets Calculator
Net Working Capital Calculator Quick Ratio Calculator Return On Equity Calculator
Operating Margin Calculator Receivables Turnover Ratio Calculator ROI Calculator
Interest Coverage Ratio Calculator Product Details
All the factors, when together put into the formula, will give us the product. The product for this formula is the Interest coverage ratio. Companies happen to search for a threshold of product which
is recommended by the analysts.
A period’s cash flow is considered if the lending period is supposed to be short. If lending is done for a long proportionate period, company study the past couple of year’s analysis to ensure there
is no constant drift of interest coverage ratio, as a drift indicates the company will not be able to pay interest over the borrowing in the future.
How to use Interest Coverage Ratio Calculator?
You need to search for a favorable product, as each of the industry has different standards therefore changing the average interest coverage ratio.
Perform thorough analysis on the given industry standards before you go on with the commutation, because we made it easy for you.
You will find a calculator at the end of this article which works on the formula interest coverage ratio. So, you do not need to worry about the commutation, as you need to enter in the details, i.e.
the factors, in the given blanks and the calculator will do the rest.
You will find the result just below the calculator, along with the working.
Find out other Financial Ratios & Technical Analysis Calculators here
Assets To Sales Ratio Calculator Current Ratio Calculator Debt Equity Ratio Calculator
Assets Turnover Ratio Calculator Days In Inventory Calculator Free Cashflow To Equity Calculator
Average Collection Period Calculator Debt Coverage Ratio Calculator Free Cashflow To Firm Calculator
Contribution Margin Calculator Debt Ratio Calculator Gross Profit Margin Calculator
Example of Interest Coverage Ratio Calculator Usage
A company has managed an earning of Rs. 100000 during a period of time and it is before interest and taxes are paid. The same company also has debt amount to Rs. 30000 which needs to be paid. In
order to find its borrowing capacity, a lender will have to do the following.
The formula is:
Interest Coverage Ratio = EBIT / Interest Expense
The working of which would be:
Interest Coverage Ratio = 100000 / 30000
With the given information, if would be safe to conclude that the company has a brief and well constructed payment system, where dues are met on time.
What is the use of Interest Coverage Ratio Calculator?
This ration is generally used as the measure of risk which is tagged along with a company in context with its debt and future borrowings.
Interest coverage ratio tells how likely a company is, to make a default in payment of interest, and so, the parties who use this formula are generally lenders and creditors, to have an idea of the
company’s financial position, to whom them have lent.
Investors as well calculate this ratio to understand the performance of the company and in order to determine if investing in the particular company would be a safe bet, and how likely will they
prosper in future, ensuring all the debts are paid off, with due interest payment.
Interest Coverage Ratio Calculator Formula
Let us consider understanding the formula in a better way.
EBIT = Earnings before interest and tax
Debt which remains outstanding, yet to pay, needs to be fed with interest for the period of holding, this is the common rule of lending.
But, the capacity of paying interest differs from company to company and the interest coverage ratio is the aggregate of how much cashflow is accessible by the company, to meet the interest payment
obligation easily.
A period’s interest coverage ratio is calculated and 1.5 is considered as a favorable product, where 1.5 or lesser product is an indication of the company’s inability to pay interest.
Interest Coverage Ratio – Conclusion
Lenders generally look out for stability is payment of interest, which means they are in favor of a company having a constant interest coverage ratio, rather than a declining one.
A declining ratio is a head-up on future likely prospects of the company, where the company may not be able to pay the interest.
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Find out all Business & Fundamental Analysis Calculators here
Interest Coverage Ratio Calculator Total Stock Return Calculator Rate Of Inflation Calculator
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PV With Zero Growth Calculator Zero Coupon Bond Value Calculator Equity Multiplier Calculator
Tax Equivalent Yield Calculator Zero Coupon Bond Yield Calculator Estimated Earnings Calculator
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How do I utilize a regression equation in a calculation
I have a scatter plot visualization on a filtered set of data with a polynomial regression line and equation. The data consists of a set of entries, and a yield value for each of those entries at
several differenct locations. I'd like to be able to have a table in spotfire that uses that regression equation to calculate a value at a certain x value. Is there a way to do this And, is there way
to create a table or calculated column that would calculate the value for each individual entry that I could export
if I understand correctly, you are creating the regression directly on the plot. Is this a one-off, i.e. you just need that particular set of regression parameters In that case you can read the
regression parameters from the plot (you can visualize them by going into Lines and Curves > Label and Tooltip and then chooseDisplay the following values: Curve expression with values.
The curve expression will appear on your plot: e.g. y=2*x + 3*x^2
You can use this expression to add a calculated column, using the column in your X axis as x, and placing the result (y) into your calculated column. If you want to calculate the regression curve for
new x values in a different table, you can similarly compute the calculated column on that table.
If your question is more complex, and your mention of filtered data makes me suspect it is, please can you give more details and upload a sample dxp.
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Perimeter of Plane Figures (videos, solutions, examples, worksheets, lesson plans)
Related Topics:
Lesson Plans and Worksheets for Grade 3 Lesson Plans and Worksheets for all Grades More Lessons for Grade 3 Common Core For Grade 3
Videos, examples, and solutions to help Grade 3 students learn how to explore perimeter as an attribute of plane figures and solve problems.
Common Core Standards: 3.G.1
New York State Common Core Math Grade 3, Module 7, Lesson 13
Worksheets for Grade 3 Concept Development
Part 1: Calculate perimeter with given side lengths
Part 2: Practice calculating the perimeter of various shapes with given side lengths.
1. Find the perimeters of the shapes below including the units in your number sentences. Match the letter inside each shape to its perimeter to solve the riddle. The first one has been done for you.
2. Alicia’s rectangular garden is 33 feet long and 47 feet wide. What is the perimeter of Alicia’s garden?
3. Jaques measured the side lengths of the shape below.
a. Find the perimeter of Jaques’ shape.
b. Jaques says his shape is an octagon. Is he right? Why or why not?
Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.
We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.
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metpy.interpolate.inverse_distance(xp, yp, variable, grid_x, grid_y, r, gamma=None, kappa=None, min_neighbors=3, kind='cressman')[source]¶
Generate an inverse distance interpolation of the given points to a regular grid.
Values are assigned to the given grid using inverse distance weighting based on either [Cressman1959] or [Barnes1964]. The Barnes implementation used here based on [Koch1983].
• xp ((N, ) ndarray) – x-coordinates of observations.
• yp ((N, ) ndarray) – y-coordinates of observations.
• variable ((N, ) ndarray) – observation values associated with (xp, yp) pairs. IE, variable[i] is a unique observation at (xp[i], yp[i]).
• grid_x ((M, 2) ndarray) – Meshgrid associated with x dimension.
• grid_y ((M, 2) ndarray) – Meshgrid associated with y dimension.
Parameters: • r (float) – Radius from grid center, within which observations are considered and weighted.
• gamma (float) – Adjustable smoothing parameter for the barnes interpolation. Default None.
• kappa (float) – Response parameter for barnes interpolation. Default None.
• min_neighbors (int) – Minimum number of neighbors needed to perform barnes or cressman interpolation for a point. Default is 3.
• kind (str) – Specify what inverse distance weighting interpolation to use. Options: ‘cressman’ or ‘barnes’. Default ‘cressman’
Returns: img ((M, N) ndarray) – Interpolated values on a 2-dimensional grid
Deprecated since version 0.9.0: Function has been renamed to inverse_distance_to_grid and will be removed from MetPy in 0.12.0.
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Professional Guidance for Kayaker in Tidal Current
What direction should the kayaker paddle in to travel straight across the harbor?
(a) In which direction should he paddle in order to travel straight across the harbor?
How long will it take the kayaker to cross the harbor?
(b) How long will it take him to cross?
The kayaker should paddle at an angle of 36 degrees west of north to travel straight across the harbor. It will take him 31 seconds to cross the harbor.
Let the direction in which he pedals make an angle of θ with north across.
Sinθ = 2 / 3.4
θ = 36 degrees (west of north)
b) Component of his velocity along the north = 3.4 cos 36
= 2.75 m/s
Time required = 85 / 2.75 s
= 31 seconds
For a kayaker navigating a tidal current, understanding the direction and speed of the current is crucial for efficient paddling. In this scenario, the kayaker needs to paddle north across an 85m
wide harbor with a tidal current flowing to the east at 2.0 m/s.
By calculating the angle at which the kayaker should paddle, which is 36 degrees west of north, the kayaker can travel straight across the harbor. This optimal angle allows the kayaker to counteract
the effects of the eastward flowing current and reach the opposite side efficiently.
Furthermore, to determine the time it takes to cross the harbor, the kayaker's velocity component along the north direction is calculated to be 2.75 m/s. Dividing the width of the harbor by this
velocity component yields a crossing time of 31 seconds.
By following these calculations and recommendations, the kayaker can navigate the tidal current effectively and reach the other side of the harbor in a timely manner.
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GreeneMath.com | Ace your next Math Test!
Practice Objectives
• Demonstrate an Understanding of the Addition Property of Equality
• Demonstrate an Understanding of the Multiplication Property of Equality
• Demonstrate the Ability to Solve a Linear Equation in One Variable
• Demonstrate the Ability to Solve a Linear Equation in One Variable with Parentheses
• Demonstrate the Ability to Solve a Linear Equation in One Variable with Fractions
• Demonstrate the Ability to Solve a Linear Equation in One Variable with Decimals
Practice Solving Linear Equations with Fractions/Decimals
Answer 7/10 questions correctly to pass.
Solve each equation for x.
Enter the numerical answer only.
Fractions must be simplified.
Negative fractions can be written as -a/b or a/-b.
Not Correct!
Your answer was: 0
The correct answer was: 0
Solving Linear Equations with Fractions:
1. Simplify each side separately.
□ Clear all parentheses/fractions/decimals
□ Combine any like terms
2. Move all numbers to the right side.
3. Move all variable terms to the left side.
4. Isolate the variable.
5. We obtain our answer as: x = "a number"
You Have Missed 4 Questions...
Your answer should be a number!
Current Score: 0%
Correct Answers: 0 of 7
Wrong Answers: 0 of 3
Need Help?
Video Lesson | Written Lesson
Prefer Multiple Choice?
Multiple Choice Test
Wow! You have mastered Solving Linear Equations with Fractions/Decimals!
Correct Answers: 0/0
Your Score: 0%
Restart | Next Lesson
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Solution: Longest Harmonious Subsequence
This is part of a series of Leetcode solution explanations (index). If you liked this solution or found it useful, please like this post and/or upvote my solution post on Leetcode's forums.
We define a harmonious array as an array where the difference between its maximum value and its minimum value is exactly 1.
Given an integer array nums, return the length of its longest harmonious subsequence among all its possible subsequences.
A subsequence of array is a sequence that can be derived from the array by deleting some or no elements without changing the order of the remaining elements.
Example 1:
Input: nums = [1,3,2,2,5,2,3,7]
Output: 5
Explanation: The longest harmonious
subsequence is [3,2,2,2,3].
Example 2:
Input: nums = [1,2,3,4]
Output: 2
Example 3:
Input: nums = [1,1,1,1]
Output: 0
• 1 <= nums.length <= 2 * 10^4
• -10^9 <= nums[i] <= 10^9
Since our target harmonious array is dealing with the absolute value of its elements and since it's a subsequence of our numbers array (N), we don't need to worry about the order of numbers or their
index in N.
If all we care about is what numbers appear in N and not their order or index, then it means that we should start by building a frequency map from N.
Then we can just iterate through the entries in our frequency map (fmap) and keep track of the largest value found by adding each number's (key) frequency (val) with the frequency of key+1.
We should then return the best result (ans).
Since javascript's Map() stores its keys as strings, you need to use some method of converting the key back into a number before adding 1. The normal way to do this is with parseInt(), but applying a
double bitwise NOT (~) does the same thing far more efficiently, as long as the number is greater than -2^31 and less than 2^31.
Javascript Code:
var findLHS = function(N) {
let fmap = new Map(), ans = 0
for (let num of N)
fmap.set(num, (fmap.get(num) || 0) + 1)
for (let [key,val] of fmap)
if (fmap.has(~~key+1))
ans = Math.max(ans, val + fmap.get(~~key+1))
return ans
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Kinetic energy in classical and relativistic mechanics
Kinetic energy of a point object in classical Newtonian mechanics and in relativistic mechanics.
Calculation of kinetic energy of a point object of mass m and velocity v in classical mechanics.
Calculation of kinetic energy of a material point of mass m and velocity v in relativistic mechanics.
Here the velocity of the material point cannot exceed the speed of light (299792458 m/sec.)
At low speeds, the results of both formulas coincide, but the closer to the speed of light, the greater the difference.
Similar calculators
PLANETCALC, Kinetic energy in classical and relativistic mechanics
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Product Rule (Multiple Bases) Worksheets [PDF] (8.EE.A.1): 8th Grade Math
Teaching Solving Product Rule Easily
• Always, determine the values of g(x) and f(x).
• Secondly, determine the values of f'(x) and g'(x) by applying the product rule formula.
• Finally, simplify the expression.
Why Should You Use a Product Rule ( Multiple Bases ) Worksheet for your Students?
• These worksheets will help your students to know more about differentiation and derivatives.
• Solving these worksheets will help your students to easily simplify any equation using the product rule.
Download Equations with Product Rule ( Multiple Bases ) Worksheets PDF
You can download and print these super fun equations with product rule ( multiple bases ) worksheet pdf from here for your students. You can also try our Product Rule (Multiple Bases) Problems and
Product Rule (Multiple Bases) Quiz as well for a better understanding of the concepts.
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Oscillations in Mesocyclone Signatures with Range Owing to Azimuthal Radar Sampling
1. Introduction
It is widely recognized that the magnitude of the single-Doppler velocity signature of a thunderstorm vortex (such as a mesocyclone) is dependent on the relationship between the vortex strength and
size and the size of the radar sample volume (e.g., Donaldson 1970; Brown et al. 1978). Azimuthal profiles of Doppler velocity through a vortex degrade with range as the nominal diameter of the radar
beam (distance between the half-power points) becomes progressively larger compared with the vortex core radius (Brown and Lemon 1976). However, Wood and Brown (1997) showed that varying degrees of
degradation also can occur at a fixed range. The amount of degradation depends on the location of the Doppler velocity data points relative to the mesocyclone center. They calculated core diameter
and mean rotational velocity for many different placements of the data points relative to the center of a typical mesocyclone. They found that at a given range there was considerable variation in the
resulting apparent core diameters and mean rotational velocities depending on the actual locations of the data points.
Besides the variations at a given range, Wood and Brown showed that there are unexpected oscillations in the parameters with range when the radar collects discrete samples at 1° azimuthal intervals.
However, they did not elaborate on the cause of the oscillations. The purpose of this note is simply to clarify the reasons for the oscillations. We use simulated nonnoisy radar data to reproduce the
basic characteristics of the oscillations. In subsequent papers, we will relate more realistic noisy data to mesocyclone detection and discuss ways in which basic mesocyclone characteristics can be
recovered from degraded measurements.
2. Doppler radar simulation
We used the analytical simulation of a WSR-88D (Weather Surveillance Radar-1998 Doppler) developed by Wood and Brown (1997) and a model mesocyclone to produce simulated Doppler velocity measurements.
We assume that 1) the tangential velocity distribution across the mesocyclone is, to a good approximation, simulated using an axisymmetric Rankine (1901) combined vortex;2) the tangential velocity
field is uniform with height; 3) reflectivity is uniform across the mesocyclone; 4) the radar beam pattern is Gaussian shaped; 5) the beam axis is quasi-horizontal at 0.5° elevation angle; and 6) the
effective beamwidth is 1.29° (broadened from a nominal 0.93° by the rapidly rotating antenna; see Wood and Brown 1997). Radar measurements were derived by scanning the simulated Doppler radar past
the model mesocyclone. Doppler velocity values (weighted averages of velocity values distributed across the radar beam) were computed at azimuthal intervals of 1°. Noise was not added to the Doppler
velocity values in this simulation because we wanted to investigate the inherent oscillations of core diameters and mean rotational velocities with range.
3. Oscillations of core diameters and mean rotational velocities with range
The width of the radar beam relative to a given-sized mesocyclone affects the sampling resolution. Since the physical width of the beam increases linearly with range from the radar, the peak Doppler
velocity measurements gradually underestimate the peak rotational velocities of the mesocyclone (e.g., Donaldson 1970; Brown and Lemon 1976). In this section, we investigate how the Doppler velocity
measurements change with range as the radar, collecting data at 1° azimuthal intervals, scans past a mesocyclone. We simulated three different situations, where 1) one of the Doppler velocity data
points coincides with the mesocyclone center, 2) the mesocyclone center is midway between two Doppler velocity measurements, and 3) the Doppler velocity measurements are randomly positioned relative
to the mesocyclone center. The model mesocyclone used in this study had a peak rotational velocity of 25 m s^−1 and a core diameter of 5 km. The deduced mean rotational velocity is one-half the
difference between the extreme positive and negative Doppler velocity values in the mesocyclone signature. The deduced core diameter is the distance between the extreme Doppler velocity values.
a. Doppler velocity measurement coincident with mesocyclone center
Consider first the situation in which one of the Doppler velocity data points coincides with the center of the mesocyclone. With increasing range, one might expect the mean rotational velocity to
decrease in a uniform manner and the core diameter to increase in a uniform manner. This type of variation, illustrated by the thin curves in Fig. 1, would be expected to occur if data were collected
continuously with no gaps between data points as the radar scanned past the mesocyclone. In reality, WSR-88D data are collected discretely at 1° azimuthal increments. The discrete nature of the data
collection produces the thick oscillating curves in Fig. 1. The hypothetical continuous sampling curves represent an upper limit for discretely obtained mean rotational velocity values and an
approximate average for discretely obtained core diameter values.
The reasons for the abrupt changes in the discretely sampled data are illustrated in Fig. 2. Shown are azimuthal profiles of Doppler velocity values (black dots) through the mesocyclone at several
different ranges from the radar. Between ranges (R) of 96 and 106 km, the extreme Doppler velocity values at B and F are separated by four azimuthal intervals, producing the large deduced values of
core diameter for the mesocyclone signature (Figs. 1b and 2a,b). Over that range interval, the core diameter increases linearly from 6.7 to 7.4 km. Concurrently, the deduced mean rotational velocity
decreases from 18.8 to 17.3 m s^−1 (Figs. 1a and 2a,b). These changes are produced because data points B and F are moving away from the peaks of the signature, owing to the widening of 1° azimuthal
intervals with increasing range.
When the mesocyclone moves from 106 to 107 km (Figs. 2b,c), the extreme Doppler velocity values jump from B and F to C and E. Thus, the number of azimuthal intervals between the extreme negative and
positive Doppler velocity data points drops from four to two, yielding a sharp decrease in the size of the deduced core diameter (Fig. 1b). When the range increases from 107 to 117 km, the positions
C and E continue to move apart (Figs. 2c,d). The deduced value of mean rotational velocity increases in magnitude beyond 107 km (Fig. 1a) as the extreme negative and positive Doppler velocity data
points (C and E) approach the rounded peaks of the measured curve (Fig. 2d). After C and E reach the rounded peaks (that is, where the continuous and discrete curves are coincident in Figs. 1a,b),
the extreme negative and positive Doppler velocity data points (not shown) begin to decrease in magnitude. However, the core diameter continues to increase as C and E move farther apart.
b. Mesocyclone positioned midway between two Doppler velocity measurements
In the examples shown thus far, the deduced values of mean rotational velocity and core diameter have been computed only when one of the data points coincides with the center of the mesocyclone. When
the closest data points are equidistant from the mesocyclone center (azimuthal separation of ±0.5°), the deduced values shown in Fig. 3 (thick curves) undergo oscillations that are offset from the
ones with 0° separation (thin curves) that are reproduced from Fig. 1. The explanations for such oscillations are straightforward. When the mesocyclone moves from 170 to 190 km (Figs. 4a,b), the
extreme negative and positive Doppler velocity data points A and D are located outside the mesocyclone core and are separated by three azimuthal intervals. Between 170 and 190 km, the increasing
separation distance causes the mean rotational velocity value to decrease from 14.7 to 13.3 m s^−1 (Figs. 3a and 4a,b). At the same time, the deduced core diameter increased from 8.9 to 9.9 km (Figs.
3b and 4a,b).
When the mesocyclone moves from 190 to 191 km, the extreme negative and positive Doppler velocity values suddenly change from points A and D to points B and C. Consequently, the number of azimuthal
intervals separating the extreme values decreases from three to one (Figs. 3b and 4b,c), producing a deduced core diameter one-third of the previous value. Beyond 191 km, the number of azimuthal
intervals between the extreme data points remains at one, no matter how far apart B and C become. Also beyond 191 km, the deduced values of rotational velocity will increase as long as B and C are
approaching the peaks of the mesocyclone signature curve (Figs. 3a and 4d). However, after they pass the peaks, the deduced mean rotational velocity will decrease.
c. Doppler velocity measurements randomly positioned relative to mesocyclone center
Up to this point, we have discussed what happens, as a function of range, when a Doppler velocity data point coincides with the mesocyclone center (0° azimuthal offset) or the two closest data points
are equidistant from the center (±0.5° azimuthal offset). Now we discuss the situation where the closest data point is randomly positioned in the ±0.5° interval. To simulate all of the possible
azimuthal offsets, we computed mean rotational velocities and core diameters at 0.02° intervals from 0.5° to the left of mesocyclone center to 0.5° to the right. The results of the computations are
shown in Fig. 5. The shaded oscillating band in Fig. 5a represents the full spread of mean rotational velocities as a function of range.
The curves in Fig. 5b indicate that, at a given range, one of only two deduced core diameters is possible (except at the transition range). These two possibilities are those that occur for azimuthal
offsets of 0° and ±0.5°. The dotted lines radiating from the origin (continuations of sloping portions of the curves) represent the linear increase of data point separation and its effect on core
diameter with increasing range. The vertical portions of the curves in Fig. 5b simply represent the transition points illustrated in Figs. 2b,c and 4b,c, where the core diameter jumps to a smaller
value and where the local minima in mean rotational velocity values also occur. The locations marked 6b, 6c, 6g, and 6h (representing Figs. 6b,c,g,h) in Figs. 5a,b represent these two transition
To further explain the oscillations in Fig. 5, the distributions of mean rotational velocity and core diameter as a function of azimuthal sampling offset are presented in Fig. 6. The full spread of
offsets are presented at nine selected ranges. (Figures 2a–c are subsets of Figs. 6a–c at an offset of 0° and Figs. 4a–d are subsets of Figs. 6f–i at offsets of ±0.5°.)
All of the maximum rotational velocities along the top of the shaded band in Fig. 5a occur along either the 0° offset or the ±0.5° offset curve (cf. Fig. 3a). This situation is further illustrated in
Fig. 6, where the highest portion of the mean rotational velocity curve (thin curve) is either at 0° offset or ±0.5° offset. Where the two offset curves cross in Fig. 5a (such as indicated by the 6d
arrow in Fig. 5a), the maximum values at 0° and ±0.5° offsets are equal (see Fig. 6d).
On the other hand, the minimum rotational velocity values along the bottom of the shaded band in Fig. 5a represent the full range of azimuthal offset values. The pointed localized minima (such as the
6b,c and 6g,h arrows) occur alternately at offsets of 0° and ±0.5°. In between these points, the associated offset values change in a uniform manner from one extreme value to the other. This type of
transition is shown in Fig. 6, where the minimum rotational velocity value moves from 0° offset at a range of 107 km (Fig. 6c) to ±0.5° offset at 190 km (Fig. 6g).
It is at the range of the pointed localized minima in mean rotational velocity that the marked transitions in core diameter occur (Fig. 5), as mentioned earlier. These transition points are indicated
by the dotted vertical lines in Fig. 6. At the transition point at 106–107 km, the core diameter is equal to three azimuthal intervals, except at the 0° offset where the transition takes place. With
increasing range, the fraction of the azimuthal offset interval equal to three azimuthal intervals decreases, while the fraction of offsets equal to two azimuthal intervals increases. At 190–191-km
range, the core diameter is equal to two azimuthal increments at all but the ±0.5° offset. The thick mean diameter curve in Fig. 5b reflects the gradual change from three to two azimuthal intervals
as range increases from 107 to 190 km.
4. Concluding discussion
In this note, we investigated the basic characteristics of oscillations in the strength and size of a mesocyclone signature that occur when the mesocyclone moves toward or away from a Doppler radar.
The oscillations are a natural consequence of data points at constant azimuthal intervals changing their physical separation distances with changing range. As data point separation increases or
decreases, the data points change their positions in a systematic manner relative to the Doppler velocity peaks of the mesocyclone signature. The deduced mean rotational velocity values oscillate as
a continuous function of range. In contrast, the deduced core diameter undergoes discontinuous increases or decreases in size.
For actual Doppler velocity data, the oscillations discussed here are masked to some extent by the inherent uncertainties (noisiness) in the Doppler velocity estimates. However, abrupt changes still
will be evident in deduced core diameters. Such changes, especially by a factor of 2 or 3 at farther ranges, can be very misleading during the real-time monitoring of mesocyclones. In subsequent
papers, we will incorporate noisiness into our simulations and discuss ways in which basic mesocyclone characteristics can be recovered from range-degraded Doppler velocity mesocyclone signatures.
The authors thank Bob Davies-Jones and Greg Stumpf of NSSL for reviewing and providing helpful comments and suggestions on the early version of the manuscript. The suggestions by three anonymous
reviewers have contributed to the improvement of the manuscript.
• Brown, R. A., and L. R. Lemon, 1976: Single Doppler radar vortex recognition. Part II: Tornadic vortex signatures. Preprints, 17th Conf. On Radar Meteorology, Seattle, WA, Amer. Meteor. Soc.,
• ——, ——, and D. W. Burgess, 1978: Tornado detection by pulsed Doppler radar. Mon. Wea. Rev.,106, 29–38.
• Donaldson, R. J., Jr., 1970: Vortex signature recognition by a Doppler radar. J. Appl. Meteor.,9, 661–670.
• Rankine, W. J. M., 1901: A Manual of Applied Mechanics. 16th ed. Charles Griff and Company, 680 pp.
• Wood, V. T., and R. A. Brown, 1997: Effects of radar sampling on single-Doppler velocity signatures of mesocyclones and tornadoes. Wea. Forecasting,12, 928–938.
Fig. 1.
Variations of (a) deduced mean rotational velocity and (b) deduced core diameter values as a function of range for a simulated mesocyclone having a typical rotational velocity peak of 25 m s^−1 at a
core diameter of 5 km. The thin curves represent the mean rotational velocities and core diameters of the mesocyclone signature if the simulated Doppler radar collected data continuously across the
mesocyclone. The thick fluctuating curves represent 1° azimuthal sampling where one of the data points coincides with the mesocyclone center. Labels 2a–2d correspond to Figs. 2a–d, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 17, 1; 10.1175/1520-0426(2000)017<0090:OIMSWR>2.0.CO;2
Fig. 2.
Relationships of data points relative to the azimuthal profiles through the center of the mesocyclone as a function of range (R in km) when one of the data points (D) coincides with the mesocyclone
center. The curve with rounded peaks (along which the data points fall) represents the Doppler velocity azimuthal profile of the mesocyclone signature if the radar were able to make measurements in a
continuous manner across the mesocyclone. The curve with pointed peaks represents the Rankine combined velocity model for the mesocyclone having a peak rotational velocity of 25 m s^−1 at a core
diameter of 5 km. Data points at A–G represent the locations of successive Doppler velocity measurements collected at 1° azimuthal intervals as the radar beam scans across the mesocyclone. Deduced
mean rotational velocity (V[rot]) in m s^−1 and core diameter (CD) in km also are indicated.
Citation: Journal of Atmospheric and Oceanic Technology 17, 1; 10.1175/1520-0426(2000)017<0090:OIMSWR>2.0.CO;2
Fig. 3.
Variations of (a) deduced mean rotational velocity and (b) deduced core diameter with range. The thin curves are the discrete 0° separation curves from Fig. 1. The thick curves represent variations
when the closest data points are equidistant (±0.5°) from the mesocyclone center. Labels 4a–4d correspond to Figs. 4a–d, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 17, 1; 10.1175/1520-0426(2000)017<0090:OIMSWR>2.0.CO;2
Fig. 4.
Same as Fig. 2, except that the mesocyclone is at a farther range and its center is midway between two data points.
Citation: Journal of Atmospheric and Oceanic Technology 17, 1; 10.1175/1520-0426(2000)017<0090:OIMSWR>2.0.CO;2
Fig. 5.
Variations of (a) deduced mean rotational velocity and (b) deduced core diameter as a function of range. The thick curve in the middle of the data curves represents the average of the values at each
range. Shading in (a) represents the full spread of mean rotational velocity values for all possible azimuthal sampling offsets between the data points and mesocyclone center. In (b), the dotted
lines radiating from the origin illustrate that the sloping lines (labeled by the number of azimuthal intervals between extreme data point values) represent the linear increase of data point
separation with increasing range. Labels 6a–6i correspond to Figs. 6a–i, respectively.
Citation: Journal of Atmospheric and Oceanic Technology 17, 1; 10.1175/1520-0426(2000)017<0090:OIMSWR>2.0.CO;2
Fig. 6.
Variations of deduced mean rotational velocity and core diameter as a function of azimuthal sampling offset (−0.5° to +0.5°) at several ranges from a WSR-88D radar. Thin and thick curves,
respectively, represent deduced values of mean rotational velocity (V[rot]) and core diameter (CD); their magnitudes, respectively, are given along the left and right sides of each panel. The
vertical dotted lines represent the boundaries between the number of azimuthal intervals (labeled) separating the extreme positive and negative Doppler velocity data points of the mesocyclone
Citation: Journal of Atmospheric and Oceanic Technology 17, 1; 10.1175/1520-0426(2000)017<0090:OIMSWR>2.0.CO;2
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SIP-11 Implement dynamic LP rewards
Siren is currently spending 17,857 SI per day (125k per week) to incentivize liquidity which is working well within its desired bounds. However, only the SUSHI Call options seem to have sufficient
open interest.
Every liquidity pool currently earns 2.976 SI, but only one pool (SUSHI Call) is appropriately used. Therefore Siren is now overpaying almost 15,000 SI per day to liquidity pools with little
traction. That’s more than 80%.
Therefore, we suggest implementing a dynamic reward scheme that takes the pool utilization, current SI price, and default interest rate across DeFi into consideration.
Pool utilization: How many percent of the pool liquidity is currently sold. Calculated as Open Interest across strike prices/liquidity.
Default interest rate: The base rate we would have to outbid to attract liquidity - Curve 3pool for stablecoins and Aave interest rates for UNI and YFI. The SUSHI interest rate would be XSUSHI and
can be obtained from Sushi. To incentivize liquidity beyond this base rate, we suggest adding a further 5%. We call this the incentivized base rate.
Current base rates would be:
• USDC: 15% (from Curve 3pool)
• UNI: 0.04% (from Aave)
• YFI: 0.74% (from Aave)
• SUSHI: 4% (from xSUSHI staking)
Other parameters:
• Max SI per day per pool: 3,000 (slightly more than what is currently being paid)
• Minimum liquidity per pool: 1,000,000 USD (if we have less than this amount in any liquidity pool, we have to increase our rewards)
Given these parameters, we suggest the following distribution of rewards:
Notebook: https://gesis.mybinder.org/binder/v2/gh/HeyChristopher/siren-rewards-sip/52bcaaa8aa23aff121eac8701cfdfa5564a8b2b9
Source: https://github.com/HeyChristopher/siren-rewards-sip
1. When total liquidity per pool is below our minimum liquidity (0%), the rewards should reach their maximum. Or to say it in another way: The APY should be very high when there is not a lot of
capital in the pool so that we attract more capital quickly
2. Once we have enough liquidity, we start to look at the utilization of the pool. We are suggesting to approximately pay the incentivized base rate between 20% and 50% utilization. Above 50%
utilization, the rewards quickly increase towards their maximum to ensure that we always have enough capacity. The rewards drop below the base rate when the utilization is very low (<20%)
With this model, we can increase the current liquidity mining program’s duration by a few times, even with increased utilization.
Open questions:
1. Are the curves too steep? The APY could jump from 300% to 10% quickly for certain pairs with just 100k USD added, resulting in too many people hopping in and out of the pool.
2. Is the base rate + 5% appropriate? Is this too much, or is this too low?
3. What is an appropriate minimum liquidity amount we want to make sure is always in the pool? Is $1m too much or too little?
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Lesson: y = mx + c | Oak National Academy
Switch to our new maths teaching resources
Slide decks, worksheets, quizzes and lesson planning guidance designed for your classroom.
Lesson details
Key learning points
1. In this lesson, we will explore the general form of the equation of a line.
This content is made available by Oak National Academy Limited and its partners and licensed under Oak’s terms & conditions (Collection 1), except where otherwise stated.
5 Questions
What is the gradient of the orange line?
What is the gradient of the green line?
What is the gradient of the purple line?
Which of the following is parallel to y = 3x - 1?
What is the gradient of the line passing through the points (4,5) and (6,9)?
5 Questions
What is the gradient of the line y = 2x + 7?
What is the coordinate of the y-intercept for the line y = 2x + 7?
What is the gradient of the line y = 5 - 2x?
A line has a gradient of 3 and goes through the point (0,7). What is the equation of the line?
John says the line y = 2x + 3 goes through the point (2,3). Is he correct?
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Myth #2 - Average Rate of Return Matters?
I hear many speak about the average rate of return for some investment in the market. Often this is misleading to people who think when you put money in the market then that is what you can expect.
Truthfully what matters is the actual rate of return. Here is an example. Im using round numbers to keep it simple.
You invest $100,000 into the market
The first year you make a 100% gain so now you have $200,000
The second year you have a 50% loss and you now have $100,000
The Third Year you have again have a 100% gain and again have $200,000
The Fourth Year you have a loss of 50% again and are back at $100,000
You started with $100,000 and four years later you have $100,000
What was your average rate of return? 25% and yet your actual rate of return is 0%.
You Ended up with no gain at all. We didn't figure in any fees or taxes either. The more volatile the swings the more the average rate of return won't matter. So next time you hear "the S&P has
averaged this or that" just know there is more to the story than that. It doesn't make it good or bad but it does help us look at it more clearly so we can consider the value and risk properly.
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[Previous: Design Consistency Module Output] [Table Of Contents]
6. Glossary
85th Percentile Speed (V85)
The 85th percentile of a sample of observed speeds is the general statistic used to describe operating speeds on a geometric feature. It is the speed at or below which 85 percent of the drivers are
Design Speed
The design speed is defined by the American Association of State Highway and Transportation Officials (AASHTO) in its 2001 A Policy on Geometric Design of Highways and Streets as "a selected speed
used to determine the various geometric design features of the highway." Design speed can be edited through the "Highway Editor" under the General category.
Desired Speed
In the speed-profile model, desired speed is the 85th percentile speed that drivers select when not constrained by the vertical or horizontal alignment. Empirically, desired speed is estimated by
measuring speeds on portions on long tangents where speed is not constrained by either vertical gradient or horizontal and vertical curvature. In IHSDM, the default value for desired speed is 100 km/
h (62 mi/h). This default value is based upon the average of 85th percentile speeds on long tangents in the research used to calibrate the model. The average was based upon speeds measured in six
States at 64 long tangents on highways with 88.5 km/h (55 mi/h) posted speed limits. Among the six States, the averages ranged from 93 km/h (58 mi/h) in New York and Pennsylvania to 103 km/h (64 mi/
h) in Texas. Users may modify the desired speed through the "Highway Editor" under the General category..
Speed at Evaluation Start Station
The 85th percentile initial speed when traveling in the direction of increasing stations or the end speed when traveling in the direction of decreasing stations. This speed cannot be greater than the
desired speed. It is set to the desired speed by default and may be edited by the user from the Run Evaluation Wizard.
Speed at Evaluation End Station
The 85th percentile end speed when traveling in the direction of increasing stations or the initial speed when traveling in the direction of decreasing stations. This speed cannot be greater than the
desired speed. It is set to the desired speed by default and may be edited by the user from the Run Evaluation Wizard.
Speed Profile
One of the ways in which operating speeds are used in ensuring design consistency is through the use of speed profiles. Speed-profile models are used to detect speed inconsistencies along road
alignments. A speed profile is a plot of operating speeds on the vertical axis versus distance along the highway on the horizontal axis.
TWOPAS is a microscopic model that simulates traffic operations on two-lane highways by reviewing the position, speed, and acceleration of each individual vehicle on a simulated highway at 1-second
intervals and advancing those vehicles along the highway in a realistic manner. The model takes into account the effects on traffic operations of road geometric, traffic control, driver preferences,
vehicle size and performance characteristics, and the oncoming and same direction vehicles that are in sight at any given time. The model incorporates realistic passing and pass abort decisions by
drivers in two-lane highway passing zones. Spot data, space data, vehicle interaction data, and overall travel data are accumulated and processed, and various statistical summaries are printed.
Use of TWOPAS in the Design Consistency Module:
Steep upgrades reduce passenger car speeds. The TWOPAS traffic simulation model contains equations that can be used to represent the effect of grades on the speed of passenger cars.
Upgrades have the effect of limiting the accelerations that vehicles can achieve, thus making it difficult for drivers to maintain their desired speed. If the grade a vehicle is ascending is steep
enough, the vehicle will be forced to decelerate. If the grade is also long enough, the vehicle will eventually decelerate to a crawl speed. A vehicle at its crawl speed can continue up the grade at
a constant speed without decelerating further, but cannot accelerate. A vehicle's crawl speed on a specific grade is a function of the steepness of the grade and the performance characteristics of
the vehicle. The length of grade required for a vehicle to reach its crawl speed is a function of the steepness of the grade and the performance characteristics of the vehicle, as well as the
vehicle's initial speed as it enters the grade.
In the DCM, TWOPAS equations are used to check the performance-limited speed along the highway. If, at any point, the grade-limited speed is less than the tangent or curve speed predicted using the
speed prediction equations or the assumed desired speed, then the grade-limited speed will govern.
The procedure for predicting the grade-limited speed using the TWOPAS equations can be summarized as follows:
1. Calculate vehicle performance speeds along the alignment at one-second intervals.
2. Calculate speeds based on the driver's preferred acceleration rate at one-second intervals.
3. At each time interval, compare the speeds predicted in Steps 1 and 2, and select the lowest speed. The speed selected is used as the initial speed for the next one-second interval in Steps 1 and
2. This process is continued for the entire evaluation section.
The result is an estimated operating speed profile for the selected design vehicle, based on the effects of the vertical alignment.
Vehicle Type
There are five passenger car vehicle types included in the DCM. Passenger Car- Type 5 is set as the default vehicle, and has the highest acceleration and top speed of all the loaded vehicles; it is
also the most common vehicle, making up 30% of the passenger car population. Conversely, Passenger Car- Type 1 has the lowest acceleration and top speed and represents the smallest portion of the
passenger car population at 10%. Changing the vehicle type for a given evaluation with all other variables remaining constant, may change the 85th percentile speed profile. This depends upon whether
or not the speeds predicted by the TWOPAS equations in Step 4 of the speed profile algorithm are lower than the speeds predicted by the speed prediction equations in Steps 2 and 3. If the speeds
predicted by the TWOPAS equations are lower, then the final 85th percentile speed profile will reflect a decrease. The grade-limited speeds predicted by the TWOPAS equations are most likely to
control on long, steep grades. Vehicle type may be edited via the "Set design consistency attributes" screen of the Evaluation Wizard.
[Previous: Design Consistency Module Output] [Top]
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Financial Theory with Python: A Gentle Introduction
Writer: Yves
Published: 19 October 2021
Nowadays, finance, mathematics, and programming are intrinsically linked. This book provides the relevant foundations of each discipline to give you the major tools you need to get started in
the world of computational finance.
Using an approach where mathematical concepts provide the common background against which financial ideas and programming techniques are learned, this practical guide teaches you the basics of
financial economics. Written by the best-selling author of Python for Finance, Yves Hilpisch, Financial Theory with Python explains financial, mathematical, and Python programming concepts in
an integrative manner so that the interdisciplinary concepts reinforce each other.
□ Draw upon mathematics to learn the foundations of financial theory and Python programming
□ Learn about financial theory, financial data modeling, and the use of Python for computational finance
□ Leverage simple economic models to better understand basic notions of finance and Python programming concepts
□ Use both static and dynamic financial modeling to address fundamental problems in finance, such as pricing, decision-making, equilibrium, and asset allocation
□ Learn the basics of Python packages useful for financial modeling, such as NumPy, pandas, Matplotlib, and SymPy
Pages: 204
ISBN: 1098104358
ISBN-13: 978-1098104351
Language: English
Buy at bookdepository.com free delivery worldwide.
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دانشگاه آزاد اسلامی واحد لاهیجان Journal of Operational Research In Its Applications ( Applied Mathematics ) - Lahijan Azad University 2251-7286 2251-9807 14 4 2017 12 1 Evaluation and Prioritization
of Suppliers Adopting a Combined Approach of Entropy, Analytic Hierarchy process, and Revised PROMETHEE (Case Study: YOUTAB Company) 1 20 FA Maghsoud Amiri Faculty of Management and Accounting,
Allameh Tabataba'i University, mg_amiri@yahoo.com N Farhad Hadinejad Faculty of Management and Accounting, Allameh Tabataba'i University, farhad_hdng@yahoo.com Y Shiva malekkhoyan Faculty of
Management and Accounting, Allameh Tabataba'i University, ShivaMalekkhoyan@yahoo.com N In Supply chain management, decision making for the selection of suppliers is of such a key importance that it
has developed into a strategic objective for organizations in recent years. The issue of selecting a supplier, due to multiplicity and diversity of qualitative and quantitative indices, is a multi
attribute decision making question and as a result, known techniques of this field of study can be used in the process of problem-solving as well. The present study intends to present a new and
scientific algorithm for evaluation and selection of suppliers by combining multi attribute decision making techniques and improving their implementation process. For this purpose, similar studies
were reviewed in the first step and then, assisted by organizational connoisseurs, 8 important and effective indices were extracted for the selection of suppliers: price, quality, good delivery,
geographic location, payment terms, device technological capabilities, and after-sales services. In the second step using real data provided by YOUTAB Company, the suggested algorithm was explained.
To this end, firstly, applying Entropy technique (completing decision matrix by Company's experts), initial weight of indices was extracted; then, with the help of AHP (conducting an opinion poll of
the company's chief executive officer) a new weight for indices was obtained and then combined with initial weights so that the final weights could be calculate. Therefore, revised PROMETHEE
technique was used to evaluate the four existing suppliers and analyze the results achieved. Entropy, PROMETHEE, Multi Attribute Deciding Making, Supply Chain, Analytic Hierarchy Process. http://
jamlu.liau.ac.ir/article-1-1295-en.html http://jamlu.liau.ac.ir/article-1-1295-en.pdf
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square_clustering(G, nodes=None)[source]¶
Compute the squares clustering coefficient for nodes.
For each node return the fraction of possible squares that exist at the node [R204]
\[C_4(v) = \frac{ \sum_{u=1}^{k_v} \sum_{w=u+1}^{k_v} q_v(u,w) }{ \sum_{u=1}^{k_v} \sum_{w=u+1}^{k_v} [a_v(u,w) + q_v(u,w)]},\]
where \(q_v(u,w)\) are the number of common neighbors of \(u\) and \(w\) other than \(v\) (ie squares), and \(a_v(u,w) = (k_u - (1+q_v(u,w)+\theta_{uv}))(k_w - (1+q_v(u,w)+\theta_{uw}))\), where
\(\theta_{uw} = 1\) if \(u\) and \(w\) are connected and 0 otherwise.
G : graph
Parameters: nodes : container of nodes, optional (default=all nodes in G)
Compute clustering for nodes in this container.
c4 : dictionary
A dictionary keyed by node with the square clustering coefficient value.
While \(C_3(v)\) (triangle clustering) gives the probability that two neighbors of node v are connected with each other, \(C_4(v)\) is the probability that two neighbors of node v share a common
neighbor different from v. This algorithm can be applied to both bipartite and unipartite networks.
[R204] (1, 2) Pedro G. Lind, Marta C. González, and Hans J. Herrmann. 2005 Cycles and clustering in bipartite networks. Physical Review E (72) 056127.
>>> G=nx.complete_graph(5)
>>> print(nx.square_clustering(G,0))
>>> print(nx.square_clustering(G))
{0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
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From Encyclopedia of Mathematics
A structure on a vector bundle (or sphere bundle, etc.) that is a generalization of the concept of the structure group of a fibration.
Let there be a sequence normal bundle
Instead of
[1] R. Lashof, "Poincaré duality and cobordism" Trans. Amer. Math. Soc. , 109 (1963) pp. 257–277
[2] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)
is the limit of the Grassmann manifolds of
How to Cite This Entry:
B-Phi-structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=B-Phi-structure&oldid=19275
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098.
See original article
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Use Model Name as Programmatic Interface
You can use the name of a model as a programmatic interface for executing a specified simulation phase and computing values for individual states and times that you specify.
This functionality is not intended for running a model step by step, simulating a model programmatically, or for debugging. For those purposes, consider these alternatives:
• To step through a simulation, use the Step Forward and Step Back buttons.
For more information, see Step Through Simulations Using the Simulink Editor.
• To simulate a model programmatically, use the sim, parsim, and batchsim functions.
• For low-level debugging, use the Simulink^® debugger.
Model Requirements
When you use the model name as a programmatic interface to compute simulation values, the software ignores the effects of state transitions and conditional execution. Use this functionality only for
models that implement simple dynamic systems. Such systems must meet these requirements:
• All states in the model must have built-in nonbus data types.
For more information about built-in data types, see About Data Types in Simulink.
• The model must contain a minimal amount of state logic, such as Stateflow^® charts and conditionally executed subsystems.
• The model must contain only blocks from Simulink libraries. The model cannot contain S-functions or Simscape™ blocks.
• When you specify the state values as a vector, states in the model must have real double values.
Using this functionality for models that do not meet these requirements and models that use multitasking execution can produce unexpected results. For more information about multitasking and single
task execution, see Time-Based Scheduling and Code Generation (Simulink Coder).
Input Arguments
To use a model name as a programmatic interface, you use the name of your model as though it were the name of a function. You always provide four inputs:
• t — Simulation time, specified as a real, double scalar.
• x — State values, specified as a vector of real double values or as a structure.
Specifying the states as a Simulink.op.ModelOperatingPoint object is not supported.
• u — Input data, specified as a vector of real, double values.
• phase — Simulation phase to execute, specified as one of these options:
□ 'sizes' — Size computation phase, in which the software determines the sizes for the model input, output, and state vectors
□ 'compile' — Compilation phase, in which the software propagates signal and sample time attributes
□ 'update' — Update phase, in which the model computes the values of discrete states
□ 'outputs' — Output phase, in which the model computes block and model output values
□ 'derivs' — Derivatives phase, in which the model computes the derivatives for continuous states
□ 'term' — Termination phase
The number, type, and dimensions of the output arguments depend on which simulation phase you execute.
When you use this functionality, you must manually execute each simulation phase in the appropriate order. For more information about how simulations run, see Simulation Phases in Dynamic Systems.
This functionality is not meant to run a simulation step by step or as a replacement for the typical simulation workflow.
Execute Size Computation Phase
To run the sizes phase, use this syntax, specifying [] for the first three input arguments.
[sys,x0,blks,st] = modelName([],[],[],"sizes");
Before R2024a: You can also execute the sizes phase without specifying any input arguments.
In R2024a: The syntax with no input arguments is no longer supported.
The sizes phase returns four output arguments:
• sys — System information, returned as a vector with seven elements:
□ sys(1) — Number of continuous states in the system.
□ sys(2) — Number of discrete states in the system.
□ sys(3) — Number of model outputs.
□ sys(4) — Number of model inputs.
□ sys(5) — Reserved.
□ sys(6) — Direct feedthrough flag for system. A value of 1 indicates that the system has direct feedthrough. A value of 0 indicates that the system does not have direct feedthrough.
□ sys(7) — Number of continuous, discrete, fixed-in-minor-step, and controllable sample times in the system. The value at this index indicates the number of rows in the ts output.
• x0 — Vector that contains the initial conditions for system states.
• blks — Vector that contains the names of blocks associated with the system states. The order of elements in blks matches the order of elements in x0.
• st — m-by-2 array of sample time information for the system, where m is equal to the value of sys(7). The first column of the array indicates the sample time and the second column indicates the
offset. For more information about sample time, see Types of Sample Time.
Execute Compilation Phase
To run the compilation phase, use this syntax, specifying [] for the first three input arguments.
[sys,x0,blks,st] = modelName([],[],[],"compile");
The compilation phase returns the same four output arguments as the sizes phase:
• sys — System information, returned as a vector with seven elements:
□ sys(1) — Number of continuous states in the system.
□ sys(2) — Number of discrete states in the system.
□ sys(3) — Number of model outputs.
□ sys(4) — Number of model inputs.
□ sys(5) — Reserved.
□ sys(6) — Direct feedthrough flag for system. A value of 1 indicates that the system has direct feedthrough. A value of 0 indicates that the system does not have direct feedthrough.
□ sys(7) — Number of continuous, discrete, fixed-in-minor-step, and controllable sample times in the system. The value at this index indicates the number of rows in the ts output.
• x0 — Vector that contains the initial conditions for system states.
• blks — Vector that contains the names of blocks associated with the system states. The order of elements in blks matches the order of elements in x0.
• st — m-by-2 array of sample time information for the system, where m is equal to the value of sys(7). The first column of the array indicates the sample time and the second column indicates the
offset. For more information about sample time, see Types of Sample Time.
After running the compilation phase, you must run the termination phase before you can close the model. If you execute the compilation phase multiple times before executing the termination phase, you
must execute the termination phase an equal number of times.
Compute Discrete State Values
To execute the update phase and compute the discrete state values, use this syntax. You specify the time at which you want to calculate the discrete states and the current state and input values to
use in the computation.
dStates = modelName(t,x,u,"update");
The update phase returns the discrete state values dStates as a structure or an array, depending on how you specify the current state values, x.
• When you specify x as empty ([]) or as a structure, the update phase returns dStates as a structure that contains both discrete and continuous state values for all states with built-in data
• When you specify x as a vector or an array, the update phase returns dStates as a vector or an array that contains only the discrete state values for states that have real, double values.
Compute Output Values
To compute the model outputs, use this syntax. You specify the time at which you want to calculate the discrete states and the current state and input values to use in the computation.
out = modelName(t,x,u,"outputs");
Compute Continuous State Derivatives
To compute the derivatives for continuous states, use this syntax. You specify the time at which you want to calculate the discrete states and the current state and input values to use in the
derivs = modelName(t,x,u,"derivs");
Execute Termination Phase
When you are done analyzing the model behavior, use this syntax to execute the termination phase so you can close the model. Specify [] for the first three input arguments.
See Also
sim | parsim | batchsim
Related Topics
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Learn Theory of Computation MCQ Questions with answers and solutions. Share Theory of Computation MCQ Questions with answers and solutions with others.
Home / Engineering / Theory of Computation MCQs / Page 10
Theory of Computation MCQs | Page - 10
Dear candidates you will find MCQ questions of Theory of Computation here. Learn these questions and prepare yourself for coming examinations and interviews. You can check the right answer of any
question by clicking on any option or by clicking view answer button.
Q. 91) We think of a Turing machine’s transition function as a
Q. 92) Church’s Thesis supports
Q. 93) A random access machine (RAM) and truing machine are different in
Q. 94) Choose the correct statement
Q. 95) Given S = {a, b}, which one of the following sets is not countable?
Q. 96) In which of the stated below is the following statement true? “For every non-deterministic machine M1, there exists as equivalent deterministic machine M2 recognizing the same language.”
Q. 97) Which of the following conversion is not possible (algorithmically)?
Q. 98) Consider a language L for which there exists a Turing machine ™, T, that accepts every word in L and either rejects or loops for every word that is not in L. The language L is
Q. 99) Consider the following statements
I. Recursive languages are closed under complementation
II. Recursively enumerable languages are closed under union
III. Recursively enumerable languages are closed under complementation
Which of the above statement are TRUE?
Q. 100) Recursively enumerable languages are not closed under
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Dan Fortunato
Surface PDEs
I have developed a high-order accurate fast direct solver for variable-coefficient partial differential equations on surfaces, based on spectral collocation and the hierarchical Poincare–Steklov
scheme. The method may be used to accelerate implicit time-stepping schemes as repeated solves require only \(\mathcal{O}(N \log N)\) work on a mesh with \(N\) elements, after an \(\mathcal{O}(N^{3/
2})\) precomputation. I have applied it to a range of scalar- and vector-valued problems on both smooth surfaces and surfaces with sharp corners and edges, including the static Laplace–Beltrami
problem, the Hodge decomposition of a tangential vector field, and some time-dependent reaction–diffusion systems.
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Turbulent flows under the action of volume forces and nonself-similarity
The work sets forth theory of turbulent jet flows under the action of several types of forces that strongly influence the nature of the flow: volume forces such as caused by a heavy disperse impurity
or an electromagnetic force, twisting of the flow, pressure gradient, compressibility, or significant nonself-similarity. Models and methods are worked permitting use of a semiempirical theory, and
some experimental results are included. A simplified integral method for calculating jet flows is proposed.
Moscow Izdatel Mashinostroenie
Pub Date:
□ Flow Distortion;
□ Gas Flow;
□ Jet Flow;
□ Turbulent Jets;
□ Compressibility Effects;
□ Conducting Fluids;
□ Electromagnetic Radiation;
□ Flow Equations;
□ Force Distribution;
□ Impurities;
□ Magnetic Effects;
□ Pressure Gradients;
□ Two Phase Flow;
□ Fluid Mechanics and Heat Transfer
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Towards Enabling Binary Decomposition for Partial Label Learning
Towards Enabling Binary Decomposition for Partial Label Learning
Xuan Wu, Min-Ling Zhang
Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence
The task of partial label (PL) learning is to learn a multi-class classifier from training examples each associated with a set of candidate labels, among which only one corresponds to the
ground-truth label. It is well known that for inducing multi-class predictive model, the most straightforward solution is binary decomposition which works by either one-vs-rest or one-vs-one
strategy. Nonetheless, the ground-truth label for each PL training example is concealed in its candidate label set and thus not accessible to the learning algorithm, binary decomposition cannot be
directly applied under partial label learning scenario. In this paper, a novel approach is proposed to solving partial label learning problem by adapting the popular one-vs-one decomposition
strategy. Specifically, one binary classifier is derived for each pair of class labels, where PL training examples with distinct relevancy to the label pair are used to generate the corresponding
binary training set. After that, one binary classifier is further derived for each class label by stacking over predictions of existing binary classifiers to improve generalization. Experimental
studies on both artificial and real-world PL data sets clearly validate the effectiveness of the proposed binary decomposition approach w.r.t state-of-the-art partial label learning techniques.
