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I don't churn monero. It's easier when you just use it how it works. - monero.town
Just send it to