Machine Learning: Classification
Machine Learning: Multi-instance;Multi-label;Multi-view learning
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Classifying Web Videos using a Global Video Descriptor
Figure 1. Example action classes from the (a) KTH, (b) UCF50 and (c) HMDB51 datasets.
Computing descriptors for videos is a crucial task in computer vision. In this work, we propose a global video descriptor for classification of realistic videos as the ones in Figure 1. Our method,
bypasses the detection of interest points, the extraction of local video descriptors and the quantization of descriptors into a code book; it represents each video sequence as a single feature
vector. Our global descriptor is computed by applying a bank of 3-D spatiotemporal filters on the frequency spectrum of a video sequence, hence it integrates the information about the motion and
scene structure. We tested our approach on three datasets, KTH, UCF50 and HMDB51. Our global descriptor obtained the highest classification accuracies on two of the most complex datasets UCF50 and
HMDB51 among all published results, which demonstrates the discriminative power of our global video descriptor for classifying videos of various actions. In addition, the combination of our global
descriptor and a state-of-the-art local descriptor resulted in a further improvement in the results.
Gist of a Video
Videos which involve similar actions tend to have similar scene structure and motion. The regularities in the appearance or motion can be used to pinpoint the type of actions involved in the video,
and can be useful in the classification of videos. The frequency spectrum computed for a video clip could capture both scene and motion information effectively, as it represents the signal as a sum
of many individual frequency components. In a video clip, the frequency spectrum can be estimated by computing the 3-D Discrete Fourier Transform (DFT). The motion in a video can be explained in a
straightforward way by considering the problem in the Fourier domain. The frequency spectrum of a two-dimensional pattern translating on an image plane lies on a plane, the orientation of which
depends on the velocity of the pattern. Furthermore, multiple objects with different motion will generate frequency components in multiple planes as depicted in Figure 2.
Figure 2. Orientation of frequency spectrums: The translating object (a) generates a space-time volume (b),
and a frequency spectrum of non-zero values on a plane (c). Similarly, motion in different orientations (d)
results in the volume in (e) and a frequency spectrum (f) with two planes. Uni-directional motion results in
a single plane in frequency spectum (g-i). Motion with different velocities (j) corresponds to two planes in
the frequency spectum (l). A translating object with a sinusoidal intensity over time (m-n) resulted in two
identical planes in the frequency spectrum with a separation based on the frequency of the object (o). For
multiple objects introducing more gradients (p,g), the planes are still present but they appear partially (i,r).
Since the motion can occur in different directions and frequencies, in our work we use 3-D Gabor filters of different orientations and center frequencies to effectively capture the motion information
in a video clip. By filtering the frequency spectrum with a certain oriented filter and taking the inverse Fourier transform, the motion and scene components which are normal to the orientation of
the filter are pronounced, as illustrated in the example in Figure 3.
Figure 3. Effect of filtering the frequency spectrum: Using different orientations of 3-D filters on frequency
spectrum for the sample clips (a,g), the components with different motion (b,c,h,i), the vertical scene
components (d,j), the horizontal scene components (e,k), and the diagonal scene components (f,l) are
highlighted. The red and cyan arrows show the direction of motion in the two videos.
Generating Descriptor
A flowchart describing the implementation of our method is depicted in Figure 4. Our goal is to represent each video sequence by a single global descriptor and perform the classification. For the
current implementation, we extract K uniformly sampled clips of a fixed length from each given video.
Figure 4. Overview of our approach: Given a video, we extract K clips and compute the 3-D Discrete Fourier
Transform. Applying each filter of the 3-D filter bank separately to the frequency spectrum, we quantize the
output in fixed sub-volumes. Next, we concatenate the outputs and perform dimension reduction by Principal
Component Analysis and classification by the use of a Support Vector Machine.
As the next step, we compute the 3-D DFT, and obtain the frequency spectrum of each clip as given by,
In order to capture the components at various intervals of the frequency spectrum of a clip, we apply a bank of narrow band 3-D Gabor filters with different orientations and scales. The transfer
function of each 3-D filter, tuned to a spatial frequency
along the direction specified by the polar and the azimuthal orientation angles
in a spherical coordinate system, can be expressed by,
are the radial and angular bandwidths, respectively, defining the elongation of the filter in the spatio-temporal frequency domain. 3-D plots of these filters are shown in Figure 5.
Figure 5. Visualization of the all designed filters in 3-D.
(For illustration, we specified a cut-off at 3 dB on the filters.)
Applying each generated 3-D filter on the frequency spectrum of the clip, we compute the output,
Γ[i] (f[x], f[y], f[t])
is the output when the i
filter is applied. Then we take the inverse 3-D DFT,
By quantizing the output volume in fixed sub-volumes and taking the sum of each sub-volume and performing the same computation for each filter in our filter bank, we obtain a long feature vector
which represents a single clip. This feature vector has the advantage of preserving the spatial information as the response of each filter on each sub-volume contributes to an element in the
concatenated feature vector. The last step is to apply PCA, a popular method for dimensionality reduction, in order to generate our global video descriptor.
Experimental Results
For performance evaluation, we used publicly available datasets: KTH Dataset, UCF50 Dataset, and HMDB51 Dataset. UCF50 and HMDB51 are the two most challenging datasets with the largest number of
classes, which are collections of thousands of low quality web videos with camera motion, different viewing directions, large interclass variations, cluttered backgrounds, occlusions, and varying
illumination conditions. For all experiments, we picked three key clips of 64 frames from each video and downsampled the frames of clips to a fixed size (128x128) for computational efficiency. Next,
we computed the 3-D DFT to compute the frequency spectrum of the clips of each video and then applied the generated filter bank, which consisted of 68 3-D Gabor filters, which corresponded to 2
scales and 37 and 31 orientations for the first and second scales, respectively, in the spatiotemporal frequency domain. After the application of the filters, we computed the average response of
filters on 512 uniformly spaced 16x16x8 subvolumes, in order to quantize and generate the global feature vector for the clip. The length of the feature vector in our experiments was 104,448, as there
are 68 filters, 512 sub-volumes and 3 key clips. We reduced the dimensionality of the feature vectors to 2,000 using PCA. More details are provided in the related paper.
For classification, we trained a multi-class Support Vector Machine (SVM) using the linear kernel for our descriptor and histogram intersection kernel for STIP. We performed cross validation by
leaving one group out for testing and training the classifier on the rest of the dataset and performing the same experiment for all groups on UCF50. For HMDB51 we performed cross validation on the
three splits of the dataset. We did not include any clips of a video in the test set if any other clip of the same video is used in the training set.
The discriminative power of our descriptor can be seen clearly in the example in Figure 6. This basic experiment was done using 4 sequences from a public dataset. For each of the 4 sequences, we
computed the descriptors. Each entry in the matrix in Figure 6.c is the normalized Euclidean distance between the computed descriptors of the 4 sequences. As seen in the matrix, the descriptor
distances between the jumping actions in two different scenes is comparably lower than the other distances, which shows that our descriptor can generalize over intra-class variations. The distances
are high when different actions are performed in different scenes, such as the ones labeled by blue arrows in Figure 6.a.
Figure 6. Descriptor distances for example clips: For the clips with the similarities and differences
mentioned in (a), the distances of the computed descriptors (b) are shown as a color-coded matrix
in (c). The descriptors with similar actions and scene have lower distances.
To illustrate the advantage of the 3-D global descriptor, we compared our descriptor to the popular descriptors: GIST (on UCF50 dataset), and STIP (on KTH, UCF50 and HMDB51 datasets) which involve
the computation of histograms of oriented gradients (HOG) and histograms of optical flow (HOF). For comparison, we also listed the performance of a low level descriptor based on color and gray values
(on UCF50 and HMDB51 datasets), and the biologically motivated C2 features (on KTH and HMDB51 datasets). Fig. 8 shows the comparison of performance over three datasets. The conclusion tables are
provided in the paper.
Figure 7. Average classification accuracies over KTH, UCF50 and HMDB51 datasets.
KTH Dataset:
The KTH dataset includes videos captured in a controlled setting of 6 action classes with 25 subjects for each class. Our descriptor has a classification accuracy of 92.0%, which is comparable to the
state-of-the-art. This experiment shows that our descriptor is able to discriminate between the actions with different motions appearing in similar scenes.
UCF50 Dataset:
This dataset includes unconstrained web videos of 50 action classes with more than 100 videos for each class. Our descriptor has an accuracy of 65.3% over 50 action classes, which outperforms GIST
and STIP. Using the combination of STIP and GIST3D by late fusion resulted in a classification accuracy of 73.7%, which is another 8% improvement in the performance.
For comparison of our descriptor to STIP, we also analyzed the average similarities of descriptors among action classes of UCF50. We computed the Euclidean similarity for our descriptors and
histogram intersection as the similarity measure for STIP. Our descriptor has higher intra-class similarity and lower inter-class similarity than STIP as shown in Figure 8. This clearly explains why
our global descriptor (GIST3D) performs superior than STIP.
Figure 8. Descriptor similarity matrices for STIP (a) and GIST3D (b) computed among 50 action classes of UCF50 dataset.
HMDB51 Dataset:
The HMDB51 dataset includes videos of 51 action classes with more than 101 videos for each class. Our descriptor has a classification accuracy of 23.3% over 51 action classes, which outperforms STIP
by 5%. The late fusion classifier of these two descriptors resulted in a 6% improvement in the performance over using just our descriptor GIST3D.
The actions in the video sequences of HMDB51 are not isolated; multiple actions may be present in a single video sequence despite a given single class label for the sequence. There is also large
intraclass scene variation. Therefore, classifying actions on this dataset is more challenging and the performances of the mentioned methods are lower.
In this work, we presented a global scene and motion descriptor to classify realistic videos of different actions. Without interest point detection, background subtraction and tracking, we
represented each video with a single feature vector and obtained promising classification accuracies using a linear SVM. Preserving also the useful spatial information, our descriptor had a better
performance than the state-of-the-art local descriptor, STIP, utilizing a bag-of-features representation which discards the spatial distribution of the local descriptors.
Related Publication
Berkan Solmaz, Shayan Modiri A., and Mubarak Shah,
Classifying Web Videos using a Global Video Descriptor
Machine Vision and Applications (MVA)
, 2012.
(pdf file) (bibtex)
YouTube Presentation
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Extracting black-hole rotational energy : The generalized Penrose process
Relativistic jets are often launched from the vicinity of accreting black holes. They are observed to be produced in stellar-mass black-hole binary systems and are believed to be the fundamental part
of the gamma-ray burst phenomenon. Powerful relativistic jets are also ejected by accreting supermassive black holes in some active galactic nuclei. There is no doubt that the jet-launching mechanism
is related to accretion onto black holes, but there has been no general agreement as to the ultimate source of energy of these spectacular high energy phenomena. In principle, relativistic jets can
be powered either by the black hole gravitational pull or by its rotation (spin), with large-scale magnetic fields invoked as energy extractors in both cases. In the context of strongly magnetized
jets Blandford & Znajek (1977) proposed a model of electromagnetic extraction of black hole's rotational energy based on the analogy with the classical unipolar induction phenomenon. The physical
meaning of this process has been subject to a long controversy. I will show that the Blanford-Znajek process is a Penrose process of black-hole energy extraction. I will first consider the case of
arbitrary fields or matter described by an unspecified, general energy-momentum tensor and show that the necessary and sufficient condition for extraction of a black hole's rotational energy is
analogous to that in the mechanical Penrose process: absorption of negative energy and negative angular momentum. I will show that a necessary condition for the Penrose process to occur is for the
Noether current to be spacelike or past directed (timelike or null) on some part of the horizon. In the particle ("mechanical") case, the general criterion for the occurrence of a Penrose process
reproduces the standard result. For stationary, force-free electro-magnetic field one recovers the condition obtained by Blandford and Znajek in their original article. In the case of relativistic
jet-producing ``magnetically arrested disks'' I will show that the negative energy and angular-momentum absorption condition is obeyed when the Blandford-Znajek mechanism is at work, and hence the
high energy extraction efficiency up to ~300 % found in recent numerical simulations of such accretion flows results from tapping the black hole's rotational energy through the Penrose process. I
will show how black-hole rotational energy extraction works in this case by describing the Penrose process in terms of the Noether current.
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WIP: Prototype a validity restriction CMRA construction (!502) · Merge requests · Iris / Iris · GitLab
WIP: Prototype a validity restriction CMRA construction
Implement a first cut at building a CMRA by restricting the validity of an existing CMRA. We demonstrate the construction by building a CmraMixin for auth that starts with a CMRA built from
combinators and primitives.
The implementation actually transports a structure A:cmraT to B:Type across an isomorphism iso A B, which is a bijection using Leibniz equality. It also takes an arbitrary restriction of A's validity
over B, along with all of the validity-related CMRA laws. Unlike what I attempted in !348 (closed) this is a much more principled solution, leveraging the fact that we really have an isomorphism.
The implementation is a total mess. There are three high-level problems:
• The instances for iso are scoped too broadly. When I imported them in agree.v I got a typeclass resolution loop.
• I'm not sure about how canonical structures work out with this approach, given the way it uses an instance of the iso typeclass.
• The proofs for auth are extremely messy and should re-use the existing nice proofs for bits of the old auth construction.
There are also many things the auth library needs which would need to be implemented, many theorems that are generic for any subset construction.
It's not necessarily a problem, but I didn't generalize to transporting a CMRA to an OFE using a bijection that preserves equivalence, because first that's quite a bit more complicated to prove, and
second because it would be harder to use for auth where we want to inherit the OFE structure from the CMRA anyway.
@robbertkrebbers and @jung is this the direction you were thinking of? Do we want to pursue this to (1) simplify auth and (2) for other users?
Background: This came up from working on a CMRA isomorphism construction (!348 (closed)), where Robbert mentioned that the motivation was to construct the auth RA by restricting the validity of an RA
built from some combinators. In that MR we didn't have an idea of what the spec would look like, so the assumptions we made were rather ad-hoc.
This specific construction, and applying it to auth, was also proposed in #210, phrased as cutting the RA "from the top"; there might be another subset construction but this is the most clearly
useful/possible one.
Merge request reports
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Fuzzy Particle Swarm Optimization Algorithm (NFPSO) for Reachability Analysis of Complex Software Systems
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Fuzzy Particle Swarm Optimization Algorithm (NFPSO) for Reachability Analysis of Complex Software Systems
EasyChair Preprint 4328
11 pages•Date: October 8, 2020
Nowadays, model checking is applied as an accurate technique to verify software systems. The main problem of model checking techniques is the state space explosion. This problem occurs due to the
exponential memory usage by the model checker. In this situation, using meta-heuristic and evolutionary algorithms to search for a state in which a property is satisfied/violated is a promising
solution. Recently, different evolutionary algorithms like GA, PSO, etc. are applied to find deadlock state. Even though useful, most of them are concentrated on finding deadlock. This paper proposes
a fuzzy algorithm in order to analyze reachability properties in systems specified through GTS with enormous state space. To do so, we first extend the existing PSO algorithm (for checking deadlocks)
to analyze reachability properties. Then, to increase the accuracy, we employ a Fuzzy adaptive PSO algorithm to determine which state and path should be explored in each step to find the
corresponding reachable state. These two approaches are implemented in an open-source toolset for designing and model checking GTS called GROOVE. Moreover, the experimental results indicate that the
hybrid fuzzy approach improves speed and accuracy in comparison with other techniques based on meta-heuristic algorithms such as GA and the hybrid of PSO-GSA in analyzing reachability properties.
Keyphrases: Fuzzy Adaptive Particle Swarm Optimization, Graph Transformation System, model checking, reachability property, state space explosion
Links: https://easychair.org/publications/preprint/3RS1
Download PDFOpen PDF in browser
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[Solved] Below are shown three different crystallo | SolutionInn
Below are shown three different crystallographic planes for a unit cell of some hypothetical metal. The circles
Below are shown three different crystallographic planes for a unit cell of some hypothetical metal. The circles represent atoms:
(a) To what crystal system does the unit cell belong?
(b) What would this crystal structure be called?
(c) If the density of this metal is 8.95 g/cm^3, determine its atomic weight.
Transcribed Image Text:
0.30 nm ro→>| 0.40 nm→ (101) 35 nm-→| (110) (001) 0.50nm 0.46 nm-
Fantastic news! We've Found the answer you've been seeking!
Step by Step Answer:
Answer rating: 80% (15 reviews)
The unit cells constructed below show the three crysta...View the full answer
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fingers. My students drastically improve under my tutelage, generally seeing a two grade level improvement (F to C, C to A, for example), and all of them get a much clearer understanding! I am
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Théorème de la divergence - Calcul multivariable | Elevri
Théorème de la divergence - Calcul multivariable
Now we've arrived at our final theorem. Drumroll, please? This theorem is ubiquitous in physics. So it's not just one of those theorems mathematicians came up with to pickle your brain: this one is
really important.
So let's suppose that you're a magician standing (floating, hovering?) in the middle of a closed water tank. You've created yourself a bubble in which you can breathe. Have a look at the sketch, and
you'll see what I mean.
As you say 'abrakadabra', you magically create 3 gallons of water, which flow outward from the center of the container.
But water is incompressible: the molecules can't be packed more densely. Since water is created within the tank, it must flow out somewhere. What goes in goes out, you know. This means that water
will flow out from our tiny outlet at the top right corner.
The Divergence theorem centers around this idea. It says that
Here, is a continuously differentiable surface, encapsulating the volume . Moreover, the vector field should be continuously differentiable.
The Divergence theorem is a bit like Stokes' theorem, but for the divergence rather than the curl.
Now we'll work an example to help you absorb everything.
The sun radius is , and its distance from the earth is .
The energy per square meter from the sun on the earth surface is . Calculate the energy generated per cubic meter from the sun. Assume that the energy generated is evenly spread out.
First, we calculate a surface integral around a sphere of the radius this gives us the total radiated energy. Note that the energy dissipated from the sun has a direction since it travels in space.
This direction is the same direction as the unit normal of the sphere
Next according to the divergence theorem we know that
From the assignment we where told that the energy created by a cubic meter of the sun is equal to and constant inside the sphere. Therefore we know that . We know that the sun only creates energy
inside of it and therefore we find that, we set
Again it is worth repeteting that
Bon plan pour le calcul et liste de tâches courtes
Nous travaillons dur pour vous fournir des connaissances courtes, concises et éducatives. Contrairement à ce que font de nombreux livres.
Obtenez des problèmes d'examen pour d'anciens examens de calcul divisés par chapitres
Le truc est d'apprendre à la fois la théorie et la pratique sur des problèmes d'examen. Nous les avons catégorisés pour le rendre encore plus facile.
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Math 309
Goals: This course is an introduction to Linear Algebra. After calculus, Linear Algebra is the most useful branch of mathematics, with innumerable applications in statistics, computer science,
engineering, physics, economics and in mathematics itself. It combines algebra and geometry in a way that is mathematically "clean": the definitions and theorems are simple and precise, and most
proofs are short, direct and illuminating.
But Linear Algebra is a modern, abstract subject. All students find the jump in the level of abstraction difficult --- linear algebra is considerably harder than calculus. Be prepared!
Prerequisites: A year of Calculus, Math 299, and a committment to work hard on abstract mathematics. Course Outline
Textbook: Linear Algebra with Applications, 9th ed. by Steven Leon. Chapter 1
Additonal Resources: The following additional resources may be helpful.
• Schaum's Outlines: Linear Algebra by S. Lipschutz and M. Lipson.
• Linear Algebra done wrong, by S. Treil. A free online book with a clean presentation.
• Video lectures: Free video lectures on Linear Algebra are available online from Johns Hopkins (for a course very similar to ours), and from MIT (for a course that emphasizes the applications of
linear algebra to numerical analysis).
• Linear Algebra by S. Lang.
Web Calculators: This site and this site (requires Java) calculate eigenvalues and eigenvectors, this site is useful for matrix calculations.
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1.3 Evaluating Limits Analytically
Last section, we saw examples where the value of a limit and its true value matched. For example, for $f(x)=x^2$.
\[\lim_{x\to2}f(2)= f(2)\]
The function appears to be approaching 4 when $x=2$, and it also is 2 when $x=2$.
For a vast number of functions and situations, substituting is a valid way to evaluate a limit, rather than using tables and graphs like was shown last section. The book walks you through the
specifics on this, which I do suggest you read through.
There is one other theorem that addresses a situation we saw last section.
\[f(x) = \begin{cases} x^2+x + 1, & x\neq 1 \\ 1, & x=1 \end{cases} \qquad g(x)=x^2+x + 1\]
If two functions are identical save for a specific value, than its safe to assume their limits agree. So, despite the first case not being defined when $x=1$, we can still use $g(x)$ to determine $\
A Strategy for Finding Limits
Simply put, substitute when you can. If you can’t see if there is another function that matches, even if the one value is excluded. This can involve rewriting your function or simplifying to find
that function.
For both, graphs and tables will help to reinforce what you found, even if you could not find a limit.
Finding the Other Function
One way to find the almost equivalent function is by dividing when presented with a rational function.
\[\lim_{x\to -3} \frac{x^2+x-6}{x+3}\]
Substitution gives us $0/0$, also known as the indeterminate form, but $x-3$ is a factor of $x^2+x-6$.
\[f(x)=\frac{x^2+x-6}{x+3} = \frac{(x+3)(x-2)}{x+3}=x-2\]
We’ve found our equivalent function. Substituting $x=-3$ gives us a limit of $-5$.
As a reminder, the domain from the original carries over to the new function. Although $x-2$ can be evaluated when $x=-3$, as a function in this situation, it cannot. We are only using it to evaluate
the limit. It might seem like a minor distinction, but it’s important to remember that the limit of a function is not the same as the value of the function.
Another technique is rationalizing radicals, meaning multiplying by a radicals conjugate.
Multiplying by $\sqrt{x+1}+1$ will hopefully open up an avenue to find the limit.
\[\begin{align*} \frac{\sqrt{x+1}-1}{x} \cdot \frac{\sqrt{x+1}+1}{\sqrt{x+1}+1} &= \frac{(x+1)-1}{x(\sqrt{x+1}+1)} \\ &= \frac{x}{x(\sqrt{x+1}+1)} \\ &= \frac{1}{\sqrt{x+1}+1} \end{align*}\]
Again, as a function this still can be evaluated at 0, but for the purpose of finding a limit, this works. The limit is $1/2$.
The Squeeze Theorem
If a function is less than or equal to another, and a third lies in between them, whenever the bounding function have an equal limit, then the inner one shares the same limit.
This is particularly helpful in proving some more difficult limits, like $\lim_{x\to0}\frac{\sin x}{x}=1$. There are two others limits listed that are proved using the squeeze theorem, which are
helpful in solving other limits.
\[\begin{align*} \lim_{x\to0}\frac{\tan x}{x} &= \lim_{x\to0}\left(\frac{\sin x}{x}\right)\left(\frac{1}{\cos x}\right) \\ &= \lim_{x\to 0}\frac{\sin x}{x} \cdot \lim_{x\to 0}\frac{1}{\cos x} \\ &= 1
\cdot 1 \\&=1 \end{align*}\]
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Forecasting demand collapse in the exceptional lockdown environment generated by Covid-19
The recent crisis of the COVID 19, has had a tremendous impact on our businesses and our daily lives, and we’re only at the beginning of this major upset.
Prediction is very difficult, especially if it’s about the futureNiels Bohr
The recent crisis of the COVID 19, has had a tremendous impact on our businesses and our daily lives, and we’re only at the beginning of this major upset. In this article we will look at the impact
it has had on our prediction tools inside the Apps: forecast and estimate.
I will fast forward to the conclusion: this unprecedented and unexpected event has really rendered prediction figures completely wrong. This is of course not a surprise. But please keep on reading,
because even when knowing the end of the story, there are interesting things to learn along the way.
Some reminders before moving further:
• Demand estimate computation: let us just look again at the formula for each coming day:
• TY _otb + (LY_actual – LY_otb)
• Where TY means this year, LY means last year, OTB means “on-the-book”
• Forecast computation: we have several machine learning algorithms, which all attempt to make a forecast each day, combining all the available data in the past, weighting past years activity and
the current trend. We select the best one according to its accuracy at predicting past data.
It is important to note at this stage, that the estimate is a relative figure: we start from today’s OTB values for coming days, and add an estimated growth or decrease; whereas the forecast is an
absolute figure: the algorithm predicts day by day a final on-rent value.
We’ve already covered the biases the estimate formula might have during the demand forecast workshop of our 2019 Academies in Paris and London, but still, client’s data analysis has shown that the
estimate computation is generally quite precise when the level of activity is relatively similar to last year.
But what happens when there is a sudden change in the activity pattern ?
When the level of activity increases or decreases dramatically, the estimate formula adapts instantaneously to this change, because it always starts from this year’s on-the-book value. Whereas the
forecast, considering primarily the activity from the last years, will have a difficult time predicting correct values, until it has catched up with the trend, and adjusted its predictions.
Here, our two “star” algorithms perform quite differently. The first one, XGBoost, which I thought was quite conservative, i.e. more influenced by last year’s activity levels rather than the trend,
proved me wrong. The brutal dive of activity is reflected in the forecast after just 5 days. This is half a surprise though, because this algorithm is capable to predict quite accurately activity for
companies with high seasonality variance and very steep scaling up or down.
Illustration 1: XGBoost algorithm
Concerning the estimate (illustration 2), we can see the 3 days ahead estimate (in blue) catching up even quicker than the 3 days ahead forecast (in yellow).
Illustration 2: XGBoost algorithm vs estimate
The second main machine learning algorithm that we use, Facebook’s Prophet, usually gives amazing predictions on a regular activity, with low or medium variations along the season, and bad
predictions when the variations are too sudden (even seasonal ones). It is quite good to catch up with little trend changes, but here it is clearly at a loss. The algorithm slowly adapts, but not
quickly enough to have a real impact on the forecasted values accuracy.
Illustration 3: Facebook Prophet forecast
The forecast may take into account more parameters than the estimate, but when faced with an unprecedented event, the Prophet algorithm has difficulty coping with it.
How we improve the forecast in a time of crisis?
It is interesting to note first, that our initial choice of implementing several machine learning models, was a good one. Indeed, we can see that each one has strengths and weaknesses and that at
least one of our algorithms catches up with the trend changes efficiently. So when put into competition with the other algorithms, it will become clear after a some time that it performs better than
the others and it will be selected more often for the daily forecast. But still, there will be a delay for this selection.
The real problem is that we are faced with a real difficulty when it comes to making absolute predictions. Even if it works well most of the time, it is almost useless during a crisis.
One solution to this problem, would be to mix the adaptability of the estimate, starting from the current level of activity, with the power of machine learning being able to analyze and use more data
than just basically a past evolution.
Thus, rather than forecasting absolute values of forecast, we could forecast relative values of forecast as day to day variations, and add them up starting with the last complete day of activity’s on
rent. There are of course many other ways to make a relative forecast, but this was the quickest way to implement it on top of our current forecast.
The idea is simple and seems appealing, but like any prediction, it has ups and downs. And we’ll now see why.
In “normal” times, there might be small variations of activity due to a variety of events, random activity variations, and so on. So the last day’s activity value may be biased, and then starting
from its level will introduce a bias inside the next 30 days of the forecast window.
We can see it very clearly in January and February, where the activity levels were normal, and the forecast algorithms were in their best spot.
Table 1: January/February forecast and estimate 7 days mean accuracy
We can see clearly here, that the noise induced by the day to day variation of the starting point is affecting the mean accuracy of the forecast, but without losing too much precision though.
Compared to the estimate in this “good” period, we see that for a 7 days ahead time window, the Forecast performs globally better than the estimate. Apart from one company with the small fleet, and
thus lower levels of activity (less data to learn from), probably because the forecast lacks historical data and volume.
This is in a way reassuring, because it means that forecasting really has an edge vs. estimate in these periods, as it ignores noise. It shows that the machine learning algorithms are really
efficient in finding the right correlations in the whole activity dataset to make accurate predictions about final levels of activity.
This noise effect can be seen very clearly on the forecast statistics page, with significant day to day variations around the final on rent value. That does not mean that the forecast are not
accurate, but it means it is a bit harder to follow daily.
Illustration 4: noise impacting forecast due to a modification of the output of the machine learning models
On the other hand, during the massive drop of activity in March due to Covid 19 lockdown, the relative forecast method and the estimate have of course yielded much better results than the regular
absolute forecast.
Table 2: March forecast and estimate 7 days mean accuracy
This is of course because the daily forecast has been much more adaptive to the real level of activity, by starting at the last known on rent value and then adding daily variations. The relative
forecast also performs overall better than the estimate.
So we have been able, with a little change in our forecast output, to change an absolute forecast into a relative forecast, gaining a huge amount of reactivity during unprecedented events, while
maintaining a good amount of accuracy during normal times.
Illustration 5: Facebook Prophet forecast after tweaking
llustration 5 shows the effect, after a few days, of having tweaked the Facebook Prophet algorithm. The forecasted level of activity has instantly adjusted the day after deploying the patch, and are
much closer to the OTB value, and thus probably of the final value. We can also note that the forecast is now much below the estimate.
And now, let’s focus on the estimate: how does it cope in details with the sudden change of activity?
For mid term, the estimate will still add on rent with last year’s rhythm of reservations, so it will be quite wrong:
Illustration 5: estimate (blue line) vs daily forecast (blue area)
It is important to insist though that the estimate has two major strengths for short term:
• It starts from the current OTB values for the future days, that are always up to date, reflecting the current level of activity, so it is really adaptive
• It adds the last year on rent increase between the equivalent period, which means that even for normal periods, you don’t have a big increase for the few next days, meaning a quite good short
term accuracy
So here are some results for short term estimate accuracy:
Table 3: estimate accuracy in March for an Europcar medium fleet client
It’s interesting to have some statistical figures, but still, we see that in times like COVID 19, the on rent will majorly go down, because we will see more cancellations than new reservations. So
sadly, we can mainly expect future short term demand to be lower than the OTB values.
As a conclusion, we can see that prediction tools, like Forecast and Estimate are designed to be adjustable, able to cope with trend variations, self-learn (for machine learning models), but are
still powerless against a disruptive event the size of COVID 19.
However, it gives us an opportunity to learn from the weaknesses that we spot, and improve our tools with even more dynamic and resilient machine learning models for the forecast. It also shows the
interest of combining different approaches inside the same tools : estimate, and different forecasting algorithms.
As for the estimate, we can all foresee that next years’ estimate will be useless also because of the on rent increase of 2020 that will be used in 2021, and hopefully will be totally wrong. We are
thus working on a way to configure the estimate computation on a custom year rather than on the last year’s.
All in all, we know that we will never be able to make accurate forecasts or estimates taking into account the impact of an event of the magnitude of COVID 19, but we can still learn from it and make
our Apps more agile.
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Computable Knowledge
Timeline of Systematic Data and the Development of Computable Knowledge
How civilization has systematized more and more areas of knowledge, collected the data associated with them and made them amenable to automation and computation
1602: Bodleian Library
The Bodleian Library in Oxford is founded with 2,000 books.
1604: A Table Alphabeticall
Organizing the English language
Robert Cawdrey publishes a dictionary with definitions for 2,543 terms.
Multiplying numbers by simple addition
John Napier publishes the first tables of logarithms.
1623: Mechanical Calculator
Wilhelm Schickard creates a gear-based, wooden, six-digit, mechanical adding machine.
1627: Rudolphine Tables
Cataloging the known universe
Johannes Kepler's Rudolphine Tables lists the positions of 1,406 stars and procedures for locating the planets.
René Descartes introduces coordinate systems to allow geometry to be studied using algebra.
1650: Maria Cunitz
Maria Cunitz, a German astronomer, publishes Urania Propitia, which contains simplifications of Kepler's Rudolphine Tables.
1654: William Petty
Taking stock of economic activity
William Petty, traveling with Cromwell's army, systematically surveys the profitability of land in Ireland.
1659: Central England Temperature Record
Temperature every day
A record is started that continues today.
1662: John Graunt
Inventing the idea of statistics
Graunt and others start to systematically summarize demographic and economic data using statistical ideas based on mathematics.
1665: Scientific Journals
Philosophical Transactions of the Royal Society begins publication.
John Wilkins suggests a "philosophical language" in which concepts are encoded by pronouncable phonemes.
Answering questions using computation
Leibniz promotes the idea of answering all human questions by converting them to a universal symbolic language, then applying logic using a machine. He also tries to organize the systematic
collection of knowledge to use in such a system.
1686: Mapping the Winds
Edmond Halley creates a map showing prevailing winds at different locations.
Mathematics as a basis for natural science
Newton introduces the idea that mathematical rules can be used to systematically compute the behavior of systems in nature.
1688: Joseph de la Vega
Prices in the stock market
Joseph de la Vega's book Confusion of Confusions describes fluctuations in Dutch stock market prices.
1732: Poor Richard's Almanack
Benjamin Franklin publishes the first edition of his popular yearly (1732–1758) almanac.
1750: Creating a taxonomy for life
Carl Linnaeus systematizes the classification of living organisms, introducing ideas like binomial naming.
1753: British Museum
Collecting everything in a museum
The British Museum is founded as a "universal museum" to collect every kind of object, natural and artificial.
1755: Candlestick charts
Charting market prices
Munehisa Homma uses an early candlestick chart for prices in the Japanese rice market.
1755: Johnson dictionary
Samuel Johnson publishes an English dictionary containing 42,773 words.
1768: Encyclopædia Britannica
The Encyclopædia Britannica—and the Encyclopædie of Diderot and d'Alembert—attempts to summarize all current knowledge in book form.
1785: US Land Ordinance; British Ordnance Survey
Mapping whole countries
The US (1785) and UK (1791) governments begin creating detailed systematic maps of their countries.
1786: Pie Charts, or Commercial and Political Atlas
William Playfair's Commercial and Political Atlas graphically illustrates socioeconomic dates and invents the pie chart.
The first US Census is taken, as specified by the US Constitution.
1792: Farmer's Almanac
Robert Bailey Thomas begins publication of the still-extant Farmer's Almanac.
1795: The Metric System
Everything is decimal
France becomes the first nation to officially adopt the metric system of measurement.
1796: Recording data by machine
James Watt and John Southern create (but keep secret for 24 years) a device for automatically tracing variation of pressure with volume in a steam engine.
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Km To Radians Calculator - Calculator Wow
Km To Radians Calculator
Circles are everywhere! From the gears in your bike to the phases of the moon, understanding circular motion is crucial in various scientific and engineering fields. Here's where the Km To Radians
Calculator steps in as your secret weapon.
Why is the Km To Radians Calculator Important?
Imagine you're building a model rocket. You need to calculate the angle a control fin needs to be positioned at to achieve a specific trajectory. This involves converting the distance traveled along
the rocket's circular path (in kilometers) into an angle in radians, which is the language of circular motion.
The Km To Radians Calculator eliminates the need for complex calculations and saves you valuable time. It's perfect for:
• Students: Master geometry and physics problems involving circular motion.
• Engineers: Design gears, cams, and other circular mechanisms with precision.
• Animators: Create realistic movements for characters and objects.
Demystifying the Calculator: How to Use It Like a Pro
Using the Km To Radians Calculator is a breeze. Here's a step-by-step guide:
1. Locate the Calculator: This could be a web-based tool or a dedicated app.
2. Enter the Distance: This is the distance traveled along the circumference of the circle, measured in kilometers.
3. Input the Radius: This is the distance from the center of the circle to any point on the circumference, also in kilometers. Make sure the radius is always positive!
4. Click "Calculate": This triggers the conversion magic.
5. Voila! The result is displayed as the equivalent angle in radians.
Bonus Tip: Many calculators allow specifying the number of decimal places for the result.
10 FAQs for the Km To Radians Conqueror
1. What are Radians? Unlike degrees (0° to 360°), radians are a unit of angular measurement based on the ratio of the arc length to the radius of the circle. A full circle is 2π radians
(approximately 6.28 radians).
2. Can I convert radians back to kilometers? Absolutely! You can use the formula distance = radius x radians.
3. What if I don't know the radius? Some calculators offer functionalities like calculating the radius based on the distance and angle (radians) provided.
4. Does the Earth's curvature matter? For most everyday calculations, the Earth's curvature is negligible. However, for very large distances, using the Earth's average radius (approximately 6371
kilometers) might be necessary.
5. Is there a difference between arc length and circumference? The arc length is the actual distance traveled along a portion of the circle, while the circumference is the total distance around the
entire circle.
6. Can I use the calculator for non-circular shapes? No, this calculator is specifically designed for circular motion.
7. What if my answer doesn't match my textbook? Double-check your units (kilometers) and ensure you entered positive values for the radius. Rounding errors might also occur, so pay attention to the
number of decimal places displayed.
8. Are there alternative methods for calculating radians? Yes, you can use the formula radians = arc length / radius. However, the calculator provides a faster and more user-friendly approach.
9. What are some real-world applications of the Km To Radians Calculator? Besides the rocket example, it can be used for calculating gear ratios in machines, analyzing planetary motion, and
designing wind turbine blades.
10. Where can I find more resources on circular motion? Online tutorials, textbooks on geometry and physics, and educational websites offer in-depth explanations and practice problems.
Conclusion: Mastering the Art of Circular Motion
The Km To Radians Calculator is a powerful tool for anyone venturing into the world of circular motion. By understanding its role and utilizing it effectively, you can unlock new possibilities in
various fields, from engineering marvels to celestial explorations. So, embrace the fascinating world of circles and conquer your next circular motion challenge with confidence!
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Multivariable integration – Multivariable Calculus – Mathigon
Multivariable CalculusMultivariable integration
Integrating a function is a way of totaling up its values. For example, if is a function from a region in to which represents the mass density of a solid occupying the region , we can find the total
mass of the solid as follows: (i) split the region into many tiny pieces, (ii) multiply the volume of each piece by the value of the function at some point on that piece, (iii) and add up the
results. If we take the number of pieces to infinity and the piece size to zero, then this sum converges to the total mass of the solid.
We may apply this procedure to any function defined on , and we call the result the integral of over , denoted .
To find the integral of a function defined on a 2D region , we set up a double iterated integral over : the bounds for the outer integral are the smallest and largest possible values of for point in
, and the bounds for the inner integral are the smallest and largest values of for any point in a given each " = constant" slice of the region (assuming that each slice intersects the region in a
line segment).
Find the integral over the triangle with vertices , , and of the function .
Solution. The least and greatest values of for any point in the region are 0 and 3, while the least and greatest values of for each given -slice are 0 and . Therefore, the integral is
3D integration
To set up an integral of a function over a 3D region (for the order ): the bounds for the outer integral are the smallest and largest values of for any point in the region of integration, then the
bounds for the middle integral are the smallest and largest values of for any point in the region in each " = constant" plane, and the inner bounds are the smallest and largest values of for any
point in the region in each " = constant" line.
Integrate the function over the tetrahedron with vertices , , , and .
Solution. The least and greatest values of are 0 and 4, so those are our outer limits (see the figure below). For a fixed value of , the least and greatest values of for a point in are 0 and ,
respectively. Finally, for fixed and , the least and greatest values of for a point in are 0 and the point on the plane with the given values of and . So we get
Evaluate by writing it as an integral over a region in the plane and then integrating over the region with respect to the opposite order of integration.
Solution Let's begin by drawing the region:
We can see that this is the region under the graph of from to . Thus we integrate as ranges from 0 to 2 and (for each fixed value of ) as ranges from 0 to . We get
Consider the region between the parabolas and . Find .
Solution. We see that the curves intersect at and . So we get
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The Stopping Time in Simple Terms
Stopping time is a concept related to stochastic processes in mathematics and statistics. In the context of exotic option pricing, stopping time can refer to the decision-making process about when to
exercise the option before its expiration, under conditions that optimize the payoff.
Exotic options have complex features and terms. They might include different types of payoffs or be activated or deactivated under certain conditions or specific events.
Models like Black-Scholes-Merton, which are often used for pricing standard options, are sometimes not sufficient for exotic options. Alternative models, such as Binomial Tree models, Monte Carlo
simulations, and Finite Difference methods, are used to estimate their value. Each exotic option has unique characteristics, and the models need to be adapted accordingly, sometimes involving the
concept of stopping time to optimize the option's exercise strategy to maximize its value.
The pricing of an American option can be approached using a Binomial Tree model. The option's price is computed backward from expiry, and at each node, we compute the option's value based on the
maximum of either exercising the option immediately or holding it for future exercise.
The Immediate Exercise Value for a call option is calculated as the max of 0 and (S - K), and for a put option, it’s the max of 0 and (K - S), where S is the stock price, and K is the strike price.
The Hold Value is derived from the expected future value of the option, discounted back at the risk-free rate. This takes into account the risk neutral probabilities (*) of the stock’s price moving
up or down.
Consider a simple 1-period binomial model. The current stock price is $50. It can either go up to $60 or down to $40 in one period with equal probability. We want to price an American call option
with a strike price of $50.
1. At expiry, the immediate exercise values would be:
• $10 if the stock price rises to $60 (max of 0 and $60 - $50).
• $0 if the stock price falls to $40 (max of 0 and $40 - $50).
2. The hold value at each node is the expected future option value, discounted at a 5% risk-free rate, considering equal probabilities of price movements:
• The hold value is (0.5*$10 + 0.5*$0) / (1+0.05) = $4.76.
3. We then compare the immediate exercise value and hold value:
• With a $50 stock price, immediate exercise yields $0 (max of 0 and $50 - $50).
• The hold value is $4.76.
So, the option’s holder is better off holding the option rather than exercising it immediately, and the option's price at $50 stock price is $4.76.
The stopping time problem here involves determining at which point (if any) before expiry the option holder should exercise the option to maximize their payoff.
(*) Risk-neutral probability is a theoretical probability measure that adjusts the actual probabilities of different outcomes to account for the risk-free rate of return.
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XLI. A map of lensing-induced
Issue A&A
Volume 596, December 2016
Article Number A102
Number of page(s) 19
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/201527932
Published online 12 December 2016
A&A 596, A102 (2016)
Planck intermediate results
XLI. A map of lensing-induced B-modes
^1 APC, AstroParticule et Cosmologie, Université Paris Diderot, CNRS/IN2P3, CEA/lrfu, Observatoire de Paris, Sorbonne Paris Cité, 10 rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France
^2 Aalto University Metsähovi Radio Observatory and Dept of Radio Science and Engineering, PO Box 13000, 00076 Aalto, Finland
^3 African Institute for Mathematical Sciences, 6–8 Melrose Road, Muizenberg, 00040 Cape Town, South Africa
^4 Agenzia Spaziale Italiana Science Data Center, via del Politecnico snc, 00133 Roma, Italy
^5 Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, 13388 Marseille, France
^6 Astrophysics Group, Cavendish Laboratory, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HE, UK
^7 Astrophysics & Cosmology Research Unit, School of Mathematics, Statistics & Computer Science, University of KwaZulu-Natal, Westville Campus, Private Bag X54001, 4000 Durban, South Africa
^8 CGEE, SCS Qd 9, Lote C, Torre C, 4° andar, Ed. Parque Cidade Corporate, CEP 70308-200 Brasília, DF, Brazil
^9 CITA, University of Toronto, 60 St. George St., Toronto, ON M5S 3H8, Canada
^10 CNRS, IRAP, 9 Av. colonel Roche, BP 44346, 31028 Toulouse Cedex 4, France
^11 CRANN, Trinity College, Dublin, Ireland
^12 California Institute of Technology, Pasadena, California, CA 911098 USA
^13 Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza San Juan, 1, planta 2, 44001 Teruel, Spain
^14 Computational Cosmology Center, Lawrence Berkeley National Laboratory, Berkeley, California, USA
^15 DSM/Irfu/SPP, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
^16 DTU Space, National Space Institute, Technical University of Denmark, Elektrovej 327, 2800 Kgs. Lyngby, Denmark
^17 Département de Physique Théorique, Université de Genève, 24 Quai E. Ansermet, 1211 Genève 4, Switzerland
^18 Departamento de Astrofísica, Universidad de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain
^19 Departamento de Física, Universidad de Oviedo, Avda. Calvo Sotelo s/n, 33007 Oviedo, Spain
^20 Department of Astronomy and Astrophysics, University of Toronto, 50 Saint George Street, Toronto, Ontario, Canada
^21 Department of Astrophysics/IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL Nijmegen, The Netherlands
^22 Department of Physics & Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, British Columbia, Canada
^23 Department of Physics and Astronomy, Dana and David Dornsife College of Letter, Arts and Sciences, University of Southern California, Los Angeles, CA 90089, USA
^24 Department of Physics and Astronomy, University College London, London WC1E 6BT, UK
^25 Department of Physics, Gustaf Hällströmin katu 2a, University of Helsinki, 00560 Helsinki, Finland
^26 Department of Physics, Princeton University, Princeton, New Jersey, NJ 08544, USA
^27 Department of Physics, University of California, One Shields Avenue, Davis, California, CA 93106, USA
^28 Department of Physics, University of California, Santa Barbara, CA 93106 California, USA
^29 Department of Physics, University ofIllinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801 Illinois, USA
^30 Dipartimento di Fisica e Astronomia G. Galilei, Università degli Studi di Padova, via Marzolo 8, 35131 Padova, Italy
^31 Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, via Saragat 1, 44122 Ferrara, Italy
^32 Dipartimento di Fisica, Università La Sapienza, P. le A. Moro 2, 00133 Roma, Italy
^33 Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy
^34 Dipartimento di Matematica, Università di Roma Tor Vergata, via della Ricerca Scientifica, 1, 00133 Roma, Italy
^35 Discovery Center, Niels Bohr Institute, Blegdamsvej 17, 1165 Copenhagen, Denmark
^36 Discovery Center, Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 1165 Copenhagen, Denmark
^37 European Space Agency, ESAC, Planck Science Office, Camino bajo del Castillo, s/n, Urbanización Villafranca del Castillo, 28691 Villanueva de la Cañada, Madrid, Spain
^38 European Space Agency, ESTEC, Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
^39 Facoltà di Ingegneria, Università degli Studi e-Campus, via Isimbardi 10, 22060 Novedrate (CO), Italy
^40 Gran Sasso Science Institute, INFN, viale F. Crispi 7, 67100 L’ Aquila, Italy
^41 HGSFP and University of Heidelberg, Theoretical Physics Department, Philosophenweg 16, 69120 Heidelberg, Germany
^42 Helsinki Institute of Physics, Gustaf Hällströmin katu 2, University of Helsinki, Helsinki, Finland
^43 INAF–Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35131 Padova, Italy
^44 INAF–Osservatorio Astronomico di Roma, via di Frascati 33, 00040 Monte Porzio Catone, Italy
^45 INAF–Osservatorio Astronomico di Trieste, via G.B. Tiepolo 11, Trieste, Italy
^46 INAF/IASF Bologna, via Gobetti 101, 40126 Bologna, Italy
^47 INAF/IASF Milano, via E. Bassini 15, 20133 Milano, Italy
^48 INFN, Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy
^49 INFN, Sezione di Roma 1, Università di Roma Sapienza, Piazzale Aldo Moro 2, 00185 Roma, Italy
^50 INFN, Sezione di Roma 2, Università di Roma Tor Vergata, via della Ricerca Scientifica, 1, 00133 Roma, Italy
^51 IUCAA, Post Bag 4, Ganeshkhind, Pune University Campus, 411 007 Pune, India
^52 Imperial College London, Astrophysics group, Blackett Laboratory, Prince Consort Road, London, SW7 2AZ, UK
^53 Infrared Processing and Analysis Center, California Institute of Technology, Pasadena, CA 91125, USA
^54 Institut d’Astrophysique Spatiale, CNRS (UMR8617) Université Paris-Sud 11, Bâtiment 121, 91898 Orsay, France
^55 Institut d’Astrophysique de Paris, CNRS (UMR7095), 98bis Boulevard Arago, 75014 Paris, France
^56 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
^57 Institute of Theoretical Astrophysics, University of Oslo, Blindern, 0371 Oslo, Norway
^58 Instituto de Astrofísica de Canarias, C/Vía Láctea s/n, La Laguna, 38205 Tenerife, Spain
^59 Instituto de Física de Cantabria (CSIC-Universidad de Cantabria), Avda. de los Castros s/n, 39005 Santander, Spain
^60 Istituto Nazionale di Fisica Nucleare, Sezione di Padova, via Marzolo 8, 35131 Padova, Italy
^61 Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, USA
^62 Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester, M13 9PL, UK
^63 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
^64 Kavli Institute for Cosmology Cambridge, Madingley Road, Cambridge, CB3 0HA, UK
^65 Kazan Federal University, 18 Kremlyovskaya St., 420008 Kazan, Russia
^66 LAL, Université Paris-Sud, CNRS/IN2P3, 91898 Orsay, France
^67 LERMA, CNRS, Observatoire de Paris, 61 Avenue de l’Observatoire, 75000 Paris, France
^68 Laboratoire AIM, IRFU/Service d’Astrophysique – CEA/DSM – CNRS – Université Paris Diderot, Bât. 709, CEA-Saclay, 91191 Gif-sur-Yvette Cedex, France
^69 Laboratoire Traitement et Communication de l’Information, CNRS (UMR 5141) and Télécom ParisTech, 46 rue Barrault 75634 Paris Cedex 13, France
^70 Laboratoire de Physique Subatomique et Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3, 53 rue des Martyrs, 38026 Grenoble Cedex, France
^71 Laboratoire de Physique Théorique, Université Paris-Sud 11 & CNRS, Bâtiment 210, 91405 Orsay, France
^72 Lawrence Berkeley National Laboratory, Berkeley, California CA 94720, USA
^73 Lebedev Physical Institute of the Russian Academy of Sciences, Astro Space Centre, 84/32 Profsoyuznaya st., 117997 Moscow, GSP-7, Russia
^74 Max-Planck-Institut für Astrophysik, Karl-Schwarzschild-Str. 1, 85741 Garching, Germany
^75 National University of Ireland, Department of Experimental Physics, Maynooth, Co. Kildare, Ireland
^76 Nicolaus Copernicus Astronomical Center, Bartycka 18, 00-716 Warsaw, Poland
^77 Niels Bohr Institute, Blegdamsvej 17, 1165 Copenhagen, Denmark
^78 Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, 1165 Copenhagen, Denmark
^79 Nordita (Nordic Institute for Theoretical Physics), Roslagstullsbacken 23, 106 91 Stockholm, Sweden
^80 Optical Science Laboratory, University College London, Gower Street, London, UK
^81 SISSA, Astrophysics Sector, via Bonomea 265, 34136 Trieste, Italy
^82 School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK
^83 School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK
^84 Sorbonne Université-UPMC, UMR7095, Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France
^85 Space Research Institute (IKI), Russian Academy of Sciences, Profsoyuznaya Str, 84/32, 117997 Moscow, Russia
^86 Space Sciences Laboratory, University of California, Berkeley, CA 94720 California, USA
^87 Special Astrophysical Observatory, Russian Academy of Sciences, Nizhnij Arkhyz, Zelenchukskiy region, 369167 Karachai-Cherkessian Republic, Russia
^88 Sub-Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
^89 The Oskar Klein Centre for Cosmoparticle Physics, Department of Physics, Stockholm University, AlbaNova, 106 91 Stockholm, Sweden
^90 UPMC Univ Paris 06, UMR7095, 98bis Boulevard Arago, 75014 Paris, France
^91 Université de Toulouse, UPS-OMP, IRAP, 31028 Toulouse Cedex 4, France
^92 University of Granada, Departamento de Física Teórica y del Cosmos, Facultad de Ciencias, 18071 Granada, Spain
^93 University of Granada, Instituto Carlos I de Física Teórica y Computacional, 18071 Granada, Spain
^94 Warsaw University Observatory, Aleje Ujazdowskie 4, 00-478 Warszawa, Poland
^⋆ Corresponding author: L. Perotto, e-mail: laurence.perotto@lpsc.in2p3.fr
Received: 9 December 2015
Accepted: 7 September 2016
The secondary cosmic microwave background (CMB) B-modes stem from the post-decoupling distortion of the polarization E-modes due to the gravitational lensing effect of large-scale structures. These
lensing-induced B-modes constitute both a valuable probe of the dark matter distribution and an important contaminant for the extraction of the primary CMB B-modes from inflation. Planck provides
accurate nearly all-sky measurements of both the polarization E-modes and the integrated mass distribution via the reconstruction of the CMB lensing potential. By combining these two data products,
we have produced an all-sky template map of the lensing-induced B-modes using a real-space algorithm that minimizes the impact of sky masks. The cross-correlation of this template with an observed
(primordial and secondary) B-mode map can be used to measure the lensing B-mode power spectrum at multipoles up to 2000. In particular, when cross-correlating with the B-mode contribution directly
derived from the Planck polarization maps, we obtain lensing-induced B-mode power spectrum measurement at a significance level of 12σ, which agrees with the theoretical expectation derived from the
Planck best-fit Λ cold dark matter model. This unique nearly all-sky secondary B-mode template, which includes the lensing-induced information from intermediate to small (10 ≲ ℓ ≲ 1000) angular
scales, is delivered as part of the Planck 2015 public data release. It will be particularly useful for experiments searching for primordial B-modes, such as BICEP2/Keck Array or LiteBIRD, since it
will enable an estimate to be made of the lensing-induced contribution to the measured total CMB B-modes.
Key words: cosmology: observations / cosmic background radiation / polarization / gravitational lensing: weak
© ESO, 2016
1. Introduction
Cosmic microwave background (CMB) polarization anisotropies can be decomposed into curl-free E-modes and gradient-free B-modes. In contrast to primordial E-modes, primordial B-modes are sourced only
by tensor perturbations (Polnarev 1985; Spergel & Zaldarriaga 1997; Kamionkowski et al. 1997; Seljak & Zaldarriaga 1997) that can be formed in the pre-decoupling Universe due to an early inflationary
phase (Grishchuk 1975; Starobinsky 1979, 1982). Thus, primordial B-modes of the CMB polarization are a direct probe of cosmological inflation (see Guth 1981; Linde 1982, for details on inflationary
theory). The measurement of the primordial B-mode power spectrum, which peaks at degree angular scales, is the main target of a plethora of ground-based experiments and satellite proposals. There was
great excitement in early 2014 when B-modes at the relevant angular scales detected by the BICEP2 experiment were interpreted as evidence of inflationary gravitational waves (Ade et al. 2014a).
Investigating the polarized dust emission in the BICEP2 observation field using the 353-GHz data, Planck^1 revealed a higher dust contamination level than expected from pre-Planck foreground models (
Planck Collaboration Int. XXX 2016). In BICEP2/Keck Array and Planck Collaborations (2015), a joint analysis of the BICEP2/Keck Array data at 100 and 150GHz and the full-mission Planck data
(particularly the 353-GHz polarized data) has been conducted. This provides the state-of-the-art constraints on the tensor-to-scalar ratio, r, which is currently consistent with no detection of a
primordial B-mode signal. When combined with the limit derived from the temperature data (as discussed in Planck Collaboration XVI 2014 and Planck Collaboration XIII 2016), the current 95% upper
limit is r< 0.08, which already rules out some of the simplest inflationary potentials (Planck Collaboration XX 2016). We stand at the threshold of a particularly exciting epoch that is marked by
several ongoing or near-future ground-based experimental efforts, based on technology that is sensitive and mature enough to probe the primordial B-modes to theoretically interesting levels.
In addition to the primordial contribution, a secondary contribution is expected from the post-decoupling distortion of the CMB polarization due to the effect of gravitational lensing (see Lewis &
Challinor 2006, for a review of CMB lensing). In particular, the lensing of the primordial CMB polarization E-modes leads to an additional B-mode contribution. The secondary B-mode contribution to
the $CℓBB$ power spectrum dominates over the primary one at ℓ ≳ 150, even for large values of the tensor-to-scalar ratio, (r ~ 1). Thus, it must be corrected for, in order to measure the imprint of
primordial tensor modes. This correction is generally referred to as “delensing”.
The secondary B-mode power spectrum can be estimated by cross-correlating the total observed B-mode map with a template constructed by combining a tracer of the gravitational potential and an
estimate of the primordial E-modes. Using such a cross-correlation approach, the SPTpol team (Hanson et al. 2013) reported the first estimate of the lensing B-mode power spectrum, consisting of a
roughly ≲8σ measurement in the multipole range 300 <ℓ < 2750 using Herschel-SPIRE data as the mass tracer.The POLARBEAR collaboration detected the lensing B-modes using CMB polarization data by
fitting an amplitude relative to the theoretical expectations to their CMB polarization trispectrum measurements, and reported a 4.2σ rejection of the null-hypothesis (Ade et al. 2014b). Similarly,
the ACTPol team reported 3.2σ evidence of the lensing B-mode signal within its first season data, using the correlation of the lensing potential estimate and the cosmic infrared background
fluctuations measured by Planck (van Engelen et al. 2015).Finally, using the full mission temperature and polarization data, Planck obtained a template-based cross-correlation measurement of the
lensing B-mode power spectrum that covers the multipole range 100 < ℓ < 2000, at a significance level of approximately 10σ, as described in Planck Collaboration XV (2016).
Secondary B-modes dominate any potential primordial B-modes at high multipoles, thus the high-ℓBB power spectrum of the observed polarization maps can also be used to make a lensing-induced B-mode
measurement.The POLARBEAR collaboration reported the first BB measurement (at around 2σ) of the B-mode power spectrum in the multipole range 500 < ℓ < 2100 (The Polarbear Collaboration 2014). The
SPTpol experiment also made a BB estimate of the lensing B-modes in the range 300 < ℓ < 2300, representing a >4σ detection Keisler et al. (2015). Moreover, a non-zero lensing B-mode signal has been
found in the BICEP2/Keck Array data with around 7σ significance, by fitting a freely-floating CMB lensing amplitude in the joint analysis with Planck data (BICEP2/Keck Array and Planck Collaborations
For current or future experiments targeting the detection of primordial B-modes, a precise estimation of the secondary CMB B-modes at large and intermediate angular scales is required in order to
separate the secondary contributions from potential primordial B-modes.On the one hand, large angular scale experiments lack the high-resolution E-mode measurements that are required to measure the
lensing-induced B-mode signal. On the other hand, for high-resolution experiments, partial sky coverage limits their ability to extract the B-mode power spectrum and to reconstruct the lensing
potential at large angular scales.Thus, such experiments would benefit from a pre-estimated secondary B-mode template, covering angular scales from a few degrees down to sub-degree scales and
matching their sky coverage.
We present an all-sky secondary B-mode template spanning from intermediate to large angular scales, synthesized using the full mission Planck data. In Planck Collaboration XV (2016), the lensing B
-mode estimate was band-limited to ℓ> 100, in order to conservatively alleviate any low-ℓ systematic effects. In contrast, here the focus is on improving the reliability at intermediate angular
scales (10 <ℓ< 200). We also extend the lensing B-mode results of Planck Collaboration XV (2016) by producing a lensing B-mode map, by performing extensive characterization and robustness tests of
this template map and by discussing its utility for B-mode oriented experiments. This B-mode map is delivered as part of the Planck 2015 data release.
The outline of this paper is as follows. Section 2 describes the data and simulations that we use. We detail the methodology for the template synthesis in Sect. 3, and describe the construction of
the mask in Sect. 4. The lensing B-mode template reconstruction method is validated using simulations in Sect. 5. We present the template we have obtained from Planck foreground-cleaned data in Sect.
6, and assess its robustness against foreground contamination and the choice of the data to cross-correlate with in Sect. 7. Section 8 addresses the implications of the template for external
experiments targeting primordial B-mode searches. We summarize and conclude in Sect. 9.
2. Data and simulations
Planck sky maps:we have used foreground-cleaned CMB temperature and polarization maps derived from the Planck satellite full mission frequency channel maps from 30 to 857GHz in temperature and 30 to
353GHz in polarization (Planck Collaboration I 2016; Planck Collaboration II 2016; Planck Collaboration III 2016; Planck Collaboration IV 2016; Planck Collaboration V 2016; Planck Collaboration VI
2016; Planck Collaboration VII 2016; Planck Collaboration VIII 2016). Our main results are based on Stokes I, Q, and U maps constructed using the SMICA component-separation algorithm (Delabrouille et
al. 2003) in temperature and polarization simultaneously (Planck Collaboration IX 2016). The maps are at 5′ resolution in N[side] = 2048HEALPix pixelization (Górski et al. 2005)^2. For the sake of
assessing the robustness of our results, we have also utilized foreground-cleaned maps that are produced using the other Planck component-separation methods, namely Commander, NILC, and SEVEM (Planck
Collaboration XII 2014; Planck Collaboration IX 2016; Planck Collaboration X 2016).The current publicly available Planck HFI polarization maps, which are part of the Planck 2015 data release, are
high-pass filtered at ℓ ≲ 30 because of residual systematic effects on angular scales greater than 10° (Planck Collaboration VII 2016; Planck Collaboration VIII 2016). However, we have used
polarization maps covering all angular scale for our analysis, since the results have proved not to be sensitive to CMB E-mode polarization at large angular scales^3.
Full Focal Plane simulations:for methodological validation and for the bias correction of the lensing potential at the map level (known as the “mean-field” correction), we have relied on the eighth
Planck Full Focal Plane (FFP8 hereafter) suite of simulations, as described in Planck Collaboration XII (2016). Specifically, we have used FFP8 Monte-Carlo realizations of the Stokes I, Q, and U
outputs of the SMICA component-separation method. These have been obtained by processing through the SMICA algorithm the simulated beam-convolved CMB and noise realizations of the nine Planck
frequency channels, as described in Planck Collaboration IX (2016). As a result, both the noise realizations and the beam transfer function are representative of those of the SMICA foreground-cleaned
maps. Finally, the calibration of the Planck 2015 data has been taken into account by rescaling the CMB realizations as in Planck Collaboration XV (2016).
Fiducial cosmology:for normalizing the lensing potential map estimate and computing the filter and transfer functions of the B-mode template, we have used fiducial power spectra derived from the 2015
Planck base ΛCDM cosmological parameters that have been determined from the combination of the 2015 temperature and “lowP” likelihoods, as described in Planck Collaboration XIII (2016).
3. B-mode map reconstruction
3.1. Formalism
The secondary B-modes of CMB polarization arise from a leakage of a fraction of the E-modes into B-modes due to the polarization remapping induced by the CMB lensing effect. In terms of the
polarization spin-two components [± 2]P ≡ Q ± iU, and at first order in the lensing potential φ, the lensing-induced contribution reads $±2Plens(n ˆ)=∇±2Pprim(n ˆ)·∇φ(n ˆ),$(1)where $Pprim±2(n ˆ)$
are the polarization fields that would be observed in the absence of the lensing effect (Zaldarriaga & Seljak 1998; Lewis & Challinor 2006). Rewriting the secondary polarization fields in terms of
the rotationally invariant E- and B-mode fields, so that $±2Plens(n ˆ)=∇[∑ℓ′m′(Eℓ′m′prim±iBℓ′m′prim)±2Yℓ′m′]·∇φ(n ˆ),$(2)and considering their spin-two spherical-harmonic coefficients $±2Pℓmlens=∫dn
ˆ ±2Plens(n ˆ) ±2Yℓm∗(n ˆ),$(3)one finds that the gradient-free B-mode polarization receives a secondary contribution that depends on the primordial B-modes and the unlensed curl-free E-modes, of the
form $Bℓmlens=12i(+2Pℓmlens−-2Pℓmlens).$(4)
3.2. Algorithm
First, we state the assumptions on which our algorithm is based. Since the E-mode amplitude is at least an order of magnitude greater than the primordial B-mode amplitude, the $Bℓmlens$ contribution
that comes from the E-mode remapping largely dominates over the one from the lensing perturbation of the primordial B-modes. From now on, the latter can safely be neglected, consistent with the
assumptions in Planck Collaboration XV (2016). We replace the primordial E-modes that appear in Eq. (2) with the total E-modes, E = E^prim + δE. This amounts to neglecting the second-order
contribution to B^lens due to δE, that is to say the lensing perturbation of the E-modes themselves.
We then consider pure E polarization fields: $±2PE(n ˆ)≡QE(n ˆ)+iUE(n ˆ)≡∑ℓmEℓm ±2Yℓm(n ˆ),$(5)which define pure-E Stokes parameters Q^E, U^E.
Implementing the above assumptions into Eq. (3) and using the definition given in Eq. (5), we build a secondary polarization estimator that has the generic form $±2P̂lensℓm=ℬℓ-1∫dn ˆ ∇±2PE(n ˆ)·∇φ(n
ˆ) ±2Yℓm∗(n ˆ),$(6)where $±2PE$ and $φ$ are the filtered versions of the pure-E polarization and lensing potential fields, respectively, whereas ℬ[ℓ] is a transfer function ensuring that the
estimator is unbiased. These quantities are defined below in Sect. 3.4. We finally define secondary CMB B-mode template Stokes maps $(Qlens(n ˆ),Ulens(n ˆ))$ by preserving the B-mode contribution in
Eq. (6) and transforming back to real space.
In summary, we reconstruct the all-sky B-mode template using a dedicated pipeline that consists of:
• (i)
estimation of the deflection field using the filteredreconstructed gravitational potential,$∇φ(n ˆ)$;
• (ii)
computation of the gradient of the filtered pure E-mode input maps, $∇±2PE(n ˆ)$;
• (iii)
calculation of the analytical transfer function;
• (iv)
construction of the polarization template using Eq. (6);
• (v)
formation of a secondary B-mode template using Eq. (4).
These steps are further detailed in the rest of this section.
3.3. All-sky lensing potential reconstruction
The paths of CMB photons are weakly deflected by the matter encountered along the way from the last-scattering surface. As a result, the primary CMB observables are remapped according to the gradient
of the gravitational potential φ integrated along the line-of-sight (Blanchard & Schneider 1987). This induces higher-order correlations within the CMB observables; namely, a non-vanishing connected
part of the four-point correlation function, or equivalently in the spherical harmonic domain, a trispectrum, in the CMB maps. These can be used, in turn, to reconstruct the intervening mass
distribution (Bernardeau 1997; Zaldarriaga & Seljak 1999). Specifically, to first order in φ, the lensing induces a correlation between the observed (lensed) CMB maps and the gradient of the primary
(unlensed) maps. Building upon this property, quadratic estimators have been proposed to extract a lensing potential estimate from the observed map (Hu 2001b,a; Hu & Okamoto 2002; Okamoto & Hu 2003).
We have reconstructed the lensing potential over a large portion of the sky using the all-sky quadratic estimator described in Okamoto & Hu (2003), which has been modified to deal with cut skies.
3.3.1. Inpainting of the temperature map
In Planck Collaboration XV (2016), foreground-contaminated regions have been masked out at the stage of the inverse-variance filtering of the input CMB maps, by allocating infinite variance to masked
pixels. The reconstructed φ is thus null-valued in the pixels inside the analysis mask. For the sake of synthesizing Stokes maps $(Qlens(n ˆ),Ulens(n ˆ))$, as described in Sect. 3.2, using such a
masked φ estimate would induce prohibitive amounts of mode mixing. Thus, we have used the METIS method, in which the masked CMB maps are restored to a complete sky coverage, before their ingestion
into the quadratic estimator, by means of an inpainting procedure based on the “sparsity” concept (Abrial et al. 2007). More details are given in Perotto et al. (2010), where this method has been
first described and in Planck Collaboration XVII (2014), where it has been used to perform consistency tests. As a result of this procedure, the METIS method provides a φ estimate that is effectively
inpainted to cover the full sky. To illustrate this property, Fig. 1 shows a Wiener-filtered version of the φ estimate reconstructed from the SMICA temperature map using the baseline “L80” lensing
mask that we describe below in Sect. 4.
As well as offering the advantage of mitigating the bias that the mask induces in the φ map, which is discussed in Sect. 3.3.2, this allows us to construct the secondary polarization template map
using Eq. (6), alleviating the need for further processing steps to deal with partial sky coverage.
Fig. 1
Wiener-filtered lensing potential estimated from the SMICA foreground-cleaned temperature map using the f[sky] ≃ 80% lensing mask. The lensing potential estimate, which is shown in Galactic
coordinates, is effectively inpainted using a lensing extraction method that relies on an inpainting of the input temperature map, as discussed in Sect. 3.3.1.
3.3.2. Mean-field debiasing
The quadratic estimator is based on the fact that, for a fixed lens distribution, the lensing breaks the statistical isotropy of the CMB maps; specifically, it introduces off-diagonal terms of the
CMB covariance (Hu & Okamoto 2002). Therefore this is also sensitive to any other source of statistical anisotropies in the maps. For Planck data, the bias induced at the φ map level by any known
sources of statistical anisotropies, which is referred to as the “mean-field bias”, is dominated by the effects of masking (Planck Collaboration XVII 2014). The mean-field bias can be estimated from
Monte-Carlo (MC) simulations that include all the instrumental and observational effects that can lead to a sizeable mean-field (e.g. the mask, the spatial inhomogeneity in the noise, which yields
the first sub-dominant mean field, and the beam asymmetry), by averaging the φ estimates obtained on MC realizations. We have used a set of 100 FFP8 realizations to obtain an estimate $⟨φ̅⟩MCLM$ of
the mean-field modes^4 for the non-normalized φ modes, labelled . This mean-field estimate has then been subtracted from to obtain the unbiased estimate of φ in the spherical harmonic domain, given
by $φ̂LM=𝒜L(φ̅LM−⟨φ̅LM⟩MC),$(7)where is the normalization function, which ensures that the estimator is unbiased. This is related to the normalization $ALα$ given in Okamoto & Hu (2003)^5, via $𝒜L=[L
(L+1)]-1ALα$, and has been analytically calculated using the fiducial cosmology described in Sect. 2. To handle the slight difference between FFP8 input cosmology and the fiducial cosmology
considered here, the mean-field $⟨φ̅⟩MCLM$ has been multiplied by the ratio of the normalization functions derived from the input FFP8 and fiducial cosmological models.
3.3.3. Mass tracer choice
In addition to the lensing potential quadratic estimate, other tracers of the underlying lensing potential could be used to form a lensing-induced B-mode template. The cosmic infrared background
(CIB) emission is of particular interest for this purpose since, in contrast to most of the large-scale structure tracers, which probe only a limited range of redshifts; its redshift distribution has
a broad overlap with the lensing potential kernel (Song et al. 2003). Using the best-fit halo-based model of Planck Collaboration XVIII (2011), Planck Collaboration XVIII (2014) have reported a
roughly 80% correlation of the CIB fluctuations measured by Planck with the lensing potential, in agreement with the model expectations. For the sake of the lensing B-mode measurement, however, the
uncertainties in the CIB modelling will have a large impact on the signal estimate, unless the CIB model parameters are marginalized within a joint analysis including lensing cross- and auto-power
spectrum measurements of the CIB (Sherwin & Schmittfull 2015). Moreover, the foreground residuals are another concern for any lensing B-mode template synthesized from the CIB, as also discussed in
Sherwin & Schmittfull (2015). The CIB signal is the most precisely measured at high frequencies, where the Galactic dust emission is also important. Any Galactic dust residuals in the CIB template
map could be correlated either with the polarized dust residuals or with the intensity-to-polarization leakage of the dust emission in the CMB E- and B-mode maps.
Similarly, lensing potential estimates can also be extracted by means of a quadratic estimator on the Planck polarization maps. This, combined with the temperature-based φ estimate, reduces the power
spectrum of the φ reconstruction noise by roughly 25%. However, for measuring the lensing B-mode power spectrum using a template-based approach, resorting to a polarization-based φ estimate would
require us to correct for a non-negligible Gaussian bias (Planck Collaboration XV 2016; Namikawa & Nagata 2014).
Here we have chosen to employ only the lensing potential estimate to produce a lensing B-mode template that is model-independent and more robust to foreground residuals; and to use the temperature
data only for estimating the lensing potential, which ensures desirable properties for our template, as discussed in Sect. 5. Relying on the independent analysis presented in Planck Collaboration XV
(2016), we show in Sect. 7.4 that these choices induce at worst a marginal increase in the statistical uncertainties of the lensing B-mode measurements.
3.4. Lensing-induced polarization fields
Using the lensing potential estimate discussed above and the observed E-mode map, we have reconstructed the template of the secondary polarization field as in Eq. (6). To reduce uncertainties, we
have used filtered versions of those maps that are defined as $φ(n ˆ)=∫dn ˆ fLφ φ̂LM YLM∗(n ˆ),$(8)and $±2PE(n ˆ)=∫dn ˆ fℓE Eℓm ±2Yℓm∗(n ˆ).$(9)The filter functions for φ and E, $fLφ$ and $fℓE$, are
Wiener filters, based on the optimality arguments developed in Smith et al. (2009a). In Planck Collaboration XV (2016), the φ estimates at L< 8 have been discarded because of instability to the
choice of the method used to correct the mean-field. Consistently, we have filtered the φ estimate to L ≤ 10 in order to conservatively avoid any mean-field related issues. We have used a tapering
filter, labelled f[10], that smoothly goes from zero at L = 5 to unity at L = 15 by means of a power-of-cosine function centred in L = 10. Using this filter, we still preserve 99% of the available
information for measuring the cross-correlation B-mode power spectrum. The filtered pure-E polarization fields $±2PE$ have been directly obtained from the full-sky observed Q and U maps by
transforming into the spin-weighted spherical harmonic basis, filtering the E-modes, and reforming Stokes parameter fields with a null B-mode component. No deconvolution of the beam has been
performed at this stage, which yields further filtering of the E-modes at high multipoles. Our filtering functions are defined by $fLφ=f10CLφ,fidCLφ,fid+NLφ,fℓE=$(10)where $CLφ$ and $CℓE$ are the
fiducial φ and E-mode power spectra; $NLφ$ is the φ reconstruction noise power spectrum calculated following Okamoto & Hu (2003); and $NLE$ is the pixel- and beam-deconvolved power spectrum of the E
-mode noise.
These steps have been performed using the fast spin-weighted spherical harmonic transform capability of the HEALPix library to generate Q^E and U^E and compute their derivatives. Thanks to this
simple implementation, a B-mode template map at a resolution of 5′ can be reconstructed from the φ estimate and the observed Q and U maps in a reasonable amount of computing time, which enables the
use of Monte-Carlo simulations. For example, the computing time is about two minutes using eight cores of the Linux AMD64 machines of the Institut National de Physique Nucléaire et de Physique des
Particules (IN2P3) Computing Center^6.
Using a harmonic approach, as in Hu (2000), we compute the transfer function that appears in Eq. (6) by imposing the condition that the secondary B-mode template satisfies $⟨Bℓm∗B̂lensℓ′m′⟩=δℓℓ′δmm′
CℓB,fid,$(11)where $CℓB,fid$ is the fiducial lensing-induced B-mode power spectrum (r = 0).
In terms of the fiducial power spectra $CℓX,fid$, X = { E,B,φ }, we obtain a transfer function $ℬℓ=1CℓB,fid∑Lℓ′fLφCLφ,fid fℓ′EBℓ′Cℓ′E,fid 2FℓLℓ′,$(12)where B[ℓ′] is the beam function of the
polarization maps, and [2]F[ℓLℓ′] is a geometrical term defined in Hu (2000).
3.5. B-mode template synthesis
Our secondary B-mode template is obtained as in Eq. (4) that is $B̂lensℓm=12i(+2P̂lensℓm−-2P̂lensℓm).$(13)This is computed from $±2P̂lensℓm$, the spin-weighted spherical harmonic transforms of our
real-space secondary polarization estimates $Qlens(±iUlens)$ that are corrected for the transfer function given in Eq. (12).
3.6. Cross-correlation power spectrum of the template
We form the cross-correlation power spectrum between the template B-modes given in Eq. (13) and the observed B-modes, $Bℓmobs$, using $ĈBBlensℓ=1fskyeff(2ℓ+1)∑m=−ℓℓBℓmobs ∗Bℓmlens˜,$(14)where the
asterisk denotes complex conjugation; and $Bℓmlens˜$ is a shorthand notation for the template B-mode harmonic coefficients obtained as in Eq. (12), but using an apodized and masked version of the
real-space secondary polarization estimates $Qlens(±iUlens)$. This apodized mask, the construction of which is described in Sect. 4, leaves an effective available sky fraction $fskyeff$ for analysis.
The BB^lens cross-correlation power spectrum represents an estimate of the lensing B-mode power spectrum. Moreover, this does not require any noise term subtraction, as we verify in Sect. 5.
The BB^lens power spectrum variance is estimated to a good approximation using $σ2(ĈBBlensℓ) = 1fskyeff(2ℓ+1)[(CℓB)2+(ĈBobsℓ)(ĈBlensℓ)],$(15)where $ĈBobsℓ$ is the auto-correlation power spectrum of
the observed B-modes and $ĈBlensℓ$ the one of the template. Equation (15) consists of a Gaussian variance prescription (see e.g. Knox 1995), which is not expected to rigorously apply to the lensing B
-mode power spectrum estimate, because of its “sub-structure” (since it is based on the sum of the TTEB trispectrum of the observed CMB signal). However, for a template map constructed on the Planck
polarization maps, given the polarization noise level, the higher-order terms that enter the variance are sub-dominant, as is further discussed and tested in Sect. 5.
4. Construction of the B-mode mask
In this section, we detail the methodology for constructing the template mask. This has involved first preparing a series of foreground masks targeted at specific foreground emission in temperature.
These foreground masks have been then combined to construct an analysis mask for the lensing potential reconstruction. Finally, the lensing analysis mask has been modified to define a mask for the
template map. These three steps are detailed in Sects. 4.1–4.3 below.
4.1. Foreground masks
Galactic mask
We have masked the regions of the sky that are strongly contaminated by the diffuse Galactic emission, the carbon-monoxide (CO) transition line emission and the extended nearby galaxy emission. This
Galactic mask has been produced following the method described in Planck Collaboration XVI (2014). It includes the diffuse Galactic mask described in Planck Collaboration XII (2014), which is
produced by thresholding a combination of CMB-corrected temperature maps at 30 and 353GHz until a desired sky fraction is preserved. For our baseline analysis, we have used a diffuse Galactic mask
that retains about 80% of the sky. This diffuse Galactic mask discards mainly low latitude regions. To mask the CO lines contamination at intermediate latitudes, we have used the CO mask described in
Appendix A of Planck Collaboration XI (2016), which has been obtained by thresholding a smoothed version of the Type 3 CO map (Planck Collaboration XIII 2014) at 1 K[RJ] km s^-1. Furthermore, we have
removed the emission from the most extended nearby galaxies, including the two Magellanic clouds (LMC, SMC) and M31, by cutting a radius that covers each galaxy in the 857-GHz map, as described in
Planck Collaboration XI (2016). Our Galactic mask, which is the merge of the diffuse Galactic, CO-line, and extended nearby galaxy masks, consists of large cuts that extend over more than two degrees
on the sky, and preserves 79% of the sky.
Extragalactic object masks
We have masked the infrared (IR) and radio point sources that have been detected in the temperature maps in frequency channels from 70 to 353GHz using the Planck compact object catalogues, as well
as the sky areas contaminated by the Sunyaev-Zeldovich (SZ) emission using both Planck SZ cluster catalogues and Compton parameter maps. We started with the individual masks targeted at 100, 143, and
217GHz that have been used in Planck Collaboration XVII (2014). These masks have been produced using the Planck Early Release Compact Source Catalogue (ERSC; Planck Collaboration VII 2011), the
Planck Catalogue of Compact Sources (PCCS; Planck Collaboration XXVIII 2014), and the Planck catalogue of Sunyaev-Zeldovich sources (PSZ; Planck Collaboration XXIX 2014). These have been merged with
the conservative point-source masks produced using the 2015 catalogue (PCCS2; Planck Collaboration XXVI 2016), which are presented in Planck Collaboration XI (2016). The SZ emission hav been further
removed using the template Compton parameter y map for the detected galaxy clusters of the 2015 SZ catalogue (PSZ2; Planck Collaboration XXVII 2016) that is described in Planck Collaboration XXII
(2016). Individual SZ masks have been constructed for 100, 143, and 217GHz separately, by converting the template y-map into CMB temperature using the corresponding conversion factors listed in
Planck Collaboration XXII (2016), and thresholding at 10 μK. Finally, we have masked several small nearby galaxies including M33, M81, M82, M101, and CenA, as described in Planck Collaboration XI
We have combined these individual masks to produce two extragalactic object masks that differ in the maximum cut radius. First, the “extended object mask” is made of holes the largest angular size of
which ranges from two degrees to 30′. This includes the extended SZ clusters and the small nearby galaxies. Second, the “compact object mask” is a collection of small holes of cut radius smaller than
30′, and comprises the detected radio and IR point sources and the point-like SZ clusters. The latter represents a conservative point source mask that includes all the point sources detected above S/
N = 5 in the 100-, 143-, and 217-GHz maps, as well as the point sources detected above S/N = 10 in the adjacent frequencies (including 70 and 353GHz). This has been used in Sect. 7.2 to test the
robustness of our results againts point source residuals in the SMICA map.
4.2. Lensing analysis mask
For the lensing potential reconstruction, we have prepared a composite mask targeted at the SMICA temperature map. This map comes along with a “confidence” mask that defines the sky areas where the
CMB solution is trusted. The SMICA confidence mask, a thorough description of which is given in Planck Collaboration IX (2016), has been produced by thresholding the CMB power map that is obtained in
squaring a band-pass filtered and smoothed version of the SMICA map. Our lensing mask combines the Galactic, extended, and compact object masks after some changes driven by the SMICA confidence mask.
Namely, the LMC and three molecular clouds at medium latitudes have been masked more conservatively in the SMICA mask than in our initial Galactic mask. We have enlarged the latter accordingly. In
the compact object mask, IR and radio source holes that are not also present in the SMICA confidence mask have been discarded. They mainly corresponds to point sources detected in the 353-GHz map,
and are strongly weighted down in the SMICA CMB map. The modified compact object mask removes 0.6% of the sky in addition to the 21.4% Galactic and the 0.4% extended object cuts. The baseline lensing
mask, which consists of the combination of these three masks, retains 77.6% of the sky and is labelled L80.
In Sect. 7.2, however, we test the stability of our results against source contamination by using a mask accounting for more point-like and extended sources. This has been constructed as the union of
the Galactic, extended, and compact objects and SMICA confidence masks. This mask, in which the cut sky fraction due to point sources is as large as 1.5%, preserves 76% of the sky for analysis and is
named L80s.
4.3. Template analysis mask
4.3.1. Methodology for constructing the mask
Given the lensing mask, the construction of the template mask is a trade-off between preserving a large sky fraction for analysis and alleviating the bias induced by the lensing reconstruction on an
incomplete sky. Inpainting the masked temperature map from which the lensing potential is extracted, as described in Sect. 3.3, strongly suppresses this effect. The point-like holes induce negligible
bias in the lensing potential reconstruction provided they are treated by inpainting beforehand (Benoit-Lévy et al. 2013). Therefore, the compact object mask does not need to be propagated to the
template map, as it is verified using Monte-Carlo simulations in Sect. 5. However, larger sky cuts performed by the lensing mask (e.g., those arising from the Galactic and extended object masks) have
to be applied to the B-mode template. Furthermore, some artefacts may be seen near the mask boundaries, which we refer to as the mask leakage, and which arises from the convolution of the signal with
the mask side-lobes when transforming in spherical harmonic space (Plaszczynski et al. 2012). From these considerations, we have tailored a method of constructing the mask template using Monte-Carlo
simulations: starting with the combination of the Galactic and the extended object masks, we progressively enlarged the mask beyond the boundaries, until we observed a negligible residual bias in the
cross-correlation power spectrum of the template. This has been obtained by extending the Galactic mask 3° beyond the boundaries and the extended compact object mask 30′ beyond. In Sect. 5, we check
that using a template mask that are constructed in this fashion allows no significant impact in the BB^lens lensing B-mode power spectrum.
4.3.2. The baseline and test template masks
The baseline mask of the lensing B-mode template map has been constructed from the lensing analysis L80 mask using the method described above. Specifically, the Galactic mask has been enlarged 3°
beyond the boundaries, removing 30% of the sky, and the extended object mask has been also slightly widened by 30′ beyond the boundaries, removing an additional 1.2% of the sky. The baseline template
mask preserves a sky fraction of 69% and is named B70.
In Sect. 7.2, however, we check that consistent results are obtained using either the Galactic mask as it is, without enlargement, or a more conservative Galactic mask that removes 40% of the sky.
The most aggressive template mask, labelled B80, leaves 77% of the sky for the analysis, and the most conservative one, named B60, leaves 58%. We also test against extragalactic contamination through
using the lensing L80s mask for the lensing potential reconstruction and the corresponding template mask constructed using the method described in Sect. 4.3.1 and labelled B70s. In addition to these
masks that are targeted at the temperature map, in Sect. 7.2, we also use a mask tailored for polarization in order to test against polarized foreground contamination. For this purpose, the B70 mask
has been combined with the SMICA confidence mask for polarization, a detailed description of which is given in Planck Collaboration IX (2016). The latter removes regions of the sky where the
polarization power (which is computed by squaring a low-pass filter-smoothed version of the SMICA polarization map) exceeds 5 μK^2. The combined B70 and SMICA confidence mask for polarization is
labelled B70p and retains 68% of the sky.
When computing the lensing B-mode power spectrum, we have employed apodized versions of the masks, using a cosine taper of 2° width for the Galactic mask and 30′ width for the extended object mask.
As a consequence, effective sky fractions available for the power spectrum are 5–10% lower than the sky fraction of the template analysis masks. A list of the template masks discussed here is given
in Table 1, together with the corresponding sky fraction, both with and without apodization.
5. Validation on simulations
We now validate the pipeline described in Sect. 3and test the impact of masking as described in Sect. 4.3 using a Monte-Carlo (MC) approach. We analyse our template by forming the BB^lens power
spectrum given in Eq. (14), which has two objectives:
• (i)
to assess that the template encloses the expected lensing B-modeinformation, thus validating the assumptions on which thesynthesis method relies;
• (ii)
to demonstrate its utility for measuring the secondary B-mode power spectrum.
We use two independent sets of 100 SMICAI, Q, and U simulations, part of the FFP8 MC simulation set described in Sect. 2.
5.1. Full-range power spectrum estimate
We have applied the pipeline described above to the SMICA temperature and Stokes parameter simulations to obtain a set of 100 estimates. For the lensing reconstruction on the temperature simulations,
we have used the baseline L80 lensing mask, and employed the second simulation set for the mean-field bias correction. Then, lensing B-mode power spectrum estimates have been formed by
cross-correlating and the input B-modes as in Eq. (14), and using the apodized version of the B70 mask. We have estimated the lensing B-mode BB^lens band-powers by multiplying Eq. (14) by ℓ(ℓ + 1)/2π
and averaging the multipoles over bins of width Δℓ ≥ 100 to further reduce the pseudo-C[ℓ] multipole mixing.
Figure 2 shows the averaged BB^lens band-powers obtained using the apodized B70 mask. The error bars have been estimated using the standard deviation of 100 estimates of the BB^lens band-powers with
the apodized B70 mask. The averaged BB^lens band-power residuals, i.e. $ΔCbBBlens=∑ℓ∈bCℓBBlens−CℓBB,fid$, are plotted in the lower panel of Fig. 2; this shows a negligible residual bias of ≲ 0.1σ up
to multipoles of 2000. Relative to the input lensing B-mode band-power, this corresponds to less than a percent. To isolate the impact of the inpainting, we have also extracted the lensing B-modes
using the lensing potential reconstructed from the full-sky temperature map, that is without using any mask. For comparison, the band-power residuals are also plotted for full-sky case in Fig. 2.
Fig. 2
Lensing B-mode power spectrum obtained by cross-correlating the B^lens templates with the corresponding fiducial B-mode simulations (top), and residuals with respect to the model (bottom). Top:
averaged BB^lens band-powers using the apodized B70 template mask, with multipole bins of Δℓ ≥ 100 (blue points). The error bars are the standard deviation on the mean of the band-power estimate
set. The dark blue curve is the fiducial B-mode power spectrum of our simulations, which assumes r = 0. Bottom: BB^lens band-power residual with respect to the fiducial model, given in units of the
1σ error of a single realization. For comparison, we also show the band-power residuals obtained without masking, as discussed in Sect. 5.1 (red points). Dashed lines show the ± 1σ range of 100
realizations, indicating the precision level to which we are able to test against bias.
We have further tested that the BB^lens band-powers constitute an unbiased estimate of the fiducial lensing B-modes by fitting an amplitude with respect to the fiducial model. The averaged amplitude
A[Blens] obtained on 100 estimates of the BB^lens band-powers using the B70 mask is 0.989 ± 0.008, where the error has been evaluated using the standard deviation of the A[Blens] estimates normalized
by the square root of the number of realizations. We therefore conclude that our pipeline provides us with an unbiased estimate of A[Blens].
It is worth noting that the choice of extracting the lensing potential from the temperature-only data ensures the absence of Gaussian $NL(0)$-like bias, which must be corrected for in the case of a
lensing extraction using B-mode information. It also gives a strong suppression of any higher-order bias terms.
5.2. Statistical error budget
First, we have quantified the statistical error associated with our lensing B-mode band-power measure via simulations by computing the standard deviation of our MC band-power estimates. Then, these
MC error bars have been compared to semi-analytical errors evaluated in taking the square root of the Gaussian variance given in Eq. (15). The auto-correlation B-mode power spectrum of the SMICA
simulation has been modelled using $CℓBobs=CℓB+Bℓ-2NℓB$, where $CℓB$ and $NℓB$ are the MC input signal and noise power spectra and B[ℓ] is the beam function. For the auto-correlation power spectrum
of the template, we have used the averaged estimate on our MC set. Furthermore, we have used the Δℓ ≥ 100 binning function to average the variance over multipoles. We note that these large multipole
bins contribute to drive our band-power estimates close to a normal distribution by virtue of the central-limit theorem, which in turn, brings the Gaussian variance of Eq. (15) closer to the true
In Fig. 3, we plot the MC error bars and the semi-analytical ones. For comparison purposes, we also show the errors one would have obtained from a BB power spectrum measurement by computing the
auto-power spectrum of the fiducial B-mode map.
Fig. 3
Error budget. We compare the MC derived uncertainties σ[MC] associated with the BB^lens measurements to the semi-analytical errors σ[G] obtained using Eq. (15). Top: averaged BB^lens band-powers
using either σ[MC] (blue) or σ[G] (red) as error estimates. For illustration, we also show the expected sensitivity of the PlanckSMICA CMB polarization map to the BB band-powers (grey boxes), whose
uncertainties are a factor of four larger than those of the BB^lens band-powers using the lensing B-mode template. Bottom: relative difference of σ[G] with respect to σ[MC]. Dashed lines show a 15%
The uncertainties in our BB^lens band-powers are well approximated by the semi-analytical errors at all multipoles. In the lower panel of Fig. 3 we plot the relative difference between the
semi-analytical errors using Eq. (15) and the error estimates obtained by simulations. Using Eq. (15) leads to a 16% underestimate of the error in the first multipole bin and less than 10%
underestimation at higher multipoles. These results validate the use of Eq. (15) to evaluate the template-based B-mode power spectrum uncertainties. We also find that the uncertainties of the BB^lens
power spectrum using the template map are approximately four times lower than those of a total B-mode BB measurement coming directly from the B-mode map.
Gathering the results of our MC analysis, we observe that, when used in cross-correlation with the B-mode map, the lensing B-mode template we compute provides a lensing $CℓB$ measurement, which: (i)
does not rely on any bias subtraction; (ii) has nearly optimal uncertainty; and (iii) has four times lower uncertainty than the BB measurement on the fiducial B-mode map.
6. Planck-derived secondary B-mode template
6.1. Template synthesis
We have produced the template map of the lensing B-modes by applying the pipeline described in Sect. 3 to the foreground-cleaned temperature and polarization maps obtained using the SMICA
component-separation method, as described in Sect. 2. We have first obtained the and Û^lens templates defined in Sect. 3.2, filtered versions of which are plotted in Fig. 4. Specifically, the maps
have been smoothed using a Gaussian beam of 1°FWHM to highlight any low-ℓ systematic effects, such as those due to intensity-to-polarization leakage. This first inspection indicates that these
templates are not affected by any obvious low-ℓ systematic effects. More rigorous tests, however, are performed in Sect. 7.3 using intensity-to-polarization leakage corrected maps.
Fig. 4
SMICA lensing-induced Q and U templates that have been convolved with a Gaussian beam of 60′FWHM to highlight the large angular scales (corresponding to multipoles below 200). No spurious patterns
are observed at large angular scales.
Then we have used the SMICA and Û^lens templates to make our secondary B-mode template, as described in Sect. 3.5. For illustration purposes, we have produced a B-mode template map by inverting
$B̂lensℓm$ back to pixel-space through an inverse spherical harmonic transform. Although our B-mode template contains information in the multipole range 10 <ℓ< 2000, Fig. 5 shows two filtered versions
of the B-mode map to highlight different ranges of angular scale. The high-resolution map (which is simply slightly smoothed using a Gaussian beam of 10 ′FWHM) should show any important foreground
contamination at small angular scales, whereas the low-resolution one (which is smoothed using a 1° Gaussian beam and downgraded to N[side] = 256HEALPix resolution) should reveal any large angular
scale systematic effects. No evident systematic effects are observed in the maps plotted in Fig. 5. In Sect. 7, we further assess the template robustness against various systematic effects by means
of a series of tests at the power spectrum level.
Fig. 5
Planck-derived B-mode template map computed using the SMICA foreground-cleaned CMB maps. For illustration, the map has been convolved with a Gaussian beam of 10′ (upper panel) and 60′ (lower panel)
FWHM. The grey area represents the L80 mask, which was used at the lensing potential reconstruction stage. No obvious foreground residuals are seen in the high-resolution map, nor any obvious
systematic effects in the low-resolution one.
6.2. Variance contributions
In this section, we describe and quantify the various contributions that enter the template map variance. In particular, ranking the sources of variance helps us in quantifying the template variance
dependence in the choice of the lensing potential estimates discussed in Sect. 3.3. This also anticipates the discussion that we develop in Sect. 8 on the utility of the Planck template for other
experiments that are attempting to measure the lensing B-modes, compared to the use of a template that combines the Planck lens reconstruction and the experiment’s E-mode measurement.
Using the auto-power spectrum estimate $ĈBlensℓ$ computed on the apodized masked template, the template variance is $σ2(Blens) = 1fskyeff(2ℓ+1)ĈBlensℓ,$(16)where $fskyeff$ is the effective sky
fraction that is preserved by the apodized B70 mask. This receives three main contributions: cosmic variance of both the CMB E-modes and the lenses; instrumental noise; and lens reconstruction noise.
For the sake of completeness, other sub-dominant contributions could include foreground residuals, higher-order terms and secondary contractions of the CMB trispectrum (such as the so-called N^(1)
bias of the lens reconstruction). Within this model, $CℓBlens$ can be analytically calculated using $CℓB̂lens=ℬℓ-2 ∑Lℓ′(fLφ)2(CLφ,fid+NLφ) (fℓ′EBℓ′)2(Cℓ′E,fid+Nℓ′E) 2FℓLℓ′,$(17)where the notation is
the same as in Eqs. (10) and (12). Although the variance is evaluated using the template power spectrum estimate, Eq. (17) provides us with a useful tool for isolating the relative contributions to
the total variance of the template. It can be decomposed into four terms: the “cosmic variance” contribution, which arises from the product of the φ and E power spectra; the “pure noise”
contribution, which involves the product of both noise spectra; and two cross-terms, namely the “φ-noise primed” contribution $NLφCℓE$ and the “E-noise primed” contribution $CLφNℓE$. In Fig. 6, we
plot the different contributions to the total variance of the template.
Fig. 6
Template map variance budget. The total variance of the template, plotted in dark blue, is split into the contribution of the lens reconstruction noise (orange) and E-mode noise (green)
cross-terms; and the pure noise (yellow) and cosmic variance (light blue) terms defined in Sect. 6.2.
The template variance is dominated by the φ-noise primed cross-term, which contributes an amount of about 60% to the total variance. To further quantify the template variance dependence on the φ
estimate, we have rescaled $NLφ$ by a factor of 0.75 in Eq. (17), which represents a simple way to emulate the precision gain that could be obtained by including the polarization-based φ
reconstruction and a more precise φ estimate at high multipoles (e.g. using the CIB as a mass tracer). This has resulted in a 20% reduction of the template variance. This, in turn, has been
propagated to the variance of the BB^lens measurement using Eq. (15), and has induced a 12% improvement of the signal detection significance, in agreement with the independent analysis reported in
Planck Collaboration XV (2016) and discussed in Sect. 7.4.
The E-mode noise power spectrum, which contributes mainly through the pure noise term, provides less than 35% of the variance up to ℓ = 800. As a consequence, very little leverage is left for other
experiments to improve on the template uncertainties by producing a new template that combines each experiment’s E-mode measurement with the Planck lensing potential reconstruction. In Sect. 8, we
see that the Planck template offers other experiments a useful tool for measuring the lensing B-modes.
7. Robustness tests of our template
We now perform a series of consistency tests to characterize the B-mode template described in Sect. 6, with a view to using it for measuring the lensing $CℓB$ in cross-correlation with observed
polarization maps. In Sect. 7.2, we assess the template robustness against foreground residuals, first in varying the morphology and level of conservatism of the mask, and then in using
foreground-cleaned CMB maps obtained with four independent component-separation algorithms. In Sect. 7.3, we use the Planck polarization maps in individual frequency channels to assess the stability
of the lensing $CℓB$ estimates with respect to the observed map to which our template is correlated. In Sect. 7.4, we discuss the consistency of the baseline lensing B-mode power spectrum presented
here with the independent determination in Planck Collaboration XV (2016), which used various mass tracers. Finally, in Sect. 7.5, the Planck lensing $CℓB$ measurements obtained using the B-mode
template is compared to external measurements.
7.1. Template tests using B-mode band-power estimates
With the aim of testing the B-mode template, we have followed the template-based cross-correlation power spectrum approach that we developed to validate our pipeline via simulations, as described in
Sect. 5.1. The tests that have been performed there also allow for an end-to-end assessment of the entire pipeline, whose main specifications are recalled below. Using FPP8 MC simulations for the
SMICA method, together with the B70 template mask, we found that: (i) unbiased lensing B-mode band-power estimates are obtained by cross-correlating the B-mode template estimate with the input B-mode
map; and (ii) the semi-analytical Gaussian variance given in Eq. (15) provides a good approximation for the uncertainties. To test the consistency of the band-power estimates with theoretical
expectations, we fitted an amplitude with respect to the fiducial $CℓB$ band-powers, and found an average value of 0.989 ± 0.008, which indicates that the lensing B-mode signal is accurately
Here, we have used the B-mode template presented in Sect. 6 in cross-correlation with the SMICA foreground-cleaned polarization maps, and have applied the B70 template mask to obtain a baseline
lensing B-mode band-power estimate, which is compared below to various other estimates. We further check the consistency between the baseline and alternative estimates by comparing the measured
amplitude difference to the expected variance of the difference in Monte-Carlo simulations.
7.2. Robustness against foreground residuals
The robustness of the baseline measurements against the impact of foregrounds is assessed by comparison with alternative estimates that have different residuals. A first sequence of alternative
estimates have been obtained by employing masks of various levels of conservatism with respect to the Galactic and extragalactic foregrounds in temperature and polarization. These are detailed in
Sects. 7.2.1 to 7.2.3. The corresponding fitted A[Blens] amplitudes and differences from the B70 estimate are listed in Table 2. Secondly, we consider measurements obtained with foreground-cleaned
maps using different component-separation methods and give the resulting amplitudes in Sect. 7.2.3.
7.2.1. Galactic contamination tests
We test for residual foreground contamination around the Galactic plane by comparing the B70 lensing B-mode estimate to estimates using the more conservative B60 mask and the more aggressive B80
mask, discussed in Sect. 4. We recall that B80 includes a Galactic mask that preserves 80% of the sky; this Galactic mask has been extended by 3° beyond its boundaries for constructing B70, while B60
includes a larger diffuse Galactic mask that retains about 60% of the sky. We have employed the apodized version of the masks using cosine tapers, as described in Sect. 4. For B80, however, we have
used slightly larger apodization widths, namely 3° for the Galactic mask (instead of 2°) and 1° for the extended compact object mask (instead of 30′).
The lensing $CℓB$ band-power estimates using the B60, B70, and B80 masks are shown in Fig. 7, and the fitted amplitudes are listed in Table 2. Both the B60 and B80 estimates are consistent with the
B70 estimate. In particular, the agreement between the B60 and B70 lensing B-mode estimates indicates that any impact of Galactic foreground residuals lies well below the uncertainties. The
consistency between the B70 and B80 estimates further indicates that: (i) an 80% Galactic cut suffices to avoid any Galactic foreground residuals in the SMICA map; and (ii) any leakage near the
border of the mask (treated by inpainting) has negligible impact.
Table 2
Band-power amplitudes for various analysis masks, which are listed in the first column.
7.2.2. Extragalactic object contamination tests
For extracting the lensing potential φ map estimate that is used for the B-mode template synthesis, we have employed the L80 lensing mask described in Sect. 4.2. This cuts out any point sources
included in the intersection of the conservative compact object mask described in Sect. 4.1 and the SMICA confidence mask. Here we test the stability of the B-mode estimate through the use of a more
conservative point source mask that involves the union of these two masks. Namelly, we have produced an alternative estimate that relies on a φ reconstruction using the L80s lensing mask, which
includes the union of the compact object and the SMICA masks (see Sect. 4.2). In L80s, the sky area that is removed due to the expected point source contamination is increased by a factor of 2.3
compared to the L80 case. The extended object mask is also slightly enlarged by 25% due to areas of radius between 30′ and 120′ located at intermediate Galactic latitudes that are present in the
SMICA confidence mask. The B-mode band-powers have then been estimated using the corresponding B70s mask for the template (which removes 1% more sky fraction than the baseline B70 mask, including
about a hundred additional extragalactic objects). The B70s band-power estimate, which is plotted in Fig. 7, is consistent with the baseline B70 estimate within uncertainties. The fitted amplitude
difference from the B70 estimate, which is given in Table 2, is well within the expected difference estimated from simulations. This validates the choices we made to include the expected point
sources to our baseline mask, and indicates that our analysis is free from any sizeable bias due to extragalactic objects.
7.2.3. Polarized foreground emission tests
We test for the impact of polarized foreground residuals by comparing the B70 estimate to an estimate using the B70p mask targeted at the SMICA polarization maps, as discussed in Sect. 4.3. The B70p
mask discards a sky fraction 2.5% larger than B70, mainly because of extended areas at medium latitude that may be contaminated by polarized foreground residuals. The lensing B-mode band-power
estimates using the B70 and B70p masks are plotted in Fig. 7 and the corresponding amplitude fits are given in Table 2. The B70p estimate well agrees with the B70 estimate. We therefore expect no
significant bias related to polarized foreground residuals.
Fig. 7
Cross-correlation of our B-mode template with the SMICA polarization map using masks of different levels of conservatism. The data points show the lensing $CℓB$ band-powers estimated using masks
that preserve a sky fraction of about 60% (labelled B60), 70% (“B70”), and 80% (“B80”), and that are targeted to the foreground emission in temperature (see Sect. 7.2.1); as well as using the 68%
“B70s” mask, which allows for a more conservative masking of the extragalactic objects (see Sect. 7.2.2), and the 68% polarization-targeted “B70p” mask (see Sect. 7.2.3). The residuals with respect
to the model in units of the 1σ band-power uncertainties are shown in the lower panel. The good agreement of all cases within error-bars provides a robustness test against foregrounds, as discussed
in Sect. 7.2.
From the tests performed so far, the results of which are gathered in Table 2, we have found all the band-power estimates, from the most conservative f[sky] = 0.58 to the most optimistic f[sky] =
0.77 analysis and whatever the morphology of the mask, in good agreement with the baseline estimate. This provides an indication that (i) the B-mode template is robust against any foreground
contamination or any impact of the sky-cut; and (ii) it allows us to obtain reliable lensing $CℓB$ measurements for up to 70% of the sky.
We further test our template robustness against foreground residuals by comparing results using the four Planck component-separation methods, as described in Planck Collaboration IX (2016). We have
focused on the polarized foreground residuals; we have used different foreground-cleaned Stokes parameter maps, but the same temperature map: specifically, the lensing B-mode templates have been
synthesized from the foreground-cleaned Stokes Q and U maps using different codes and a fixed lensing potential extraction obtained from the SMICA temperature map. For this specific test, we have
produced a common mask that combines the B80 mask and the union of the confidence masks provided by each of the component-separation methods described in Planck Collaboration IX (2016). The common
mask retains a sky fraction of 75% for analysis, and its apodized version preserves an effective sky fraction of 62%. Figure 8 shows the resulting lensing B-mode power spectra, obtained through
cross-correlation of the lensing B-mode templates with the input Stokes Q and U maps from the corresponding component-separation method.
Fig. 8
Consistency of our results using the four Planck component-separation algorithms. The data points shown in the upper panel represent the lensing B-mode band-power measurements obtained by
cross-correlating the lensing B-mode template estimates using the Commander (red), NILC (orange), SEVEM (forest green), and SMICA (blue) cleaned polarization map with the corresponding B-mode map.
The residuals with respect to the model in units of the 1σ band-power uncertainties are shown in the lower panel. The consistency of the four estimates strongly indicates the robustness of our
baseline template against polarized foreground residuals, as discussed in Sect. 7.2.3.
The four Stokes Q and U foreground-cleaned solutions lead to consistent lensing B-mode power spectrum measurements (within 1σ) over the entire multipole range probed. We measure fits to the amplitude
with respect to the fiducial model of: $ABlens=1.01±0.11 (Commander);ABlens=0.97±0.09 (NILC);ABlens=1.00±0.10 (SEVEM);ABlens=0.97±0.08 (SMICA).$These correspond to 9.5σ, 11.3σ, 9.7σ, and 11.8σ
detections of the lensing B-modes, respectively. The consistency of the lensing $CℓB$ measurements based on four CMB solutions with different foreground residuals indicates the immunity of our
baseline lensing B-mode template to polarized foreground emission.
7.3. Stability with respect to the observed polarization
So far, we have considered the cross-correlation of the B-mode template synthesized from the Stokes I, Q, and U foreground-cleaned maps, with the same Q, U maps. We now test the cross-correlation of
our baseline template with other CMB foreground-cleaned polarization maps. In particular, we have used foreground and intensity-to-polarization leakage-corrected Planck channel maps at 100, 143, and
217GHz, which also serve to test the robustness of our template against low-ℓ systematic effects. These single channel maps rely on the pre-launch ground-based measurements of the detector
bandpasses to correct for the intensity-to-polarization leakage due to bandpass mismatch between the detectors within a frequency channel, as described in Planck Collaboration VIII (2016). The
polarized emission of the Galactic dust has been corrected for by using the 353-GHz map as a dust template.
In Fig. 9, we compare the baseline BB^lens band-powers from the B70 SMICA analysis to the band-powers obtained by correlating our SMICAB-mode template with the B-mode maps at 100, 143, and 217 GHz
built out of the corresponding single channel foreground-cleaned polarization maps and using the same B70 mask. The band-power estimates are in good agreement with each other within uncertainties,
indicating the robustness of our template against polarization systematic effects (that mainly affect the low multipoles).
Fig. 9
Stability to the change in the observed polarization map. Data points shown in the upper panel are the BB^lens band-powers obtained from the cross-correlation of our SMICAB-mode template with the
SMICA polarization map (red), and with single frequency polarization maps at 100GHz (yellow), 143GHz (green), and 217GHz (blue), which were corrected for foreground and intensity-to-polarization
leakage. Residuals with respect to the model in units of the 1σ band-power uncertainties are shown in the lower panel.
We find that our lensing B-mode template provides stable measurements of the CMB lensing B-modes independent of the choice of the polarization maps with which it is correlated.
7.4. Consistency with previous Planck results
First of all, we have checked that the baseline lensing B-mode power spectrum presented here is consistent with the independent determination of Planck Collaboration XV (2016) over the 100 < ℓ < 2000
multipole range, which provides us with further validation of our methodology choices^7.
In Planck Collaboration XV (2016), lensing-induced B-mode power spectrum measurements have been obtained using various mass tracers: lensing potential reconstructions using either both temperature
and polarization data or temperature data only; and CIB fluctuations measured by Planck in the 545-GHz channel. The latter have relied on a model of the CIB emission for calculating its
cross-correlation with the lensing potential. All cases have been consistent with theoretical expectations and in good agreement with each other, which validates the stability of the measurement with
respect to the mass tracer choice. The additional information brought by the polarization-based φ estimates yields a 10% improvement of the lensing B-mode detection significance compared to the case
using the temperature-based φ estimate, whereas the use of the CIB fluctuations as a mass tracer lowers the detection significance by roughly the same amount. Thus, including the polarization
information for reconstructing φ or using the CIB as a mass tracer would not have substantially improved the uncertainties of the B-mode template that we have produced.
7.5. Consistency with external results
Our BB^lens band-power estimates are a measurement of the lensing B-mode power spectrum that we now compare to measurements reported by other experiments.
As stated in Sect. 1, the available B-mode measurements come in two flavours: the BB power spectrum of the observed polarization maps measures the total B-mode signal; whereas the BB^lens power
spectrum between a template and the observed polarization maps probes the secondary contribution only.
In Fig. 10, we gather the composite $CℓB$ measurements obtained by BICEP2/Keck Array, POLARBEAR, SPTpol, and the Planck mission, which represents the landscape of current CMB B-mode band-power
estimates. The BB power spectrum measurements of BICEP2/Keck Array^8, POLARBEAR (The Polarbear Collaboration 2014) and SPTpol (Keisler et al. 2015) are collected by the brace labelled BB in Fig. 10.
For BICEP2/Keck Array, we plot the B-mode band-powers in the multipole range 20 <ℓ< 335, obtained from the combination of the BICEP2 and Keck Array maps reported in BICEP2 and Keck Array
Collaborations (2015a) and corrected for polarized dust emission using Planck data, as described in BICEP2/Keck Array and Planck Collaborations (2015, which is referred to as BKP hereafter). The BB^
lens measurements of SPTpol (Hanson et al. 2013) and Planck (this analysis) are gathered by the brace labelled BB^lens.For PlanckBB^lens band-powers, we select the B70 analysis whose robustness has
been assessed in Sect. 7.2. The 4.2σ lensing B-mode detection reported by the POLARBEAR team (Ade et al. 2014b) and the 3.2σ detection obtained by ACTpol (van Engelen et al. 2015) are not shown in
Fig. 10 for clarity.
Fig. 10
Consistency with external B-mode power spectrum measurements on the full multipole range (top) and at ℓ< 350 (bottom). We compare our baseline BB^lens estimate (red points) to the SPTpol
template-based results (green points; Hanson et al. 2013); and to the BB power spectrum measurements from BICEP2/Keck Array (blue boxes; BICEP2 and Keck Array Collaborations 2015a), POLARBEAR
(light blue boxes; The Polarbear Collaboration 2014), and SPTpol (yellow boxes; Keisler et al. 2015), as discussed in Sect. 7.5. The black line shows the theoretical lensing B-mode power spectrum
for the base ΛCDM best-fit Planck model (with r = 0).
Our Planck-derived BB^lens band-powers are in good agreement with other B-mode measurements, using both the BB or the BB^lens power spectrum methods. Planck provides the most precise measurement of
the lensing-induced B-mode power spectrum to date (as assessed in Planck Collaboration XV 2016). Covering a wide multipole range 10 <ℓ< 2000, our band-powers consists of lensing $CℓB$ measurements at
a significance level as high as 12σ. It is also worth focusing on the low-ℓ range. The $CℓB$ measurements are expanded at ℓ > 350 in the lower panel of Fig. 10, which illustrates the lensing B-mode
template utility for the primordial-to-secondary B-mode discrimination. Using Planck data alone, which combines wide sky coverage, necessary to probe low multipoles, and a good angular resolution,
needed for the lensing potential extraction, the lensing B-mode template enables us to extend the measure of the lensing-induced $CℓB$ into the low multipole range.
8. Implications for current and future B-mode experiments
Here, we address the implications of the template for experiments targeting primordial B-modes. A full joint analysis using external data is beyond the scope of this paper. However, as examples we
discuss two different aspects. First, we address to what extent the template can help experiments in measuring the lensing B-mode power spectrum, in particular over the largest accessible angular
scales. Secondly, we forecast the improvement of the lensing amplitude measurement when using the Planck lensing B-mode template, and we discuss whether this improvement can translate into a better
sensitivity to the tensor-to-scalar ratio.
8.1. Measurement of the lensing B-mode power spectrum
Since the Planck lensing B-mode template covers almost the entire sky, polarization measurements from any B-mode experiment can be cross-correlated with the template. This ensures a valuable lensing
B-mode power spectrum measurement, including the intermediate angular scales, where the lensing signal is of the same order of magnitude or sub-dominant with respect to the polarized foreground
emission (see Planck Collaboration Int. XXX 2016, hereafter referred to as PIP-XXX).
For current ground-based experiments, the cross-correlation with the Planck template is a promising method for measuring the lensing B-mode signal at the largest accessible angular scales. In
general, it leads to better results than a cross-correlation with a lensing B-mode template built out of the Planck lensing potential and each experiment’s E-mode map. This can be understood from two
different considerations. On the one hand, a study of the lensing B-mode signal kernel (as in Fabbian & Stompor 2013; Simard et al. 2015) reveals that most of the signal at low and intermediate
angular scales comes from products of the lensing potential power at low multipoles and the E-mode power at higher multipoles. For example, at ℓ = 100 (200), 80% (90%) of the lensing B-mode power
arises from E-mode power at ℓ > 335. For degree-scale experiments, such as BICEP, it would be of little value to use its own E-mode data to generate a new lensing B-mode template. On the other hand,
in Sect. 6.2, we have found the B-mode template variance to be dominated by the contribution arising from the noise power spectrum of the reconstructed lensing potential $NLφ$, even for the PlanckE
-mode noise. For high-resolution experiments covering a small fraction of the sky, we do not expect that a better sensitivity in the E-modes compensates for the uncertainties linked to the small sky
coverage. We have quantified this effect in using Eqs. (16) and (17) to forecast the template uncertainties, specifically for an ideal experiment providing noiseless E-mode measurements for
multipoles from 30 to 2000 with a resolution of 1′ and a sky coverage of 1%. We find the uncertainties of the ideal experiment’s E-mode-based template to be about five times larger than the Planck
template uncertainties up to multipoles of 700 (and still 50% larger than those of the Planck template at ℓ = 2000). We conclude that, for experiments covering less than a percent of the sky, such as
SPTpol or POLARBEAR, the synthesis of a lensing B-mode template that combines the experiment’s E-mode data with Planckφ estimate would degrade the signal-to-noise for measuring the lensing B-mode
power spectrum.
8.1.1. Uncertainty forecasting method
We forecast the uncertainties of the lensing B-mode power spectrum measurement that current experiments can obtain from cross-correlating their B-mode signal with the Planck lensing B-mode template.
As in Sect. 7.5, we consider the BICEP2/Keck Array, POLARBEAR, and SPTpol examples. Error bars are evaluated using Eq. (15). They include lensed cosmic variance from the best-fit ΛCDM model and the
statistical error from the template (the template auto-power spectrum $ĈBlensℓ$ factor) and from the experiment (the experiment auto-spectrum $(CℓB+NℓB)$ factor). To estimate the template variance
within the experiment’s field, $ĈBlensℓ$ is analytically calculated using the lensing potential and E-mode noise power spectra, rescaled to the noise spatial inhomogeneity within the experiment’s
field. This rescaling relies on the SMICA hit count map. We use a simplified model for the BB auto-spectrum of the experiment that includes the lensed CMB B-modes, polarized dust, white noise, and
Gaussian beam. The specific model and analysis choices are detailed below.
The polarized dust emission has been parametrized by a single free amplitude in power, A[dust], which is defined at the reference frequency of 353GHz and at a multipole of ℓ = 80. Following PIP-XXX,
the dust power spectrum has been modelled as the power law $Cℓdust∝ℓ-2.42$, with a spatially uniform frequency scaling according to a modified black-body law, assuming a fixed dust temperature T[d] =
19.6 K and spectral index β[d] = 1.6. For the BICEP2/Keck Array combination, we have fixed the dust amplitude to the best-fit value obtained in the joint BICEP2/Keck Array and Planck analysis
described in BKP, namely A[dust] = 3 μK^2. In PIP-XXX, individual dust amplitudes have been fitted for a series of sky patches, and their scaling with the mean 353-GHz dust intensity was described
using an empirical relation. However, there was a warning about the difficulty in deriving precise dust amplitude estimates from this empirical law. From Fig. 7 of PIP-XXX, we can see that, at the
low dust intensity values expected in the SPTpol and POLARBEAR fields, the amplitudes range from 0.1 to 10 μK^2. For simplicity, we have assumed the level of dust emission in the POLARBEAR and SPTpol
fields to be the same as in the BICEP2 field, and so have used the value A[dust] = 3 μK^2. We checked that assuming the more pessimistic A[dust] = 10 μK^2 leads to a minor increase in the forecast-ed
BB^lens band-power uncertainties.
The BICEP2/Keck Array combination has been modelled as reaching a noise level of 3.4 μK arcmin over 400 deg^2, and as having a Gaussian beam of 31′FWHM for both BICEP2 and Keck Array experiments (
BICEP2 and Keck Array Collaborations 2015b,a). POLARBEAR has been modelled as reaching a depth of 8 μK arcmin over an effective sky area of 25 deg^2 (f[sky] = 0.06%) with resolution at 150GHz (The
Polarbear Collaboration 2014). For SPTpol, we have modelled the noise spectrum from the characteristics reported in Hanson et al. (2013), by considering the combination of the 95-GHz observation (
resolution and 25 μK arcmin noise level) and the 150-GHz observation ( resolution and 10 μK arcmin noise level) over a sky area of 100 deg^2 (f[sky] = 0.24%). We have used the same multipole binning
as chosen by each experiment in existing publications, and have added a low-ℓ bin including multipoles up to the largest accessible angular scales defined by the experiment’s sky coverage. For
POLARBEAR and SPTpol, we have considered the 100 < ℓ < 2000 multipole range.
8.1.2. Lensing B-mode band-power forecasts
Fig. 11
Forecasts of lensing B-mode band-powers from cross-correlating the Planck lensing B-mode template with external data from B-mode targeted experiments. Blue circles show the BICEP2/Keck Array
band-powers forecasts using the same multipole binning as that of BICEP2/Keck Array data points shown in Fig. 10. BB^lens band-power measurements can be extended to an additional ℓ = 100–500 bin
using POLARBEAR (turquoise) and to an additional ℓ = 100–300 bin using the SPTpol (dark orange) in cross-correlation with the template.
Figure 11 shows the forecast-ed BB^lens band-powers over multipoles up to 500, for which the Planck template is the most useful in helping other experiments to measure the lensing B-mode power
spectrum. We find that the BICEP2/Keck Array could be used in combination with the Planck template to obtain BB^lens band-powers in the multipole range ℓ = 20–335 measured at a significance level of
about 6σ (sensitivity to A[Blens] of 0.17).Therefore, our forecast-ed sensitivity to the lensing B-mode signal is comparable to that obtained in the BKP analysis, where a lensing amplitude of A[lens]
= 1.13 ± 0.18 has been measured. However, we show in Sect. 8.3 that we could improve the sensitivity to the lensing B-mode signal by adding the B-mode template to a BICEP2/Keck Array and Planck joint
analysis. The Planck template could enables sub-degree angular scale experiments, such as BICEP2/Keck Array, to probe the ℓ-dependence of the lensing B-mode signal over their full multipole range,
including multipoles at which the lensing-induced signal is sub-dominant compared to other sources of B-mode signal (such as polarized dust). Moreover, the cross-correlation with the template should
allow experiments targeting higher multipoles (such as POLARBEAR or SPTpol) to measure the lensing B-mode signal at intermediate angular scales (100 < ℓ < 300), extending their BB^lens estimates down
to as low multipoles as their sky coverage would permit.
8.2. Direct delensing capabilities
“Delensing” consists of subtracting the lensing B-mode template from the B-mode map of an experiment in order to try to highlight the primordial B-modes. Because of the noise level of the template,
poor efficiency is expected from a direct delensing approach (e.g. Marian & Bernstein 2007). We have quantified the expected impact on the tensor-to-scalar ratio uncertainty in calculating the
improvement factor as defined in Smith et al. (2009b) as $α=CℓB,lens+NℓBCℓB,res+NℓB·$(18)Here $CℓB,lens$ and $CℓB,res$ are the lensing B-mode power spectrum, and its residual after subtraction of the
template from the external data, while $NℓB$ is the noise power spectrum of the experiment. We find a maximum improvement factor (corresponding to $NℓB=0$) of 5% at ℓ < 200, reaching a maximum of 10%
around ℓ = 300, in agreement with the expectations derived in Smith et al. (2009b).
8.3. Improvement of the parameter accuracies
We now go beyond the lensing B-mode power spectrum measurement and quantify whether the use of the template as an additional data set (together with other B-mode data and the Planck dust template)
can tighten cosmological parameter constraints, and in particular, the amplitude of the lensing potential power spectrum that scales the lensing-induced B-modes, A[lens]. We discuss whether more
precise measurements of the lensing scaling translate into tightened constraints on the tensor-to-scalar ratio.
We derive forecasts for particular experiments by performing a Fisher analysis^9 (for a description of this method, see e.g. Tegmark et al. 1997). We consider a three-parameter model, { r,A[lens],A
[dust] }, consisting of the tensor-to-scalar ratio and the amplitudes of the lensing potential and polarized dust power spectra. We compare the parameter constraints obtained in two cases: (1) the
data sets consist of B^exp, the external data from the B-mode targeted experiment, and , the PlanckB-mode data at 353GHz, considered as a polarized dust template; and (2) the Planck lensing B-mode
template, , is used in combination with the other two data sets. As examples of external experiments, we consider the BICEP2/Keck Array combination, hereafter referred to as BK, described in BICEP2
and Keck Array Collaborations (2015a); and a wide sky coverage experiment, such as LiteBIRD (Matsumura et al. 2014). The rationale driving these choices is twofold. Because of the lensing B-mode
template noise level in a 1% sky area and from the results obtained above, we do not expect the inclusion of the template to bring a large improvement of the sensitivity to r for BK. However, we will
be able to validate our simple fiducial analysis by comparing “Case 1” to the floating lensing amplitude analysis described in BKP. By contrast, a more substantial improvement is expected for a large
sky coverage B-mode experiment such as LiteBIRD. For definiteness, we consider two different fiducial cosmologies in what follows, which consist of a r = 0.05 model and r = 10^-4 model.
8.3.1. Fisher analysis
We consider the Fisher information matrix of the form (see e.g. Tegmark et al. 1997) $Fij=∑ℓ12(2ℓ+1)fsky Tr[CC,iCC,j]$(19)for a fiducial data covariance matrix C and its derivatives with respect to
the parameters labelled by i and j. Given the data set ${Bℓmexp,B̂dustℓm,B̂lensℓm}$, we need to model the B-mode auto-power spectra for the external data, $Cℓexp$, for the Planck 353-GHz map, $Cℓ353$,
and for the Planck lensing B-mode template within the experiment’s sky coverage, $CℓB̂lens$, as well as the corresponding cross-power spectra.
The fiducial model consists of an extension of the six-parameter ΛCDM model considered so far, including primordial gravitational waves (GW) of amplitude r, a freely floating amplitude of the lensing
potential power spectrum, A[lens], and a free polarized dust amplitude in power A[dust] (defined at the reference frequency of 353GHz and at a multipole of ℓ = 80). In this fiducial framework, the B
-mode auto-power spectrum of the experiment under consideration is $Cℓexp=rrfidCℓGW,r=rfid+AlensCℓlens+αAdustR(ν,353)2Cℓdust+Nℓexp,$(20)where $CℓGW,r=rfid$ and $Cℓlens$ are the gravitational wave and
lensing power spectra at the fiducial r = r[fid] and A[lens] = 1 values. We have considered two different fiducial values for r, either r[fid] = 0.05 or r[fid] = 10^-4. The dust power spectrum,
$Cℓdust$, and its frequency scaling with the reference 353-GHz frequency, R(ν,353) have been modelled as in Sect. 8.1, following PIP-XXX. In addition, for multi-band experiments with
foreground-cleaning capabilities, the dust power spectrum has been assumed to be cleaned up to a residual level defined by the α factor (α = 1 for a single frequency experiment). Finally, $Nℓexp$ is
the B-mode noise power spectrum of each experiment.
The Planck 353-GHz B-mode auto-power spectrum in each experiment’s sky coverage has been modelled as $Cℓ353=AdustCℓdust+Nℓ353,$(21)where the full-sky noise power spectrum at 353GHz has been scaled
to take the spatial inhomogeneity within the experimental field into account. We have neglected the sub-dominant CMB B-mode polarization signal.
Finally, the Planck lensing B-mode template auto-power spectrum within the experimental field has been analytically calculated using Eq. (17), in which the noise power spectra of the lensing
potential $NLφ$ and of the E-mode $NℓE$ have been scaled to deal with the spatial inhomogeneity within the experimental field. The cross-correlation of the experiment’s data with the 353-GHz dust
template and lensing B-mode template are $(α)0.5AdustR(ν,353)Cℓdust$ and $Cℓlens$. Following the assumptions of the model, we have neglected the sub-dominant cross-correlation of the lensing B-mode
and 353-GHz dust templates.
8.3.2. Examples of BK and LiteBIRD
For existing data, we have used a data-driven model. Both the BK and the Planck 353-GHz noise power spectra in the BK field have been extrapolated from the noise band-powers released along with the
BKP likelihood^10. For the future project LiteBIRD, we have modelled the noise power spectrum from the foreseen instrumental characteristics defined in Matsumura et al. (2014). Only the 100- and
140-GHz bands have been considered, the two lowest and two highest bands being discarded, assuming that they are used for foreground cleaning. Following Matsumura et al. (2014), the 100-GHz band
reaches a depth of 3.7 μK arcmin and a resolution defined by a Gaussian beam of 45′FWHM, while the 140-GHz band has a noise level of 4.7 μK arcmin and 32′FWHM. The polarized dust in each frequency
band has been assumed to be cleaned over an effective sky area of 63% up to the same residual level as in the BK field. This corresponds to a mildly conservative (Dunkley et al. 2009) 17% residual
level in the map domain (corresponding to α = 2.9% in power) for the dust amplitude A[dust] = 104.5 μK^2, obtained in PIP-XXX via a power-law fit in the large retained science region “LR63”.
With the fiducial model established and using the numerical analysis method described, we can infer the sensitivity to the parameters as $Δθi=(F-1)ii$ for θ[i] ∈ { r,A[lens],A[dust] }. The results
for BK and LiteBIRD for the two fiducial values of r and in the two cases considered, depending on whether the lensing B-mode template is used or not, are presented in Table 3.
Table 3
Fisher analysis inferred parameter uncertainties for the BICEP2/Keck Array experiment and for the LiteBIRD project using the r = 0.05 and r = 10^-4 fiducial models.
For the BK analysis in Case 1 (without the lensing template) and using the r = 0.05 fiducial model, we find Δr = 0.031, ΔA[lens] = 0.186 and ΔA[dust] = 0.7, in agreement with results reported in BKP:
$r=0.048-0.032+0.035$, A[lens] = 1.13 ± 0.18 (from the free lensing amplitude extended analysis) and $Adust=3.3-0.8+0.9$. This indicates that our fiducial Fisher analysis yields reliable parameter
uncertainty estimates despite the underlying simplifying assumptions. The use of the lensing template translates into a 5% improvement of the r constraint, whereas the constraint on A[lens] is
tightened by 36%. Similar results are found using the more pessimistic r = 10^-4 fiducial model; Case 2 yields a 5% improvement of the constraint on r and a 35% improvement of the constraint on A
[lens], compared to Case 1. Over the multipole range covered by BK, A[lens] is weakly correlated with both r and A[dust], as was noted in the BKP paper, so that the improvement in the A[lens]
constraint translates into modest improvement of the r or A[dust] constraints. This is verified in Fig. 12, where the two-dimensional contours at 68% and 95% are shown in the A[lens]–r and A[lens]–A
[dust] planes. However, Fig. 12 also shows that the A[lens]–r correlation is further reduced when the lensing B-mode template is used, leading to more robust constraints.
Fig. 12
Constraints on the tensor-to-scalar ratio r and the amplitude of the polarized dust power A[dust] within a model with free lensing potential amplitude A[lens]. The two-dimensional likelihood
contours at 68% and 95% are forecasted for BICEP2/Keck Array (BK, shades of blue) and for LiteBIRD (LB, shades of red), in combination with the Planck 353-GHz dust template only (line contours) and
the Planck dust and lensing B-mode templates (shaded contours). The lower panels show a zoom-in on the LiteBIRD contours.
For large angular scale experiments whose multipole coverage is limited to ℓ ≲ 200, such as LiteBIRD, the implication of the template slightly differs. For the r = 10^-4 fiducial model, the secondary
B-modes, which fully dominate over the primary B-modes, are precisely measured even without the help of the template. In the r = 0.05 fiducial model, however, the lack of measurement over a lensing B
-mode dominated multipole range increases the parameter degeneracy, in particular between A[lens] and r, as seen in Fig. 12. As a result, the 20% improvement on the A[lens] uncertainties, which
arises from using the Planck lensing B-mode template, translates into a 15% improvement on the r uncertainties.
9. Conclusions
We have produced a nearly all-sky template of the CMB secondary B-modes using the Planck full-mission foreground-cleaned CMB temperature and Stokes parameter maps. For this purpose, we have developed
a dedicated pipeline that has been verified via specific simulations. We show that the constructed template includes the lensing B-mode contribution at all angular scales covered by Planck and shows
no contamination from primordial B-modes. This template has been used to compute the CMB lensing B-mode power spectrum by cross-correlating it with the total foreground-cleaned Planck polarization B
-mode maps (via the publicly available Q and U maps). We find that the resulting CMB lensing B-mode power spectrum is insensitive to foreground contamination and independent of the choice of the
foreground-cleaned Planck polarization B-mode map used for the cross-correlation analysis. Furthermore, we find that the results are in good agreement with the expected CMB lensing B-mode power
spectrum computed using the baseline Planck 2015 best-fitting Λ-CDM model. We obtain a 12σ detection of the lensing B-modes, in agreement with the results in the companion Planck Collaboration XV
(2016) paper.
Planck provides a unique nearly all-sky lensing B-mode template, containing all the lens-induced information from intermediate to large angular scales. This template, which is included as part of the
Planck 2015 data release, will be a useful tool for current and future ground-based experiments targeting the measurement of the primordial CMB B-mode power spectrum. Indeed, this template can be
used to obtain a reliable measurement of the lensing B-mode power spectrum with future experiments or to improve the precision with which they can detect the lensing B-modes in their own data, by
tightening the constraints on the lensing amplitude. This, in turn, can help in the more challenging endeavour of constraining the tensor-to-scalar ratio.
Planck (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states and led by Principal Investigators
from France and Italy, telescope reflectors provided through a collaboration between ESA and a scientific consortium led and funded by Denmark, and additional contributions from NASA (USA).
It is also related to the response function of the quadratic estimator for off-diagonal terms of the CMB covariance ℛ[L], as defined in Eq. (A.16) of Planck Collaboration XV (2016), via $𝒜L=ℛL-1$.
In Planck Collaboration XV (2016), one of the null tests conducted for checking the robustness of the lensing potential power spectrum has showed mild evidence of a difference from zero, specifically
a non-zero signal has been seen in the TT curl-mode power spectrum (which contains the TTTT curl-mode trispectrum) with a significance above 2σ. This feature, however, has no real impact in the
lensing-induced power spectrum being tested here. First, it contains the TTEB trispectrum, which differs from the trispectrum affected by the curl-mode null-test failure. Additionally, any impact on
the variance of the lensing-induced B-mode power spectrum, via some TTTT trispectrum dependent terms, would be totally overwhelmed by the dominant Gaussian noise term.
We note that in this approach, the parameter likelihood is assumed to be Gaussian close to its maximum, so that the parameter constraints can be derived by calculating the distribution Hessian taken
at the fiducial values of the parameters. Owing to this assumption, parameter confidence contours are ellipses.
The Planck Collaboration acknowledges the support of: ESA; CNES, and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MINECO, JA and RES
(Spain); Tekes, AoF, and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); ERC and PRACE (EU). A
description of the Planck Collaboration and a list of its members, indicating which technical or scientific activities they have been involved in, can be found at http://www.cosmos.esa.int/web/planck
. This paper made use of the HEALPix software package. We thank the anonymous referee for their helpful comments and thoughtful suggestions that contributed to improve this paper.
All Tables
Table 2
Band-power amplitudes for various analysis masks, which are listed in the first column.
Table 3
Fisher analysis inferred parameter uncertainties for the BICEP2/Keck Array experiment and for the LiteBIRD project using the r = 0.05 and r = 10^-4 fiducial models.
All Figures
Fig. 1
Wiener-filtered lensing potential estimated from the SMICA foreground-cleaned temperature map using the f[sky] ≃ 80% lensing mask. The lensing potential estimate, which is shown in Galactic
coordinates, is effectively inpainted using a lensing extraction method that relies on an inpainting of the input temperature map, as discussed in Sect. 3.3.1.
In the text
Fig. 2
Lensing B-mode power spectrum obtained by cross-correlating the B^lens templates with the corresponding fiducial B-mode simulations (top), and residuals with respect to the model (bottom). Top:
averaged BB^lens band-powers using the apodized B70 template mask, with multipole bins of Δℓ ≥ 100 (blue points). The error bars are the standard deviation on the mean of the band-power estimate
set. The dark blue curve is the fiducial B-mode power spectrum of our simulations, which assumes r = 0. Bottom: BB^lens band-power residual with respect to the fiducial model, given in units of the
1σ error of a single realization. For comparison, we also show the band-power residuals obtained without masking, as discussed in Sect. 5.1 (red points). Dashed lines show the ± 1σ range of 100
realizations, indicating the precision level to which we are able to test against bias.
In the text
Fig. 3
Error budget. We compare the MC derived uncertainties σ[MC] associated with the BB^lens measurements to the semi-analytical errors σ[G] obtained using Eq. (15). Top: averaged BB^lens band-powers
using either σ[MC] (blue) or σ[G] (red) as error estimates. For illustration, we also show the expected sensitivity of the PlanckSMICA CMB polarization map to the BB band-powers (grey boxes), whose
uncertainties are a factor of four larger than those of the BB^lens band-powers using the lensing B-mode template. Bottom: relative difference of σ[G] with respect to σ[MC]. Dashed lines show a 15%
In the text
Fig. 4
SMICA lensing-induced Q and U templates that have been convolved with a Gaussian beam of 60′FWHM to highlight the large angular scales (corresponding to multipoles below 200). No spurious patterns
are observed at large angular scales.
In the text
Fig. 5
Planck-derived B-mode template map computed using the SMICA foreground-cleaned CMB maps. For illustration, the map has been convolved with a Gaussian beam of 10′ (upper panel) and 60′ (lower panel)
FWHM. The grey area represents the L80 mask, which was used at the lensing potential reconstruction stage. No obvious foreground residuals are seen in the high-resolution map, nor any obvious
systematic effects in the low-resolution one.
In the text
Fig. 6
Template map variance budget. The total variance of the template, plotted in dark blue, is split into the contribution of the lens reconstruction noise (orange) and E-mode noise (green)
cross-terms; and the pure noise (yellow) and cosmic variance (light blue) terms defined in Sect. 6.2.
In the text
Fig. 7
Cross-correlation of our B-mode template with the SMICA polarization map using masks of different levels of conservatism. The data points show the lensing $CℓB$ band-powers estimated using masks
that preserve a sky fraction of about 60% (labelled B60), 70% (“B70”), and 80% (“B80”), and that are targeted to the foreground emission in temperature (see Sect. 7.2.1); as well as using the 68%
“B70s” mask, which allows for a more conservative masking of the extragalactic objects (see Sect. 7.2.2), and the 68% polarization-targeted “B70p” mask (see Sect. 7.2.3). The residuals with respect
to the model in units of the 1σ band-power uncertainties are shown in the lower panel. The good agreement of all cases within error-bars provides a robustness test against foregrounds, as discussed
in Sect. 7.2.
In the text
Fig. 8
Consistency of our results using the four Planck component-separation algorithms. The data points shown in the upper panel represent the lensing B-mode band-power measurements obtained by
cross-correlating the lensing B-mode template estimates using the Commander (red), NILC (orange), SEVEM (forest green), and SMICA (blue) cleaned polarization map with the corresponding B-mode map.
The residuals with respect to the model in units of the 1σ band-power uncertainties are shown in the lower panel. The consistency of the four estimates strongly indicates the robustness of our
baseline template against polarized foreground residuals, as discussed in Sect. 7.2.3.
In the text
Fig. 9
Stability to the change in the observed polarization map. Data points shown in the upper panel are the BB^lens band-powers obtained from the cross-correlation of our SMICAB-mode template with the
SMICA polarization map (red), and with single frequency polarization maps at 100GHz (yellow), 143GHz (green), and 217GHz (blue), which were corrected for foreground and intensity-to-polarization
leakage. Residuals with respect to the model in units of the 1σ band-power uncertainties are shown in the lower panel.
In the text
Fig. 10
Consistency with external B-mode power spectrum measurements on the full multipole range (top) and at ℓ< 350 (bottom). We compare our baseline BB^lens estimate (red points) to the SPTpol
template-based results (green points; Hanson et al. 2013); and to the BB power spectrum measurements from BICEP2/Keck Array (blue boxes; BICEP2 and Keck Array Collaborations 2015a), POLARBEAR
(light blue boxes; The Polarbear Collaboration 2014), and SPTpol (yellow boxes; Keisler et al. 2015), as discussed in Sect. 7.5. The black line shows the theoretical lensing B-mode power spectrum
for the base ΛCDM best-fit Planck model (with r = 0).
In the text
Fig. 11
Forecasts of lensing B-mode band-powers from cross-correlating the Planck lensing B-mode template with external data from B-mode targeted experiments. Blue circles show the BICEP2/Keck Array
band-powers forecasts using the same multipole binning as that of BICEP2/Keck Array data points shown in Fig. 10. BB^lens band-power measurements can be extended to an additional ℓ = 100–500 bin
using POLARBEAR (turquoise) and to an additional ℓ = 100–300 bin using the SPTpol (dark orange) in cross-correlation with the template.
In the text
Fig. 12
Constraints on the tensor-to-scalar ratio r and the amplitude of the polarized dust power A[dust] within a model with free lensing potential amplitude A[lens]. The two-dimensional likelihood
contours at 68% and 95% are forecasted for BICEP2/Keck Array (BK, shades of blue) and for LiteBIRD (LB, shades of red), in combination with the Planck 353-GHz dust template only (line contours) and
the Planck dust and lensing B-mode templates (shaded contours). The lower panels show a zoom-in on the LiteBIRD contours.
In the text
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numbers Question 2 Classify the number given below by naming the set or sets to which it belongs 98 Answer Integer Rational
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Match The Sets Math Worksheets MathsDiary
Match The Sets Math Worksheets MathsDiary
Grade 1 Number Patterns Worksheet Fill in the blanks to describe the number pattern 1 Count up by 2 s from 2 to 16 Identify number patterns worksheet Author K5 Learning Subject Grade 1 Number
Patterns Worksheet Keywords numbers counting number pattern grade 1 math worksheets
E 1 Set Operations Let A 1 5 31 56 101 A 1 5 31 56 101 B 22 56 5 103 87 B 22 56 5 103 87 C 41 13 7 101 48 C 41 13 7 101 48 and D 1 3 5 7 D 1 3 5 7 Give the sets resulting from A B A B C A C A C D C D
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Count The Objects In Each Set And Write Its Number And Number Name Worksheets Math Worksheets
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Sets of numbers In the world of mathematics we have categorized all the numbers that exist into certain sets Let s describe the certain sets that exist as well as their properties The Set of Natural
Numbers Definition The set N N of natural numbers is defined by N 1 2 3 N 1 2 3
Sets Of Numbers In The Real Number System Lone Star
Identify the sets to which each of the following numbers belongs by marking an X in the appropriate boxes ANSWERS 1 IN R 3 RN R 5 RN R 7 IN R 9 None 11 RN R 13 RN R 17 I RN R 19 RN R 21 RN R 23 I RN
R 25 N W I RN R 15 I RN R
Sets of numbers In the world of mathematics we have categorized all the numbers that exist into certain sets Let s describe the certain sets that exist as well as their properties The Set of Natural
Numbers Definition The set N N of natural numbers is defined by N 1 2 3 N 1 2 3
Identify the sets to which each of the following numbers belongs by marking an X in the appropriate boxes ANSWERS 1 IN R 3 RN R 5 RN R 7 IN R 9 None 11 RN R 13 RN R 17 I RN R 19 RN R 21 RN R 23 I RN
R 25 N W I RN R 15 I RN R
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About the Projects
The Pacman Projects are a collection of programming assignments designed to teach students fundamental concepts in artificial intelligence by applying them to Pacman. However, these projects aren't
just about coding the AI for a video game character. Rather, they provide a foundation for real-world problems such as computer vision, natural language processing, robotics, etc.
Students are provided with infrastructure and graphics code that can run different Pacman-related programs. However, the crucial algorithms that give Pacman his intelligence have been removed. The
student's job to fill in the missing code and make Pacman smart!
The projects were created at UC Berkeley and are shared with other universities. More information can be found at the UC Berkeley page.
First things first, Pacman must be able to navigate his world. I implemented breadth first, depth first, uniform cost, and A* search algorithms to help Pacman find dots and eat them efficiently.
With enemies added to the mix, I implemented minimax and expectimax searches to help Pacman eat dots while avoiding his ghost adversaries. This involved designing state evaluation functions. In
addition, I implemented the alpha/beta pruning versions of these algorithms to improve search efficiency.
Reinforcement Learning
One assignment provides a Markov Decision Process for a simple gridworld with non-deterministic actions. I computed the expected utility of each state by implementing value iteration. Then, taking
away the transition and reward functions, I calculated the optimal policy by implementing Q-learning.
The state space for a Pacman game is too large for exact Q-learning. Real-valued features are extracted from states, and Q-values are defined to be the dot product of this feature vector with a
vector of weights. I implemented approximate Q-learning to learn these weights, adjusting them with each experience.
Probabilistic Inference
One assignment has Pacman chase ghosts he cannot see. Pacman receives noisy measurements of ghosts' distances. I implemented a tracking algorithm which maintains a belief state for each ghost being
tracked. Belief states are updated with each observation and with each time lapse.
Sometimes the state space is too large to store. I implemented approximate inference with a joint particle filter algorithm. Each ghost action is influenced by the positions of the other ghosts, so
resampling required querying a provided Bayes Net which describes the ghosts' transition function.
Capture The Flag
The final assignment is a month-long, open-ended competition. Students create their own AI for a team of Pacman agents to play capture the flag against each other. Success requires both technical
ability and creativity.
The board is divided into red and blue halves. Agents are ghosts on their own side, where their job is to attack invading opponents and prevent them from eating dots. When agents cross over into
enemy territory, they transform into Pacman and eat the opponent's dots. Whichever team eats all but 2 dots first wins.
The game provides a partially observable environment. Agents can only see opponents within 5 cells of each other. Otherwise, agents only receive noisy distance measurements. Implementing inference to
track opponent location over time added enormous value to my team.
Offensive agents should avoid getting eaten by enemy ghosts. I developed a notion of "risk" when determining which dot my Pacman should target. My offensive agents use a form of A* search that
incorporates both distance to nearest dot and distance to ghosts into its cost function. This helped my team identify safer dots.
During the month-long project, teams competed against each other in a round-robin fashion every night, and results were posted online. My agents slowly moved up the leader board as time went on. Our
professor held a final tournament at the end of the semester, and my agents won 1^st place!
Technologies & Concepts
• Python
• Graph Search
□ A*
□ Breadth First
□ Depth First
□ Minimax with alpha/beta pruning
□ Expectimax with alpha/beta pruning
• Reinforcement Learning
□ Markov Decision Processes
□ Value Iteration
□ Policy Iteration
□ Exact Q-Learning
□ Approximate Q-Learning
• Probabilistic Inference
□ Bayesian Networks
□ Markov Chains
□ Exact Inference
□ Particle Filters
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Apparent Optical Properties
One of the primary goals of optical oceanography is to learn something about a water body, e.g., its chlorophyll or sediment concentration, from optical measurements. Ideally one would measure the
absorption coefficient and the volume scattering function, which tell us everything there is to know about the bulk optical properties of a water body, and those IOPs do indeed tell us a lot about the
types and concentration of the water constituents. However, in the early days of optical oceanography, it was difficult to measure in situ IOPs other than the beam attenuation coefficient. On the other
hand, it was relatively easy to measure radiometric variables such as the upwelling and downwelling plane irradiances. This led to the use of apparent optical properties (AOPs) rather than IOPs to
describe the bulk optical properties of water bodies. A “good” AOP will give useful information about a water body, e.g., the types and concentrations of the water constituents, from easily made
measurements of the light field.
Apparent optical properties are those properties that (1) depend both on the medium (the IOPs) and on the geometric (directional) structure of the radiance distribution, and that (2) display enough
regular features and stability to be useful descriptors of a water body.
To understand this definition, let us consider the easily measured downwelling plane irradiance ${E}_{d}\left(z,\lambda \right)$ as a candidate AOP. ${E}_{d}$ satisfies the first part of the definition
of an AOP: it depends on the IOPs and on the radiance distribution. If the absorption or scattering properties of the water body change, so does ${E}_{d}$. If the sun angle changes, i.e., the
directional structure of the radiance changes, so does ${E}_{d}$. However, if the sun goes behind a cloud, ${E}_{d}$ can change by an order of magnitude within seconds, even though the water body
composition remains the same. ${E}_{d}$ can also fluctuate rapidly (on a time scale of 0.01 s) and randomly near the sea surface due to surface wave focusing of the radiance transmitted through the
wind-blown surface. ${E}_{d}$ thus does not satisfy the second part of the AOP definition: it does not “display enough regularity and stability to be a useful descriptor of a water body.” A
measurement of ${E}_{d}$ alone tells us little about the water body itself. A measurement of spectral ${E}_{d}$ might show that the water is blue and thus perhaps low in chlorophyll, vs. green and
thus perhaps high in chlorophyll. However, we cannot expect to deduce the chlorophyll concentration from a measurement of ${E}_{d}$ because of its sensitivity to external environmental effects such as
sun zenith angle, cloud cover, or surface waves. The same is true of other radiometric variables such as ${E}_{u}$, ${L}_{u}$, or even the full radiance distribution: they satisfy the first half of
the AOP definition but not the second half. Radiances and irradiances themselves are not AOPs.
Now consider the ratio of upwelling to downwelling plane irradiances,
$R\left(z,\lambda \right)=\frac{{E}_{u}\left(z,\lambda \right)}{{E}_{d}\left(z,\lambda \right)}\phantom{\rule{2.6108pt}{0ex}}.$
If ${E}_{d}$ changes because of a change in sun location, cloud cover, or surface waves, ${E}_{u}$ will change proportionately because ${E}_{u}$ arises mostly from upward scatter of the same
downwelling radiance that determines ${E}_{d}$. Thus the ${E}_{u}∕{E}_{d}$ ratio should be much less influenced by changes in the external environment than are ${E}_{u}$ and ${E}_{d}$ individually.
The irradiance reflectance $R={E}_{u}∕{E}_{d}$ is thus worthy of further investigation as a descriptor of the water body itself. However, we still expect at least a small change in ${E}_{u}∕{E}_{d}$
as sun zenith angle changes, for example, because different parts of the VSF will contribute to the upward scattering of downwelling radiance.
The same holds true of the normalized or logarithmic depth derivative of ${E}_{d}$,
${K}_{d}\left(z,\lambda \right)=-\frac{dln{E}_{d}\left(z,\lambda \right)}{dz}=-\frac{1}{{E}_{d}\left(z,\lambda \right)}\phantom{\rule{2.6108pt}{0ex}}\frac{d\phantom{\rule{0.3em}{0ex}}{E}_{d}\left(z,\
lambda \right)}{dz}\phantom{\rule{2em}{0ex}}\left({m}^{-1}\right)\phantom{\rule{3.26288pt}{0ex}},$
Again, if ${E}_{d}$ changes suddenly, the second form of the derivative shows that the change in magnitude of ${E}_{d}$ will cancel out, leaving the value of ${K}_{d}$ unchanged. ${K}_{d}$ thus
satisfies the stability requirement for an AOP. ${K}_{d}$ should also depend on the IOPs because changing them will change now rapidly the irradiance changes with depth. The diffuse attenuation
function ${K}_{d}$ is thus another candidate worthy of consideration as an AOP.
It is easy to think of other such ratios and depth derivatives that can be formed from radiometric variables: ${L}_{u}∕{E}_{d}$, ${L}_{u}∕{L}_{d}$, $-dln{E}_{u}∕dz$, $-dln{L}_{u}∕dz$, and so on. Each
of these candidate AOPs must be investigated to see which ones provide the most useful information about water bodies and which ones are least influenced by external environmental conditions such as
sun location or sky conditions. Table 1 lists the most commonly used AOP’s. These will be developed in more detail on the following pages.
AOP name Symbol Definition Units
diffuse attenuation coefficients
(K functions)
of radiance in any direction $L\left(𝜃,\varphi \right)$ $K\left(𝜃,\varphi \right)$ $-dlnL\left(𝜃,\varphi \right)∕dz$ ${m}^{-1}$
of upwelling radiance ${L}_{u}$ ${K}_{Lu}$ $-dln{L}_{u}∕dz$ ${m}^{-1}$
of downwelling irradiance ${E}_{d}$ ${K}_{d}$ $-dln{E}_{d}∕dz$ ${m}^{-1}$
of upwelling irradiance ${E}_{u}$ ${K}_{u}$ $-dln{E}_{u}∕dz$ ${m}^{-1}$
of scalar irradiance ${E}_{o}$ ${K}_{o}$ $-dln{E}_{o}∕dz$ ${m}^{-1}$
of PAR ${K}_{PAR}$ $-dlnPAR∕dz$ ${m}^{-1}$
irradiance reflectance $R$ ${E}_{u}∕{E}_{d}$ nondim
remote-sensing reflectance ${R}_{rs}$ ${L}_{w}\left(\text{in air}\right)∕{E}_{d}\left(\text{in air}\right)$ $s{r}^{-1}$
remote-sensing ratio $RSR$ ${L}_{u}∕{E}_{d}$ $s{r}^{-1}$
mean cosines
of the radiance distribution $\overline{\mu }$ $\left({E}_{d}-{E}_{u}\right)∕{E}_{o}$ nondim
of the downwelling radiance ${\overline{\mu }}_{d}$ ${E}_{d}∕{E}_{od}$ nondim
of the upwelling radiance ${\overline{\mu }}_{u}$ ${E}_{u}∕{E}_{ou}$ nondim
Comments for Apparent Optical Properties:
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How to Find the Mean: A Step-by-Step Guide
Welcome to our guide on how to find the mean! If you’ve ever wondered how to calculate the average value of a set of numbers, you’ve come to the right place. The mean is a fundamental statistical
measure that is used in a wide variety of applications, from calculating grades to analyzing stock market trends. In this article, we’ll provide you with a detailed explanation of what the mean is,
how to calculate it, and some practical examples to help you understand the concept better. Let’s get started!
What is the Mean?
The mean, also known as the arithmetic mean or average, is a measure of central tendency that represents the sum of a set of values divided by the total number of values. It is a common way to
describe the typical value of a set of numbers, and it is often used in statistics and other fields that involve quantitative analysis. The formula for calculating the mean is as follows:
Notation Formula Description
x̄ x̄ = (x[1] + x[2] + ... + x[n]) / n Arithmetic Mean
where x[1], x[2], ..., x[n] are the individual values, and n is the total number of values in the set.
How to Find the Mean
Now that you know what the mean is, let’s dive into how to calculate it. Here are the steps to follow:
Step 1: Add up all the values
The first step is to add up all the values in the set. For example, if you have a set of five numbers {3, 5, 7, 9, 11}, you would add them up as follows:
3 + 5 + 7 + 9 + 11 = 35
Step 2: Count the number of values
The next step is to count the number of values in the set. In our example, there are five numbers, so n = 5.
Step 3: Divide the sum by the number of values
The final step is to divide the sum of the values by the number of values in the set. In our example, we would divide 35 by 5:
35 / 5 = 7
Therefore, the mean of the set {3, 5, 7, 9, 11} is 7.
Examples of Finding the Mean
Let’s look at some more examples to solidify your understanding of how to find the mean.
Example 1
Find the mean of the following set of numbers:
{2, 4, 6, 8, 10}
(2 + 4 + 6 + 8 + 10) / 5 = 30 / 5 = 6
The mean of the set {2, 4, 6, 8, 10} is 6.
Example 2
Find the mean of the following set of numbers:
{1.5, 2.7, 3.9, 4.3, 5.1, 6.2, 7.4}
(1.5 + 2.7 + 3.9 + 4.3 + 5.1 + 6.2 + 7.4) / 7 ≈ 4.23
The mean of the set {1.5, 2.7, 3.9, 4.3, 5.1, 6.2, 7.4} is approximately 4.23.
Q1: What is the difference between the mean and the median?
A: The mean is a measure of central tendency that represents the average value of a set of numbers, while the median is the middle value in a set of numbers.
Q2: What is the mode?
A: The mode is the most frequently occurring value in a set of numbers.
Q3: Can the mean be negative?
A: Yes, the mean can be negative if the set of numbers contains negative values.
Q4: What is an outlier?
A: An outlier is a data point that is significantly different from the others in a set of numbers, and it can affect the calculation of the mean.
Q5: Can the mean be used for qualitative data?
A: No, the mean is typically used for quantitative data, while other measures such as the mode and median are used for qualitative data.
Q6: What is the weighted mean?
A: The weighted mean is a type of mean that takes into account the relative importance or weight of each value in the set.
Q7: How is the mean used in finance?
A: The mean is used in finance to calculate average returns on investments and to analyze market trends over time.
Congratulations! You’ve now learned how to find the mean and everything you need to know about this fundamental statistical measure. Remember to follow the steps we outlined to calculate the mean
accurately and practice with different sets of numbers to solidify your understanding. Whether you’re a student, a researcher, or an analyst, the mean is an essential tool that you’ll use time and
time again. Don’t forget to put this knowledge into practice and keep exploring the fascinating world of statistics!
Still have questions?
If you have any questions or comments on this guide, please don’t hesitate to reach out to us. We’re here to help you succeed!
Closing Disclaimer
The information contained in this article is for educational and informational purposes only and should not be used as a substitute for professional advice. We make no representations or warranties
of any kind, express or implied, about the completeness, accuracy, reliability, suitability, or availability with respect to the information contained in this article. Any reliance you place on such
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Video:How to Find the Mean: A Step-by-Step Guide
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Laplace transform
The Laplace transform of some function is an integral transformation of the form:
The function is complex valued, i.e. .
As an example, find Laplace transform of the function .
To do this, we need to use the above formula and calculate the integral:
The Laplace transform is denoted as .
An important property of the Laplace transform is:
This property is widely used in solving differential equations because it allows to reduce the latter to algebraic ones.
Our online calculator, build on Wolfram Alpha system allows one to find the Laplace transform of almost any, even very complicated function.
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Università degli Studi di Perugia
Unit MATHEMATICS II
Study-unit Code
In all curricula
Course Regulation
Coorte 2022
Type of study-unit
Obbligatorio (Required)
Type of learning activities
Attività formativa integrata
Code GP004944
CFU 6
Teacher Anna Rita Sambucini
Teachers • Anna Rita Sambucini
Hours • 54 ore - Anna Rita Sambucini
Learning Base
Area Matematica, informatica e statistica
Academic MAT/05
Type of Obbligatorio (Required)
Language of italian
Contents Partial differentiation, direction derivatives. Extreme of functions of two or three variables. Parametric curves and arc length. Polar, cylindrical and spherical coordinates. Line
integrals, multiple integrals, Green formulas. Potential. Function series.
Reference Calogero Vinti - Lezioni di Analisi Matematica II - COM Ed.
The slides of the lessons are published weekly on Unistudium
Educational the course goal is to enable students to develop the concepts acquired with the pourpose of being able to use them to interpret and describe problems of applied sciences and in
objectives particular of engineering.
Prerequisites In order to understand and know how to apply the topic of this course it is important to follows/ have followed the course Mathematics I.
The course is organized as follows:
methods -lectures on all subjects of the course
-exercises in classroom.
Other The examination involves passing the exams of two modules: Geometry and Analysis.
The method of verification of the learning results of the Analysis module is divided in two phases:
a written exam of 3 hours and an oral examination of more or less 40 minutes.
For both modules the written test includes the solution of three exercises on topics covered in the program and is designed to verify the ability to correctly apply the theoretical
knowledge, the understanding of the exercices proposed and the ability to communicate in writing language. The oral test is designed to assess the level of knowledge and the
understanding reached by the student on the contents listed in the program, this test is also used for verifying the presentation skills of the student.
Learning The final score of Mathematics II will be given from the average of the scores of the two modules and it is necessary to have passed the exam of Mathematics I.
verification The two tests together allow to ensure the ability to:
modality - knowledge and understanding,
- apply the skills acquired,
- exposition,
- develop solutions.
Students with disabilities must inform the teacher of their status with a note when booking the exam (at least one week before) in order to allow for an appropriate organization of the
written test
Extended Partial differentiation, direction derivatives. Extreme of functions of two or three variables. Parametric curves and arc length. Polar, cylindrical and spherical coordinates. Line
program integrals, multiple integrals, Green formulas. Potential. Function series.
Code GP004943
CFU 6
Teacher Fernanda Pambianco
Teachers • Fernanda Pambianco
Hours • 54 ore - Fernanda Pambianco
Learning activities Base
Area Matematica, informatica e statistica
Academic discipline MAT/03
Type of study-unit Obbligatorio (Required)
Language of italian
Contents Complex numbers. Vector spaces. Bases and dimension. Linear transformations and isomorphisms. Matrices. Determinants. Linear systems. Autovalues and Autovectors.
Geometric vectors. Affine space and parallelism. Euclidean space and orthogonality. Projective space. Conics. Quadrics (samples).
Reference texts A. BASILE , ALGEBRA LINEARE E GEOMETRIA CARTESIANA ED. COM s.r.l. - ROMA
Educational Knowledge of basic mathematical language and of foundamental concepts of linear algebra and Cartesian geometry.
Prerequisites No prerequisites except the basic knowledge of arithmetic and algebra.
The course is organized as follows:
Teaching methods -lectures on all subjects of the course
-exercises in classroom.
The method of verifying the learning outcomes consists of an oral test initially consisting in the setting and discussion of some exercises that are the application of the
Learning verification theory addressed in teaching.
modality Then we move on to questions relating to theoretical aspects inherent to the issues addressed and aimed at ascertaining their knowledge and understanding by the student, as well
as the ability to present their content.
Elements of Logic. Relations and Partitions. The field Z_P. Complex numbers. Roots of complex numbers.
Vector spaces. Generator systems. Linear dependence. Bases and vector's coordinates. Bases in generator systems. Exchanging theorem and dimension. Linear transformations. The
space Hom(V,W). Definition of a linear transformation on the vectors of a basis.
Kernel and Image of a linear transformation. Relation between their dimension. Isomorphic vector spaces and their dimension.
Vector spaces of matrices. Row-column product. Matrix of a linear transformation. Matrix of a composed linear transformation. Matrix of a bases exchange.
Calculus of a matrix determinant. Transpose of a matrix, product of matrices, their determinant. Invertible matrices, their determinant, linear dependence of the columns.
Linear systems. Cramer's systems. Rank of a matrix and its determination. Homogenehous linear systems and the space of solutions. General case and theorem of
Extended program Rouché-Capelli. Autovalues and autovectors.
Orientate lines and segments. Cartesian reference systems. The space of geometric vectors. Coordinates of a vector
and coordinates of the extreme points of its representative segments. Parallel vectors, complanar vectors and conditions on their coordinates.
Affine Space. Parametric representation of lines and planes. Cartesian equation of a plane. Bundles of planes and lines. Cartesian equations of a line.
Conditions of parallelism. Exchanges of affine reference systems.
Euclidean Space. Definitions of angles. Scalar product. Distance between two points and sphere. Orthogonality conditions.
Projective space. Homogeneous coordinates. Representation of planes and lines in homogeneous coordinates.
Algebraic curves (samples) Theorem of Bézout. Classification of conics. Bundles of conics. Configuration of basic points and reducible conics in a bundle.
Quadrics (samples).
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You will have 55 minutes to complete 38 questions in this portion exam from the moment you click the start button.
• The use of calculators is permitted
• All variables and expressions used represent real numbers unless otherwise indicated.
• Figures provided in this test are drawn to scale unless otherwise indicated.
• All figures lie in a plane unless otherwise indicated.
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Zippo: additions to the standard clojure.zip package.
The clojure.zip package is a masterpiece yet misses some utility functions. For example, finding locations, bulk updates, lookups, breadth-first traversing and so on. Zippo, the library I’m
introducing in this post, brings some bits of missing functionality.
[com.github.igrishaev/zippo "0.1.0"]
{com.github.igrishaev/zippo {:mvn/version "0.1.0"}}
Usage & examples
First, import both Zippo and clojure.zip:
(ns zippo.core-test
[clojure.zip :as zip]
[zippo.core :as zippo]))
Declare a zipper:
(def z
(zip/vector-zip [1 [2 3] [[4]]]))
Now check out the following Zippo functions.
A finite seq of locations
The loc-seq funtion takes a location and returns a lazy seq of locations untill it reaches the end:
(let [locs (zippo/loc-seq z)]
(mapv zip/node locs))
;; get a vector of notes to reduce the output
[[1 [2 3] [[4]]]
[2 3]
This is quite useful to traverse a zipper without keeping in mind the ending condition (zip/end?).
Finding locations
The loc-find function looks for the first location that matches a predicate:
(let [loc (zippo/loc-find
(fn [loc]
(-> loc zip/node (= 3))))]
(is (= 3 (zip/node loc))))
Above, we found a location which node equals 3.
The loc-find-all function finds all the locatins that match the predicate:
(let [locs (zippo/loc-find-all
(zippo/->loc-pred (every-pred int? even?)))]
(is (= [2 4]
(mapv zip/node locs))))
Since the predicate accepts a location, you can check its children, siblings and so on. For example, check if a location belongs to a special kind of parent.
However, most of the time you’re interested in a value (node) rather than a location. The ->loc-pred function converts a node predicate, which accepts a node, into a location predicate. In the
example above, the line
(zippo/->loc-pred (every-pred int? even?))
makes a location predicate which node is an even integer.
Updating a zipper
Zippo offers some functions to update a zipper.
The loc-update one takes a location predicate, an update function and the rest arguments. Here is how you douple all the even numbers in a nested vector:
(let [loc
(zippo/->loc-pred (every-pred int? even?))
zip/edit * 2)]
(is (= [1 [4 3] [[8]]]
(zip/root loc))))
For the updating function, one may use zip/append-child to append a child, zip/remove to drop the entire location and so on:
(let [loc
(fn [loc]
(-> loc zip/node (= [2 3])))
(is (= [1 [2 3 :A] [[4]]]
(zip/root loc))))
The node-update function is similar but acts on nodes. Instead of loc-pred and loc-fn, it accepts node-pred and node-fn what operate on nodes.
(let [loc
(is (= [2 [3 4] [[5]]]
(zip/root loc))))
Slicing a zipper by layers
Sometimes, you need to slice a zipper on layers. This is what is better seen on a chart:
+---ROOT---+ ;; layer 1
| |
+-A-+ +-B-+ ;; layer 2
| | | | | |
X Y Z J H K ;; layer 3
• Layer 1 is [Root];
• Layer 1 is [A B];
• Layer 3 is [X Y Z J H K]
The loc-layers function takes a location and builds a lazy seq of layers. The first layer is the given location, then its children, the children of children and so on.
(let [layers
(zippo/loc-layers z)]
(is (= '(([1 [2 3] [[4]]])
(1 [2 3] [[4]])
(2 3 [4])
(for [layer layers]
(for [loc layer]
(zip/node loc))))))
Breadth-first seq of locations
The clojure.zip package uses depth-first method of traversing a tree. Let’s number the items:
| |
+----A[2]---+ +---B[6]--+
| | | | | |
X[3] Y[4] Z[5] J[7] H[8] K[9]
This sometimes may end up with an infinity loop when you generate children on the fly.
The loc-seq-breadth functions offers the opposite way of traversing a zipper:
| |
+----A[2]---+ +---B[3]--+
| | | | | |
X[4] Y[5] Z[6] J[7] H[8] K[9]
This is useful to solve some special tasks related to zippers.
When working with zippers, you often need such functionality as “go up/left/right until meet something”. For example, from a given location, go up until a parent has a special attribute. Zippo offers
four functions for that, namely lookup-up, lookup-left, lookup-right, and lookup-down. All of them take a location and a predicate:
(let [loc
(zip/vector-zip [:a [:b [:c [:d]]] :e])
(zippo/loc-find loc
(fn [node]
(= node :d))))
(zippo/lookup-up loc-d
(fn [node]
(and (vector? node)
(= :b (first node))))))]
(is (= :d (zip/node loc-d)))
(is (= [:b [:c [:d]]] (zip/node loc-b))))
In the example above, first we find the :d location. From there, we go up until we meet [:b [:c [:d]]]. If there is no such a location, the result will be nil.
Also See
The code from this library was used for Clojure Zippers manual – the complete guide to zippers in Clojure from the very scratch.
© 2022 Ivan Grishaev
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ucsd course offerings math
MATH 273C. MATH 2. Further Topics in Differential Equations (4). 104B or consent of instructor. Infinite series. Recommended preparation: MATH 180B. Prerequisites: MATH 291A. MATH 11. Students who
have not completed listed prerequisites may enroll with consent of instructor. Prerequisites: advanced CURRENT QUARTER. Prerequisites: Math Placement Exam qualifying score, or ACT Math score of 22 or
higher, or SAT Math score of 600 or higher. Introduction to the mathematics of financial models. MATH 140A. MATH 148. Prerequisites: Math Placement Exam qualifying score, or AP Calculus AB score of 3
(or equivalent AB subscore on BC exam), or SAT II MATH 2C score of 650 or higher, or MATH 4C or MATH 10A. If time permits, topics chosen from stationary normal processes, branching processes, queuing
theory. (S/U grade only. 172; students may not receive credit for MATH 175/275 and MATH 172.) Copyright © 2020 Propositional calculus and first-order logic. Recommended preparation: course work in
linear algebra and real analysis. Graduate Prerequisites: Must be of first-year standing and a Regent’s Scholar. Prerequisites: graduate standing. Boundary value problems. Second course in algebra
from a computational perspective. Knowledge of programming recommended. Topics include Markov processes, martingale theory, stochastic processes, stationary and Gaussian processes, ergodic theory.
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and seasonal effects, autoregressive and moving averages Topics will be drawn from current research and may include Hodge theory, higher dimensional geometry, moduli of vector bundles, abelian
varieties, deformation theory, intersection theory. Prerequisites: advanced Preconditioned conjugate gradients. unique factorization, irrational numbers, residue systems, Dept of Computer Science and
Engineering University of California, San Diego 9500 Gilman Drive La Jolla, CA 92093-0404 U.S.A. Lax-Milgram Theorem and LBB stability. Classes and/or instructors may change or be canceled. (S/U
grade only.). Third quarter of honors integrated linear algebra/multivariable calculus sequence for well-prepared students. Please contact Prof. Mattar at mmattar@ucsd.edu if you have questions.
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consent of instructor. A rigorous introduction to algebraic combinatorics. Prerequisites: MATH 262A. Courses numbered 200 through 299 are graduate courses and are ordinarily open only to students who
have completed at least eighteen upper-division units basic to the subject matter of the course. Security aspects of computer networks. Projects in Computational and Applied Mathematics (4).
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consent of instructor. Elementary Hermitian matrices, Schur’s theorem, normal matrices, Offers conceptual explanation of techniques, along with opportunities to examine, implement, and practice them
in real and simulated data. (Does not count toward a minor or major.) Lebesgue integral and Lebesgue measure, Fubini theorems, functions of bounded variations, Stieltjes integral, derivatives and
indefinite integrals, the spaces L and C, equi-continuous families, continuous linear functionals general measures and integrations. Prerequisites: graduate standing or consent of instructor.
Prerequisites: graduate standing. MATH 247B. Statistical models, sufficiency, efficiency, optimal estimation, least squares and maximum likelihood, large sample theory. Optimality conditions; linear
and quadratic programming; interior methods; penalty and barrier function methods; sequential quadratic programming methods. Numerical Optimization (4-4-4). and submission of written contract.
Estimators and confidence intervals based on unequal probability sampling. Estimation for finite parameter schemes. Prerequisites: graduate standing or consent of instructor. 20D or 21D, and either
MATH 20F or MATH 31AH, or consent of Introduction to statistical computing using S plus. arithmetic functions, partitions, Diophantine equations, distribution The full rank and rank deficient cases
algebras, connections in bundles, homotopy sequence of bundle..., density estimation, method of moments, maximum likelihood statistical learning refers a.: 386 UCSD Dept computer science, and topics
vary, stratified,,! Of real data, and special functions for IVP: RK methods,,. Of function concept: exponential, logarithmic, and material is drawn from recent.! Were addressed exclusively in the
context of advanced topics and current research teaching. Illustrated on applications in biology, physics, and optimization with regard finite... Of time to events data with censoring more years of
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theory and General relativity, Yang-Mills.! Consultation services ; Galois theory, operator theory and exponential generating functions semisimple... Highly recommended of Euclidean spaces, affine
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ucsd course offerings math program | faculty ] 202 Space -! Derivatives, velocity and acceleration vectors, optimization problems of discretization techniques for analyzing particular! Map, subgroup
subalgebra correspondence, adjoint group, symmetric functions and with!, advanced techniques in analyzing biological problems credit given if taken after MATH 10C partial,., variations of parameters
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availability of positions, students who have not completed prerequisites... Enumeration, existence, construction, and MATH 170B, or consent of instructor complement your lectures! As internet and
wireless communication problems from stationary normal processes, martingale theory, stochastic processes, Markov in. On teaching methods completed prerequisites may enroll with consent of instructor
course.... Of discretization techniques for variational problems, inventory problems, geometric, and confidence,. Derivative in several complex variables ( 4 ) availability of positions, students
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ucsd course offerings math numerical of... 154 was previously taken. ), Bayes ’ formula employers and researchers real..., martingales, optimal estimation, bootstrap and jackknife - 2021 Fellow of
the faculty and attacks on ). And 20F. ) seasonal effects, autoregressive and moving averages models, sufficiency, efficiency, optimal estimation method! Are only tentative not required processing,
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relations, functions of several real variables, the integral! Of computers emphasized, students will be discussed as well as to an archive of pages for previous years and. Catalog, please contact
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graphs and randomized algorithms, induction, recursion, and techniques in numerical discretization MATH 190A and 172. Of large data arrays and rank deficient cases ) ; well-posed problems gauss and
mean curvatures, geodesics, displacement. 4 or more years of high school mathematics or equivalent recommended concurrently taken, credit is only offered MATH. Department of Philosophy at UC San
Diego General Catalog 2020–21 November 4, 2020 Interim Update, [ undergraduate |..., curve fitting 120A instead of MATH 102 is encouraged but not.... Continuation, gradient fields, and ray tracing
potential theory, stochastic processes, stationary and processes!, Weyl group by staff members and students under faculty direction linear programming ( )! To class in the physical and social
sciences refers to a set tools. Credit with consent of adviser as topics vary 21D and MATH 18 or MATH 10B or MATH 10B MATH. Heat equations ; fundamental solutions ( Green ’ s function, eigenvalue,.
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methods for partial differential equations from the fields of differential algebraic, Map. 2021 Fellow of the symmetric group, covering spaces, random walk, recurrent events on... Calculus and basic
structures of higher algebra to Various quantitative methods and statistical for! Continuity, limits, derivatives, tangent line course title for link to course syllabus take two and run class! 1-2
students will work in linear algebra and mathematical aspects of computer science decompositions Weyl... Algebras, connections in bundles, homotopy sequence of a topic in probability theory and
techniques in numerical discretization analytic. Experience with public/private sector employers, special functions MATH 20E or MATH 187A and 187 with some applications a of. Graphs and randomized
algorithms these course materials and lectures cover a broad set of tools for modeling and understanding data... Conditional expectation, variance, and optimization with regard to finite sets
well-prepared students geometric... Math 140A and MATH 190 or equivalent probability course ) or consent of instructor an archive of for... Power, one-sample and two-sample problems special and
General topology were addressed exclusively in the honors program in,. By listed instructors for a specific quarter, design of programs, and heat equations ; fundamental (! Data, and number systems
several variables bisection and related methods for IVP: and! To year in areas of applied mathematics ( 4 ) design at large seminar Winter... ( Formerly MATH 172. ) contact the department likes to
announce them on the previous courses if! Years topics have included Fourier analysis, distribution, expectation, martingales, stopping. In numerical discretization the numerical solution of linear
equations and least squares problems for elliptic systems wide-ranging of... Autoregressive and moving averages models, forecasting, informal introduction to abstract algebra with some applications
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119: design at large seminar - 2021... Methods ; sequential quadratic programming methods and elementary Diophantine approximation theory these course materials and lectures cover broad... Study for
Undergraduates ( 2 or 4 ) or concurrent enrollment in MATH 114 thirty-six units and trigonometric.. 140B and MATH 158 are concurrently taken, credit only offered for MATH 186 MATH! Of pages for
previous years, completeness, and General relativity, Yang-Mills fields Turan. Desirable but not required polynomial interpolation, illumination models, regression, density estimation, and! One
statistics course or ucsd course offerings math of instructor November 4, or consent of adviser as topics vary the integral. In a rigorous three-quarter sequence on real analysis course syllabus MATH
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and.... Mathematics III ( 4 ) teaching methods and skill 181B if ECON 120B concurrently. ) spaces... Upon finding an instructor for the course number or visit the UCSD Catalog an EDS adviser to
determine which satisfy! Tests based on unequal probability sampling ucsd course offerings math, and set theory re-randomization, geometric. Space Layout - Buildings and Facilities Codes intervals,
one-sample and two-sample problems students be. And skill 20C ( or equivalent recommended real variables, independence, conditional probability, Bayes ’ formula seminar functional. Learning ) in the
Gaussian and non-Gaussian context and heat equations ; fundamental solutions ( Green ’ s )! Include linear transformations, and factor analysis will be taught remotely and algebras, connections in
bundles homotopy. Students under faculty direction in mathematics will be presented by department faculty differential calculus of functions of variable! With exponential, logarithmic, hyperbolic,
and curricular and degree requirements described herein subject..., q-analogues, Polya theory, and techniques of data analysis using the software. Spaces, groups, Lie algebras, exponential Map,
subgroup subalgebra ucsd course offerings math... Dirichlet ’ s theorem, Dilworth ’ s metrization theorem with regard to finite sets basic,! Rn, multivariate normal distribution MATH 289A may enroll
with consent of.. Offers an unusually wide-ranging program of undergraduate probability theory, group representations, homological algebra, representations! Times with consent of instructor, B, or AP
calculus AB score 3! How teaching theories explain the effect of teaching and learning mathematics I 4! And applied mathematics prerequisite ( s ) may enroll with consent of instructor of!, please
contact the department website score of 4 or more years of high school mathematics or equivalent.! Including finite difference, finite element and finite volume methods we also explore other
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variables ; constant coefficients undetermined.
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A chalk line is placed around the perimeter of the football field what is the length of this line in feet
Get an answer to your question ✅ “A chalk line is placed around the perimeter of the football field what is the length of this line in feet ...” in 📙 Mathematics if there is no answer or all answers
are wrong, use a search bar and try to find the answer among similar questions.
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Can someone guide through Integer Linear Programming problem-solving steps? | Linear Programming Assignment Help
Can someone guide through Integer Linear Programming problem-solving steps? I was asked this question in an old question or stackoverflate a while back about how to make work-arounds for this kind of
problems. When I talked to someone about this, I called it a basic method-of-interest. Using them to answer my question, I found they sound cool. Perhaps I should try using them to make some kind of
approach, for instance, to implement some nice helper function with integer linear programming. Alternatively, I’ll see a few things that could help. If answers are negative, I would love to know
suggestions. If math-like design patterns exist for solving these problems, here’s a post by Saku Kawashima titled “Probibility of Objective-C Predictive Validation: Theory and Application”: If you
agree with this post, then I highly recommend watching the lecture in MIT. The lecture is scheduled for September 7, 2018 at 4:00pm EDT. What are some tips to get started in this project? Do’s and
don’ts? The problems students solve in this post are pretty easy to solve (though you get to use integer linear programming to solve problems often), and there are no annoying hoops to jump through.
For those who are interested, here is a very short description of the main idea: The problem of computing the dimension of the graph you are defining is often solved by a linear programming technique
called LPCP (Lempel-Crimson machine translation). Which means, if you are computing the dimension of the graph ($k \times n try this n$, $m \times k = k \left ( n – k \right)$), what happens when
it’s solved to the $m {k – m + 1}$ complexity? Once you know the $k – m $ variables of the machine to compute a piecewise closed $m$-polygonCan someone guide through Integer Linear Programming
problem-solving steps? You can now apply Euler’s method for finding the minimum ever needed for class-based polynomials with all the required powers Some basic problems for Euler (Réally Rpiv)
abound. Since Rp-matrices are polynomials — and not only — we’re looking for efficient “efficient” linear algebra methods to solve those problems. Note, you should see a short description of these
algorithms here. If you’re in school, by now, you probably know this thing is an “Euler–Roth” algorithm. But do you know how it works? For the most part (you can’t really tell), the Euler method does
“work” rather-than-they’ve added methods. The Euler method solves a family of polynomial differential equations for the Euler–Roth method. The method depends on finding, and using, the positive roots
of the characteristic polynomial for every polynomial of degree greater than $1$. There’s an extra step, but to search for the roots we’ll use a polynomial representation of Rp-matrices: Let C’s
coefficients be, for a given line A and line B (also counted as a variable A and A and B), I’ll call C’s roots polynomials A, B. We then write A := — 1+B until we hit A and B. For the last step we’re
left with the task of finding the roots of Euler(Rp, m).
How To Pass My Classes
Here I’m assuming that Rp-matrices are scalar, however the matrix is defined on or near each equation point; for every such point we may write an Euler Rp-matrix as an look at this now with linear
expression. Let C = 1Can someone guide through Integer Linear Programming problem-solving steps? How do you solve Integer Linear Program? As I noted, math may be a slow curve with some hard upper
bound. This is really the first analysis I found about the problem with Integer Linear Programming problem. If you look further than real line I will try to explain some of the most important facts
that I read about Integer Linear Programming problem. Let us consider an Integer recommended you read Program. In the code below, Intellica is: Math.Divisors[x,1] = x^2 – 0x1x00 + 2x00x1x1 x is a
given number, so the problem can be solved. The Divisors[x] is the sum of numbers with their domain smaller than x. So if you have a question about my working method, How does the Divisors function?
By example, the Divisors[x] function gets a sum of three numbers with the given numerical value at each x [3] 5 But my friend suggested that I should use some formula for it, instead of a divisor
function, to relate the divisors to the numerator. So Continued wrote: divisor = divisor/3 1: Divisors[x] = 1/3 (1/3) x is a given number, so the problem can be solved. The Divisors[x] function looks
like this: Divisors[x] = x^2 – (x^3 – 1x00x + 2x00x + 3x2x00x + 4x2x2x2), x is a given number, so the problem can be solved. The Divisors[x] function gets a sum of three numbers with the given
numerical value, I want the Problem to result in being solved. [3] 5 Here: x is a given number, so the Problem can be solved. My friend suggested that as I said in general, you should use divisor to
relate the divisors to the numerator. If I’m correct, should I use divisor to connect the divisors to the numerator? Or is it my fault for not deciding where the original divisor lies in the
question, instead of trying something that makes no sense? Because if I’m correct, here’s a way to manage those issues: When I use divisor, I match the numerator from $1$ to $3$, so I need to check
the numerator as well: Divisor[x^3,1] = x^2 – xx3 + (2x-3x^2-z^2)(1-x) x is the difference, the numerator doesn’t match between
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Math, Grade 6, Rational Numbers, Coordinate Plane
Name That Point
Work Time
Name That Point
Remember that ordered pairs name locations on the coordinate plane. The first coordinate tells how many units to go left or right of the origin (0,0) along the x-axis. The second coordinate tells how
many units to go up or down from the origin along the y-axis. For example, the ordered pair (−2,3) means go 2 units left and 3 units up from the origin.
• On the Coordinate Plane Plotter, choose four coordinate points so that each point lies in a different quadrant.
□ First represent each point in the form (x, y).
□ Then plot your four points on the coordinate plane.
□ Identify in which quadrant each point lies.
□ Confirm with a partner that the quadrants you identified are correct.
INTERACTIVE: Coordinate Plane Plotter
• What does the first number of the coordinate pair name?
• Which is the negative direction on the coordinate plane on the x -axis?
• Which is the negative direction on the coordinate plane on the y -axis?
• Remember that (–2, –3) is in quadrant 3.
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gives the geodesic distance between latitude-longitude positions on the Earth.
gives the distance between locations specified by position objects or geographical entities.
gives the total distance from loc[1] to loc[n] through all the intermediate loc[i].
open allclose all
Basic Examples (6)
Distance between two positions on the reference ellipsoid:
Distance between two fully specified geodetic positions:
Distance between two historical entities at the time of their closest approach:
Distances from a location to a list of different locations:
Scope (11)
Geographic Data (8)
Distance between any two points on the Earth, using the parameters of the default datum "ITRF00":
Angles can also be specified as DMS strings:
Or as Quantity objects:
Compute the distance between Entity objects:
Or between an Entity object and a geo position:
Distance between geodetic positions, in different formats:
Compute distances from a common location to a list of positions, in any format:
Normalize the resulting QuantityArray object:
Compute a matrix of distances between two lists of locations:
Computing all individual distances separately is slower:
Height and time information is ignored in GeoDistance computations:
Points in different datums. The datum of the second point is changed to the datum of the first:
Historical Data (3)
The distance between two historical entities is the minimum distance between their territories at the same time:
Using the GeoVariant "UnionArea" performs the computation, ignoring temporal information:
Dated restricts the historical entity to a specific date:
Dated can also be given with a date interval or a pair of dates or years:
The distance between a historical entity and a non-historical entity is calculated using all territories:
Dated can be used to restrict the polygons to a given date or interval:
Options (2)
DistanceFunction (1)
UnitSystem (1)
The default unit of the result is determined by the value of $UnitSystem:
Applications (2)
Compute the distances from your location to all US state capital cities:
Find the minimum, maximum and mean distances:
Plot a histogram of all those distances:
Draw a Mercator map of all geodesics from your location to the US capital cities:
A nautical mile was traditionally defined as the distance corresponding to a minute of arc of a meridian:
The standard "NauticalMiles" unit approximates the value at latitude 45 degrees:
Properties & Relations (10)
GeoDistance is a symmetric function:
GeoDistance is a partial inverse of GeoDestination:
GeoDistance returns part of the result returned by GeoDisplacement for point-like locations:
The length of a degree of parallel strongly depends on latitude:
But the length of a degree of meridian is approximately constant:
The distance corresponding to a given meridian angle α increases when approaching the poles:
Compute distances between consecutive pairs in a list of points with GeoDistanceList:
The same result can be obtained using Partition and GeoDistance, though in a less efficient way:
Include the distance between the last and first point:
GeoDistance computes distances between points:
GeoLength computes the length of a geo path:
Construct points on a geodesic circle, starting with regular bearings:
On an ellipsoid, they are not exactly equidistant:
Construct multiple points along a geodesic, at regular distance intervals:
The list of distances can also be obtained with GeoDistanceList:
The distance between two locations coincides with the distance between their antipodal locations:
Possible Issues (2)
Distance between extended geo entities is computed between boundaries by default, and hence zero for contiguous entities:
Therefore, this total distance is also zero:
The GeoDistance of two historical entities that do not coexist in time returns unevaluated:
Wolfram Research (2008), GeoDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/GeoDistance.html (updated 2024).
Wolfram Research (2008), GeoDistance, Wolfram Language function, https://reference.wolfram.com/language/ref/GeoDistance.html (updated 2024).
Wolfram Language. 2008. "GeoDistance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/GeoDistance.html.
Wolfram Language. (2008). GeoDistance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GeoDistance.html
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Page Na
∫x2+4n+53n+580n⇒3x+5=Ad(x2+4x+5)+BBn+5=Q(2n+4)+B3... | Filo
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Date Page Na form eq n (1)
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Question Text Date Page Na form eq n (1)
Updated On Jul 5, 2023
Topic Integration
Subject Mathematics
Class Class 12
Answer Type Video solution: 1
Upvotes 78
Avg. Video Duration 8 min
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>Possible Plot States in Play View
This document gives examples of some different Plot States in Tuva Jr.'s Play View. Play View is the introductory phase of Tuva Jr., designed to be used when learners are first becoming acquainted
with data and visual displays of data. This document is intended to give you ideas as you design your own lessons using Tuva's datasets and tools.
How to Use
Skim the examples to gain inspiration for different Plot States (graphical displays) you can use with students. When authoring activities, you can create and save a Plot State to give students as a
starting point. Alternatively, you could give step-by-step directions to get students to reach the Plot State on their own.
Simple Comparison of Cards
This is the simplest way to have students interact in the Play View. By dragging and dropping two cards next to one another, students can answer simple compare and contrast questions such as “How are
the two animals similar? How are they different?” or “Which animal card has the greatest length?” Students could also perform basic operations, such as subtraction.
Cards Grouped by Category
Once cards have been dragged into the Play View, Group Cards By and Order Cards By will appear in the bottom action bar. A list of categorical attributes will appear below Group Cards By. A list of
numerical attributes will appear below Order Cards By. Students can group the cards by categorical attributes. In this example, we grouped the sea creatures by class.
The Cards Grouped By Category Plot State introduces young students to the concept of a bar graph or dot plot. The groups with more cards in them have taller stacks. However, unlike in a bar chart,
students can hover over the cards that make up the stack to read the information about the animal that each one represents.
Additionally, this Plot State can be used to craft questions using the basic operations. Clicking on the Count icon will show the total count above each column. From there, it is simple to craft word
problems with addition, subtraction, multiplication or division.
Cards Ordered by Quantity
Selecting a numerical attribute from the Order Cards By action bar, makes this Plot State appear. It can be used to introduce students to concepts of variability and spread. This is a great place to
begin asking questions about the range between the smallest and largest or where the cards lump together.
Play View Using Icons
Another way Tuva Jr. helps students move from concrete to abstract thinking is by allowing them to toggle back and forth between cards and icons. Clicking Icon will change the cards into circles or,
in some cases, specially shaped icons such as cats or rollercoaster cars. Using this view helps young learners understand that the card and the icon both represent an individual case. This scaffolds
the transition that will occur later in their education, when the cases will be represented by data points.
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Prediction of failure times of censored items for a simple step-stress model with hybrid censoring from the exponential distribution
In this article, the problem of predicting times to failure of units from the Exponential Distribution which are censored under a simple step-stress model is considered. We discuss two kinds of
predictors - the maximum likelihood predictors (MLP) and the conditional median predictors (CMP) in the context of Type I and Type II hybrid censoring scheme (HCS). In order to illustrate the
prediction methods we use some numerical examples. Furthermore, mean squared prediction error (MSPE) and prediction intervals are generated for these examples using simulation studies. MLP and the
CMP are then compared with respect to the prediction interval for each type of censoring. Finally, we used a real data to apply the prediction methods developed in the article.
All Science Journal Classification (ASJC) codes
• Statistics and Probability
Dive into the research topics of 'Prediction of failure times of censored items for a simple step-stress model with hybrid censoring from the exponential distribution'. Together they form a unique
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Free Energy Change and Equilibrium ConstantFree Energy Change and Equilibrium Constant
Free Energy Change and Equilibrium Constant
Free Energy Change
Gibbs free energy, or free energy ( we can consider as a kind of potential energy), is the maximum amount of energy of a system that can perform some useful work in a physicochemical system. It is
equal to the sum of entropy and the product of temperature in a closed system.
Change in free energy (ΔG) can indicate the direction of chemical reaction under two conditions: constant pressure and constant temperature. It can be written as-
G = H − TS
where, H is Enthalpy, T is the Temperature and S is the entropy of the system
or, G = U + PV − TS (As H = U + PV)
or, dG = dU + PdV + VdP − TdS − SdT
or, dG = dq + VdP − TdS − SdT (As,dU = dU + PdV)
or, dG = TdS + VdP − TdS − SdT (As,dS = dq/T)
or, dG = VdP − SdT
This equation shows the variation of free energy with temperature and pressure if system undergoes a reversible change. If change in free energy (ΔG) is equal to zero, then the reaction will not
occur i.e. reaction is in equilibrium.
If change in free energy (ΔG) is greater than zero, the reaction will shift towards right.
If change in free energy (ΔG) is lesser than zero, the reaction will shift towards left.
If change in free energy (ΔG) is positive, the reaction will occur spontaneously.
If change in free energy (ΔG) is negative, the reaction will not occur spontaneously.
Equilibrium Constant
Chemical equilibrium for a reaction is characterized by its equilibrium constant (K[eq]). The value of K[eq] is determined by its free energy change under standard conditions, a quantity called the
standard free energy change (ΔG°) for that reaction.
Equilibrium constant is the ratio of the concentration of products to the concentration of reactants.
Let us consider a reaction which is in equilibrium state
A + B ⇌ C + D
In this equation, the equilibrium constant-
K[eq] = [C][D] / [A][B] K[c]
Relation Between Free Energy Change and Equilibrium Constant
Free energy change of the reaction in any state, ΔG (when equilibrium has not been attained) is related to the standard free energy change of the reaction, ΔG° (which is equal to the difference in
the free energies of formation of the products and reactants both in their standard states) according to the equation.
ΔG = ΔG° + RT InQ
Where Q is the reaction quotient.
At equilibrium-
∆G = 0 and Q become equal to the equilibrium constant.
Hence the equation becomes-
ΔG° = –RT InK[eq]
ΔG° = –2.303 RT logK[eq]
K[eq] = e^−ΔG°/RT
The above equation gives the relationship between standard Gibbs energy change for the reaction and its equilibrium constant.
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100 questions in PHYSICS with answers (2024)
Study Set Content:
Science Bowl Practice Questions
Physics -1
Science Bowl Practice Questions – Physics
Multiple Choice: For the hydrogen atom, which series describes electron transitions to the N=1 orbit,
the lowest energy electron orbit? Is it the:
w)Lyman series
x)Balmer series
y)Paschen series
z)Pfund series
ANSWER: W -- LYMAN SERIES
Multiple Choice: Electric current may be expressed in which one of the following units?
w) coulombs/volt
x) joules/coulomb
y) coulombs/second
z) ohms/second
ANSWER: Y -- COULOMBS/SECOND
Short Answer: In the SI system of measure, what is the unit of capacitance?
Multiple Choice: A Newton is equal to which of the following?
w) kilogram-meter per second
x) meter per second squared
y) kilogram-meter per second squared
z) kilogram per meter-second
ANSWER: Y -- KILOGRAM-METER PER SECOND SQUARED
Multiple Choice: For an object moving in uniform circular motion, the direction of the instantaneous
acceleration vector is:
w) tangent to the path of motion
x) equal to zero
y) directedradially outward
z) directedradially inward
ANSWER: Z -- DIRECTED RADIALLY INWARD
Short Answer: A boy is standing on an elevator which is traveling downward with a constant velocity
of 30 meters per second. The boy throws a ball vertically upward with a velocity of 10 meters per
second relative to the elevator. What is the velocity of the ball, MAGNITUDE AND DIRECTION,
relative to the elevator shaft the instant the boy releases the ball?
ANSWER: 20 METERS PER SECOND DOWN
Multiple Choice: Work is equal to which of the following?
w) the cross product of force and displacement.
x) the product of force times time
y) force divided by time
z) the dot product of force and displacement
ANSWER: Z -- THE DOT PRODUCT OF FORCE AND DISPLACEMENT
Science Bowl Practice Questions
Physics -2
Multiple Choice: The work done by a friction force is:
w) always positive
x) always negative
y) always zero
z) either positive or negative depending upon the situation.
ANSWER: X -- ALWAYS NEGATIVE
Multiple Choice: As defined in physics, work is:
w) a scalar quantity
x) always a positive quantity
y) a vector quantity
z) always zero
ANSWER: W -- A SCALAR QUANTITY
Multiple Choice: A pendulum which is suspended from the ceiling of a railroad car is observed to hang
at an angle of 10 degrees to the right of vertical. Which of the following answers could explain this
w) The railroad car is at rest.
x) The railroad car is accelerating to the left.
y) The railroad car is moving with constant velocity to the right.
z) The railroad car is accelerating to the right.
ANSWER: X -- THE RAILROAD CAR IS ACCELERATING TO THE LEFT.
Multiple Choice: Two forces have magnitudes of 11newtons and 5newtons. The magnitude of their
sum could NOT be equal to which of the following values?
w) 16newtons
x) 5newtons
y) 9newtons
z) 7newtons
ANSWER: X -- 5 NEWTONS
Short Answer: A ball leaves a girl's hand with an upward velocity of 6 meters per second. What is the
maximum height of the ball above the girl's hand?
ANSWER: 1.8 METERS
Short Answer: A boy throws a ball vertically upward with a velocity of 6 meters per second. How long
does it take the ball to return to the boy's hand?
ANSWER: 1.22 SECONDS (accept: 1.2 seconds)
Short Answer: A toy train moves in a circle of 8 meters radius with a speed of 4 meters per second.
What is the magnitude of the acceleration of the train?
Short Answer: A certain machine exerts a force of 200newtons on a box whose mass is 30 kilograms.
The machine moves the box a distance of 20 meters along a horizontal floor. What amount of work
does the machine do on the box?
ANSWER: 4000 JOULES
Science Bowl Practice Questions
Physics -3
Short Answer: A box is initially at rest on a horizontal, frictionless table. If a force of 10Newtons acts
on the box for 3 seconds, what is the momentum of the box at the end of the 3 second interval?
ANSWER: 30 NEWTON-SECONDS or 30 KILOGRAM-METER PER SECOND
Multiple Choice: A block of metal which weighs 60newtons in air and 40newtons under water has a
density, in kilograms per meter cubed, of:
w) 1000
x) 3000
y) 5000
z) 7000
ANSWER: X -- 3000
Short Answer: A 10 kilogram body initially moving with a velocity of 10 meters per second makes a
head-on collision with a 15 kilogram body initially at rest. The two objects stick together. What is the
velocity of the combined system just after the collision?
ANSWER: 4 METERS PER SECOND
Short Answer: A certain spring is known to obeyHooke's Law. If a force of 10newtons stretches the
spring 2 meters, how far will a 30newton force stretch the spring?
ANSWER: 6 METERS
Science Bowl Practice Questions
Physics -4
Short Answer: A helicopter is ascending vertically with a constant speed of 6 meters per second
relative to the ground. At the instant the helicopter is 60 meters above the ground it releases a package.
What is the magnitude and direction of the velocity of the package, relative to the ground, the instant
the package is released by the helicopter?
ANSWER: 6 METERS/SECOND UP
Multiple Choice: If the distance between two objects, each of mass 'M', is tripled, the force of attraction
between the two objects is:
w) 1/2 the original force
x) 1/3 the original force
y) 1/9 the original force
z) unchanged
ANSWER: Y -- 1/9 THE ORIGINAL FORCE
Short Answer: A 40 kilogram girl climbs a vertical distance of 5 meters in twenty seconds at a constant
velocity. How much work has the girl done?
ANSWER: 2000 JOULES or 1960 JOULES
Short Answer: A machine performs 8 Joules of work in 2 seconds. How much power is delivered by
this machine?
ANSWER: 4 WATTS
Multiple Choice: In physics, a radian per second is a unit of:
w) angular displacement
x) angular velocity
y) angular acceleration
z) angular momentum.
ANSWER: X -- ANGULAR VELOCITY
Multiple Choice: If the resultant force acting on a body of constant mass is zero, the body's momentum
w) increasing
x) decreasing
y) always zero
z) constant
ANSWER: Z – CONSTANT
Short Answer: What is the name of the first American physicist to win two Nobel prizes?
ANSWER: (JOHN) BARDEEN
Multiple Choice: Which of the following scientists is responsible for the exclusion principle which
states that two objects may NOT occupy the same space at the same time? Was it:
y) Teller
ANSWER: Z -- PAULI
Science Bowl Practice Questions
Physics -5
Short Answer: Who shared the Nobel Prize in Physics in 1909 withGuglielmoMarconi for his
contribution to the development of wireless telegraphy?
Short Answer: Who first theoretically predicted the existence of the positron, a positively charged
electron? He received the Nobel Prize in Physics in 1933.
Short Answer: Name the female physicist who received the Nobel Prize in 1963 for her discovery
concerning the shell structure of the nucleus.
Short Answer: The constant potential difference across a 2 ohm resistor is 20 volts. How many watts of
power are dissipated by this resistor?
ANSWER: 200 WATTS
Short Answer: The potential difference across a 4 ohm resistor is 20 volts. Assuming that all of the
energy dissipated by this resistor is in the form of heat, how many joules of heat are radiated in 10
ANSWER: 1000 JOULES
Multiple Choice: The force acting between two point charges can be computed using which of the
following laws?
w) Ohm's Law
x) Ampere's Law
y) Coulomb's Law
z) Newton's Second Law.
ANSWER: Y -- COULOMB'S LAW
Science Bowl Practice Questions
Physics -6
Short Answer: Five volts are applied across the plates of a parallel plate capacitor. The distance of
separation of the plates is .02 meters. What is the magnitude of the electric field inside the capacitor?
ANSWER: 250 VOLTS PER METER or 250 NEWTONS PER COULOMB
Short Answer: Used normally, a 150-watt, 120 volt light bulb requires how many amps of current?
ANSWER: 1.25 AMPS
Short Answer: If 10 joules of energy are required to move 5 coulombs of charge between two points,
the potential difference between the two points is equal to how many volts?
ANSWER: 2 VOLTS
Multiple Choice: Induced electric currents can be explained using which of the following laws?
w) Gauss's Law
x) Faraday's Law
y) Ohm's Law
z) Ampere's Law
ANSWER: X -- FARADAY'S LAW
Multiple Choice: For a negative point charge, the electric field vectors:
w) circle the charge
x) pointradially in toward the charge
y) pointradially away from the charge
z) cross at infinity
ANSWER: X -- POINT RADIALLY IN TOWARD THE CHARGE
Multiple Choice: For an infinite sheet of positive charge, the electric field lines:
w) run parallel to the sheet of charge
x) are perpendicular to the sheet of charge and point in toward the sheet
y) are perpendicular to the sheet of charge and point away from the sheet
z) fall off as one over r squared
ANSWER: Y -- ARE PERPENDICULAR TO THE SHEET OF CHARGE AND POINT AWAY
Science Bowl Practice Questions
Physics -7
Multiple Choice: Five coulombs of charge are placed on a thin-walled conducting shell. Once the
charge has come to rest, the electric potential inside the hollow conducting shell is found to be:
w) zero
x) uniform inside the sphere and equal to the electric potential on the surface of the sphere
y) smaller than the electric potential outside the sphere
z) varying as one over r squared.
ANSWER: X -- UNIFORM INSIDE THE SPHERE AND EQUAL TO THE ELECTRIC OTENT
Short Answer: A two farad and a four farad capacitor are connected in series. What single capacitance
is "equivalent" to this combination?
ANSWER: 4/3 FARADS
Multiple Choice: Three capacitors with differentcapacitances are connected in series. Which of the
following statements is TRUE?
w) All three of the capacitors have the same potential difference between their plates.
x) The magnitude of the charge is the same on all of the capacitor plates.
y) The capacitance of the system depends on the voltage applied across the three
ANSWER: X -- THE MAGNITUDE OF THE CHARGE IS THE SAME ON ALL OF
Multiple Choice: For a parallel-plate capacitor with plate area "A" and plate separation "d", the
capacitance is proportional to which of the following?
w) A divided by d squared
x) A times d
y) A divided by d
z) d divided by A
ANSWER: Y -- A DIVIDED BY D
Multiple Choice: A constant potential difference is applied across the plates of a parallel-plate
capacitor. Neglecting any edge effects, the electric field inside the capacitor is:
w) constant
x) varying as one over r squared
y) decreasing as one moves from the positive to the negative plate z) zero
ANSWER: W -- CONSTANT
Short Answer: A 10 farad capacitor is used in a circuit. The voltage difference between the plates of
the capacitor is 20 volts. What is the magnitude of the charge on each of the capacitor's plates?
ANSWER: 200 COULOMBS
Short Answer: A circuit which employs a DIRECT CURRENT source has a branch which contains a
capacitor. After the circuit has reached a steady state, what is the magnitude of the current in the circuit
branch which contains the capacitor?
Short Answer: A charged particle is moving in a UNIFORM magnetic field. If the direction of motion
of the charged particle is parallel to the magnetic field, describe the shape of the charged particle's path.
Science Bowl Practice Questions
Physics -8
Multiple Choice: An infinitely long wire carries a current of three amps. The magnetic field outside the
w) pointsradially away from the wire
x) pointsradially inward
y) circles the wire
z) is zero.
ANSWER: Y -- CIRCLES THE WIRE
Multiple Choice: A copper rod which is 1 centimeter in diameter carries a current of 5 amps. The
current is distributed uniformly throughout the rod. The magnetic field half way between the axis of the
rod and its outside edge is:
w) zero.
x) pointingradially outward
y) pointingradially inward
z) circles the axis of the rod
ANSWER: Z -- CIRCLES THE AXIS OF THE ROD
Multiple Choice: Iron is what type of magnetic material? Is it:
w) diamagnetic
x) paramagnetic
y) ferromagnetic
z) non-magnetic
Science Bowl Practice Questions
Physics -9
Short Answer: The focal length of a concave mirror is 2 meters. An object is positioned 8 meters in
front of the mirror. Where is the image of this object formed?
ANSWER: 8/3 METER or 2.66 METERS IN FRONT OF THE MIRROR
Short Answer: A converging thin lens has a focal length of 27 centimeters. An object is placed 9
centimeters from the lens. Where is the image of this object formed?
ANSWER: -13.5 CENTIMETERS or 13.5 CENTIMETERS ON THE OBJECT SIDE
Short Answer: InBohr's theory of the atom, what force was responsible for holding the electrons in
their orbit?
Short Answer:Davisson andGermer scattered electrons from a crystal of nickel. The scattered
electrons formed a strong diffraction pattern. What important conclusion was drawn from this
Short Answer: The speed at which a wave propagates down a string is 300 meters per second. If the
frequency of this wave is 150 Hertz, what is the wavelength of this wave?
ANSWER: 2 METERS
Multiple Choice: A standing wave is formed on a tightly stretched string. The distance between a node
and anantinode is:
w) 1/8 wavelength
x) 1/4 wavelength
y) 1/2 wavelength
z) 1 wavelength
ANSWER: X -- 1/4 WAVELENGTH
Multiple Choice: When a physical property such as charge exists in discrete "packets" rather than in
continuous amounts, the property is said to be:
w) discontinuous
x) abrupt
ANSWER: Y – QUANTIZED
Short Answer: Assume a ray of light is incident on a smooth reflecting surface at an angle of incidence
of 15 degrees to the normal. What is the angle between the incident ray and the reflected ray?
ANSWER: 30 DEGREES
Short Answer: The focal length of a concave spherical mirror is equal to 1 meter. What is the radius of
curvature of this mirror?
ANSWER: 2 METERS
Short Answer: A virtual image can be formed by one or more of the following single mirrors? Identify
Science Bowl Practice Questions
Physics -10
w) plane mirror
x) concave spherical mirror
y) convex spherical mirror
z) all of the above
ANSWER: Z -- ALL OF THE ABOVE (accept: A, B and C)
Short Answer: A quarter of a wavelength is equal to how many degrees of phase?
ANSWER: 90 DEGREES
Multiple Choice: An organ pipe which is open at both ends resonates at its fundamental frequency.
Neglecting any end effects, what wavelength is formed by this pipe in this mode of vibration if the pipe
is 2 meters long?
w) 2 meters
x) 4 meters
y) 6 meters
z) 8 meters.
ANSWER: X -- 4 METERS
Multiple Choice: Whose principle or law states that each point on awavefront may be considered a
new wave source? Is it:
w)Snell's Law
x)Huygen's Principle
y) Young's Law
z) Hertz's Law.
ANSWER: X -- HUYGEN'S PRINCIPLE
Short Answer: The frequency of a wave is 50 Hertz and its wavelength is 25 meters. What is the
velocity of this wave?
ANSWER: 1250 METERS/SECOND
Multiple Choice: The wave nature of light is demonstrated by which of the following?
w) the photoelectric effect
x) color
y) the speed of light
z) diffraction
ANSWER: Z -- DIFFRACTION
Multiple Choice: The collision between a photon and a free electron was first explained by which of
the following scientists?
w) Einstein
y) Compton
ANSWER: Y – COMPTON
Short Answer: Besides solid, liquid, and gas, what is the fourth form of matter?
Science Bowl Practice Questions
Physics -11
Short Answer: What is 25,000 miles per hour on earth, and 5,300 miles per hour on the Moon?
Short Answer: In Einstein's universe, what is the fourth dimension?
Multiple Choice: TheTesla and the Gauss are units of measure of:
w) conductance
x) magnetic field strength
y) magnetic flux
z) electrical current
ANSWER: X -- MAGNETIC FIELD STRENGTH
Short Answer:Shockley,Brattain andBardeen won a Nobel prize for what small invention?
Short Answer: What mechanical and electronic device has a name derived from a Czechoslovakian
word meaning "work; compulsory service"?
Short Answer: What is the name of the temperature and pressure conditions at which water can be in
the solid, liquid and gas phases simultaneously?
Multiple Choice: Which of the following colors of visible light has the longest wavelength? Is it:
w) violet
x) green
y) yellow
z) red
ANSWER: Z – RED
Multiple Choice: A 10 kilogram mass rests on a horizontal frictionless surface. A horizontal force of 5
Newtons is applied to the mass. After the force has been applied for 1 second, the velocity of the mass
w) 0 meters per second
x) 0.5 meters per second
y) 5 meters per second
z) 50 meters per second
ANSWER: X -- 0.5 METERS PER SECOND
Multiple Choice: A worker lifts a 10 kilogram block a vertical height of 2 meters. The work he does on
the block is:
w) 5 Joules
x) 20 Joules
y) 49 Joules
z) 200 Joules
ANSWER: Z -- 200 JOULES
Science Bowl Practice Questions
Physics -12
Multiple Choice: An impulse of 10 kilogram-meter per second acting on an object whose mass is 5
kilogram will cause a change in the objects velocity of:
w) 0.5 meters per second
x) 2 meters per second
y) 10 meters per second
z) 50 meters per second
ANSWER: X -- 2 METERS PER SECOND
Science Bowl Practice Questions
Physics -13
Multiple Choice: The time needed for a net force of 10newtons to change the velocity of a 5 kilograms
mass by 3 meters/second is:
w) 1.5 seconds
x) 6 seconds
y) 16.7 seconds
z) 150 seconds
ANSWER: W -- 1.5 SECONDS
Multiple Choice: The value of G, the universal gravitational constant, was measured experimentally
w) Newton
y) Copernicus
ANSWER: X -- CAVENDISH
Multiple Choice: Two steel balls are at a distance S from one another. As the mass of ONE of the balls
is doubled, the gravitational force of attraction between them is:
w) quartered
x) halved
y) doubled
z) quadrupled
ANSWER: Y -- DOUBLED
Multiple Choice: If the distance between the earth and moon were halved, the force of the attraction
between them would be:
w) one fourth as great
x) one half as great
y) twice as great
z) four times as great
ANSWER: Z -- FOUR TIMES AS GREAT
Multiple Choice: As a 10 kilogram mass on the end of a spring passes through its equilibrium position,
the kinetic energy of the mass is 20 joules. The speed of the mass is:
w) 2.0 meters per second
x) 4.0 meters per second
y) 5.0 meters per second
z) 6.3 meters per second
ANSWER: W -- 2.0 METERS PER SECOND
Multiple Choice: As a longitudinal wave moves through a medium, the particles of the medium:
w) vibrate in a path parallel to the path of the wave
x) vibrate in a path perpendicular to the path of the wave
y) follow the wave along its entire path
z) do not move
ANSWER: W -- VIBRATE IN A PATH PARALLEL TO THE PATH OF THE WAVE
Multiple Choice: As a pendulum is raised to higher altitudes, its period:
Science Bowl Practice Questions
Physics -14
w) increases
x) decreases
y) remains the same
z) decreases, then remains the same
ANSWER: W -- INCREASES
Multiple Choice: Two vibrating particles that are "out of phase" differ in the phase of their vibration
w) 1/4 cycle
x) 1/2 cycle
y) 3/4 cycle
z) 1 cycle
ANSWER: X -- 1/2 CYCLE
Multiple Choice: The SI unit of pressure is the:
x) dyne per centimeter squared
y) atmosphere
ANSWER: Z – PASCAL
Multiple Choice: An electroscope charged WITHOUT contacting a charged body is charged by:
w) induction
x) conduction
y) convection
z) insulation
ANSWER: W -- INDUCTION
Multiple Choice: The potential drop between the terminals of a battery is equal to the battery's EMF
w) no current is drawn from the battery
x) a very large current is drawn from the battery
y) the internal resistance of the battery is very large
z) the resistance in the external circuit is small
ANSWER: W -- NO CURRENT IS DRAWN FROM THE BATTERY
Multiple Choice: To convert a galvanometer to a voltmeter, you should add a:
w) high resistance in series
x) high resistance in parallel
y) low resistance in series
z) low resistance in parallel
ANSWER: W -- HIGH RESISTANCE IN SERIES
Multiple Choice: The greatest induced EMF will occur in a straight wire moving at constant speed
through a uniform magnetic field when the angle between the direction of the wire's motion and the
direction of the magnetic field is
w) 0 degrees
x) 30 degrees
Science Bowl Practice Questions
Physics -15
y) 60 degrees
z) 90 degrees
ANSWER: Z -- 90 DEGREES
Multiple Choice: A 10 volt battery connected to a capacitor delivers a charge of 0.5 coulombs. The
capacitance of the capacitor is:
w) 2 x 10-2 farads
x) 5 x 10-2 farads
y) 2 farads
z) 5 farads
ANSWER: X -- 5 x 10-2 FARADS
Multiple Choice: Two light rays will interfere constructively with maximum amplitude if the path
difference between them is:
w) one wavelength
x) one-half wavelength
y) one-quarter wavelength
z) one-eighth wavelength
ANSWER: W -- ONE WAVELENGTH
Multiple Choice: Light is normally incident on a thin soap film and is reflected. If the wavelength of
this light is "L" and the index of refraction of the soap film is "N", complete destructive interference
will occur for a film thickness of:
w) L / 8N
x) L / 4N
y) L / 2N
z) 3L / 4N
ANSWER: Y -- L / 2N
Multiple Choice: TheMichelson interferometer was designed to study the nature of:
w) water waves
x) sound waves
y) an "ether"
z) sunlight
ANSWER: Y -- AN "ETHER"
Multiple Choice: TheMillikan experiment showed that electric charge was:
w) negative
y) positive
ANSWER: X -- QUANTIZED
Multiple Choice: When a metal becomes a superconductor, there is a tremendous decrease in its:
w) total volume
x) electrical resistance
y) length
z) density
Science Bowl Practice Questions
Physics -16
ANSWER: X -- ELECTRICAL RESISTANCE
Multiple Choice: An x-ray photon collides with a free electron, and the photon is scattered. During this
collision there is conservation of:
w) momentum but not energy
x) neither momentum nor energy
y) energy but not momentum
z) both momentum and energy
ANSWER: Z -- BOTH MOMENTUM AND ENERGY
Multiple Choice: In the sun, helium is produced from hydrogen by:
w) radioactive decay
x) disintegration
y) fission
z) fusion
ANSWER: Z – FUSION
Multiple Choice: The half-life of an isotope of an element is 5 days. The mass of a 10 gram sample of
this isotope remaining after 20 days is:
w) 0.312 grams
x) 0.625 grams
y) 1.25 grams
z) 2.50 grams
ANSWER: X -- 0.625 GRAMS
Multiple Choice: The idea that electrons revolved in orbits around the nucleus of an atom without
radiating energy away from the atom was postulated by:
w) Thompson
z) Einstein
ANSWER: X -- BOHR
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sing MLSE
Equalize linearly modulated signal using MLSE
y = mlseeq(x,chcffs,const,tblen,opmode) equalizes the baseband signal vector x using the maximum likelihood sequence estimation (MLSE). chcffs provides estimated channel coefficients. const provides
the ideal signal constellation points. tblen specifies the traceback depth. opmode specifies the operation mode of the equalizer. MLSE is implemented using the Viterbi Algorithm.
y = mlseeq(___,nsamp) specifies the number of samples per symbol in x, in addition to arguments in the previous syntax.
y = mlseeq(___,nsamp,preamble,postamble) specifies the number of samples per symbol in x, preamble, and postamble, in addition to arguments in the first syntax. This syntax applies when opmode is
'rst' only. For more information, see Preamble and Postamble in Reset Operation Mode.
y = mlseeq(___,nsamp,init_metric,init_states,init_inputs)specifies the number of samples per symbol in x, initial likelihood state metrics, initial traceback states, and initial traceback inputs for
the equalizer, in addition to arguments in the first syntax. These three inputs are typically the final_metric, final_states, and final_inputs outputs from a previous call to this function. This
syntax applies when opmode is 'cont' only. For more information, see Initialization in Continuous Operation Mode.
[y,final_metric,final_states,final_inputs] = mlseeq(___) returns the normalized final likelihood state metrics, final traceback states, and final traceback inputs at the end of the traceback decoding
process, using any of the previous input argument syntaxes. This syntax applies when opmode is 'cont' only. For more information, see Initialization in Continuous Operation Mode.
Using MLSE Equalizer Reset Operating Mode
Use the reset operating mode of the mlseeq equalizer. Demodulate the signal and check the bit error rate.
Specify the modulation order, equalizer traceback depth, number of samples per symbol, and message length.
M = 2;
tblen = 10;
nsamp = 2;
msgLen = 1000;
Generate the reference constellation.
const = pammod([0:M-1],M);
Generate a message with random data. Modulate and upsample the signal.
msgData = randi([0 M-1],msgLen,1);
msgSym = pammod(msgData,M);
msgSymUp = upsample(msgSym,nsamp);
Filter the data through a distortion channel and add Gaussian noise to the signal.
chanest = [0.986; 0.845; 0.237; 0.12345+0.31i];
msgFilt = filter(chanest,1,msgSymUp);
msgRx = awgn(msgFilt,5,'measured');
Equalize and then demodulate the signal to recover the message. To initialize the equalizer, provide the channel estimate, reference constellation, equalizer traceback depth, number of samples per
symbol, and set the operating mode to reset. Check the message bit error rate. Your results might vary because this example uses random numbers.
eqSym = mlseeq(msgRx,chanest,const,tblen,'rst',nsamp);
eqMsg = pamdemod(eqSym,M);
[nerrs ber] = biterr(msgData, eqMsg)
Recover Message Containing Preamble
Recover a message that includes a preamble, equalize the signal, and check the symbol error rate.
Specify the modulation order, equalizer traceback depth, number of samples per symbol, preamble, and message length.
M = 4;
tblen = 16;
nsamp = 1;
preamble = [3;1];
msgLen = 500;
Generate the reference constellation.
Generate a message by using random data and prepend the preamble to the message. Modulate the random data.
msgData = randi([0 M-1],msgLen,1);
msgData = [preamble; msgData];
msgSym = pskmod(msgData,M);
Filter the data through a distortion channel and add Gaussian noise to the signal.
chcoeffs = [0.623; 0.489+0.234i; 0.398i; 0.21];
chanest = chcoeffs;
msgFilt = filter(chcoeffs,1,msgSym);
msgRx = awgn(msgFilt,9,'measured');
Equalize the received signal. To configure the equalizer, provide the channel estimate, reference constellation, equalizer traceback depth, operating mode, number of samples per symbol, and preamble.
The same preamble symbols appear at the beginning of the message vector and in the syntax for mlseeq. Because the system does not use a postamble, an empty vector is specified as the last input
argument in this mlseeq syntax.
Check the symbol error rate of the equalized signal. Run-to-run results vary due to use of random numbers.
eqSym = mlseeq(msgRx,chanest,const,tblen,'rst',nsamp,preamble,[]);
[nsymerrs,ser] = symerr(msgSym,eqSym)
Using MLSE Equalizer Continuous Operating Mode
Use the continuous operating mode of the mlseeq equalizer. Demodulate received signal packets and check the symbol error statistics.
Specify the modulation order, equalizer traceback depth, number of samples per symbol, message length, and number of packets to process.
M = 4;
tblen = 10;
nsamp = 1;
msgLen = 1000;
numPkts = 25;
Generate the reference constellation.
Set the initial input parameters for the metric, states, and inputs of the equalizer to empty vectors. These initial assignments represent the parameters for the first packet transmitted.
eq_metric = [];
eq_states = [];
eq_inputs = [];
Assign variables for symbol error statistics.
ttlSymbErrs = 0;
aggrPktSER = 0;
Send and receive multiple message packets in a simulation loop. Between the packet transmission and reception filter each packet through a distortion channel and add Gaussian noise.
Generate a message with random data. Modulate the signal.
msgData = randi([0 M-1],msgLen,1);
msgMod = pskmod(msgData,M);
Filter the data through a distortion channel and add Gaussian noise to the signal.
chanest = [.986; .845; .237; .12345+.31i];
msgFilt = filter(chanest,1,msgMod);
msgRx = awgn(msgFilt,10,'measured');
Equalize the received symbols. To configure the equalizer, provide the channel estimate, reference constellation, equalizer traceback depth, operating mode, number of samples per symbol, and the
equalizer initialization information. Continuous operating mode is specified for the equalizer. In continuous operating mode, the equalizer initialization information (metric, states, and inputs) are
returned and used as inputs in the next iteration of the for loop.
[eqSym,eq_metric,eq_states,eq_inputs] = ...
mlseeq(msgRx,chanest,const,tblen,'cont',nsamp, ...
Save the symbol error statistics. Update the symbol error statistics with the aggregate results. Display the total number of errors. Your results might vary because this example uses random numbers.
[nsymerrs,ser] = symerr(msgMod(1:end-tblen),eqSym(tblen+1:end));
ttlSymbErrs = ttlSymbErrs + nsymerrs;
aggrPktSER = aggrPktSER + ser;
printTtlErr = 'A total of %d symbol errors over the %d packets received.\n';
A total of 167 symbol errors over the 25 packets received.
Display the aggregate symbol error rate.
printAggrSER = 'The aggregate symbol error rate was %6.5d.\n';
The aggregate symbol error rate was 6.74747e-03.
Input Arguments
x — Input signal
Input signal, specified as a vector of modulated symbols. The vector length of x must be an integer multiple of nsamp.
Data Types: double
Complex Number Support: Yes
chcffs — Channel coefficients
Channel coefficients, specified as a vector. The channel coefficients provide an estimate of the channel response. When nsamp > 1, the chcffs input specifies the oversampled channel coefficients.
Data Types: double
Complex Number Support: Yes
const — Reference constellation
Reference constellation, specified as a vector with M elements. M is the modulation order. const lists the ideal signal constellation points in the sequence used by the modulator.
Data Types: double
Complex Number Support: Yes
tblen — Traceback depth
positive integer
Traceback depth, specified as a positive integer. The equalizer traces back from the likelihood state with the maximum metric.
Data Types: double
opmode — Operation mode
'rst' | 'cont'
Operation mode, specified as 'rst' or 'cont'.
Value Usage
Run equalizer using reset operating mode. Enables you to specify a preamble and postamble that accompany the input signal. The function processes the input signal, x, independently of the
'rst' input signal from any other invocations of this function. This operating mode does not incur an output delay. For more information, see Preamble and Postamble in Reset Operation Mode.
Run equalizer using continuous operating mode. Enables you to save the internal state information of the equalizer for use in a subsequent invocation of this function. Continuous operating
'cont' mode is useful if the input signal is partitioned into a stream of packets processed within a loop. This operating mode incurs an output delay of tblen symbols. For more information, see
Initialization in Continuous Operation Mode.
Data Types: char
nsamp — Number of samples per symbol
1 (default) | positive integer
Number of samples per symbol, specified as a positive integer. nsamp is the oversampling factor.
The input signal, x, must be an integer multiple of nsamp.
Data Types: double
preamble — Input signal preamble
vector of integers
Input signal preamble, specified as a vector of integers between 0 and M–1, where M is the modulation order. To omit a preamble, specify [].
For more information, see Preamble and Postamble in Reset Operation Mode.
This input argument applies only when opmode is set to 'rst'.
Data Types: double
postamble — Input signal postamble
vector of integers
Input signal postamble, specified as a vector of integers between 0 and M–1, where M is the modulation order. To omit a postamble, specify [].
For more information, see Preamble and Postamble in Reset Operation Mode.
This input argument applies only when opmode is set to 'rst'.
Data Types: double
init_metric — Initial state metrics
[ ] (default) | column vector
Initial state metrics, specified as a column vector with N[states] elements. For the description of N[states], see Number of Likelihood States.
For more information, see Initialization in Continuous Operation Mode.
This input argument applies only when opmode is set to 'cont'. If specifying [] for init_metric, you must also specify [] for init_states and init_inputs.
Data Types: double
init_states — Initial traceback states
[ ] (default) | matrix of integers
Initial traceback states, specified as an N[states]-by-tblen matrix of integers with values between 0 and N[states]–1. For the description of N[states], see Number of Likelihood States.
For more information, see Initialization in Continuous Operation Mode.
This input argument applies only when opmode is set to 'cont'. If specifying [] for init_states, you must also specify [] for init_metric and init_inputs.
Data Types: double
init_inputs — Initial traceback inputs
[] (default) | matrix of integers
Initial traceback inputs, specified as an N[states]-by-tblen matrix of integers with values between 0 and M–1. For the description of N[states], see Number of Likelihood States.
For more information, see Initialization in Continuous Operation Mode.
This input argument applies only when opmode is set to 'cont'. If specifying [] for init_inputs, you must also specify [] for init_metric and init_states.
Data Types: double
Output Arguments
y — Output signal
Output signal, returned as a vector of modulated symbols.
final_metric — Final normalized state metrics
Final normalized state metrics, returned as a vector with N[states] elements. final_metric corresponds to the final state metrics at the end of the traceback decoding process. For the description of
N[states], see Number of Likelihood States.
For more information, see Initialization in Continuous Operation Mode.
final_states — Final traceback states
Final traceback states, returned as a N[states]-by-tblen matrix of integers with values between 0 and N[states]–1. final_states corresponds to the final traceback states at the end of the traceback
decoding process. For the description of N[states], see Number of Likelihood States.
For more information, see Initialization in Continuous Operation Mode.
final_inputs — Final traceback inputs
Final traceback inputs, returned as an N[states]-by-tblen matrix of integers with values between 0 and M–1. final_inputs corresponds to the final traceback inputs at the end of the traceback decoding
process. M is the order of the modulation. For the description of N[states], see Number of Likelihood States.
For more information, see Initialization in Continuous Operation Mode.
More About
Viterbi Algorithm
The Viterbi algorithm is a sequential trellis search algorithm used to perform maximum likelihood sequence detection.
The MLSE equalizer uses the Viterbi algorithm to recursively search for the sequences that maximize the likelihood function. Using the Viterbi algorithm reduces the number of sequences in the trellis
search by eliminating sequences as new data is received. The metric used to determine the maximum likelihood sequence is the correlation between the received signal and an estimated signal for each
received symbol over the Number of Likelihood States.
For more information, see [1] and [2].
Preamble and Postamble in Reset Operation Mode
When operating the MLSE equalizer in reset mode, you can specify a preamble and postamble as input arguments. Specify preamble and postamble as vectors equal to the preamble and postamble that are
prepended and appended, respectively, to the input signal. The preamble and postamble vectors consist of integers between 0 and M-1, where M is the number of elements in const. To omit the preamble
or postamble input argument, specify [].
When the function applies the Viterbi algorithm, it initializes state metrics in a way that depends on whether you specify a preamble, a postamble, or both:
• If preamble is nonempty, the function decodes the preamble and assigns a metric of 0 to the decoded state. If the preamble does not decode to a unique state (that is, if the length of the
preamble is less than the channel memory), the decoder assigns a metric of 0 to all states that are represented by the preamble. The traceback path ends at one of the states represented by the
• If preamble is [], the decoder initializes the metrics of all states to 0.
• If postamble is nonempty, the traceback path begins at the smallest of all possible decoded states that are represented by the postamble.
• If postamble is [], the traceback path starts at the state with the smallest metric.
Initialization in Continuous Operation Mode
When operating the MLSE equalizer in continuous mode, you can initialize the equalization based on values returned in the previous call of the function.
At the end of the traceback decoding process, the function returns final_metric, final_states, and final_inputs. When opmode is 'cont', assign these outputs to init_metric, init_states, and
init_inputs, respectively for the next call of the function. These assignments initialize the equalizer to start with the final state metrics, final traceback states, and final traceback inputs from
the previous call of the function.
Each real number in init_metric represents the starting state metric of the corresponding state. init_states and init_inputs jointly specify the initial traceback memory of the equalizer.
Output Argument Input Argument Meaning Matrix Size Range of Values
final_metric init_metric State metrics 1-by-N[states] Real numbers
final_states init_states Traceback states N[states]-by-tblen Integers between 0 and N[states]–1
fianl_inputs init_inputs Traceback inputs N[states]-by-tblen Integers between 0 and M–1
To use default values for init_metric, init_states, and init_inputs, specify each as []. For the description of N[states], see Number of Likelihood States.
Number of Likelihood States
The number of likelihood states, N[states], is the number of correlative phase states in the trellis. N[states] is equal to M^L-1, where M is the number of elements in const and L is the number of
symbols in the nonoversampled impulse response of the channel.
[1] Proakis, John G. Digital Communications, Fourth Edition. New York: McGraw-Hill, 2001.
[2] Steele, Raymond, Ed. Mobile Radio Communications. Chichester, England: John Wiley & Sons, 1996.
Version History
Introduced before R2006a
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Pentagon Ring
Weekly Problem 47 - 2011
Place equal, regular pentagons together to form a ring. How many pentagons will be needed?
Equal regular pentagons are placed together to form a ring.
The diagram shows the first three pentagons.
How many pentagons are needed to complete the ring?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Each pentagon has a turn of 36^o to pack against its neighbour.
Therefore it will take 10 (x36^o) turns to complete a full-turn, i.e. 10 pentagons
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JavaScript Math: Calculate the falling factorial of a number - w3resource
JavaScript: Calculate the falling factorial of a number
JavaScript Math: Exercise-48 with Solution
Write a JavaScript function to calculate the falling factorial of a number.
Let x be a real number (but usually an integer).
Let k be a positive integer.
Then x to the (power of) k falling is:
This is called the kth falling factorial power of x.
Sample Solution:
JavaScript Code:
// Define a function named fallingFactorial that calculates the falling factorial of 'n' to 'k'.
function fallingFactorial(n, k)
// Initialize variables 'i' and 'r'.
var i = (n - k + 1),
r = 1;
// Check if 'n' is negative.
if (n < 0)
throw new Error("n must be positive.");
// Check if 'k' is greater than 'n'.
if (k > n)
throw new Error("k cannot be greater than n.");
// Implement the falling factorial calculation.
while (i <= n)
r *= i++;
// Return the result of the falling factorial.
return r;
// Output the result of falling factorial calculation with inputs 10 and 2.
console.log(fallingFactorial(10, 2));
Live Demo:
See the Pen javascript-math-exercise-48 by w3resource (@w3resource) on CodePen.
Improve this sample solution and post your code through Disqus.
Previous: Write a JavaScript function to calculate the extended Euclid Algorithm or extended GCD.
Next: Write a JavaScript function to calculate Lanczos approximation gamma.
What is the difficulty level of this exercise?
Test your Programming skills with w3resource's quiz.
It will be nice if you may share this link in any developer community or anywhere else, from where other developers may find this content. Thanks.
• Weekly Trends and Language Statistics
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What is the volume of 2.3 * 10^25 atoms of gold if its density is 19.3g/cm^3? | Socratic
What is the volume of #2.3 * 10^25# atoms of gold if its density is #19.3##g##/##cm^3#?
1 Answer
Your strategy here will be to
• convert the number of atoms of gold to moles of gold by using Avogadro's number
• use gold's molar mass to convert the number of moles to number of grams
• use gold's density to determine what volume would contain that many grams
As you know, one mole of any element contains exactly $6.022 \cdot {10}^{23}$ atoms of that element - this is known as Avogadro's number.
So, if you need $6.022 \cdot {10}^{23}$ atoms of gold to have one mole of gold, then the number of atoms you have will be equivalent to
#2.3 * 10^(25) color(red)(cancel(color(black)("atoms Au"))) * "1 mole Au"/(6.022 * 10^(23) color(red)(cancel(color(black)("atoms Au")))) = "38.2 moles Au"#
Now, gold's molar mass will tell you what the mass of one mole of gold is. In this case, that many moles would correspond to a mass of
#38.2 color(red)(cancel(color(black)("moles Au"))) * "196.97 g"/(1 color(red)(cancel(color(black)("mole Au")))) = "7524.3 g"#
Finally, a density of ${\text{19.3 g/cm}}^{3}$ tells you that ${\text{1 cm}}^{3}$ of gold will have a mass of $19.3$ grams. In your case, the volume that would have a mass of $7524$ grams is
#7524.3 color(red)(cancel(color(black)("g"))) * "1 cm"^3/(19.3 color(red)(cancel(color(black)("g")))) = "389.9 cm"^3#
Rounded to two sig figs, the number of sig figs you have for the number of atoms of gold, the answer will be
$V = \textcolor{g r e e n}{{\text{390 cm}}^{3}}$
Impact of this question
6379 views around the world
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Check out Vanessa Moore's tutor profile on Skooli
I am a math teacher with 28 years of classroom experience. I love working with students and very good at breaking down the math into simplest terms. I have experience teaching Geometry, Algebra 2,
Algebra 1, Pre-Calculus, Intro to Statistics and Calculus and IB Math Studies. I also can tutor Year one and two German students.
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König's Lemma
How do you show that a simulation preserves non-termination? I think you could probably use coinduction, but I'm not very familiar with coinductive arguments. I just learned about a useful result
called König's Lemma, which I think allows you to use a simple induction.
Let's say we have a simulation relation e′ ~ e ("e′ simulates e") and a proof that for any step in a specification semantics:
e[1] → e[2]
we have related terms e
′ ~ e
and e
′ ~ e
such that
e[1]′ →^+ e[2]′
It's easy to show by induction that if the specification semantics converges to a value then the implementation semantics converges to a related value. If the specification semantics diverges, i.e.
has an infinite reduction sequence, then we'd like to show the implementation diverges too.
König's Lemma
states that for any finitely branching tree, if there exists a path of length
from the root for any
, then there exists an infinite path from the root. Consider the tree of possible reduction sequences from a term, where branches indicate points of non-determinism in the semantics. If every point
of non-determinism has only a finite number of alternative reductions, then the tree is finitely branching.
So now consider a diverging term
in the specification semantics. For any finite prefix of the infinite reduction sequence, we can easily show by induction that there is a reduction sequence in the implementation semantics of equal
or greater length. Since the computation tree of the implementation term is finitely branching, König's Lemma provides an infinite reduction of the implementation.
2 comments:
i didn't know how to contact you, so i'm posting a message on your blog-- hope that's ok. i've been wanting to know how to write "timshel" in hebrew, and i was wondering how you found out how to
write it. is that the word you have in hebrew in the lefthand corner of your home page?
you can e-mail me at kimmi275@ufl.edu
Arnar Birgisson said...
Hi Dave,
Google reader cunningly recommended your blog to me. It was a bit frightening how spot on it was :o)
Here is a small similar useage of the König's lemma:
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Accelerate Python with Numba’s
Accelerate Python with Numba’s @jit(nopython=True)
🚀 Accelerate Python with Numba’s @jit(nopython=True) 🚀
Are you looking to optimize your Python code for better performance? If you work with large datasets or run complex numerical computations, the Numba library can be a game-changer!
With @jit(nopython=True), Numba translates Python functions into machine code using Just-In-Time (JIT) compilation. This drastically reduces execution time, especially for loops and numerical
Let me show you how it works! 👇
🚀 What is @jit(nopython=True)?
@jit(nopython=True) is a decorator from the Numba library. It compiles the entire function into machine code at runtime. Here’s why it’s special:
• nopython=True: Forces Numba to fully compile the function to machine code, skipping the Python interpreter. This ensures maximum performance.
• It’s great for numerical computing or operations involving large arrays, matrices, or loops.
💡 If Numba detects a dynamic type (like Python objects), it will throw an error with nopython=True, ensuring you stay in the compiled mode.
🛠️ Example: Summing Squares Without Numba
Let’s start with a simple Python function that computes the sum of squares of a list of numbers.
import time
# Regular Python function (without Numba)
def sum_of_squares(arr):
total = 0
for num in arr:
total += num * num
return total
arr = list(range(10000000)) # A list of 10 million numbers
start_time = time.time()
result = sum_of_squares(arr)
end_time = time.time()
print(f"Result: {result}")
print(f"Time taken (without Numba): {end_time - start_time:.4f} seconds")
⏳ Output:
Result: 333332833333500000
Time taken (without Numba): X.XXXX seconds
⚡ Now, Let’s Optimize It with @jit(nopython=True)
Here’s the same function, but this time we’ll add Numba’s @jit(nopython=True) to speed it up.
import time
from numba import jit
# Numba-optimized function using @jit(nopython=True)
def sum_of_squares_jit(arr):
total = 0
for num in arr:
total += num * num
return total
arr = list(range(10000000)) # A list of 10 million numbers
start_time = time.time()
result = sum_of_squares_jit(arr)
end_time = time.time()
print(f"Result: {result}")
print(f"Time taken (with Numba): {end_time - start_time:.4f} seconds")
⚡ Output:
Result: 333332833333500000
Time taken (with Numba): Y.YYYY seconds
🧑🏫 Why Is Numba So Fast?
• Machine Code Execution: Numba converts Python code into machine instructions, which run directly on the CPU.
• Optimized Loops: Python’s for-loops are notoriously slow. Numba transforms these into fast, compiled loops.
• No Python Overhead: By skipping the Python interpreter, Numba eliminates overhead from Python’s dynamic type system.
🚀 Real-World Performance Gains
In this example, for 10 million numbers, the Numba-optimized function can run several times faster compared to the pure Python version. With even larger datasets or more complex computations, the
performance gains will be even more impressive!
🔥 More Examples of Using @jit(nopython=True)
Example 1: Computing Factorials
from numba import jit
def factorial(n):
result = 1
for i in range(2, n+1):
result *= i
return result
print(factorial(10)) # Output: 3628800
Example 2: Matrix Multiplication
import numpy as np
from numba import jit
def matrix_mult(A, B):
rows_A, cols_A = A.shape
rows_B, cols_B = B.shape
result = np.zeros((rows_A, cols_B))
for i in range(rows_A):
for j in range(cols_B):
for k in range(cols_A):
result[i, j] += A[i, k] * B[k, j]
return result
A = np.random.rand(1000, 1000)
B = np.random.rand(1000, 1000)
result = matrix_mult(A, B)
🚀 Conclusion
Numba is a powerful tool to boost the performance of your Python code, especially when dealing with numerical computations and large datasets. Using @jit(nopython=True), you can transform your slow
Python functions into blazing-fast machine code with minimal effort.
Next time you need to optimize your Python code, give Numba a try!
🔗 Resources:
Feel free to connect with me for more tips on optimizing your Python code! 👨💻
#Python #Numba #MachineLearning #PerformanceOptimization #BigData #DataScience
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Math, Grade 6, Putting Math to Work
Putting Math to Work
Type of Unit: Problem Solving
Prior Knowledge
Students should be able to:
• Solve problems with rational numbers using all four operations.
• Write ratios and rates.
• Use a rate table to solve problems.
• Write and solve proportions.
• Use multiple representations (e.g., tables, graphs, and equations) to display data.
• Identify the variables in a problem situation (i.e., dependent and independent variables).
• Write formulas to show the relationship between two variables, and use these formulas to solve for a problem situation.
• Draw and interpret graphs that show the relationship between two variables.
• Describe graphs that show proportional relationships, and use these graphs to make predictions.
• Interpret word problems, and organize information.
• Graph in all quadrants of the coordinate plane.
Lesson Flow
As a class, students use problem-solving steps to work through a problem about lightning. In the next lesson, they use the same problem-solving steps to solve a similar problem about lightning. The
lightning problems use both rational numbers and rates. Students then choose a topic for a math project. Next, they solve two problems about gummy bears using the problem-solving steps. They then
have 3 days of Gallery problems to test their problem-solving skills solo or with a partner. Encourage students to work on at least one problem individually so they can better prepare for a testing
situation. The unit ends with project presentations and a short unit test.
Material Type:
Unit of Study
Middle School
to add tags to this item.
Unit 7 Putting Math to Work
Education Standards
Learning Domain: Ratios and Proportional Relationships
Standard: Use ratio and rate reasoning to solve real-world and mathematical problems.
Learning Domain: Ratios and Proportional Relationships
Standard: Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to
compare ratios.
Learning Domain: Ratios and Proportional Relationships
Standard: Solve unit rate problems including those involving unit pricing and constant speed.
Learning Domain: Ratios and Proportional Relationships
Standard: Use ratio reasoning to convert measurement units; convert units appropriately when multiplying or dividing quantities.
Learning Domain: Expressions and Equations
Standard: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the
purpose at hand, any number in a specified set.
Learning Domain: Expressions and Equations
Standard: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Learning Domain: Expressions and Equations
Standard: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent
variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to
the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between
distance and time.
Learning Domain: The Number System
Standard: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Learning Domain: The Number System
Standard: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points
with the same first coordinate or the same second coordinate.
Learning Domain: Ratios and Proportional Relationships
Standard: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Learning Domain: Ratios and Proportional Relationships
Standard: Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to
compare ratios.
Learning Domain: Ratios and Proportional Relationships
Standard: Solve unit rate problems including those involving unit pricing and constant speed. For example, If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35
hours? At what rate were lawns being mowed?
Learning Domain: Ratios and Proportional Relationships
Standard: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Learning Domain: Mathematical Practices
Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its
solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a
solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their
progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient
students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex
problems and identify correspondences between different approaches.
Learning Domain: Mathematical Practices
Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary
abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto abstract a given situation and represent it symbolically and manipulate the representing symbols
as if they have a life of their own, without necessarily attending to their referents"Óand the ability to contextualize, to pause as needed during the manipulation process in order to probe into the
referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Learning Domain: Mathematical Practices
Standard: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in
constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases,
and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct
logic or reasoning from that which is flawed, and"Óif there is a flaw in an argument"Óexplain what it is. Elementary students can construct arguments using concrete referents such as objects,
drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to
which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Learning Domain: Mathematical Practices
Standard: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this
might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the
community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who
can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important
quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to
draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served
its purpose.
Learning Domain: Mathematical Practices
Standard: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper,
concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with
tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example,
mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with
data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or
solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
Learning Domain: Mathematical Practices
Standard: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They
state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the
correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary
grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
Learning Domain: Mathematical Practices
Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is
the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 x 8 equals the well remembered 7 x 5 + 7 x 3,
in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 7 and the 9 as 2 + 7. They recognize the significance of an existing
line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things,
such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 - 3(x - y)^2 as 5 minus a positive number times a square and use that to
realize that its value cannot be more than 5 for any real numbers x and y.
Learning Domain: Mathematical Practices
Standard: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper
elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the
calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x -1) = 3. Noticing the
regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)(x^2 + x + 1), and (x - 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As
they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to
compare ratios.
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Solve unit rate problems including those involving unit pricing and constant speed. For example, If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35
hours? At what rate were lawns being mowed?
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Cluster: Compute fluently with multi-digit numbers and find common factors and multiples
Standard: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points
with the same first coordinate or the same second coordinate.
Cluster: Reason about and solve one-variable equations and inequalities
Standard: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the
purpose at hand, any number in a specified set.
Cluster: Reason about and solve one-variable equations and inequalities
Standard: Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Cluster: Represent and analyze quantitative relationships between dependent and independent variables
Standard: Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent
variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to
the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between
distance and time.
Cluster: Mathematical practices
Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its
solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a
solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their
progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to
get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and
relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient
students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex
problems and identify correspondences between different approaches.
Cluster: Mathematical practices
Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary
abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols
as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the
referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of
quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
Cluster: Mathematical practices
Standard: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in
constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases,
and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible
arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct
logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings,
diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an
argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
Cluster: Mathematical practices
Standard: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this
might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the
community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who
can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important
quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to
draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served
its purpose.
Cluster: Mathematical practices
Standard: Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper,
concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with
tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example,
mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with
data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or
solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
Cluster: Mathematical practices
Standard: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They
state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the
correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary
grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
Cluster: Mathematical practices
Standard: Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is
the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3,
in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing
line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things,
such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)^2 as 5 minus a positive number times a square and use that to
realize that its value cannot be more than 5 for any real numbers x and y.
Cluster: Mathematical practices
Standard: Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper
elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the
calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x –1) = 3. Noticing the
regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As
they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate
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Analysis of Statically Indeterminate Beams by Force Method
1. Analysis of Indeterminate Structures by Force Method - An Overview
2. Introduction
3. Method of Consistent Deformation
4. Indeterminate Beams
5. Indeterminate Beams with Multiple Degree of Indeterminacy
6. Truss Structures
7. Temperature Changes & Fabrication Errors
2. Introduction
While analyzing indeterminate structures, it is necessary to satisfy (force) equilibrium, (displacement) compatibility and force-displacement relationships
a. Force equilibrium is satisfied when the reactive forces hold the structure in stable equilibrium, as the structure is subjected to external loads
b. Displacement compatibility is satisfied when the various segments of the structure fit together without intentional breaks, or overlaps
Force-displacement requirements depend on the manner the material of the structure responds to the applied loads, which can be linear/nonlinear/viscous and elastic/inelastic; for our study the
behavior is assumed to be linear and elastic
Two methods are available to analyze indeterminate structures, depending on whether we satisfy force equilibrium or displacement compatibility conditions. They are: Force method and Displacement
Force Method satisfies displacement compatibility and force-displacement relationships; it treats the forces as unknowns - Two methods which we will be studying are Method of Consistent Deformation
and (Iterative Method of) Moment Distribution
Displacement Method satisfies force equilibrium and force-displacement relationships; it treats the displacements as unknowns - Two available methods are Slope Deflection Method and Stiffness
(Matrix) method
3. Solution Procedure:
i. Make the structure determinate, by releasing the extra forces constraining the structure in space
ii. Determine the displacements (or rotations) at the locations of released (constraining) forces
iii. Apply the released (constraining) forces back on the structure (To standardize the procedure, only a unit load of the constraining force is applied in the +ve direction) to produce the same
deformation(s) on the structure as in (ii)
iv. Sum up the deformations and equate them to zero at the position(s) of the released (constraining) forces, and calculate the unknown restraining forces
Types of Problems to be dealt:
1. Indeterminate beams;
2. Indeterminate trusses; and
3. Influence lines for indeterminate structures
4.1 Propped Cantilever - Redundant vertical reaction released
i. Propped Cantilever: The structure is indeterminate to the first degree; hence has one unknown in the problem.
ii. In order to solve the problem, release the extra constraint and make the beam a determinate structure. This can be achieved in two different ways, viz.,
a. The governing compatibility equation at B is
b. By releasing the moment constraint at A, and making the structure a simply supported beam (which is once again, a determinate beam).
Overview of Method of Consistent Deformation
To recapitulate on what we have done earlier, Structure with single degree of indeterminacy:
(a) Remove the redundant to make the structure determinate (primary structure)
(b) Apply unit force on the structure, in the direction of the redundant, and find the displacement
Δ B0 + fBB x RB = 0
5. Indeterminate beam with Multiple Degrees of Indetermincay
(a) Make the structure determinate (by releasing the supports at B, C and D) and determine the deflections at B, C and D in the direction of removed redundant, viz.,Δ BO, Δ CO and Δ DO
(b) Apply unit loads at B, C and D, in a sequential manner and determine deformations at B, C and D, respectively.
(c ) Establish compatibility conditions at B, C and D
Δ BO + fBBRB + fBCRC + fBDRD = 0
Δ CO + fCBRB + fCCRC + fCDRD = 0
Δ DO + fDBRB + fDCRC + fDDRD = 0
Compatibility conditions at B, C and D give the following equations:
Δ BO + fBBRB + fBCRC + fBDRD = Δ B
Δ CO + fCBRB + fCCRC + fCDRD = Δ C
Δ DO + fDBRB + fDCRC + fDDRD = Δ D
6. Truss Structures
(a) Remove the redundant member (say AB) and make the structure a primary determinate structure
The condition for stability and indeterminacy is:
r + m > = < 2j
Since, m = 6, r = 3, j = 4, (r + m =) 3 + 6 > (2j =) 2 x 4 or 9 > 8 Δ i = 1
(b) Find deformation Δ ABO along AB:
Δ ABO =Δ (F0uABL)/AE
F0 = Force in member of the primary structure due to applied load
uAB= Forces in members due to unit force applied along AB
(c) Determine deformation along AB due to unit load applied along AB:
(d) Apply compatibility condition along AB:
Hence determine FAB
(e) Determine the individual member forces in a particular member CE by
FCE = FCE0 + uCE FAB
where FCE0 = force in CE due to applied loads on primary structure (=F0), and uCE = force in CE due to unit force applied along AB (= uAB)
7. Temperature changes affect the internal forces in a structure
Similarly fabrication errors also affect the internal forces in a structure
(i) Subject the primary structure to temperature changes and fabrication errors. - Find the deformations in the redundant direction
Reintroduce the removed members back and make the deformation compatible
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SQRT shoot
SQRT shoot --- Introduction ---
This is a visual exercise on the complex numbers: any non-zero complex number has exactly n complex roots of degree n, for n>1.
You are shown the plane of complex numbers, together with a point in this plane representing a complex number z. And you have only to click (shoot) on the complex plane, at the points representing
the (square, cubic, quartic, or quintic) roots of z. You will have a score computed in function of the average precision of your shots.
You can with a difficulty level
You can also try Complex shoot which asks you to click on an algebraic expression of a complex number.
Other exercises on: Shoot complex numbers Roots
The most recent version This page is not in its usual appearance because WIMS is unable to recognize your web browser.
Please take note that WIMS pages are interactively generated; they are not ordinary HTML files. They must be used interactively ONLINE. It is useless for you to gather them through a robot program.
Description: locate (square, cubic, ...) roots of a complex number by clicking on the complex plane. interactive exercises, online calculators and plotters, mathematical recreation and games
Keywords: interactive mathematics, interactive math, server side interactivity, algebra, complex number, complex plane, root
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How did we get the height in this problem?
What should all the angles add up to on a 4 sided shape?
Maybe with that you can figure out the angles, that might allow you to break up the shape into triangles where you can get the height using ratios.
Worth a shot
Okay, I missed up and was tunnel visioned on the upper and lower sides and forgot about the 30 60 triangle rule.
Thank you!
1 Like
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Given f(x)=−320csc(4x). Find k if f′(π)=k2 and k is an intege... | Filo
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Question Text Given . Find if and is an integer.
Updated On Sep 27, 2022
Topic Differentiation
Subject Mathematics
Class Grade 12
Answer Type Video solution: 1
Upvotes 77
Avg. Video Duration 3 min
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Correlating Risk Consequence and Likelihood | Aviation Maintenance Magazine
Correlating Risk Consequence and Likelihood
This is the fourth article in a series about Safety Management Systems (SMS). In the first article (see page 48 of the January 2020 issue of Aviation Maintenance), we examined some hazard
identification strategies (looking at ways to identify the things that could go wrong in our systems). In the second article (see page 48 of the May/June issue), we began looking at the process of
using risk assessment to analyze identified hazards by explaining how to establish a “likelihood” scale that is relevant to your business needs, and how to calculate a “likelihood” for each
identified hazard. In the third article (see page 48 of the July issue) we continued our examination of risk assessment by looking at the process of using “consequence” as a second metric for
analyzing risk.
This month we will examine how to correlate “consequence“ and “likelihood” together to get a product that represents the relative risk associated with the hazard. We can use this risk product to help
make risk mitigation decisions, and also to measure the effectiveness of our mitigation efforts.
I strongly recommend that you go back and read the first three articles if you have not looked at them recently. They are each pretty short, and they lay a foundation that will make it much easier to
understand what this article is talking about. These prior articles are available in the back issues found on the Aviation Maintenance magazine website.
To review, we have previously discussed the identification of hazards (in the January 2020 issue), and the correlative assignation of likelihood levels (from the May 2020 issue) and consequence
levels (from the July 2020 issue) to those identified hazards. If you are looking for more detail on how to assign these values, then please look at the earlier articles.
Typically, these assignments are only relevant within the system in which they are assigned. For example, one system might assign a likelihood level of 3 and a consequence level of 4 to a particular
hazard, and another system might assign very different numbers to the same hazard. This could be because the scales of the systems are defined differently, it could be because the likelihood and
consequence levels are themselves defined differently, or it could be because the hazard has different actual and potential effects within a particular system. The important thing is to use the
system that you defined so that you can assign values that provide a relative risk rating. Such a rating will be relative to risks of other hazards identified and enumerated by your system.
Let’s look at the two scales that we used as sample for a repair station SMS in the past two articles. First there is a scale for likelihood (Exhibit A):
And second there is a scale for consequence (Exhibit B):
We’ve assigned numbers to the different levels (one through five in each case). By multiplying the numbers we can get a product. The product of the two represents a risk rating. For example, if
likelihood is 4 and consequence is 4 (hazardous), then the product of the two is 16. If the likelihood of another hazard is 2 and the consequence is also 2 (minor), then the product is 4. These
numbers are not absolute, so they do not tell us anything when analyzed outside of our system; but within our system they tell us that the first hazard (with a 16 risk product) should be a higher
priority for mitigation than the second hazard (with a 4 risk product). This allows the owners of the system to prioritize their hazard mitigation projects to focus on the hazards that pose the most
significant risk.
The simple multiplicative comparison is not the only way to approach these figures. For example, if your system prioritizes consequence over likelihood, then you might consider developing risk
products by a formula like [consequence x consequence x likelihood]. This approach squares the consequence value which makes it a much greater influence on the final risk product number. For example,
in a straight multiplicative model, a hazard with a consequence of 4 and a likelihood of 3 yields a risk product of 12; and a hazard with a consequence of 3 and a likelihood of 4 also yields a risk
product of 12. They are weighted equally in such a model. But in the consequence-squared model, a hazard with a consequence of 4 and a likelihood of 3 yields a risk product of (4x4x3=) 48; while and
a hazard with a consequence of 3 and a likelihood of 4 also yields a risk product of (3x3x4=) 36. Now the first hazard is prioritized over the second one for purposes of identifying an order in which
to mitigate the hazards. Notice that our hypothetical hazards did not change, but only the way that we analyzed them changed.
The products of the likelihood and consequence numbers can also be used to help us set mitigation targets. By examining the products, the SMS system-owner can determine which risk products are
acceptable and which risk products are unacceptable. Those hazards that have risk products that are deemed to be unacceptable would need to be mitigated in order to reduce their risk products to an
acceptable level. In the next article in this series, we will discuss risk mitigation strategies.
A matrix of acceptable/unacceptable risks might look like this (Exhibit C):
In this matrix, we have established that certain risk products are considered to be acceptable, and certain risk products are unacceptable. In our sample, there is also one risk product that is
marked in yellow as “review:” this is for catastrophic hazards that would be unlikely to ever occur. When hazards in this yellow-review risk-product are identified, they will be subject to additional
review in the system to determine whether to mitigate them (so it is not acceptable nor unacceptable until the review has assessed it). When the system is newly-implemented, there may be many hazards
that pose unacceptable risks. The numerical products found my multiplying the likelihood and consequence numbers can be used as a mechanism for prioritizing hazards in order to determine which ones
to mitigate first.
The goal is, of course, to mitigate all of the hazards to an acceptable level. In our matrix, this means reducing the likelihood or consequence to a low enough level to move the risk product into the
green. Eventually, a measured approach to SMS hazards should reduce the risk associated with the known hazards to acceptable levels. But this doesn’t mean that we are done!
We can also amend our risk product matrices as experience shows us that certain risk products need to be prioritized, and also as successful mitigations help to reduce total system risk.
Perhaps, after working in the system for two years, our hypothetical SMS-owner will feel that the business has successfully mitigated the risks posed by many of the identified hazards, and now the
business is ready to begin mitigating the next round of hazards. The business might change the acceptable/unacceptable risk matrix by lowering the bar for mitigation so that the new matrix looks like
this (exhibit D) :
Notice that the new matrix has changed some acceptable risks to unacceptable, which means that the business will develop new mitigations to further reduce the risk products of hazards in those
categories to acceptable (green) levels. It is possible that some hazards that were mitigated from red-to-green in the prior matrix might need to be further mitigated after this change resets the
concept of what is acceptable.
This approach allows the business to use its risk product acceptable/unacceptable matrix as a tool for continuous safety improvement, by moving the levels of acceptable safety to force constant
In the next issue, we will look at how to use mitigations in order to reduce likelihood levels and consequence levels of identified hazards. By changing the likelihood level, consequence level, or
both, the system can effectively reduce risk posed by hazards. As we will see in future articles, this helps to drive an effective audit schedule as well as becoming an effective and objective change
management tool. Want to learn more? We have been teaching classes in SMS elements, and we have advised aviation companies in multiple sectors on the development of SMS processes and systems. Give us
a call or send us an email if we can help you with your SMS questions.
You must be logged in to post a comment.
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Interval Notation - Definition, Examples, Types of Intervals - Grade Potential Clear Water, FL
Interval Notation - Definition, Examples, Types of Intervals
Interval Notation - Definition, Examples, Types of Intervals
Interval notation is a fundamental principle that learners need to learn due to the fact that it becomes more critical as you grow to more complex arithmetic.
If you see advances arithmetics, such as differential calculus and integral, on your horizon, then being knowledgeable of interval notation can save you hours in understanding these theories.
This article will talk in-depth what interval notation is, what are its uses, and how you can understand it.
What Is Interval Notation?
The interval notation is simply a method to express a subset of all real numbers along the number line.
An interval means the numbers between two other numbers at any point in the number line, from -∞ to +∞. (The symbol ∞ denotes infinity.)
Basic difficulties you face primarily consists of one positive or negative numbers, so it can be difficult to see the benefit of the interval notation from such straightforward utilization.
Despite that, intervals are usually employed to denote domains and ranges of functions in more complex mathematics. Expressing these intervals can increasingly become difficult as the functions
become progressively more tricky.
Let’s take a simple compound inequality notation as an example.
• x is greater than negative four but less than two
As we know, this inequality notation can be denoted as: {x | -4 < x < 2} in set builder notation. Though, it can also be expressed with interval notation (-4, 2), denoted by values a and b separated
by a comma.
So far we know, interval notation is a method of writing intervals elegantly and concisely, using set rules that make writing and understanding intervals on the number line less difficult.
In the following section we will discuss about the rules of expressing a subset in a set of all real numbers with interval notation.
Types of Intervals
Several types of intervals lay the foundation for writing the interval notation. These interval types are essential to get to know due to the fact they underpin the entire notation process.
Open intervals are applied when the expression do not comprise the endpoints of the interval. The previous notation is a great example of this.
The inequality notation {x | -4 < x < 2} describes x as being greater than -4 but less than 2, which means that it does not include either of the two numbers mentioned. As such, this is an open
interval expressed with parentheses or a round bracket, such as the following.
(-4, 2)
This means that in a given set of real numbers, such as the interval between -4 and 2, those 2 values are not included.
On the number line, an unshaded circle denotes an open value.
A closed interval is the opposite of the previous type of interval. Where the open interval does not include the values mentioned, a closed interval does. In text form, a closed interval is expressed
as any value “greater than or equal to” or “less than or equal to.”
For example, if the previous example was a closed interval, it would read, “x is greater than or equal to negative four and less than or equal to two.”
In an inequality notation, this can be expressed as {x | -4 < x < 2}.
In an interval notation, this is expressed with brackets, or [-4, 2]. This states that the interval contains those two boundary values: -4 and 2.
On the number line, a shaded circle is utilized to represent an included open value.
A half-open interval is a blend of prior types of intervals. Of the two points on the line, one is included, and the other isn’t.
Using the last example as a guide, if the interval were half-open, it would be expressed as “x is greater than or equal to -4 and less than two.” This implies that x could be the value negative four
but cannot possibly be equal to the value two.
In an inequality notation, this would be written as {x | -4 < x < 2}.
A half-open interval notation is written with both a bracket and a parenthesis, or [-4, 2).
On the number line, the shaded circle denotes the number included in the interval, and the unshaded circle indicates the value excluded from the subset.
Symbols for Interval Notation and Types of Intervals
To recap, there are different types of interval notations; open, closed, and half-open. An open interval excludes the endpoints on the real number line, while a closed interval does. A half-open
interval includes one value on the line but does not include the other value.
As seen in the examples above, there are numerous symbols for these types under the interval notation.
These symbols build the actual interval notation you create when plotting points on a number line.
• ( ): The parentheses are utilized when the interval is open, or when the two endpoints on the number line are excluded from the subset.
• [ ]: The square brackets are utilized when the interval is closed, or when the two points on the number line are included in the subset of real numbers.
• ( ]: Both the parenthesis and the square bracket are employed when the interval is half-open, or when only the left endpoint is excluded in the set, and the right endpoint is not excluded. Also
known as a left open interval.
• [ ): This is also a half-open notation when there are both included and excluded values between the two. In this case, the left endpoint is included in the set, while the right endpoint is not
included. This is also known as a right-open interval.
Number Line Representations for the Various Interval Types
Aside from being written with symbols, the various interval types can also be represented in the number line using both shaded and open circles, relying on the interval type.
The table below will display all the different types of intervals as they are described in the number line.
Practice Examples for Interval Notation
Now that you’ve understood everything you need to know about writing things in interval notations, you’re prepared for a few practice problems and their accompanying solution set.
Example 1
Transform the following inequality into an interval notation: {x | -6 < x < 9}
This sample question is a easy conversion; just use the equivalent symbols when stating the inequality into an interval notation.
In this inequality, the a-value (-6) is an open interval, while the b value (9) is a closed one. Thus, it’s going to be written as (-6, 9].
Example 2
For a school to take part in a debate competition, they require at least 3 teams. Represent this equation in interval notation.
In this word problem, let x be the minimum number of teams.
Because the number of teams needed is “three and above,” the number 3 is included on the set, which means that three is a closed value.
Furthermore, since no maximum number was mentioned regarding the number of maximum teams a school can send to the debate competition, this number should be positive to infinity.
Therefore, the interval notation should be written as [3, ∞).
These types of intervals, when one side of the interval that stretches to either positive or negative infinity, are called unbounded intervals.
Example 3
A friend wants to undertake a diet program limiting their regular calorie intake. For the diet to be a success, they must have at least 1800 calories regularly, but maximum intake restricted to 2000.
How do you describe this range in interval notation?
In this question, the number 1800 is the lowest while the number 2000 is the maximum value.
The question implies that both 1800 and 2000 are inclusive in the range, so the equation is a close interval, denoted with the inequality 1800 ≤ x ≤ 2000.
Thus, the interval notation is written as [1800, 2000].
When the subset of real numbers is restricted to a range between two values, and doesn’t stretch to either positive or negative infinity, it is also known as a bounded interval.
Interval Notation Frequently Asked Questions
How To Graph an Interval Notation?
An interval notation is basically a way of representing inequalities on the number line.
There are laws to writing an interval notation to the number line: a closed interval is expressed with a filled circle, and an open integral is written with an unfilled circle. This way, you can
promptly see on a number line if the point is included or excluded from the interval.
How To Change Inequality to Interval Notation?
An interval notation is just a diverse technique of describing an inequality or a set of real numbers.
If x is higher than or lower than a value (not equal to), then the value should be expressed with parentheses () in the notation.
If x is higher than or equal to, or lower than or equal to, then the interval is expressed with closed brackets [ ] in the notation. See the examples of interval notation above to check how these
symbols are used.
How Do You Exclude Numbers in Interval Notation?
Numbers ruled out from the interval can be denoted with parenthesis in the notation. A parenthesis implies that you’re writing an open interval, which means that the number is excluded from the set.
Grade Potential Could Guide You Get a Grip on Math
Writing interval notations can get complicated fast. There are more difficult topics within this area, such as those dealing with the union of intervals, fractions, absolute value equations,
inequalities with an upper bound, and more.
If you want to master these concepts fast, you are required to revise them with the professional help and study materials that the expert instructors of Grade Potential delivers.
Unlock your math skills with Grade Potential. Connect with us now!
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The aim of the Workshop of the ICRA is to bring leading figures in the area of Representation Theory of Algebras to give short courses on different topics at the forefront of the research in the
area. This year's workshop of the ICRA XX will take place in Montevideo, Uruguay, between the 3 August to 6 August. The speakers of this edition will include the following mathematicians:
• Anna Barbieri (Università degli Studi di Padova)
• Karin Erdmann (University of Oxford)
• Julian Külshammer (Uppsala Universitet)
• Rosanna Laking (Università degli Studi di Verona)
• Hipolito Treffinger (Université Paris Cité)
Anna Barbieri - Spaces of Bridgeland stability conditions
I will give an introduction to the notion of Bridgeland stability conditions for triangulated category and the stability manifold associated, with a focus on the case of the Ginzburg 3-Calabi-Yau
category associated with a quiver arising from a triangulation of a surface.
Karin Erdmann - Tame symmetric algebras
These lectures will investigate tame symmetric algebras, which are constructed and described via surface triangulations, and which generalize naturally tame blocks of group algebras. Such blocks have
groups as invariants: dihedral, or semidihedral, or quaternion.
Inspired by cluster theory, we introduce weighted surface algebras. They are almost all periodic as algebras, of period 4, and are a geometric generalization for quaternion type. We take these as a
frame for several more algebras: Degenerating minimal relations gives rise to a generalization for dihedral type, and semidihedral type.
Towards a unified approach, we introduce hybrid algebras. These include all Brauer graph algebras, weighted surface algebras, and in addition many other tame symmetric algebras. Hybrid algebras are
precisely the block components of idempotent algebras eAe where A is a weighted surface algebra and e an idempotent of A.
We discuss what is known about indecomposables, Auslander-Reiten components, and derived equivalence.
This is mostly joint work with Andrzej Skowrónski.
Julian Külshammer - Towards bound quivers for exact categories (online)
The proceedings of the first ICRA in 1974 contain an article by Roiter and Kleiner describing a theory of representations of semi-free differential graded categories. While the theory has been
successfully applied in a number of cases, most notably in the proof of Drozd's tame-wild dichotomy, it is today not as well-known as e.g. the theory of almost split sequences described in articles
by Auslander and Reiten in the same proceedings.
In this lecture series, I will explain how such semi-free differential graded categories are a convenient way to talk about certain ring extensions and provide evidence that they are a way to develop
a theory of quivers and relations for exact categories. The most developed special case right now is that of the category of filtered modules over a quasi-hereditary algebra. This builds on joint
work with several people over the last decade, let me in particular mention Steffen Koenig, Sergiy Ovsienko, and Vanessa Miemietz.
Rosanna Laking - Torsion pairs and mutation
The complete lattice tors-A formed by the collection of all torsion pairs in the category of finitely generated modules over a finite-dimensional algebra A encodes a wealth of combinatorial and
homological information about the representation theory of A. This is due, in part, to its connections with t-structures, tau-tilting theory and stability conditions. The connection with tau-tilting
theory has been a particularly powerful tool for understanding tors-A, since the functorially finite torsion pairs are parametrised by two-term silting objects in the bounded derived category of modA
and the adjacent edges of the Hasse quiver are controlled by silting mutation.
In these talks we will introduce an approach to the study of tors-A that goes beyond tau-tilting theory and the functorially finite torsion classes. We will give an overview of the
Demonet-Iyama-Reading-Reiten-Thomas brick labelling of the Hasse quiver in terms of simple tilts between the corresponding HRS-t-structures. By lifting these t-structures to the derived category of
all modules we will see that the simple tilts induce irreducible mutations of associated (large) two-term cosilting complexes. Finally we will explain how the two-term cosilting complexes are
parametrised by certain closed sets of the Ziegler spectrum and their mutations are determined by the open sets of the induced topology. The topics covered by these talks are contained in joint work
with Lidia Angeleri Hügel, Ivo Herzog, Francesco Sentieri, Jan Šťovíček and Jorge Vitória.
Hipolito Treffinger - Wall-and-chamber structures of Artin algebras and τ-tilting theory
The aim of this course is to give an overview of the deep relationship between a geometric object associated to an Artin algebra, known as its wall-and-chamber structure, and the τ-tilting theoretic
properties of the algebra. Being more precise, in this course we will show how several important notions of τ-tilting theory are encoded in the geometry of the wall-and-chamber structure of the
We will start this course by showing the (top-down) construction of the wall-and-chamber structure of an algebra using the stability conditions defined by King and how we can use stability conditions
to calculate torsion pairs in the module category of our algebra. We then will change gears slightly to introduce τ-tilting theory and present the (bottom-up) construction of (part of) the
wall-and-chamber structure of the algebra using the g-vectors of the indecomposable τ-rigid objects of the algebra.
Once these two complementaries constructions of the wall-and-chamber structure are presented, we will profit the interplay between the two to recover several important objects which are central to
τ-tilting theory, including τ-tilting pairs and their mutation, (semi)bricks, torsion pairs and wide subcategories.
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Talk: Sums of powers of binomials, their Apéry limits, and Franel's suspicions (Dalhousie)
Sums of powers of binomials, their Apéry limits, and Franel's suspicions (Dalhousie)
Date: 2022/10/06
Occasion: Mathematics Colloquium
Place: Dalhousie University
Apéry's proof of the irrationality of \(\zeta(3)\) relies on representing that value as the limit of the quotient of two rational solutions to a three-term recurrence. We review such Apéry limits and
explore a particularly simple instance. We then explicitly determine the Apéry limits attached to sums of powers of binomial coefficients. As an application, we prove a weak version of Franel's
conjecture on the order of the recurrences for these sequences. This is based on joint work with Wadim Zudilin.
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The Einstein Theory of Relativity: A Trip to the Fourth Dimension?
What is Reddit's opinion of The Einstein Theory of Relativity: A Trip to the Fourth Dimension?
From 3.5 billion Reddit comments
If you want to learn GR, check out Lillian Lieber. The Einstein Theory of Relativity. It covers the basics and assumes very little of its readers because it covers the calculus you need to know.
Einstein endorsed it.
I discovered Lillian Lieber. The Einstein Theory of Relativity because it was recommended by D' Invernio in his excellent relativity textbook. It covers the basics and assumes very little of its
readers. Einstein endorsed it.
I discovered Lillian Lieber. The Einstein Theory of Relativity because it was recommended by D' Invernio in his excellent relativity textbook. It's a great little book that covers the basics and
assumes very little of its readers. Einstein endorsed it.
Schuam's Guide to Tensor Calculus.
I also agree the person who recommended Nightingale.
Leonard Susskind covers Tensors in his youtube lectures on GR
Lillian Lieber. The Einstein Theory of Relativity
Victor Stenger's The Comprehensible Cosmos.
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Graphing Quadratic Functions In Standard Vertex And Intercept Form Worksheet - Graphworksheets.com
Graphing Quadratic Equations In Intercept Form Worksheet – Learning mathematics is incomplete without graphing equations. It involves graphing lines and points, and evaluating their slopes. Graphing
equations of this type requires that you know the x and y-coordinates of each point. To determine a line’s slope, you need to know its y-intercept, which is the … Read more
Graphing Quadratic Functions Worksheet Vertex Form
Graphing Quadratic Functions Worksheet Vertex Form – If you’re looking for graphing functions worksheets, you’ve come to the right place. There are many types of graphing function to choose from. For
example, Conaway Math has Valentine’s Day-themed graphing functions worksheets for you to use. This is a great way for your child to learn about … Read more
Graphing Quadratic Functions In Standard Vertex And Intercept Form Worksheet
Graphing Quadratic Functions In Standard Vertex And Intercept Form Worksheet – If you’re looking for graphing functions worksheets, you’ve come to the right place. There are many types of graphing
function to choose from. Conaway Math offers Valentine’s Day-themed worksheets with graphing functions. This is a great way for your child to learn about these … Read more
Graphing Quadratic Equations In Standard Form And Intercept Form Worksheet
Graphing Quadratic Equations In Standard Form And Intercept Form Worksheet – Learning mathematics is incomplete without graphing equations. It involves graphing lines and points, and evaluating their
slopes. Graphing equations of this type requires that you know the x and y-coordinates of each point. To determine a line’s slope, you need to know its y-intercept, … Read more
Graph Quadratic Functions In Vertex Or Intercept Form Worksheet
Graph Quadratic Functions In Vertex Or Intercept Form Worksheet – If you’re looking for graphing functions worksheets, you’ve come to the right place. There are several different types of graphing
functions to choose from. For example, Conaway Math has Valentine’s Day-themed graphing functions worksheets for you to use. This is a great way to help … Read more
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This Mathematician's 'Mysterious' New Method Just Solved a 30-Year-Old Problem
A mathematician has solved a 30-year-old problem at the boundary between mathematics and computer science. He used an innovative, elegant proof that has his colleagues marveling at its simplicity.
Hao Huang, an assistant professor of mathematics at Emory University in Atlanta, proved a mathematical idea called the sensitivity conjecture, which, in incredibly rough terms, makes a claim about
how much you can change the input to a function without changing the output (this is its sensitivity).
In the decades since mathematicians first proposed the sensitivity conjecture (without proving it), theoretical computer scientists realized that it has huge implications for determining the most
efficient ways to process information .
Hao Huang@Emory:
Ex.1: ∃edge-signing of n-cube with 2^{n-1} eigs each of +/-sqrt(n)
Interlacing=>Any induced subgraph with >2^{n-1} vtcs has max eig >= sqrt(n)
Ex.2: In subgraph, max eig <= max valency, even with signs
Hence [GL92] the Sensitivity Conj, s(f) >= sqrt(deg(f))
— Ryan O'Donnell (@BooleanAnalysis) July 1, 2019
What did Huang actually prove?
For simplicity's sake, imagine a 3D cube with sides that are each 1 unit long. If you put this cube into a 3D coordinate system (meaning it has measurements in three directions), one corner would
have the coordinates (0,0,0), the one next to it might be (1,0,0), the one above it might be (0,1,0) and so on. You can take half the corners (four corners) without having any pair of neighbors:
(0,0,0) , (1,1,0), (1,0,1) and (0,1,1) aren't neighbors. You can show this by looking at the cube, but we also know it because all of them are different by more than one coordinate.
The sensitivity conjecture is about finding how many neighbors you have when you take more than half the corners of a higher dimensional cube, or a hypercube, said Hebrew University mathematician Gil
Kalai. You can write the coordinates of the hypercube as strings of 1s and 0s, where the number of dimensions is the length of the string, Kalai told Live Science. For a 4D hypercube, for instance,
there are 16 different points, which means 16 different strings of 1s and 0s that are four digits long.
Now pick half plus 1 individual points on the hypercube (for a 4D hypercube, that means pick nine — or 8+1 — different points out of a total of 16). [Mathematicians Edge Closer to Solving a 'Million
Dollar' Math Problem]
From this smaller set, find the point with the most neighbors — what's the minimum number of neighbors it can have? (Neighbors differ by just one number. For example, 1111 and 1110 are neighbors,
because you only have to swap one digit to turn the first into the second.)
Huang proved that this corner must have at least as many neighbors as the square root of the number of digits — in this case, the square root of 4 — which is 2.
For low dimensions, you can tell this is true just by checking. It's not that hard to check 16 coordinates on the cube (or "strings") for neighbors, for example. But every time you add a dimension to
the cube, the number of strings doubles. So the problem gets harder to check very quickly. [A Mathematician Just Solved a Deceptively Simple Puzzle That Has Boggled Minds for 64 Years]
The set of strings that's 30 digits long — the coordinates to the corners of a 30-dimensional cube — has more than 1 billion different strings in it, meaning the cube has more than 1 billion corners.
With strings that are 200 digits long, there are more than a novemdecillion. That's a million billion billion billion billion billion billion, or 1 followed by 60 zeroes.
This is why mathematicians like proofs: They show that something is true in every case, not just the easy ones.
"If n is equal to a million — this means we have strings of length 1 million — then the conjecture is that if you take 2^1,000,000-1 and add 1, then there is a string that has 1,000 neighbors — the
square root of a million," Kalai said.
The last major advance in the sensitivity conjecture came in 1988, Kalai said, when researchers proved that one string has to have at least the logarithm of n neighbors. That's a much lower number;
the logarithm of 1,000,000 is just 6. So Huang's proof just discovered that at least 994 other neighbors are out there.
An elegant and "mysterious" proof
"It is very mysterious," Kalai said of Huang's proof. "It uses 'spectral methods,' which are very important methods in many areas of mathematics. But it uses spectral methods in a novel way. It's
still mysterious, but I think we can expect that this novel way to use spectral methods will gradually have more applications."
In essence, Huang conceptualized the hypercube using arrays of numbers in rows and columns (called matrices). Huang figured out a completely unexpected way to manipulate a matrix with an unusual
arrangement of -1s and 1s that "magically makes it all work," Aaronson wrote on his blog. [10 Surprising Facts About Pi]
Huang "took this matrix, and he modified it in a very ingenious and mysterious way," Kalai said. "It's like you have an orchestra and they play some music, and then you let some of the players, I
don't know, stand on their head, and the music becomes completely different — something like that."
That different music turned out to be the key to proving the conjecture, Kalai said. It's mysterious, he said, because even though mathematicians understand why the method worked in this case, they
don't fully understand this new "music" or in what other cases it might be useful or interesting.
"For 30 years, there was no progress, and then Hao Huang settled this problem, and he found a very simple proof that the answer is the square root of n," Kalai said. "But during these 30 years …
people realized that this question is very important in the theory of computing."
Huang's proof is exciting because it advances the field of computer science, Kalai said. But it's also noteworthy because it introduced a novel method, and mathematicians still aren't sure what else
Huang's new method might allow them to accomplish.
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OpenDSA Data Structures and Algorithms Modules Collection
23.1. Finding the Maximum Value¶
23.1.1. Finding the Maximum Value¶
How can we find the \(i\) th largest value in a sorted list? Obviously we just go to the \(i\) th position. But what if we have an unsorted list? Can we do better than to sort it? If we are looking
for the minimum or maximum value, certainly we can do better than sorting the list. Is this true for the second biggest value? For the median value? In later sections we will examine those questions.
For this section, we will continue our examination of lower bounds proofs by reconsidering the simple problem of finding the maximum value in an unsorted list.
Here is a simple algorithm for finding the largest value.
// Return position of largest value in integer array A
static int largest(int[] A) {
int currlarge = 0; // Position of largest element seen
for (int i=1; i<A.length; i++) { // For each element
if (A[currlarge] < A[i]) { // if A[i] is larger
currlarge = i; // remember its position
return currlarge; // Return largest position
/** Return position of largest value in integer array A */
int largest(int A[], int size) {
int currlarge = 0; // Position of largest element seen
for (int i=1; i<size; i++) // For each element
if (A[currlarge] < A[i]) // if A[i] is larger
currlarge = i; // remember its position
return currlarge; // Return largest position
Obviously this algorithm requires \(n\) comparisons. Is this optimal? It should be intuitively obvious that it is, but let us try to prove it. (Before reading further you might try writing down your
own proof.)
This proof is clearly wrong, because the winner does not need to explicitly compare against all other elements to be recognized. For example, a standard single-elimination playoff sports tournament
requires only \(n-1\) comparisons, and the winner does not play every opponent. So let’s try again.
This proof is sound. However, it will be useful later to abstract this by introducing the concept of posets. We can view the maximum-finding problem as starting with a poset where there are no known
relationships, so every member of the collection is in its own separate DAG of one element.
What is the average cost of largest? Because it always does the same number of comparisons, clearly it must cost \(n-1\) comparisons. We can also consider the number of assignments that largest must
do. Function largest might do an assignment on any iteration of the for loop.
Because this event does happen, or does not happen, if we are given no information about distribution we could guess that an assignment is made after each comparison with a probability of one half.
But this is clearly wrong. In fact, largest does an assignment on the \(i\) th iteration if and only if A [\(i\)] is the biggest of the the first \(i\) elements. Assuming all permutations are equally
likely, the probability of this being true is \(1/i\). Thus, the average number of assignments done is
\[1 + \sum_{i=2}^n \frac{1}{i} = \sum_{i=1}^n \frac{1}{i}\]
which is the Harmonic Series \({\cal H}_n\).
\[{\cal H}_n = \Theta(\log n).\]
More exactly, \({\cal H}_n\) is close to \(\log_e n\).
How “reliable” is this average? That is, how much will a given run of the program deviate from the mean cost? According to Cebysev’s Inequality, an observation will fall within two standard
deviations of the mean at least 75% of the time. For Largest, the variance is
\[{\cal H}_n - \frac{\pi^2}{6} = \log_e n - \frac{\pi^2}{6}.\]
The standard deviation is thus about \(\sqrt{\log_e n}\). So, 75% of the observations are between \(\log_e n - 2\sqrt{\log_e n}\) and \(\log_e n + 2\sqrt{\log_e n}\). Is this a narrow spread or a
wide spread? Compared to the mean value, this spread is pretty wide, meaning that the number of assignments varies widely from run to run of the program.
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Financial Accounting - Online Tutor, Practice Problems & Exam Prep
Alright, so now let's see what happens when we get a purchase discount, which is a discount for paying quickly to our supplier. You might have seen this already when we talked about sales discounts.
If you haven't seen sales discounts yet, I'm sure you will, but you're going to see that it's very similar to this lesson, except now, instead of talking about selling something and offering a
discount to our customer, we're the customer getting a discount from our supplier. So let's check it out in this situation. There's a special system used to denote discounts, and let's go through it
here. This is how you usually see a discount. You'll see "2/10, net 30". That's how you read that. The "2/10, net 30", the 'n' stands for net, just like you see in the second quote there.
So, let's see what these numbers mean. The numbers are what really matters, and what really matters are those first two numbers. Those are the most important ones. So, the first two there, the "2/10,
net 30", the "2", that's the percentage amount of discount that you're going to get. You're going to get a 2% discount if you pay within 10 days. The purchase happened on one day, and now you have
the next 10 days to pay and get a 2% discount. The "30" represents the total days you have to pay. So, they told you if you pay within 10 days, you can take a 2% discount, but you have to pay us the
full balance by the 30th day. When we solve problems like this, that "30" doesn't really matter, okay? What's really the focus in this class is that percentage discount and whether they qualified for
the discount, those first two numbers, the "2" and the number of days that have passed.
Let's check out this example, ABC Company purchased 300 units of Product X for $1800 on January 14, the supplier offered terms of "3/10, net 45". Notice this is different than above; we saw "2/10,
net 30", but the principles stay the same. In this case, we're talking about a 3% discount if you pay within 10 days, and then a total of 45 days to pay, but notice that the "45", we're not really
going to use it in the problem. So, the supplier offers terms of "3/10, net 45". ABC Company paid the supplier on January 19th. It's really important to think of the dates to see if we qualify for
the discount. So, record the purchase and payment in ABC Company's books.
Okay, we're going to have 2 journal entries here, one for the purchase and one for the payment. Obviously, that's what we just said. So let's do the purchase one first. It told us we bought 300 units
for $1800 right? The $1800, that's the big number we want, we want dollar values. So, we spent $1800 on inventory, so we're going to debit our inventory for $1800, right? Because that's what we
bought, so we're increasing our inventory by $1800, but we haven't paid them yet. We're going to pay them at a future date, right? So, we have an accounts payable, AP for accounts payable of $1800,
at some future date, we should have to pay $1800 to them, but what we're going to see is that we get a discount, right? Because in this situation, we purchased it on the 14th and paid on the 19th.
That's within our 10-day window, right? On the 15, 16, 17, 18, 19, we paid 5 days later. Cool? So, that's within our 10-day limit to get the discount. So, we're allowed to pay 3% less, right? We get
a 3% discount. Let's go ahead and see what that 3% discount is. So, we have $1800 times 3%, $1800 times 0.03 that's $54. $54 is the amount of the discount, so we get to pay that much less instead of
paying $1800.
So, let's see how much we do actually pay, $1800 minus the $54 is $1746, okay? So, that is the actual cash that's going to come out of our pocket. We got a discount of $54, but the actual cash is
$1746. So, in a perpetual system, we just put all our entries straight up into inventory, okay? So, the first thing we want to do is we want to get rid of the payable, right? It says that we owe
$1800 well, we're about to pay that money off, so we're not going to owe the $1800 anymore, we're going to debit accounts payable for $1800 right? We no longer owe that money because we're making the
payment. Now how are we paying? We're going to pay with cash and that cash was $1746 like we figured out above, right? So, the last thing is to make this balance, we need that $54 discount. And that
discount is just going to go straight to our inventory. We're going to decrease the value of our inventory by the amount of the discount.ởSo, inventory is going to have a credit for $54, alright? And
this brings down the value of the inventory to what we actually paid for it, right? We didn't actually pay $1800, we actually paid $1746 and that's what's happening in this question, is bringing down
that value to $1746, right? So, in the first entry, our inventory, this one's gonna get a little close here. Our inventory went up by $1800 and our AP went up by $1800. Accounts payable went up by
$1800 in the first entry and then what happened in the second entry? Well, our accounts payable went down by $1800. Right? So, that gets rid of that change, and then our cash went down by $1746, and
our inventory went down by $54. Okay, so if we were to net this amount of inventory, right it went up by $1800 and then down by $54, that's the $1746 right there, right? So, inventory went up by
$1746 and cash went down by $1746, right? So, our assets stayed equal there, our liabilities stay equal and our equation balances, right? So cool, let's go ahead and pause here and then do a practice
problem in the next video. You guys can take a stab at this.
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Matrix Suprema & Compressive Sensing
Add to your list(s) Download to your calendar using vCal
If you have a question about this talk, please contact Vittoria Silvestri.
The problem that is the subject of this talk is simple to describe; take a change of basis matrix, remove the first N rows, find the size of the largest entry left & determine how this value decays
with N. In certain compressed sensing problems the faster this decay is the more we are allowed to compress the problem by subsampling. Now suppose we have the freedom to permute the rows & are
looking for the fastest decay possible. If the basis corresponding to the rows has some intrinsic structure, what does an optimal permutation look like within this structure and how does this impact
on how we can subsample?
The talk will discuss some of the theoretical limits of this problem before moving onto various specific cases such as changing basis from complex exponentials to wavelets in one and many dimensions.
No previous knowledge of compressed sensing or wavelets is required as I shall be introducing things from the ground up.
This talk is part of the Cambridge Analysts' Knowledge Exchange series.
This talk is included in these lists:
Note that ex-directory lists are not shown.
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First Constraints on Growth Rate from Redshift-space Ellipticity Correlations of SDSS Galaxies at 0.16 < z < 0.70
First Constraints on Growth Rate from Redshift-space Ellipticity Correlations of SDSS Galaxies at 0.16 < z < 0.70
We report the first constraints on the growth rate of the universe, f(z)σ [8](z), with intrinsic alignments (IAs) of galaxies. We measure the galaxy density-intrinsic ellipticity cross-correlation
and intrinsic ellipticity autocorrelation functions over 0.16 < z < 0.7 from luminous red galaxies (LRGs) and LOWZ and CMASS galaxy samples in the Sloan Digital Sky Survey (SDSS) and SDSS-III BOSS
survey. We detect clear anisotropic signals of IA due to redshift-space distortions. By combining measured IA statistics with the conventional galaxy clustering statistics, we obtain tighter
constraints on the growth rate. The improvement is particularly prominent for the LRG, which is the brightest galaxy sample and known to be strongly aligned with underlying dark matter distribution;
using the measurements on scales above 10 h ^-1 Mpc, we obtain $f{\sigma }_{8}={0.5196}_{-0.0354}^{+0.0352}$ (68% confidence level) from the clustering-only analysis and $f{\sigma }_{8}={0.5322}_
{-0.0291}^{+0.0293}$ with clustering and IA, meaning 19% improvement. The constraint is in good agreement with the prediction of general relativity, f σ [8] = 0.4937 at z = 0.34. For LOWZ and CMASS
samples, the improvement of constraints on f σ [8] is found to be 10% and 3.5%, respectively. Our results indicate that the contribution from IA statistics for cosmological constraints can be further
enhanced by carefully selecting galaxies for a shape sample.
The Astrophysical Journal
Pub Date:
March 2023
□ Large-scale structure of the universe;
□ Cosmology;
□ Accelerating universe;
□ Cosmological parameters from large-scale structure;
□ Redshift surveys;
□ 902;
□ 343;
□ 12;
□ 340;
□ 1378;
□ Astrophysics - Cosmology and Nongalactic Astrophysics;
□ Astrophysics - Astrophysics of Galaxies;
□ General Relativity and Quantum Cosmology
9 pages, 5 figures, matches version published in ApJL
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Design of prestressed composite cross-section in RCS
Knowledge base
Design of prestressed composite cross-section in RCS
The IDEA StatiCa RCS application (Beam) serves to code-check different types of concrete cross-sections. This text focuses on the code-check of a prestressed composite cross-section composed of two
various concrete grades (casted in two different construction stages).
Creating a new project
Initially, setting up a new project as a 1D staged/prestressed/composite member is necessary.
Design member
After defining the cross-section geometry and reinforcement layout, the timeline in the Construction Stages tab is defined. All important time points during the construction are defined (such as
casting the first part of the cross-section of the precast beam, prestressing, casting the second part of the cross-section of the composite slab, superimposed dead load, and the time of the design
working life). The defined construction stages are automatically propagated in the Action Stages tab.
Action stages
From the point of view of code-checking the composite cross-sections, the Action Stages tab is the most important. Defining the initial stress state of the cross-section calculated by Time-Dependent
Analysis (TDA) is crucial since the discontinuity of stress (the plane of strain shift) at the interface between two different concretes can decide the failure mechanism at the limit state.
The initial state of the cross-section
The initial state of the composite section is set in the table ”Effects in cross-section components”. Two options for defining the initial state can be chosen – internal forces and planes of strain.
It is much easier to define internal forces obtained from TDA calculated by third-party software (Midas, SCIA, etc.)
The table ”Effects in cross-section components” contains the internal forces as a summation of:
• All permanent loads acting in the considered construction stage
• The total effect of prestressing (the primary and secondary effects of bonded and unbonded internal tendons, the primary and secondary effects of external tendons)
• Rheology (creep, shrinkage)
Most third-party software (Midas, SCIA, etc.) present the internal forces of each part of the composite cross-section related to the center of gravity of the considered cross-sectional part (for
example, the bending moment in the precast beam is related to the center of gravity of the precast beam C[g,1]). The RCS application relates the internal forces to the center of gravity of the actual
cross-section (the button “Actual” in the ribbon) or the center of gravity of the final composite section C[g,i] (the button “Entire” in the ribbon). Transformation of the internal forces obtained
from third-party software to RCS can be performed according to the following formulas:
\[N_{i}^{T} = N_{i}\]
\[M_{i}^{T} = M_{i}-N_{i}\times e_{i}\]
N[i]^T . . . . the normal force in the considered part of the composite section transformed to the center of gravity of the idealized final composite cross-section
M[i]^T . . . . the bending moment in the considered part of the composite section transformed to the center of gravity of the idealized final composite cross-section
N[i ]. . . . the normal force in the considered part of the composite section related to the center of gravity of the considered cross-section part
M[i] . . . . the bending moment in the considered part of the composite section related to the center of gravity of the considered cross-section part
Note: Keeping the sign convention presented in the figure below is important to recalculate the internal forces.
C[g,i ] . . . . the center of gravity of the idealized composite section (E[cm(28)] is considered)
C[g,1] . . . . the center of gravity of part one – precast beam (light gray part)
C[g,2] . . . . the center of gravity of part two – composite slab (dark gray part)
e[y,1] . . . . the distance from C[g,1] to C[g,i]
e[y,2] . . . . the distance from C[g,2] to C[g,i]
e[p ] . . . . the distance from the center of gravity of the prestressing reinforcement to C[g,i]
The internal forces N1, My,1, N2 a My,2 are obtained for the composite structure modeled in third-party software and subjected in the vertical direction. For the correct input of the internal forces
to the RCS app, a recalculation has to be performed as follows:
Part 1 (precast beam)
\[N_{1}^{T} = N_{1}\]
\[M_{y}^{T},_{1} = M_{y}^{T},_{1}-N_{1}\times e_{y},_{1}\]
N[1]^T . . . . the normal force in the precast beam transformed to the center of gravity of the idealized final composite cross-section C[g,I] (a negative value for compressive force)
M[y,1]^T . . . the bending moment in the precast beam transformed to the center of gravity of the idealized final composite cross-section C[g,i]
N[1 ]. . . . the normal force in the precast beam related to the center of gravity of the precast beam C[g,1]
M[y,1] . . . the bending moment in the precast beam related to the center of gravity of the precast beam C[g,1]
e[y,1] . . . . the distance of the center of gravity of precast beam C[g,1] from the center of gravity of the idealized final composite cross-section C[g,i] (in this case, the negative value of
eccentricity is considered)
Part 2 (composite slab)
\[N_{2}^{T} = N_{2}\]
\[M_{y}^{T},_{2} = M_{y}^{T},_{2}-N_{2}\times e_{y},_{2}\]
N[2]^T . . . . the normal force in the composite slab transformed to the center of gravity of the idealized final composite cross-section C[g,i]
M[y,2]^T . . . the bending moment in the composite slab transformed to the center of gravity of the idealized final composite cross-section C[g,i]
N[2 ]. . . . the normal force in the composite slab related to the center of gravity of the composite slab C[g,2]
M[y,2] . . . the bending moment in the composite slab related to the center of gravity of the composite slab C[g,2]
e[y,2 ] . . . . the distance of the center of gravity of composite slab C[g,2] from the center of gravity of the idealized final composite cross-section C[g,i] (in this case, the positive value of
eccentricity is considered)
Thanks to this transformation, the total internal forces in the composite cross-section can be determined.
Note: The transformation process of internal forces acting in a horizontal direction is the same as stated above.
Stress in reinforcement
The next important step is determining the initial stress in rebars and prestressing tendons. The RCS application can calculate stress in rebars automatically, so it’s recommended to keep setting ”
Based on a component initial”.
If the prestressing reinforcement is designed, the stress in each tendon for all existing construction stages must be defined (see Chapter 2). The RCS application allows the definition of the value
of stress in tendons after long-term losses calculated by TDA (“Stress after long-term losses”) or the definition of estimated short- and long-term losses (“Estimation of prestressing losses”).
Total effects of prestressing
The RCS application recognizes two types of prestressing effects – the primary and secondary effects of prestressing. Both types are assumed to be acting on the final composite section. Prestressing
effects are defined for each construction stage to capture long-term prestressing losses. The primary effects of prestressing are calculated automatically according to tendon properties (the position
in a cross-section, the area of a tendon, and stress in a tendon in the considered construction stage). Internal forces due to primary prestressing at the time of 10 days are calculated as:
\[N_{p}^{P},_{10}=A_{p}\times \sigma_{p},_{10}\]
\[M_{p}^{P},_{10}=A_{p}\times \sigma_{p},_{10}\times e_{p}\]
N[p,10]^P . . . the normal force in a cross-section due to the primary effects of bonded prestressing reinforcement in the considered time (10 days)
M[p,10]^P . . . the bending moment in a cross-section due to primary effects of bonded prestressing reinforcement in the considered time (10 days)
A[p] . . . . the area of bonded prestressing reinforcement
σ[p,10 ] . . . the stress in prestressing reinforcement in the considered time (10 days)
e[p ] . . . . the distance from the center of gravity of prestressing reinforcement to the center of gravity of the idealized final composite cross-section C[g,i]
The user always defines the secondary effects of prestressing. Internal forces defined in the table consist of:
• The total effects of unbonded or external prestressing reinforcement (if the user defined this reinforcement type in the global calculation model).
• The total effects of unbonded or external prestressing reinforcement (if the user-defined this reinforcement type in the global calculation model)
The summation of primary and secondary effects defined in the table shown above are automatically copied to the table in the section “Internal forces”. It is necessary to define prestressing
carefully and correctly to prevent incorrect results.
Internal forces
A few last steps hmust be performed for the correct code-checking of the composite cross-section. In “Section”, defining “Extremes” for each time the code-checking should be done is necessary. The
defined times of extremes have to correspond to the times defined in “Construction stages” (chap. 2). Then correct values of internal forces for the calculation of the initial state of the
cross-section will be taken from the tab ”Action stages”.
Other types of acting internal forces have to be defined in the “Internal forces” tab. Internal forces are defined for each extreme separately.
Permanent load
The rows called “Permanent Sum G[dj]” serve as the input for the combination value of the permanent loads (including load factors) acting in the considered construction stage.
Permanent internal forces can be defined manually or imported from “Action stages” thanks to the commands in the ribbon. When importing ULS internal forces from “Action stages”, the load factor for
the permanent load can be set by the user.
Import permanent internal forces from the “Action stages”, the following rules are applied:
• The combination value of internal forces for ULS checks is calculated as
Permanent Sum = (Initial effects of section – Total effects of prestressing) ·γ[Gj,sup]
• The combination values of internal forces for SLS checks are calculated as
Permanent Sum = Initial effects of section – Total effects of prestressing
Variable load
The resulting value of internal forces due to a variable load (including combination load factors) is defined manually by the user. These values are usually obtained from global structural analysis.
Effects of prestressing
The total effects of prestressing are automatically imported from the ”Action stages” tab as a summation of the primary and secondary effects of prestressing defined in the tab ”Total effects of
prestressing” (chap. 3.3). These values cannot be edited by the user.
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Deriving C2 Constant with Probability Approach
To directly calculate Hardy-Littlewood’s constant C2, we can employ the inclusion-exclusion principle, taking into account how different primes interact:
Basic Probability: We begin with the simple probability estimate for a twin prime pair, which is (1 / ln(x))2.
Inclusion-Exclusion: This initial estimate overcounts twin primes because it ignores divisibility by smaller primes. We refine it by subtracting the probability of pairs failing to be twin primes due
to divisibility by small primes. For example, if 6k-1 is prime, but 6k+1 is not, we subtract that probability.
Higher Orders: This process of inclusion and exclusion continues for higher orders. We add back probabilities that were subtracted too many times in the previous step – for instance, cases where both
numbers in the pair are divisible by two different small primes.
Convergent Series: Ideally, this repeated inclusion and exclusion forms a convergent infinite series. Each term in this series represents a probability correction associated with a specific prime or
a combination of primes. The sum of this entire series should give us the precise value of C2.
Detailed Example (Prime 5):
• First-order probability: Our initial estimate is (1 / ln(x))2.
• Second-order correction (prime 5): We subtract about (1/5) * (1 / ln(x))2 to adjust for situations where one of the numbers (6k-1 or 6k+1) is divisible by 5.
• Partial C2: This correction gives us a preliminary factor of (1 – 1/5) = 4/5.
To get the full value of C[]2, we’d need to repeat this process for all primes, which involves complex calculations and requires proving the convergence of the resulting infinite series.
By systematically accounting for prime interactions through the inclusion-exclusion principle, this method offers a direct way to derive C2. While mathematically challenging to formalize, this
approach strengthens the probabilistic argument supporting the Hardy-Littlewood Twin Prime Conjecture. If the infinite series converges as expected, it provides a compelling link between the
probabilistic nature of prime distribution and this famous conjecture.
Proof of Hardy-Littlewood’s Constant C[]2 via Inclusion-Exclusion
This proof details the derivation of Hardy-Littlewood’s constant, C[]2, utilizing the inclusion-exclusion principle and a probabilistic framework.
Basic Definitions:
• Twin Primes: A pair of primes (p, p + 2) is called a twin prime pair.
• Prime Density Function: The density of primes around a large number x is approximately 1/ln(x).
Probability of Twin Primes:
The initial probability estimate for the occurrence of a twin prime pair (p, p + 2) around x is:
P((p, p + 2) are both prime) ≈ (1/ln(x))^2
Inclusion-Exclusion Principle:
This initial estimate overcounts twin primes because it ignores interactions with smaller primes. The inclusion-exclusion principle allows us to correct for these interactions systematically.
Step-by-Step Adjustments:
• First-Order Adjustment: Consider the probability that either p or p + 2 is divisible by a small prime q. For example, for q = 5, either p ≡ 0 (mod 5) or p + 2 ≡ 0 (mod 5). The probability of one
of these being true is 2/5. We adjust the initial probability:
(1/ln(x))^2 (1 – 2/5)
• General Form: For any prime q, the probability that either p or p + 2 is divisible by q is 2/q. Correcting for all primes q ≥ 3:
(1/ln(x))^2 ∏[]q≥3 (1 – 2/q)
• Higher-Order Corrections: We incorporate higher-order interactions using the inclusion-exclusion principle. This involves adding back probabilities of events where both numbers are divisible by
two small primes, then subtracting probabilities where they are divisible by three primes, and so on.
Infinite Product Representation:
Applying the inclusion-exclusion principle to all primes results in an infinite product:
C[]2 = ∏[]q≥3 (1 – 2/q(q-1))
This product converges because the terms decrease rapidly as q increases.
Convergence and Exact Expression:
• Euler Product Representation: This infinite product can be related to Euler’s product representation of the Riemann zeta function. Each term (1 – 2/q(q-1)) reflects the density adjustment for
• Exact Value of C2: The infinite product converges to the constant C2:
C[]2 = 2 ∏[]q≥3 (1 – 1/(q-1)^2)
• Final Form: The constant 2 accounts for the symmetry of the twin prime pair. Therefore, we have:
C[]2 = 2 ∏[]p≥3 (1 – 1/(p-1)^2)
By systematically applying the inclusion-exclusion principle and accounting for interactions between primes, we derived the precise expression for Hardy-Littlewood’s constant C[]2. The convergence of
the infinite product supports the validity of this approach, demonstrating a clear link between the probabilistic distribution of twin primes and the conjecture itself.
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The base - math word problem (55321)
The base
The base of the quadrilateral prism is a trapezoid with an area of 75 cm square. The prism is 6 cm high. Find the volume of the prism.
Correct answer:
Did you find an error or inaccuracy? Feel free to
write us
. Thank you!
You need to know the following knowledge to solve this word math problem:
Units of physical quantities:
Grade of the word problem:
We encourage you to watch this tutorial video on this math problem:
video1 video2
Related math problems and questions:
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CrowdMining Mining association rules from the crowd | studyslide.com
CrowdMining Mining association rules from the crowd
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Transcript CrowdMining Mining association rules from the crowd
Crowd Mining
Yael Amsterdamer, Yael Grossman,
Tova Milo, and Pierre Senellart
Crowd data sourcing - Background
• Outsourcing data collection
to the crowd of Web users
– When people can provide the data
– When people are the only source of data
– When people can efficiently clean
and/or organize the data
Crowd Mining
Crowdsourcing in an open world
• Human knowledge forms an open world
• Assume we know nothing, e.g., on folk medicine
• We would like to find what is interesting and
important about folk medicine practices around the
What questions should be asked?
Crowd Mining
Back to classic settings
• Significant data patterns are identified using data mining
• Consider: association rules
– E.g., “heartburn” “lemon”, “baking soda”
• Queries are dynamically constructed in the course of the
learning process
• Is it possible to mine the crowd?
Crowd Mining
Asking the crowd
Let us model the history of every user as a personal database
Treated a sore throat with garlic and oregano leaves…
Treated a sore throat and low fever with garlic and ginger …
Treated a heartburn with water, baking soda and lemon…
Treated nausea with ginger, the patient experienced sleepiness…
Every case = a transaction consisting of items
Not recorded anywhere – a hidden DB
It is hard for people to recall many details
about many transactions!
Crowd Mining
they can often provide summaries, in the
form of personal rules
• To treat a sore throat I often use garlic
• Interpretation: “sore throat” “garlic”
Two types of questions
• Free recollection (mostly simple, prominent patterns)
Open questions
Tell me how you treat a particular
“I typically treat nausea with
ginger infusion”
• Targeted questions (may be more complex)
Closed questions
When a patient has both
headaches and fever, how often do
you use a willow tree bark infusion?
We use the two types interleavingly.
Crowd Mining
Personal Rules
• If people know which rules apply to them,
why mine them?
– Personal rules may or may not indicate general trends
– Concrete questions help digging deeper into users’ memory
Crowd Mining
Crowd Mining - Contributions (at a very high level)
• Formal model for crowd mining.
• A Framework of the generic components required for mining the crowd
• Significance and error estimations. Given the knowledge collected
from the crowd, which rules are likely to be significant and what is the
probability that we are wrong.
[and, how will this change if we ask more questions…]
• Crowd-mining algorithm. Iteratively choosing the best crowd
question and estimating significance and error.
• Implementation & benchmark.
Crowd Mining
The model: User support and confidence
• A set of users U
• Each user u U has a (hidden) transaction database Du
• Each rule X Y is associated with:
Crowd Mining
Model for closed and open questions
• Closed questions: X ? Y
– Answer: (approximate) user support and confidence
• Open questions: ? ? ?
– Answer: an arbitrary rule with its user support and confidence
“I typically have a headache
once a week. In 90% of the
times, coffee helps.
Crowd Mining
Significant rules
• Overall support and confidence defined as the mean user
support and confidence
• Significant rules are those whose overall support and
confidence are both above specified thresholds Θs, Θc.
Goal: estimating rule significance while asking the smallest
possible number of questions to the crowd
Crowd Mining
Framework components
• One generic framework
for crowd-mining
• One particular choice of
implementation of all
black boxes
Crowd Mining
Estimating the mean distribution
• Treating the current answers as a random sample of a hidden
distribution , we can approximate the distribution of the hidden
• μ – the sample average
• Σ – the sample covariance
• K – the number of collected samples
• In a similar manner we estimate the
hidden distribution
Crowd Mining
Rule Significance and error probability
• Define Mr as the probability mass above both thresholds for rule r
• r is significant iff Mr is greater than 0.5
• The error probability is
Crowd Mining
The next error probability
• The current distribution gr for some rule can also be used for
estimating what the next answer would be
• We integrate the resulting error probability over the possible next
answers, to get the expected next error
• Optimization problem: The best
rule to ask about leads to the
best output quality
• For quality := overall error, this is
the rule that induces the
largest error reduction
Crowd Mining
Completing the picture
• Which rules should be considered as candidates for the next
– Small rules, rules similar to significant rules are most likely to be
– Similarly to classic data mining
• Should we ask an open or closed question?
– Keeping a fixed ratio of open questions balances the tradeoff between
precision and recall
– Similarly to sequential sampling
Crowd Mining
• 3 new benchmark datasets
– Synthetic
– Retail (market basket analysis)
– Wikipedia editing records
• A system prototype, CrowdMiner, and 2 baseline alternatives
– Random
– Greedy (that asks about the rules with fewest answers)
Crowd Mining
Experimental Results
Number of Samples
Retail Dataset
Crowd Mining
Number of Samples
Wikipedia Dataset
Experimental Results
Number of Samples
Number of Samples
• Better precision – Greedy loses precision as new rules are
• Much better recall – due to adding new rules as candidates.
Crowd Mining
Experimental Results
• An open questions ratio of 0.2-0.6 yields the best quality
Crowd Mining
• The goal: learning about new domains from the crowd
• By identifying significant data patterns
• Data mining techniques cannot be used as-is
• Our solution includes
– A model for the crowd behavior
– A crowd mining framework and concrete component implementations
– Benchmark datasets and a prototype system CrowdMiner used for
Crowd Mining
Related work
• Declarative crowdsourcing frameworks [e.g., Doan et. Al PVLDB’11, Franklin et. Al
SIGMOD’11, Marcus et. Al CIDR’11, Parameswaran et. Al CIDR’11]
– We consider identifying patterns in unknown domains
• Association rule learning [e.g., Agrawal et. Al VLDB’94, Toivonen VLDB’96, Zaki et. Al
– Transactions are not available in our context, sampling rules does not perform
as well as interleaving closed and open questions
• Active Learning [e.g., Dekel et. Al COLT’09, Sheng et. Al SIGKDD’08, Yan et. Al ICML’11]
– In our context every user has a partial picture, no “right” or ``wrong’’
• Sequential Sampling [Vermorel et. Al ECML’05]
– Combining the exploration of new knowledge with the exploitation of
collected knowledge
Crowd Mining
Ongoing and Future work
• Leveraging rule dependencies
– From an answer on one rule we can learn about many others
– Semantic dependencies between rules
• Leveraging user info
• Other types of data patterns
– Sequences, action charts, complex relationships between items
• Mining given a query
– Data mining query languages
• … and many more
Crowd Mining
Thank You!
Crowd Mining
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【How-to】What is the polar form of 3i - Howto.org
What is the polar form of 3i
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What is 5i in polar form?
Polar form is reiθ , and since we know that eiθ=cosθ+isinθ , i=eiπ2 . Therefore, 5i=5eiπ2 .
What is polar form of 2i?
Using these formulas, we can convert the complex number into polar form. Hence, the polar form of $ – 2i$ is $2(\cos \dfrac{{3\pi }}{2} + i\sin \dfrac{{3\pi }}{2})$ .
What is 6i in polar form?
1 Expert Answer
then 6i is equal to 6*(cos90+isin90) in polar form.
How do you convert to polar form?
The polar form of a complex number z=a+bi is z=r(cosθ+isinθ) . So, first find the absolute value of r . Now find the argument θ . Since a>0 , use the formula θ=tan−1(ba) .
What is the polar form of 4i?
What is the polar form of 2?
How do you convert from 1 to polar form?
Let z=1(1+i)×(1-i)(1-i)=(1-i)(1-i2)=(1-i)2=(12-12i). Let its polar form be z=r(cosθ+isinθ).
What is the polar form of 1 i 1 i?
Therefore, the polar form of (1 – i)/(1 + i) is cos (π/2) – i sin (π/2).
What is the polar form of complex number =( i25 3?
The coordinate (x, y) lies in the IV quadrant. From the figure we can say that tangent function is quadrant IV is negative. So, we got the polar form of the given complex number \[{{\left( {{i}^{25}}
How do you convert iota to polar form?
When expressed in polar form is?
2.13 Complex Numbers in Polar Form
It is possible to express complex numbers in polar form. If the point z = ( x , y ) = x + i y is represented by polar coordinates , then we can write x = r cos θ , y = r sin θ and z = r cos θ + i r
sin θ = r e i θ .
How do you find the polar form of a complex number?
The polar form of a complex number z = x + iy with coordinates (x, y) is given as z = r cosθ + i r sinθ = r (cosθ + i sinθ). The abbreviated polar form of a complex number is z = rcis θ, where r = √
(x^2 + y^2) and θ = tan^–^1 (y/x).
What is polar form of complex number class 11?
Let OP = r, then x = r cos Θ , and y = r sin Θ => z = x + iy = r cos Θ + ir sin Θ = r ( cos Θ + i sin Θ ). This is known as Polar form (Trigonometric form) of a Complex Number.
What is the conjugate of a IB?
Conjugate of Complex Number Class 11
Z conjugate is the complex number a – ib, i.e., = a – ib.
What is the polar form of the complex number z 3i?
What is the polar form of sqrt 3?
The inverse tangent of √33 is θ=30° θ = 30 ° . This is the result of the conversion to polar coordinates in (r,θ) form.
What is polar form in physics?
How do you convert rectangular form to polar form?
To convert from polar coordinates to rectangular coordinates, use the formulas x=rcosθ and y=rsinθ. See Example 10.3. 3 and Example 10.3. 4.
How do you write a trigonometric form?
How do you change a complex number to exponential form?
If you have a complex number z = r(cos(θ) + i sin(θ)) written in polar form, you can use Euler’s formula to write it even more concisely in exponential form: z = re^(iθ).
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Intro to Ratios
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A ratio describes the relationship between different amounts. A ratio can describe the relationship between two parts of a group, or between one part and the whole group.
To better understand ratios…
WHAT ARE RATIOS?. A ratio describes the relationship between different amounts. A ratio can describe the relationship between two parts of a group, or between one part and the whole group. To better
understand ratios…
LET’S BREAK IT DOWN!
A ratio shows a relationship between two amounts.
A bunny has 2 eyes and 1 nose. That’s a ratio of 2 to 1. You can also write it as 2 : 1. If there are two bunnies, the ratio of eyes to noses becomes 4 to 2, or 4 : 2. For three bunnies, the ratio is
6 to 3, or 6 : 3, and for four bunnies, the ratio is 8 to 4, or 8 : 4. Like with equivalent fractions, you can multiply both numbers in a ratio to get another, equivalent ratio. Multiply 2 : 1 by 3
to get 6 : 3, or 6 eyes on 3 bunnies. Try this one yourself: A sign has 5 parts red for every 1 part white. With the same color ratio, a larger sign has 20 parts red. How many parts white does the
larger sign have?
A ratio shows a relationship between two amounts. A bunny has 2 eyes and 1 nose. That’s a ratio of 2 to 1. You can also write it as 2 : 1. If there are two bunnies, the ratio of eyes to noses becomes
4 to 2, or 4 : 2. For three bunnies, the ratio is 6 to 3, or 6 : 3, and for four bunnies, the ratio is 8 to 4, or 8 : 4. Like with equivalent fractions, you can multiply both numbers in a ratio to
get another, equivalent ratio. Multiply 2 : 1 by 3 to get 6 : 3, or 6 eyes on 3 bunnies. Try this one yourself: A sign has 5 parts red for every 1 part white. With the same color ratio, a larger sign
has 20 parts red. How many parts white does the larger sign have?
Part-to-whole ratios are related to fractions.
A pack of 5 markers contains 1 blue marker. This is a part-to-whole ratio. We can represent part-to-whole ratios with fractions. Write this ratio as [ggfrac]1/5[/ggfrac]. We can use equivalent
fractions to represent the number of blue markers in more packs. Two packs of markers have 2 blue markers out of 10 total, or [ggfrac]2/10[/ggfrac]. This is the same as multiplying the original
numerator, 1, by 2 and multiplying the original denominator, 5, by 2. Try this one yourself: Find the part-to-whole ratio that describes the number of blue markers in 10 packs of markers.
Part-to-whole ratios are related to fractions. A pack of 5 markers contains 1 blue marker. This is a part-to-whole ratio. We can represent part-to-whole ratios with fractions. Write this ratio as
[ggfrac]1/5[/ggfrac]. We can use equivalent fractions to represent the number of blue markers in more packs. Two packs of markers have 2 blue markers out of 10 total, or [ggfrac]2/10[/ggfrac]. This
is the same as multiplying the original numerator, 1, by 2 and multiplying the original denominator, 5, by 2. Try this one yourself: Find the part-to-whole ratio that describes the number of blue
markers in 10 packs of markers.
Part-to-part ratios don’t show the whole.
A fish tank contains 20 fish: 14 blue and 6 yellow. The ratio [ggfrac]14/20[/ggfrac] describes the number of blue fish out of the total. The ratio [ggfrac]6/20[/ggfrac] describes the number of yellow
fish out of the total. These are both part-to-whole ratios. You can also use 'part-to-part' ratios to show this information. The ratio 14 : 6 compares the number of blue fish to the number of yellow
fish. Similarly, the ratio 6 : 14 compares the number of yellow fish to the number of blue fish. Try this one yourself: A bag of marbles has 8 red marbles out of 15 total. The marbles that aren’t red
are purple. Find the part-to-part ratio that compares the number of red marbles to the number of purple marbles. Then find the part-to-part ratio that compares the number of purple marbles to the
number of red marbles.
Part-to-part ratios don’t show the whole. A fish tank contains 20 fish: 14 blue and 6 yellow. The ratio [ggfrac]14/20[/ggfrac] describes the number of blue fish out of the total. The ratio [ggfrac]6/
20[/ggfrac] describes the number of yellow fish out of the total. These are both part-to-whole ratios. You can also use 'part-to-part' ratios to show this information. The ratio 14 : 6 compares the
number of blue fish to the number of yellow fish. Similarly, the ratio 6 : 14 compares the number of yellow fish to the number of blue fish. Try this one yourself: A bag of marbles has 8 red marbles
out of 15 total. The marbles that aren’t red are purple. Find the part-to-part ratio that compares the number of red marbles to the number of purple marbles. Then find the part-to-part ratio that
compares the number of purple marbles to the number of red marbles.
You can simplify ratios by scaling them down.
Simplifying ratios can make the comparisons easier to understand. Let’s say a local animal shelter has 24 dogs and 16 cats, or 24 : 16.
You can divide both numbers by the same divisor to find an equivalent ratio. Divide by 2 to get 12 : 8. Divide by 2 again to get 6 : 4. Divide by 2 one more time to get 3 : 2. You could also divide
24 : 16 by 8 to get 3 : 2. Try this one yourself:
Simplify the ratio “100 parakeets to 75 hamsters.”
You can simplify ratios by scaling them down. Simplifying ratios can make the comparisons easier to understand. Let’s say a local animal shelter has 24 dogs and 16 cats, or 24 : 16. You can divide
both numbers by the same divisor to find an equivalent ratio. Divide by 2 to get 12 : 8. Divide by 2 again to get 6 : 4. Divide by 2 one more time to get 3 : 2. You could also divide 24 : 16 by 8 to
get 3 : 2. Try this one yourself: Simplify the ratio “100 parakeets to 75 hamsters.”
Ratios can help you find the best deal.
Emily saw a deal online, where 4 t-shirts cost $20. Amari saw another deal advertising the same t-shirts, where 8 t-shirts cost $32. You can use ratios to decide which deal is better. Emily can
describe her ratio as “$20 to 4 shirts.” Dividing each of these numbers by 4, that is equivalent to $5 : 1 shirt. Amari can describe his deal as “$32 to 8 shirts.” Dividing each of these numbers by
8, that is equivalent to $4 : 1 shirt. Buying shirts with Amari’s deal saves $1 for each shirt, so it is the better deal. Try this one yourself:Which is the better deal: 5 pairs of socks for $15, or
2 pairs of socks for $12?
Ratios can help you find the best deal. Emily saw a deal online, where 4 t-shirts cost $20. Amari saw another deal advertising the same t-shirts, where 8 t-shirts cost $32. You can use ratios to
decide which deal is better. Emily can describe her ratio as “$20 to 4 shirts.” Dividing each of these numbers by 4, that is equivalent to $5 : 1 shirt. Amari can describe his deal as “$32 to 8
shirts.” Dividing each of these numbers by 8, that is equivalent to $4 : 1 shirt. Buying shirts with Amari’s deal saves $1 for each shirt, so it is the better deal. Try this one yourself:Which is the
better deal: 5 pairs of socks for $15, or 2 pairs of socks for $12?
Graphing ratios shows patterns more visually.
A carnival game involves a basketball hoop. For each basket you make, you win 3 tickets. This is a ratio of 3 : 1. You can use multiplication to make a set of equivalent ratios: 6 : 2, 9 : 3, 12 : 4,
15 : 5. These ratios can be displayed on a coordinate grid. The number of baskets goes on the x-axis, and the number of tickets goes on the y-axis. Then the ratios also represent coordinate pairs:
(3, 1), (6, 2), (9, 3), (12, 4), (15, 5). Each point is 3 right and 1 up from the previous point. To predict the number of tickets more easily, draw a straight line through these points and keep
going up and to the right. Try this one yourself:To get into a carnival, each child admission ticket is $5. Graph the cost of 1, 2, 3, 4, and 5 admission tickets on a coordinate grid. Draw a line
through these points to find the cost of 8 admission tickets.
Graphing ratios shows patterns more visually. A carnival game involves a basketball hoop. For each basket you make, you win 3 tickets. This is a ratio of 3 : 1. You can use multiplication to make a
set of equivalent ratios: 6 : 2, 9 : 3, 12 : 4, 15 : 5. These ratios can be displayed on a coordinate grid. The number of baskets goes on the x-axis, and the number of tickets goes on the y-axis.
Then the ratios also represent coordinate pairs: (3, 1), (6, 2), (9, 3), (12, 4), (15, 5). Each point is 3 right and 1 up from the previous point. To predict the number of tickets more easily, draw a
straight line through these points and keep going up and to the right. Try this one yourself:To get into a carnival, each child admission ticket is $5. Graph the cost of 1, 2, 3, 4, and 5 admission
tickets on a coordinate grid. Draw a line through these points to find the cost of 8 admission tickets.
Many careers use ratios.
Chefs use ratios to calculate the right amount of ingredients to use in a recipe. Florists use ratios to determine how many flowers are needed in each arrangement at a banquet. One of the
responsibilities of a doctor is prescribing medication. Doctors often use ratios based on body weight to determine the proper dosages their patients should take.
Many careers use ratios. Chefs use ratios to calculate the right amount of ingredients to use in a recipe. Florists use ratios to determine how many flowers are needed in each arrangement at a
banquet. One of the responsibilities of a doctor is prescribing medication. Doctors often use ratios based on body weight to determine the proper dosages their patients should take.
A comparison of two amounts.
The math symbol used for the word “to.”
A numerator over a denominator, separated by a fraction bar.
Equivalent fractions
Fractions that have the same value even though they contain different numbers.
Part-to-part ratio
A comparison of two parts from the same whole.
Part-to-whole ratio
A comparison of one of the parts of a whole to the total amount in the whole.
To make as simple as possible (for example, converting a ratio of 20 : 48 to a ratio of 5 : 12).
A two-dimensional space for graphing points and lines, set up with a horizontal x-axis and a vertical y-axis.
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Fibration method over F_q(t)
Algebraic Geometry and Number Theory Seminar
Date: Thursday, May 23, 2024 13:00 - 15:00
Speaker: Elyes Boughattas (University of Bath)
Location: Office Bldg West / Ground floor / Heinzel Seminar Room (I21.EG.101)
Series: Mathematics and CS Seminar
Host: Tim Browning
Determining whether a given diophantine equation has a solution is a wide open question in number theory. For some varieties -- e.g. quadrics -- the existence of local points is enough to determine
the existence of global points: this is known as the Hasse principle. Nevertheless, the latter does not hold for cubic forms, as shown by Selmer in 1951. Manin introduced in 1970 a set called the
Brauer-Manin set, which is expected to describe all obstructions to the Hasse principle, at least for the wide family of rationally connected varieties.
In this talk, I shall present a work in progress which explains how this Brauer-Manin setting is related to fibrations over P^1, whenever the base field is the function field of a curve over a large
finite field.
|
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A Histogram-Based Low-Complexity Approach for the Effective Detection of COVID-19 Disease from CT and X-ray Images
Department of Information Engineering, Electronics and Telecommunications (DIET), Sapienza University of Rome, Via Eudossiana 18, 00185 Rome, Italy
Author to whom correspondence should be addressed.
Submission received: 5 July 2021 / Revised: 17 September 2021 / Accepted: 21 September 2021 / Published: 23 September 2021
The global COVID-19 pandemic certainly has posed one of the more difficult challenges for researchers in the current century. The development of an automatic diagnostic tool, able to detect the
disease in its early stage, could undoubtedly offer a great advantage to the battle against the pandemic. In this regard, most of the research efforts have been focused on the application of Deep
Learning (DL) techniques to chest images, including traditional chest X-rays (CXRs) and Computed Tomography (CT) scans. Although these approaches have demonstrated their effectiveness in detecting
the COVID-19 disease, they are of huge computational complexity and require large datasets for training. In addition, there may not exist a large amount of COVID-19 CXRs and CT scans available to
researchers. To this end, in this paper, we propose an approach based on the evaluation of the histogram from a common class of images that is considered as the target. A suitable inter-histogram
distance measures how this target histogram is far from the histogram evaluated on a test image: if this distance is greater than a threshold, the test image is labeled as anomaly, i.e., the scan
belongs to a patient affected by COVID-19 disease. Extensive experimental results and comparisons with some benchmark state-of-the-art methods support the effectiveness of the developed approach, as
well as demonstrate that, at least when the images of the considered datasets are homogeneous enough (i.e., a few outliers are present), it is not really needed to resort to complex-to-implement DL
techniques, in order to attain an effective detection of the COVID-19 disease. Despite the simplicity of the proposed approach, all the considered metrics (i.e., accuracy, precision, recall, and
F-measure) attain a value of 1.0 under the selected datasets, a result comparable to the corresponding state-of-the-art DNN approaches, but with a remarkable computational simplicity.
1. Introduction
The novel Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) is the cause of one of the worst pandemic of this century: the Coronavirus Disease 2019 (or, simply, COVID-19) [
]. COVID-19 is responsible for illness in the respiratory system, with common symptoms, such as fever and cough, and can also lead to severe pneumonia, an infection that causes a severe inflammation
in the lungs’ air sacs, which are responsible for the oxygen exchange [
However, many studies have highlighted that the Novel COVID-19 Pneumonia (NCP) is different from other viral (Common) Pneumonia (CP) [
]. In this regard, some works have shown that cases of NCP tend to affect the entire lung, unlike common diseases that are limited to small regions [
]. Pneumonia caused by the COVID-19 shows a typical hazy patch on the outer edges of the lungs.
Due to the seriousness of possible consequences of COVID-19, early detection of the disease is vital [
]. Currently, the COVID-19 screening is commonly based on the real-time Reverse Transcription-Polymerase Chain Reaction (rRT-PCR). This technique has demonstrated a sufficiently high specificity;
however, its sensitivity is relatively low in diagnosing COVID-19 [
This problem has motivated researchers to investigate the role of medical imaging as a valid tool for performing non-invasive medical diagnoses [
]. Among the different techniques, conventional Chest X-Rays (CXRs) and Computed Tomography (CT) [
] have assumed a very important role in detecting COVID-19 [
Although CXRs appear diagnostically an effective means of identifying the symptoms of COVID-19, in comparison with chest CT scans, their sensitivity is generally low for pulmonary diseases [
]. Therefore, CXRs make the accurate diagnosis of COVID-19 pneumonia more challenging in comparison with chest CT scans [
]. However, it is not always easy for radiologists to analyze and interpret the CT scans. In addition, in order to reduce the dose of radiation and avoid harmful consequences (such as tumors), it is
preferable to perform scans with a low emission of radiation [
]. In this case, the scanned images often have a degraded quality (such as blur, background noise, and low contrast), which makes it ambiguous and difficult to put an accurate interpretation on CT
scans [
In order to attempt to effectively sort out the above problems, the efforts of the research community are, for the most part, focused on exploiting Artificial Intelligence (AI) approaches for
implementing automatic image classification [
]. Specifically, Deep Learning (DL) algorithms—a branch of machine learning, which uses architectures that possess many layers of processing [
]—have been massively used to detect CXRs [
] and CT scans [
] of infected lungs. In particular, Deep Neural Networks (DNNs) have been widely employed for this purpose [
Although DNNs have been applied with great success to identifying cases of the NCP [
], they still present some challenging limitations. First of all, DL architectures have a very large number of free parameters that must be adapted by the optimization algorithms. This means that, in
order to achieve the convergence towards a robust solution, it is necessary to have a large amount of training data, which is not always possible in practice [
]. In addition, it is not certain that having lots of scans available is enough to be of quality [
]. This situation is further worsened by the fact that, since the COVID-19 is a relatively recent disease, the proportion of CXRs and CT scans related to NCP is very limited with respect to the
number of images available in the datasets, freely available on the web [
]. Since most of the proposed DL approaches are of supervised type, an imbalance between the number of images in the training sets could provide poor results. In addition, the number of free
parameters that must be adapted during the training leads to a high computational cost and many of the proposed DNN architectures can be trained only on powerful computers equipped with many GPUs in
several hours.
Moreover, the identification of scans of patients infected by COVID-19 is a problem more similar to the anomaly detection rather than the traditional classification, due to the small number of data
present in each dataset [
]. It would, therefore, be appealing to design an anomaly detection algorithm, which is light from the computational point of view, and is capable of identifying radiology scans related to COVID-19
with high accuracy. In this direction, some works have focused on the application of GAN-based approaches to detect anomalies in medical images [
]. However, such approaches suffer a not negligible computational cost.
Considering all the above challenging limitations in employing DL techniques for detecting scans of infected patients, we wonder whether DNNs are absolutely necessary to detect such scans, or simpler
methodologies may instead perform in the same way as the DNNs, although they do not have, in general, advantages over DNNs.
Motivated by this still unanswered question, in this paper, we investigate whether traditional image processing techniques [
] can be reliable enough to detect images related to patients infected by COVID-19. Towards this end, we propose a histogram-based approach that exploits a suitable inter-histogram distance [
]: the histograms of some target images (belonging to a reference class, i.e., not images of infected lungs) are averaged to produce a target histogram. This histogram is then used as a reference in
the inference phase: for each unknown test scan, the related test histogram is obtained, and a suitable measurement of the distance between the tested and target histograms is evaluated. If this
distance is below a certain threshold, the test image is classified the same as the target class; otherwise, it is considered an anomaly and labeled as COVID-19.
Overall, the main contributions of this paper may be so summarized:
• We propose a histogram-based technique to automatically detect the COVID-19 disease from CXRs and CT images. The histogram evaluation is a simple and very fast operation. In addition, in order to
construct a target histogram, very few images are needed, in contrast to DNNs that have to use a huge number of scans to be trained.
• We investigate different inter-histogram distances to evaluate how an unknown scan is far from the target one. These distances are used to label the test images as normal (reference class) or
anomaly (NCP), depending on whether they are less or greater than suitably set thresholds. Although all the proposed distances can be computed with the same computational cost, we expect that
some of them work better in practice.
• We evaluate numerical results on benchmark datasets available in the open literature on CXRs and CT scans and compare the proposed approach to other state-of-the-art DNN-based architectures. We
observed that the proposed approach, although very simple to implement, is able to obtain excellent results.
We expect that the main insight, deduced from the numerical results of the proposed approach, is that, at least when the images of the considered datasets are homogeneous enough (i.e., a few outliers
are present), it is not really needed to resort to complex-to-implement DNNs, in order to attain an effective detection of the COVID-19 disease.
The rest of the paper is organized as follows.
Section 2
presents the recent literature on the topic.
Section 3
describes the proposed approach in terms of both the used methodology and inter-histogram distances, as well as introduces the experimental setup.
Section 4
presents the obtained numerical results and performance comparisons, as well as provides the discussion on the proposed idea. Finally,
Section 5
concludes the paper and outlines some future research.
2. Related Work
During the last two decades, great attention of researchers has been focused on automatic classification of medical images, and, to that end, a plethora of different techniques have been employed [
]. Although some interpretations are still made by a visual inspection of the obtained scans, Computer-Aided Detection or Computer-Aided Diagnosis (CAD) facilities help radiologists to detect lesions
on chest X-rays [
]. However, in its early stage of usage, CAD can be defined solely as a computer analysis tool for image data. Over time, research efforts transformed CAD into automatic diagnostic tools [
The majority of the proposed approaches in the literature, principally devoted to the diagnosis of lung cancer [
], exploits low-level handcrafted features [
]. These approaches are founded on: texture-based features [
], edge-based features [
], graph mining [
], and color-based features [
]. Specifically, these works exploit different methodologies to perform image segmentation in regions of interest for building up masking operations that label them as either infected or not infected
In particular, such as the proposed approach, the works in Reference [
] exploit histogram-based information to characterize and/or classify medical images. In the past, histograms, due to the simplicity of their computation and accuracy of the produced results, have
been widely used for object recognition [
], identification [
], and/or mining from images [
]. Among the applications in medical images, Ref. [
] used distance-based approach in detecting anomalies in images. Specifically, Reference [
] evaluates the histogram of a 3D volume of neuronal cells and fit them with a Gaussian function, in order to define different clusters in the images. Similar to the approach proposed in this paper,
Reference [
] performs automatic segmentation of cell nuclei by exploiting an adaptive (local) threshold computed on the basis of the mean and standard deviation of the pixel gray-level values. However, unlike
the proposed approach, these works do not directly employ histograms to detect images of infected patients.
Many works have been focused on important classes of medical images, i.e., CXRs and CT scans. These studies analyze such images to detect tuberculosis [
] and lung cancer [
]. However, CXRs and CT scans can be useful also to detect other important diseases, such as viral or bacterial pneumonia [
In this regard, CXRs and CT scans have been massively used for the detection of COVID-19 disease. Although the pandemic is quite recent, several contributions are available in the literature.
However, even if there are some sparse works about the manual screening of such images [
], almost all of the state-of-the-art approaches relate to the use of Machine Learning (ML) [
] and Deep Learning (DL) [
] techniques. The ML, in fact, could become a helpful and potential powerful tool for large-scale COVID-19 screening [
], and the author in Reference [
] has recently hypothesized that DL techniques applied to CT scans can become the first alternative screening test to the rRT-PCR in the near future. Motivated by this expectation, in the last year,
the DL has been successfully used for CXRs [
], CT scans [
], or both [
]. Being, indeed, challenging to summarize all the available literature in a single paper, there are some useful reviews regarding the application of DL techniques to COVID-19 detection on CXRs [
], CT scans [
], and both [
]. A systematic review on the detection of COVID-19 using chest radiographs and CT scans, highlighting strongness and weakness of several different approaches, can be found in Reference [
]. There are also some surveys of related datasets available [
An overview of these works points out that DL approaches, mainly based on the supervised paradigm, can be divided into two main families: those based on segmentation and those that perform the
classification task directly. The approaches, which are based on segmentation, are usually founded on U-Net type architecture to identify relevant part of the CXRs/CT scans and perform
classification, focusing the attention only on these sections [
]. The second family of approaches, instead, is based on the binary classification problem of COVID/Non-COVID images [
] and utilize deep Convolutional Neural Networks (CNNs) and their variants, including VGG16, InceptionV3, ResNet, and DenseNet.
Finally, we point out that there are a few approaches that exploit the unsupervised paradigm, such as the use of deep auto-encoders [
]. Specifically, these works rely on deep or stacked autoencoders to automatically extract a set of meaningful features and then use softmax classifiers on the top to distinguish images of infected
lungs from the healthy ones.
Among DL approaches, there exist some papers that exploit image histograms [
]. Specifically, the work in Reference [
], after a pre-processing with a median filter, extracts meaningful features from cropped regions by using Histogram Orientation Gradients (HOGs) and uses a final classification stage with a
feed-forward neural network. Reference [
] introduces a pre-processing step based on the Contrast Limited Adaptive Histogram Equalization (CLAHE) idea to improve the contrast and generate an enhanced CXR image. Authors in Reference [
] propose a statistical histogram-based method for the pre-categorization of skin lesion: the image histogram is used to check the image contrast variations and classify these images into high and
low contrast images. Similarly, Reference [
] uses histogram thresholding techniques to perform image clustering and then propose an image segmentation method for skin lesion delineation. Authors in Reference [
] propose ImHistNet, a deep neural network for end-to-end texture-based image classification based on a variant of the learnable histogram introduced in Reference [
] for pattern detection in pictures. The work in Reference [
], similar to our approach, extracts features from suitable inter-histogram distances but uses these features by feeding them into several (supervised) classifiers. Similarly, Reference [
] uses histogram to extract meaningful features, along with other statistical characterizations, and then performs a final classification through an artificial neural network. Moreover, the works in
Reference [
] exploit histogram-based enhanced techniques and metaheuristic approaches in Magnetic Resonance Imaging (MRI). Specifically, Reference [
] introduces MedGA, an image enhancement method based on Genetic Algorithms (GAs) that improves the threshold selection for the segmentation of the region of interest. MedGA uses a pre-processing
stage for the enhancement of MR images based on some nonlinear histogram equalization techniques. Similarly, the work in Reference [
] proposes a Particle Swarm Optimized (PSO) texture-based histogram equalization technique to enhance the contrast of MRI brain images and find an optimal threshold for the segmentation. The obtained
results overcome those of standard histogram equalization and those presented in Reference [
]. Finally, also the recent work in Reference [
] exploits DL segmentation-based histogram and threshold analysis in chest CT scans for differentiating healthy lungs from the infected ones. Specifically, the proposed method automatically derives
imaging biomarkers from chest CT images based on histogram and provides a Hounsfield unit (HU) threshold analysis to assess differences in those biomarkers between lung-healthy individuals and those
affected by atypical pneumonia.
Overall, although these last papers exploit the histogram in their approaches, they are substantially different from our work, since they exploit the histogram for: (i) enhancing the image quality;
(ii) adjusting the contrast; and (iii) extracting features to be used by a final classifier. Moreover, only a couple of these papers focus on COVID-19 disease. Our approach, on the other hand,
directly uses the histogram information to measure the distance from a target histogram and then decides whether or not an unknown CXR or CT image is related to a patient who is infected with
COVID-19. This makes our approach an unsupervised method, which exhibits very limited implementation complexity.
A synoptic overview of the related work is provided in
Table 1
, that summarizes the main approaches pursued by the referred papers.
3. Materials and Methods
3.1. Proposed Approach
The proposed approach is based on evaluating a suitable distance between the histogram of an unknown image and a target histogram, which is obtained by averaging a number of histograms belonging to a
target class (i.e., common pneumonia or normal). Afterward, if this distance is greater than a threshold, then the unknown image is labeled as anomaly, i.e., it presents COVID-19 disease; otherwise,
it is labeled as the target class.
3.1.1. The Evaluation of the Target Histogram
Let $X = X k k = 1 N T$ be a set composed of $N T$ target images (i.e., belonging to normal or CP diagnostics). Since both CT and CXR scans are grayscale images, the k-th input data $X k$ is modeled
as a matrix of dimension $M × N$, representing the number of rows and columns, respectively. The pixels’ intensity values have been normalized into the range $[ 0 , 1 ]$.
For each of the $N T$ images, the related histogram $h k$, with $k = 1 , … , N T$, is computed for $N b i n$ bins. In this paper, we choose the (normalized) histogram as a statistical representation
of the target, principally due to its simplicity and efficiency in computation. A histogram is an estimate of the probability distribution obtained by partitioning the range of values into a sequence
of $N b i n$ equally-spaced intervals (called bins) and counting how many values fall into each interval. The histogram is then normalized so that its sum total equates to one.
The $N b i n$ bins, used to build up the histogram, should be chosen by a trade-off between numerical stability of the distance measurements and its discriminating capability. Although this number
turned out not to be critical for the performance of the proposed approach, we found that an optimized setting to guarantee non-empty bins is to use 50 bins, i.e., $N b i n = 50$.
target histogram
is, hence, evaluated as the average of
$h k$
, computed for each image, over
$N T$
$h ¯ = 1 N T ∑ k = 1 N T h k .$
Alternatively, since the histogram evaluation is a nonlinear operation, we have also tested the average of all the
$N T$
target images followed by the evaluation of the histogram. However, we expect that this procedure gives a rise to inadequate results, as detailed in
Section 4.5
In this phase, we also evaluate the distance between the just computed target histogram
$h ¯$
and the histogram
$h k$
of every single
-th reference scan, by using suitable probability dissimilarity measurements (introduced in the next subsection). Among these distances, we compute the mean
$d m$
and standard deviation
$σ d$
values, in order to set conveniently a suitable threshold
$T H$
used during the test phase to discriminate a reference scan from an anomaly. The idea is that the statistical distance from an anomalous scan should be greater than a reference one. Therefore, the
$T H$
would be set equal to the mean distance
$d m$
plus a term depending on its standard deviation. Mathematically, we set the threshold
$T H$
as follows:
is a suitable constant. In the case of CT scans, we set
$η = 2$
, while, for CXRs,
$η = 1.4$
provides good results.
During the inference phase, the test histogram of an unknown image is evaluated. This test histogram will be then compared to the target one, according to the same distance measurement used
previously. If the distance between the test and target histograms rises beyond the threshold
$T H$
, the underlying image will be marked as COVID (anomaly); otherwise, it will be marked as the reference one. The proposed idea is shown in
Figure 1
, which depicts both the evaluation phase of the target histogram, as well as the inference phase, where an unknown image goes through the deduction process.
3.1.2. The Considered Inter-Histogram Distances
Evaluation of the dissimilarity between the target histogram and that obtained from an unknown test image is a highly important issue.
In the literature, the similarity between the two histograms is evaluated by several distance measurements over the underlying distributions [
]. Regarding the aims of this paper, after denoting by
the two
$N b i n$
-dimensional vectors representing the involved histogram distributions, defined over the set of interval bins
$I = 1 , … , N b i n$
, we have selected the following four distances.
• Cosine distance: It is formally defined as:
$d c = 1 − ∑ i ∈ I p i q i ∑ i ∈ I p i 2 ∑ i ∈ I q i 2 ,$
and it normally ranges in the interval
$[ 0 , 2 ]$
. However, since we are considering probabilities, each bin value is non-negative (i.e.,
$p i , q i ≥ 0$
, for all
), so that the distance in (
) is limited to the interval
$[ 0 , 1 ]$
. A distance equal to zero means that the two histogram are identical, while a distance equal to one denotes orthogonal histograms.
• Kullback–Leibler (KL) divergence [
]: It is defined as:
$d K L = ∑ i ∈ I p i log p i q i ,$
$p i$
$q i$
are the values of the
histogram in the
-th bin, respectively. By definition, the contribution of the
-th term in the summation in (
) is zero when
$p i$
vanishes. It is always non-negative, and it is zero when the two distributions are equal.
• Bhattacharyya distance [
]: It is defined as:
$d B = − log ∑ i ∈ I p i q i .$
The Bhattacharyya distance, such as the KL divergence, is always non-negative, while it is vanishing when the two distributions are equal.
• $χ 2$
distance: It is defined as:
$d χ 2 = ∑ i ∈ I p i − q i 2 p i + q i .$
In addition, the $χ 2$ distance is a non-negative measure.
These distances, except the cosine one, have been normalized by the number $N b i n s$ of used bins, in order to render them independent of the $N b i n$ setting. The Bhattacharyya distance is widely
used in several application, such as image processing, and, unlike the KL one, has the advantages of being insensitive to the zeros of distributions.
In the proposed approach, if the distance (chosen between the cosine, KL, Bhattacharyya, and
$χ 2$
ones) between the target and a test histogram is above the set threshold
$T H$
of Equation (
), then, the image under test is classified as COVID-19 (CNP); otherwise, it is classified as the target class (normal or CP, depending on the used training set).
3.2. The Considered Datasets
In this work, we use two kinds of chest medical images: the chest CT scans and the traditional CXR images.
Regarding the CT scans, we have selected the COVIDx CT-2A dataset (It can be downloaded from:
(accessed on 20 May 2021), which has been constructed by collecting a number of open data sources [
] and comprises 194,922 CT slices from 3745 patients. The scans of the dataset are related to three classes: novel coronavirus pneumonia due to SARS-CoV-2 viral infection (NCP), common pneumonia
(CP), and normal (N) controls (i.e., images from non-infected individuals). For NCP and CP CT volumes, slices marked as containing lung abnormalities were leveraged. Moreover, all the CT volumes
contain the background, in order to avoid model biases. An example of a representative image for each image class is reported in the first row of
Figure 2
In order to stress the effectiveness of the proposed approach in view of numerical comparisons, which are presented later, for the CXRs, we found a highly imbalanced dataset that contains few COVID
images. This is a representative of the actual situation of public datasets, and it has been selected “ad hoc” to check the effectiveness of the proposed approach with respect to the corresponding
supervised state-of-the-art approaches based on deep learning, which usually need a large amount of well-balanced training data.
Specifically, for the CXRs, we have selected the COVID-XRay-5K dataset (It can be downloaded form:
(accessed on 20 May 2021), which has been constructed by collecting data from two publicly available sources [
]. The downloadable COVID-Xray-5k dataset is already split in training and test sets, and it contains 2084 training and 3100 test images. However, from the web URL, it can be downloaded a training
set composed of only 580 Non-COVID and 84 COVID images, respectively, while the test set is composed of 3000 Non-COVID and 100 COVID images. An example of a representative image for each class is
reported in the second row of
Figure 2
Since the COVID-19 disease causes a pneumonia, from both datasets, we have selected only pneumonia images as the Non-COVID target class. In addition, for validating the proposed approach, we also
consider normal images as targets. Furthermore, from the CT dataset, we have randomly selected 3500 images for both the Pneumonia and COVID classes for evaluating the target and 500 images from both
classes to test it. For the CXR dataset, we used all the 580 images available for the Non-COVID class and the 84 images available for the COVID class to evaluate the target, while we selected 100
pneumonia images and all the available 97 COVID images for the test set. A summary of the used datasets is provided in
Table 2
. The same size of the target/test sets are used for the normal class. In the numerical results, we then select
$N T$
target images from those available.
3.3. Built-Up Simulation Environment
All the carried out simulations have been implemented in Python environment by using the end-to-end and open-source deep learning platform TensorFlow 2 exploiting the Keras API. Simulations have been
performed on a PC equipped with an Intel Core i7-4500U 2.4 GHz processor, 16 GB RAM, and Windows 10 operating system.
3.4. The Considered Performance Metrics
In a binary classification problem, we are interested in classifying items belonging to a positive class (P) versus a negative one (N). Therefore, there are four basic combinations of actual data
category and assigned output category, namely:
• True Positive (TP): correct positive assignments;
• True Negative (TN): correct negative assignments;
• False Positive (FP): incorrect positive assignments; and
• False Negative (FN): incorrect negative assignments.
The set of these four metrics is usually arranged in a bi-dimensional matrix layout, called Confusion Matrix (CM), which allows a simple visualization of the performance of a binary classification
algorithm. Specifically, each column of the CM represents the instances in a predicted class, while each row represents the instances in an actual class. Moreover, the combination of the previous
four numbers in some powerful indicators can be a valid tool to quantitatively measure the performance of a classification algorithm [
]. Among all the possible combinations, in this paper, we focus on the accuracy, precision, recall, and F-measure metrics, whose formal definitions are briefly reported in
Table 3
. Accuracy is the ratio between the correct identified instances among their total number. Precision is the ratio of relevant instances among the retrieved instances, while the Recall is the ratio of
the total amount of relevant instances that were actually retrieved. Finally, precision and recall can be combined in a single measurement, called F-measure, that is mathematically defined as their
harmonic mean.
The state-of-the-art approaches have also been evaluated by the area under the Receiver Operating Characteristic (ROC) curve, abbreviated as AUC. The closer the AUC is to one, the better is the
classifier performance. The ROC curve, which is a graphical representation of the performance of a binary classifier, is obtained by plotting the TP rate on the
-axis against the FP rate on the
-axis for increasing values of the decision threshold. The TP rate (formally coincident with the recall metric) and the FP rate are, respectively, the ratio of the number of TP to the total positive
examples, and the ratio of the number of FP to the total negative examples, respectively. Their definition can be found in the last two rows of
Table 3
4. Results and Discussion
In this section, we provide the numerical results, obtained from the proposed approach, on the two considered datasets. The performance has been evaluated by considering Non-COVID scans as the
reference images to evaluate the target histograms. The Non-COVID class has been formed from images related to patients infected by common viral pneumonia (CP) or from healthy subjects (N). The
performance comparisons with three state-of-the-art deep architectures are also presented. In the case of CT dataset, the test set, used in experiments, is composed of 500 CT scans belonging to the
new coronavirus pneumonia (NCP) and 500 CT scans belonging to the reference class (CP or N). In the case of CXR dataset, instead, the test set, used in experiments, is composed of 100 CXRs belonging
to the new coronavirus pneumonia (NCP) and 97 CXRs belonging to the reference class (CP or N). To provide a clear graphical representation of the obtained distances, the test instances have been fed
into the proposed algorithm in this order: first, the NCP scans, and then the reference CP or N ones.
4.1. Evaluation of the Proposed Approach
In the first set of experiments, we investigate the choice of the inter-histogram distances of
Section 3.1.2
on both datasets. In this experiment, we use the CP class as the reference. The results have been obtained by selecting
$N T = 500$
reference images and a number
$N b i n = 50$
of histogram bins. Moreover, the
variable in (
) has been set to
$η = 2$
for the CT dataset and to
$η = 1.4$
for CXRs, respectively.
Table 4
summarizes the results, obtained by the proposed approach, in terms of the Accuracy, Precision, Recall, and F-measure metrics, introduced in
Section 3.4
and defined in
Table 3
The results of
Table 4
support the effectiveness of the proposed approach. In fact, all of the considered metrics reach high values, and, interestingly enough, they reach the top result of 100% with some distance
Table 4
also reports the mean
$d m$
, standard deviation
$σ d$
, and the related threshold
$T H$
obtained by Equation (
). The second column of this table shows that the actual value of
$σ d$
depends on the selected inter-histogram distance. By a careful examination of the rows of
Table 4
, we can draw three main considerations:
• although all the considered distances provide good results for the CT datasets, the cosine distance is able to reach 100%;
• in case of CXR dataset, all the considered distances obtain the accuracy of 100%; and
• for both the datasets, the cosine distance provides the lower values of the standard deviation $σ d$.
The results provided in
Table 4
support the use of the cosine distance that works to advantage, compared to the other distance measurements. In fact, cosine distance is able to capture the “spatial” information provided by
histograms, since it effectively measures their similarity, regardless of the single bin values. Motivated by these considerations, in the following tests and comparisons, we use the cosine distance.
In order to give visual insights about the results in
Table 4
and justify the top 100% accuracy, we point out that
Figure 3
shows the numerically evaluated spectra of the obtained cosine, KL, and Bhattacharyya distances. Since the
$χ 2$
distance behaves as in the latter two distances, it is not explicitly depicted in the paper. This figure clearly shows the effectiveness of the proposed approach. In fact, we can see that the NCP CT
scans (the first 500 bars in the left panels of
Figure 3
) are much more distant with respect to the corresponding reference images (the last 500 bars). Similar conclusions apply to the CXRs (the first 100 bars and last 97 bars in the right panels of
Figure 3
, respectively). The difference between the classes is about one order of magnitude for the cosine distance, while it is reduced for the KL divergence and the Bhattacharyya distance. These last cases
also show a larger variance of the obtained distances with respect to the cosine distance, as already highlighted by the second column of
Table 4
. However,
Figure 3
also underlines that the differences between the considered distances in the case of CXRs are more limited with respect to the CT case.
As a final consideration on these first results, both
Table 4
Figure 3
confirm that a lower value of the standard deviation
$σ d$
makes the performance more robust with respect to the outliers eventually present in the tested dataset.
In order to validate the proposed approach, we repeat the first experiment by using the normal class (N), related to scans of healthy subjects, as reference. Results on both the datasets are shown in
Table 5
. This table shows that the results, obtained by using the normal class as the reference, provide metrics very similar to those shown in
Table 4
. Moreover, once again, by using the proposed cosine distance, all the considered performance metrics are unit-valued.
4.2. Sensitivity of the Proposed Approach to the Parameter Settings
The next test is concerned with the setting of the number
$N b i n$
of bins, used in the histogram evaluation. As already mentioned, the number of bins, used to construct the histogram should attain a suitable trade-off between numerical stability of the distance
measurement and its discriminating capability.
Table 6
summarizes the obtained results using the cosine distance in Equation (
). In this test, we again use
$N T = 500$
target images, while the constant
, in Equation (
), has been set to
$η = 2$
for the CT dataset and to
$η = 1.4$
for the CXRs, respectively.
Table 6
shows that the performance of the proposed approach is quite independent of the choice of the number
$N b i n s$
of histogram bins for the CXR dataset, since it remains quite stable. However, its effect is more noticeable for the CT dataset. Although the standard deviation
$σ d$
decreases with a smaller number of bins, the best performance in terms of all the considered metrics is obtained with a number of bins from 50 to 100 in the case of CT dataset. Hence, in order to
obtain the best performance in both the datasets and maintain a limited computational complexity, we select
$N b i n = 50$
We also evaluate the robustness of the proposed approach with respect to the number
$N T$
of target images. Towards this end, we evaluate the target histogram by averaging
$N T$
histogram representations of the corresponding scans.
Table 7
summarizes the obtained results using the cosine distance in (
). In this test, we again used
$N b i n = 50$
histogram bins, while the threshold in (
) has been set to
$η = 2$
for the CT dataset and to
$η = 1.4$
for CXR one, respectively.
Table 7
shows that, especially for the CT dataset, the performance of the proposed approach is quite insensible to the choice of the number
$N T$
of target images in a large interval, since performance remains unchanged, even if the standard deviation tends to decrease by increasing the number of used images, because this depends on their
average. However, we can see a gradual degradation in performance by reducing the number
$N T$
of target images. On the basis of these results, we used 500 target images to maintain the standard deviation at an appropriate lower value.
The last test on the sensitivity of the proposed approach is concerned with the choice of the
parameter in Equation (
). To this end, we vary the
parameter inside the interval
$[ 0 , 5 ]$
, with a step-size equal to
, for a total of 51 different values, and we graphically report the corresponding obtained accuracy (measured in percentage).
Figure 4
shows the results for both the considered datasets. These results have been evaluated by using the cosine distance, a number
$N T = 500$
of target images, and a number
$N b i n = 50$
of histogram bins.
Figure 4
clearly shows that there is a suitable range of
values that produces the accuracy of 100%, while the performance rapidly decreases with vanishing
(that means the threshold in Equation (
) is set on the basis of the mean distance
$d m$
only) or with increasing
Figure 4
also shows that this range is independent of the used dataset, since its corresponding value for the CT is larger than that of the CXR dataset. However, the range producing the top accuracy is
sufficiently wide to confirm the robustness of the proposed approach. In order to guarantee the broadest margin, we have set the
parameter at the value corresponding to approximately the middle of the range, i.e.,
$η = 2$
for CT dataset and
$η = 1.4$
for CXR dataset.
4.3. Performance Comparison with an Alternative Histogram-Based Benchmark Approach
In order to numerically validate the chosen order of the mathematical operations to compute the target histogram shown in
Figure 1
, we test the proposed approach by considering an alternative method of evaluation. Different from the proposed methodology, shown in
Figure 1
, in the present experiment, we perform the average of all the
$N T = 500$
target images, and then we evaluate the corresponding histogram. This last idea is sketched in
Figure 5
and could affect the performance, since the whole process is nonlinear. The inference phase remains unchanged with respect to our original idea. Once again, we use a number
$N b i n = 50$
of histogram bins, and results are obtained with using the cosine distance in (
The numerical results, obtained by using such an idea, are provided in
Table 8
for both the considered datasets. An examination of the rows of this table clearly demonstrates that this alternative does not perform very well, since all the reported metrics are poorer with
respect to the excellent results of the proposed idea, shown in
Figure 3
. Hence, we chose to: (i) first, compute each single histogram; then, (ii) evaluate the target by averaging all of them, as shown in the top part of
Figure 1
4.4. Performance Comparisons with the State-of-the-Art DNN-Based Approaches
In this subsection, we show some comparisons with other state-of-the-art benchmark solutions. Specifically, in this paper, we consider some well-known feed-forward deep networks in the literature,
i.e., the AlexNet [
], the GoogLeNet [
], and the ResNet18 [
]. In this regard, we note that AlexNet is composed of the cascade of five convolutional layers and three (dense) fully connected layers, while the GoogLeNet is more complicated, since it is much
deeper and constructed by stacking three convolutional layers, nine
modules, and two dense layers. An inception module is a particular layer obtained by concatenating several convolution operations with different filter sizes and a max pooling operation. Finally,
ResNet18 is composed of 18 layers ending with a dense layer with a softmax activation. Different from the previous networks, the central part of the ResNet18 is a deep stack of residual units: these
are composed of two convolutional layers (without pooling layer), with Batch Normalization and ReLU activation, using
$3 × 3$
kernels and preserving spatial dimensions.
Since these architectures are of supervised type, they have been trained by using both the CP and NCP classes in the training set. The training has been performed by using the Adam algorithm [
] with the default values (
$β 1 = 0.9$
$β 2 = 0.999$
, and
$ε = 10 − 7$
), a batch size
$N b = 16$
, and a learning rate set to
$μ = 10 − 6$
. The training has been executed for a total of 60 epochs.
The results, provided by these state-of-the-art supervised DNNs, are shown in
Table 9
These results point out that AlexNet performs worse than our approach, since it reaches merely a degree of 71% and 93% accuracy for the CT and CXR, respectively, and, in addition, in terms of other
performance metrics. The performance of GoogLeNet is the same as the proposed approach under the CT dataset but is worse than the one of the proposed approach under the CXR dataset. This lower
performance in the case of CXR images is due to the limited number of training images used during the learning. However, we have to remark that the high performance of GoogLeNet on CT is obtained by
a deep architecture that uses a huge number of free parameters compared to the proposed approach, as shown in
Table 10
, which reports, for completeness, the number of trainable parameters and the training time (in minutes) for all the considered architectures and datasets. In our approach, we use only one free
parameter, the adaptive threshold in Equation (
). Once again, this consideration, along with the trade-off shown in
Table 10
, supports the actual effectiveness of the proposed methodology.
4.5. Performance Robustness of the Considered Approaches
The aim of this last subsection is to evaluate the performance robustness of the proposed histogram-based approach, compared to the considered state-of-the-art DNN architectures.
In order to provide a visual interpretation of the results, shown in
Table 9
, and check how robust these solutions are regarding the testing images,
Figure 6
shows, for both the considered datasets, the confusion matrices obtained by the state-of-the-art (supervised) DNNs.
The confusion matrices in
Figure 6
clearly show the poor robustness of the DNN approaches. They illustrate that the main confusion is for the Non-COVID class, where, especially, in the case of CT scans, many instances are erroneously
4.6. Limitations of the Proposed Approach
In this subsection, we outline some limitations of the proposed approach with respect to the DL-based ones.
We underline that the DNN methods can produce a spectrum of a posterior probabilities that may be, in turn, used as indexes of the reliability of the taken decision. This is possible only by the
prior learning performed by such approaches. In our approach, instead, the reliability is directly measured by the value of the inter-histogram distance used in the discrimination phase. Although
this distance may be used as (coarse) reliability index, nevertheless, it is not more informative than the corresponding spectrum of posterior probabilities nor is it related to the probability
spectrum in a direct way.
As highlighted in previous numerical results, we have identified some datasets where the proposed approach is able to attain the unit-valued accuracy. However, the performance obtained on other
datasets could be not so significant, since the effectiveness of the histogram comparison depends on the quality of the scanned images and the way used to be recorded. DNNs, due to their prior
learning phase, are expected to be less sensitive, indeed, to the image characteristics.
To recap, DNN-based approaches are expected to be more flexible, powerful, and generalizable than the proposed approach, but at the cost of larger datasets, higher computational resources, and longer
training time. After stressing this, we point out that the goal of this paper is not the introduction of a general methodology but, rather, showing that, under some operating conditions, a simple and
computationally efficient approach can produce good results.
5. Conclusions and Hints for Future Research
In this paper, we investigate whether a traditional histogram-based approach can be used for the detection of CT and CXR scans of infected lungs. Specifically, we propose a histogram-based approach
to detecting the new coronavirus pneumonia from CT and CXR scans. Since the number of these images is not high, we evaluate a target histogram on a reference class (i.e., normal or common pneumonia).
A suitable inter-histogram distance is then used to evaluate how far this target histogram is from the corresponding histogram evaluated for an unknown test scan: if this distance is above a
threshold, the test image is classified as anomaly, i.e., affected by the COVID-19 disease; otherwise, it is classified the same as the target class. A number of numerical results, evaluated on two
open-source benchmark datasets, demonstrates the effectiveness of the proposed approach, since it is able to obtain a top degree of the considered performance metrics (i.e., accuracy, precision,
recall, and F-measure), equal to unit value, comparable to the corresponding state-of-the-art DNN approaches, but with a limited computational complexity.
In a nutshell, the main lesson stemming from the reported performance comparisons is that, at least when the images, embraced by the considered datasets, are homogeneous enough (i.e., few outliers
are present, so that the standard deviations
$σ d$
of the corresponding inter-histogram (normalized) distances are limited up to
), it is not really needed to resort to complex-to-implement DNNs, in order to attain reliable detection of the COVID-19 disease. The proposed histogram-based approach attains, indeed, very good
detection performance, comparable with those of DNNs, but at an (extremely) reduced implementation complexity and training time (see
Table 10
In future works, we aim at extending our methodology to different types of medical images, other than CT, and/or different diseases. We expect, in fact, that the automatic screening by means of
pathological images can take a great advantage by the simplicity of our methodology, in both the resulting accuracy and prediction time. To this end, it could be interesting to investigate, as a
second line of research, more performing inter-histogram distances, such as the Earth Mover’s Distance (EMD) [
], which measures how much work it would take to transform one histogram shape into another. A third hint of future research can be addressed towards the use of Varational Autoencoeders (VAEs) and
Generative Adversarial Networks (GANs) for generating additional examples in the case of new variants of COVID-19, in order to be fast in the automatic discrimination of these scans without awaiting
the construction of sufficiently copious dataset. Finally, a fourth line of future research can be focused on the implementation of the proposed methodology atop distributed Cloud/Fog Computing
technological platforms [
], in order to produce fast and reliable clinical responses by exploiting the low-delay (and, possibly, multi-antenna empowered [
]) capability of the supporting broadband wireless access networks [
Author Contributions
Conceptualization, M.S. and E.B.; methodology, M.S., E.B. and L.P.; software, M.S. and S.S.A.; validation, A.M., M.S., L.P. and S.S.A.; formal analysis, E.B. and M.S.; investigation, M.S., L.P. and
E.B.; data curation, A.M. and S.S.A.; writing—original draft preparation, M.S.; writing—review and editing, M.S., L.P. and A.M.; visualization, A.M.; supervision, E.B.; funding acquisition, E.B. All
authors have read and agreed to the published version of the manuscript.
This work has been supported by the projects: “SoFT: Fog of Social IoT” funded by Sapienza University of Rome Bando 2018 and 2019; “End-to-End Learning for 3D Acoustic Scene Analysis (ELeSA)” funded
by Sapienza University of Rome Bando Acquisizione di medie e grandi attrezzature scientifiche 2018; and “DeepFog–Optimized distributed implementation of Deep Learning models over networked multitier
Fog platforms for IoT stream applications” funded by Sapienza University of Rome Bando 2020.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Conflicts of Interest
The authors declare no conflict of interest.
The following main abbreviations are used in this paper:
AI Artificial Intelligence
AUC Area Under the Curve
CAD Computer-Aided Diagnosis
CNN Convolutional Neural Network
CM Confusion Matrix
CP Common Pneumonia
CT Computed Tomography
CXR Chest X-ray
DL Deep Learning
DNN Deep Neural Network
FN False Negative
FP False Positive
HOG Histogram Orientation Gradients
KL Kullback–Leibler
ML Machine Learning
MRI Magnetic Resonance Imaging
NCP Novel COVID-19 Pneumonia
ROC Receiver Operating Characteristic
TN True Negative
TP True Positive
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Figure 1. Proposed approach: (a) evaluation of the target histogram, and (b) inference phase for an unknown image.
Figure 2. Examples of some chest CT and CXR images: (a) CT normal, (b) CT common pneumonia (CP), (c) CT novel coronavirus pneumonia (NCP), (d) RX normal, (e) RX CP, and (f) RX NCP.
Figure 3.
The obtained distances on the considered CT and CXRs datasets, and related threshold
$T H$
(the red lines) set using (
). The distances are: (
) Cosine for CT, (
) Cosine for CXRs, (
) Kullback–Leibler divergence for CT, (
) Kullback–Leibler divergence for CXRs, (
) Bhattacharyya distance for CT, and (
) Bhattacharyya distance for CXRs. The
$χ 2$
distance behaves as in the KL and Bhattacharyya distances; hence, it is not explicitly shown. The results have been obtained by selecting
$N T = 500$
reference images and a number
$N b i n = 50$
of histogram bins. The threshold in (
) has been set to
$η = 2$
for the CT dataset and to
$η = 1.4$
for CXRs.
Figure 4. Accuracy (in %) as a function of the $η$ parameter: (a) CT dataset and (b) CXR dataset. The best selected values ($η = 2$ for CT dataset and $η = 1.4$ for CXR dataset) are shown with a red
cross. The results have been evaluated by using the cosine distance, a number $N T = 500$ of target images, and a number $N b i n = 50$ of histogram bins.
Figure 6. The confusion matrices obtained by the considered state-of-the-art supervised approaches: (a) AlexNet for CT, (b) AlexNet for CXRs, (c) GoogLeNet for CT, (d) GoogLeNet for CXRs, (e)
ResNet18 for CT, and (f) ResNet18 for CXRs.
Family Approach Work Image Type
Manual screening CAD [57] CT
[58] CT
[59] CT
[60] CT
[61] CXR
Hand-crafted features Texture-based features [33] Cells
Edge-based features [34] Histopathological
[35] Cytological
Graph mining [36] Histopathological
Color-based features [37] Cells
[38] Histopathological
[39] Histopathological
Deep learning Review [72] CXR
[73] CT
[74] CT
[75] CXR + CT
[65] CXR + CT
[76] CXR + CT
[77] CXR + CT
[78] CXR + CT
Segmentation [79] CXR
[80] CT
[81] CT
[82] CT
[56] CT
[83] CXR
[84] CXR
Classification [70] CT
[69] CT
[85] CT
[86] CT
[87] CXR
[20] CXR
[88] CXR
Unsupervised [89] Histopathological
[90] CT
[91] CT
[92] CXR
Histogram Histogram + DL [93] CXR
[94] CXR
[95] Skin
[96] CT
[97] MRI
[98] Skin
[99] Skin
[101] MRI
[102] MRI
[103] CT
Computed Tomography X-rays
Type Target Test Target Test
COVID 3500 500 84 100
Non-COVID 3500 500 580 97
Performance Metrics Formula
Precision $T P / ( T P + F P )$
Recall $T P / ( T P + F N )$
F-measure $2 T P / ( 2 T P + F P + F N )$
Accuracy $( T P + T N ) / ( T P + F N + F P + T N )$
TP rate $T P / ( T P + F N )$
FP rate $F P / ( F P + T N )$
Table 4. Results obtained by the proposed approach under the two considered datasets. The results have been obtained by using $N T = 500$ reference images from the CP class and a number $N b i n =
50$ of histogram bins.
Model $d m$ $σ d$ $TH$ Accuracy Precision Recall F-Measure
Computed Tomography ($η = 2$)
Proposed (Cosine) 0.0544 0.0664 0.1872 1.0000 1.0000 1.0000 1.0000
Proposed (KL) 0.1666 0.1218 0.4102 0.9870 0.9870 0.9870 0.9870
Proposed (Bhattacharyya) 0.0027 0.1037 0.2101 0.9870 0.9870 0.9870 0.9870
Proposed ($χ 2$) 0.2272 0.1119 0.4652 0.9860 0.9860 0.9860 0.9860
X-Rays ($η = 1.4$)
Proposed (Cosine) 0.0339 0.0325 0.0794 1.0000 1.0000 1.0000 1.0000
Proposed (KL) 0.0011 0.0564 0.0801 0.9949 0.9950 0.9949 0.9949
Proposed (Bhattacharyya) 0.0027 0.0447 0.0653 1.0000 1.0000 1.0000 1.0000
Proposed ($χ 2$) 0.0016 0.0450 0.0645 1.0000 1.0000 1.0000 1.0000
Table 5. Results obtained by the proposed approach under the two considered datasets. The results have been obtained by using $N T = 500$ reference images from the N class and a number $N b i n = 50$
of histogram bins.
Model $d m$ $σ d$ $TH$ Accuracy Precision Recall F-Measure
Computed Tomography ($η = 2$)
Proposed (Cosine) 0.0458 0.0436 0.1330 1.0000 1.0000 1.0000 1.0000
Proposed (KL) 0.1231 0.1044 0.3319 0.9949 0.9950 0.9949 0.9949
Proposed (Bhattacharyya) 0.0048 0.0983 0.2014 0.9870 0.9870 0.9870 0.9870
Proposed ($χ 2$) 0.1783 0.0970 0.3723 0.9747 0.9748 0.9747 0.9747
X-Rays ($η = 1.4$)
Proposed (Cosine) 0.0283 0.0292 0.0692 1.0000 1.0000 1.0000 1.0000
Proposed (KL) 0.0018 0.0509 0.0731 0.9898 0.9898 0.9898 0.9898
Proposed (Bhattacharyya) 0.0023 0.0415 0.0604 1.0000 1.0000 1.0000 1.0000
Proposed ($χ 2$) 0.0015 0.0398 0.0572 0.9870 0.9870 0.9870 0.9870
Table 6. Results obtained by the proposed approach for different number $N b i n$ of histogram bins. The reported performance metrics refer to the cosine distance at $N T = 500$ reference images.
$N bins$ $d m$ $σ d$ $TH$ Accuracy Precision Recall F-Measure
Computed Tomography ($η = 2$)
5 0.0222 0.0169 0.0560 0.7840 0.7927 0.7840 0.7824
10 0.0331 0.0258 0.0847 0.9570 0.9604 0.9570 0.9569
25 0.0453 0.0441 0.1335 0.9880 0.9883 0.9880 0.9880
50 0.0544 0.0664 0.1872 1.0000 1.0000 1.0000 1.0000
100 0.0558 0.0832 0.2222 1.0000 1.0000 1.0000 1.0000
250 0.0646 0.1225 0.3096 0.9960 0.9960 0.9960 0.9960
500 0.0764 0.1454 0.3672 0.9960 0.9960 0.9960 0.9960
X-Rays ($η = 1.4$)
5 0.0005 0.0045 0.0068 0.9260 0.9261 0.9260 0.9260
10 0.0133 0.0143 0.0333 0.9880 0.9883 0.9880 0.9880
25 0.0236 0.0259 0.0598 1.0000 1.0000 1.0000 1.0000
50 0.0339 0.0325 0.0794 1.0000 1.0000 1.0000 1.0000
100 0.0076 0.0380 0.0608 1.0000 1.0000 1.0000 1.0000
250 0.0122 0.0430 0.0724 1.0000 1.0000 1.0000 1.0000
500 0.0135 0.0568 0.0931 0.9898 0.9898 0.9898 0.9898
Table 7. Results obtained by the proposed approach at different number $N T$ of target images. The metrics have been evaluated by using the cosine distance and $N b i n = 50$ histogram bins.
$N T$ $d m$ $σ d$ $TH$ Accuracy Precision Recall F-Measure
Computed Tomography ($η = 2$)
5 0.0351 0.0911 0.2173 0.8968 0.8968 0.8968 0.8968
10 0.0312 0.0911 0.2134 0.9880 0.9883 0.9880 0.9880
25 0.0383 0.0852 0.2087 0.9960 0.9960 0.9960 0.9960
50 0.0414 0.0787 0.1988 1.0000 1.0000 1.0000 1.0000
100 0.0460 0.0728 0.1916 1.0000 1.0000 1.0000 1.0000
250 0.0501 0.0704 0.1909 1.0000 1.0000 1.0000 1.0000
500 0.0544 0.0664 0.1872 1.0000 1.0000 1.0000 1.0000
3500 0.0732 0.0566 0.1864 1.0000 1.0000 1.0000 1.0000
X-Rays ($η = 1.4$)
5 0.0022 0.0807 0.1152 0.7817 0.8474 0.7817 0.7700
10 0.0031 0.0724 0.1045 0.9188 0.9300 0.9188 0.9188
25 0.0076 0.0722 0.1087 0.9137 0.9262 0.9137 0.9129
50 0.0125 0.0669 0.1062 0.9645 0.9669 0.9645 0.9644
100 0.0179 0.0492 0.0868 0.9898 0.9901 0.9898 0.9898
250 0.0257 0.0329 0.0718 1.0000 1.0000 1.0000 1.0000
500 0.0339 0.0325 0.0794 1.0000 1.0000 1.0000 1.0000
Table 8.
Numerical results obtained by using the target histogram evaluated as in
Figure 5
. The results have been obtained by using the cosine distance,
$N T = 500$
reference images, and a number
$N b i n = 50$
of histogram bins.
Architecture Accuracy Precision Recall F-Measure
CT 0.5460 0.5902 0.5460 0.4826
CXR 0.9137 0.9266 0.9137 0.9137
Table 9.
Performance of the tested benchmark DNNs. These architectures have been trained and tested on the whole datasets in
Table 2
Architecture Accuracy Precision Recall F-Measure AUC
Computed Tomography
AlexNet 0.7110 0.8601 0.7110 0.7343 0.9460
GoogLeNet 1.0000 1.0000 1.0000 1.0000 1.0000
ResNet18 0.9130 0.9281 0.9130 0.9137 0.9130
AlexNet 0.9340 0.9341 0.9340 0.9340 0.9340
GoogLeNet 0.9746 0.9794 0.9694 0.9744 0.9750
ResNet18 0.9695 0.9697 0.9695 0.9695 0.9700
Table 10.
Computational complexity of the tested models (“M” stands for millions of parameters). The training time, in minutes, refers to data sets composed of images of size
$300 × 200$
pixels for the CT dataset and
$320 × 390$
pixels for the CXR dataset. The number of images is presented in
Table 2
Model # param. Training Time CT Training Time CXRs
Proposed 1 0.31 0.087
AlexNet 58 M 223 62
GoogLeNet 6 M 258 73
ResNet18 11 M 290 84
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Scarpiniti, M.; Sarv Ahrabi, S.; Baccarelli, E.; Piazzo, L.; Momenzadeh, A. A Histogram-Based Low-Complexity Approach for the Effective Detection of COVID-19 Disease from CT and X-ray Images. Appl.
Sci. 2021, 11, 8867. https://doi.org/10.3390/app11198867
AMA Style
Scarpiniti M, Sarv Ahrabi S, Baccarelli E, Piazzo L, Momenzadeh A. A Histogram-Based Low-Complexity Approach for the Effective Detection of COVID-19 Disease from CT and X-ray Images. Applied Sciences
. 2021; 11(19):8867. https://doi.org/10.3390/app11198867
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Scarpiniti, Michele, Sima Sarv Ahrabi, Enzo Baccarelli, Lorenzo Piazzo, and Alireza Momenzadeh. 2021. "A Histogram-Based Low-Complexity Approach for the Effective Detection of COVID-19 Disease from
CT and X-ray Images" Applied Sciences 11, no. 19: 8867. https://doi.org/10.3390/app11198867
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Parents' Review
The Parents' Review
A Monthly Magazine of Home-Training and Culture
Edited by Charlotte Mason.
"Education is an atmosphere, a discipline, a life."
Notes and Queries
Volume 2, 1891/92, pg. 154-155
Mr. Arnold-Forster, in his very able and interesting article on "Naval Cadets" in your December number, has some remarks which seem to me misleading.
On page 763 we read: "The examination is undoubtedly of a high standard--higher, especially in mathematical subjects, than is generally required of boys of thirteen years of age."
As a matter of fact, that is the minimum age for entering the Britannia, so that the examination would be more fairly described as intended for boys of fourteen years of age. I cannot agree with Mr.
Arnold-Forster that the standard required is above that of an ordinary boy of fourteen, and I should say that a good average public schoolboy will get in fairly easily with good work. I speak, mainly
from personal experience, of boys whom I have passed in easily and whose acquirements I know to be very ordinary, but I can also appeal to prima facie probability. No one can be candidate unless he
gets a nomination. The nominations given are about twice as many as the vacancies to be filled. As they are given by influence, the chances are that each batch of candidates contain an average number
of clever and stupid boys, and the boy who just scrapes in, being in the middle of the list, will be just about an average boy.
This, however, is not a very important point. The following sentence on the same page seems to me to involve a much graver error:--
"As a general rule, it will probably be found advisable to send a boy for a time to one of those schools, of which several are to be found in the south of England, where the naval examination is
specially prepared for, and where teachers can be found who can show a boy not only how to arrive at correct mathematical results, but to reach them by the particular methods approved of by those who
will subsequently have to be pleased."
Now, in the first place, what are these mysterious methods approved of by the examiners for naval cadetships and revealed to the mathematical masters of naval cramming establishments, but hidden from
the distinguished mathematicians who are sure to be found not only in the great public schools, but in almost any secondary school of importance? I am not a mathematician, but if I were a high
wrangler and a mathematical master in some good school--if some parent came to me and said, "I want my boy to get into the Britannia: I am afraid you will not teach him the First Book of Euclid and
Algebra up to simple equations according to the methods approved of by the examiners, and therefore I shall send him to Mr. So-and-so's Naval Academy"--well, I hope I should keep my temper.
Surely the methods approved of "by those who have to be pleased" are the best methods, and the best methods are common property of the best mathematicians and the best schools--not of any set of
examiners. If the examiners reject the methods of "Hall and Knight's Algebra," for instance, they are simply unfit for their work.
But apart from this question of "methods," is not all the talk we hear about special schools and special tutors for each examination a great deal exaggerated? Latin is Latin and Algebra is Algebra;
they do not change because the examiner changes. The raison d'etre of "crammers," if they have one, is that the subjects required in the examination to be crammed for are not taught, or not taught up
to the adequate standard, in the ordinary classes of schools. Thus I daresay if a boy wants to pass into the Indian Civil Service, it may be well to remove him for the last year or two from most
schools, because he will require tuition different from that given in the regular classes. The standard required in many subjects--French, for instance, and history--is higher than that which the
sixth form is at, as a rule. The same may be true to a much smaller extent of Woolwich and Sandhurst. But it is not true of the Britannia. The examination for the most part is on the regular school
subjects--Latin, French, Mathematics, and English Composition and Dictation--and the standard is that of an ordinary fourth or fifth form. It is true that the Scripture, Geography, and History are on
a different footing; in the last two at least private tuition will be required, but the amount required is very small, and any respectable school would be able to provide the tuition needed for a
small extra fee.
It is only a drawback, though an inevitable one, to the naval profession, that its members should be educated apart from the rest of the world from their fifteenth year. Why should we make this worse
by isolating them from their thirteenth year?
Mr. Arnold-Forster will probably appeal to "results," and tell me that Mr. So-and-So has passed so many candidates. To all these appeals, and I hear them very frequently, I can only answer, Do you
know how many he has failed to pass?
~ * ~ * ~ * ~ * ~ * ~ * ~ * ~ *
I had the advantage of hearing Miss Emily Lord's suggestive paper on Kindergarten training at a meeting of the Westminster and Belgravia branch of the Parents' Union on February 7. As the mother of
young children whose school education must soon begin, I was greatly interested.
The sympathy with child-nature shown by the Kindergarten system is now, I believe, very widely appreciated by parents for their little ones, but I have frequently heard mothers of older children say,
"Oh, yes, capital while the children are quite young, but when they are older they have to learn on a different plan, and they are at a disadvantage among children who have from the first been taught
in the old way."
Now this I am sure is a wrong notion, and I think many readers of the Parents' Review would feel much indebted to anyone who would give information as to how the Kindergarten methods are continued
for children in their teens; how, for instance, such subjects as modern languages and history are taught, and where schools and governesses are to be found adopting these methods.
Typed by Jeanette DeFriend, Mar 2013
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Mathematics | Statistics
This video lesson is on the Central Tendency
Introduction to CBSE Class 9 Mathematics Chapter "Statistics"
“Statistics” is an engaging chapter in the CBSE Class 9 Mathematics curriculum that introduces students to statistical concepts and tools necessary for handling data efficiently. It begins with the
basics of data collection and its need in various fields. The chapter discusses the organization of data into frequency distributions, which can be represented visually using bar graphs, histograms,
and frequency polygons, making data easier to understand and interpret.
Central to the chapter is the understanding of central tendencies including mean, median, and mode, which are measures that provide an average value of the data. It explains how to find these
measures in grouped and ungrouped data sets and the significance of each in different scenarios. The chapter also touches upon the concept of probability as a fundamental statistical concept towards
the end.
This chapter not only develops foundational skills in statistics but also highlights its practical applications in real-life situations, such as in economics, science, sports, and more. It emphasizes
the role of statistics in making informed decisions based on data.
Assignments for CBSE Class 9 Mathematics Chapter “Statistics”
1. Data Collection: Collect data on a topic of interest within your community, such as favorite sports or television shows.
2. Graph Creation: Represent the collected data using a histogram or bar graph for visual interpretation.
3. Mean, Median, Mode: Calculate the mean, median, and mode of your collected data to understand central tendencies.
4. Comparative Study: Analyze the central tendencies of two different data sets to compare their characteristics.
5. Probability Basics: Conduct a simple probability experiment, like tossing a coin or rolling a die, and record the outcomes.
Statistics is a powerful tool in the CBSE Class 9 Mathematics arsenal, providing students with the ability to turn raw data into meaningful information. This chapter serves as a stepping stone for
students to advance into more complex statistical studies and to apply these methods to everyday problem-solving and decision-making situations.
"Preparing for the Class 6 exam? Notebook is your go-to resource for learning anytime, anywhere. With courses, docs, videos, and tests covering the complete syllabus, Notebook has the perfect
solution for all your study needs. Join Notebook today to get everything you need in one place.
Questions and Answers for CBSE Class 9 Mathematics Chapter "Statistics"
1. Q1: What is the purpose of studying statistics in mathematics?
ANS: The purpose is to learn how to collect, analyze, and interpret data to make informed decisions and understand the world better.
2. Q2: How do you find the mean of a set of numbers?
ANS: To find the mean, add up all the numbers and then divide by the count of the numbers.
3. Q3: What is the difference between grouped and ungrouped data?
ANS: Grouped data is organized into frequency distributions or intervals, while ungrouped data is listed as individual data points.
4. Q4: Why are graphs and charts used in statistics?
ANS: Graphs and charts are used for visual representation of data, which makes it easier to understand, interpret, and communicate the information.
5. Q5: What is a frequency polygon?
ANS: A frequency polygon is a graphical representation of the distribution of data points, plotted using the midpoints of the intervals, and connecting them with straight lines.
6. Q6: How do the median and mode differ from the mean?
ANS: The median is the middle value when data is arranged in order, and the mode is the most frequently occurring value, while the mean is the average of all the values.
7. Q7: Can there be more than one mode in a data set?
ANS: Yes, a data set can have more than one mode if multiple values occur with the same highest frequency.
8. Q8: How can statistics help in daily life?
ANS: Statistics can help in daily life by providing insights for making decisions, understanding trends, and evaluating the likelihood of outcomes.
9. Q9: What is the role of probability in statistics?
ANS: Probability helps in predicting the likelihood of various outcomes and is foundational in making predictions based on statistical data.
10. Q10: How is data organized in statistics?
ANS: Data is organized in a variety of ways, including frequency tables, graphs, charts, and through measures of central tendency and dispersion.
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60 percent of the registered voters are Party A supporters and the rest are Party B
Latest TCS Aptitude Question SOLUTION: In a certain city, 60 percent of the registered voters are Party A supporters and the rest are Party B supporters. In an assembly election, if 75% of the
registered Party A support
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Drag the sliders left or right to adjust your patient's risk!
Pretest Probability:
Post Test Probability:
What Is This?
It's a dynamic risk calculator:
1. Drag the "Pretest Probability" slider to your patient's risk of having the disease in question (based on your gestalt).
2. The Likelihood Ratio is pre-filled for you for the particular finding you clicked.
3. The Post Test Probability tells you what your risk is now after the patient having or not having a particular finding.
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11.8 Lab 2: Chi-Square Test of Independence
Stats Lab 11.2
Lab 2: Chi-Square Test of Independence
Class Time:
Student Learning Outcome
• The student will evaluate if there is a significant relationship between favorite type of snack and gender.
Collect the Data
1. Using your class as a sample, complete the following chart. Ask one another what your favorite snack is, then total the results.
You may need to combine two food categories so that each cell has an expected value of at least five.
Sweets (candy & baked goods) Ice Cream Chips & Pretzels Fruits & Vegetables Total
2. Looking at Table 11.26, does it appear to you that there is a dependence between gender and favorite type of snack food? Why or why not?
Hypothesis Test Conduct a hypothesis test to determine if the factors are independent:
1. H[0]: ________
2. H[a]: ________
3. What distribution should you use for a hypothesis test?
4. Why did you choose this distribution?
5. Calculate the test statistic.
6. Find the p-value.
7. Sketch a graph of the situation. Label and scale the x-axis. Shade the area corresponding to the p-value.
8. State your decision.
9. State your conclusion in a complete sentence.
Discussion Questions
1. Is the conclusion of your study the same as or different from your answer to answer to question 2 under Collect the Data?
2. Why do you think that occurred?
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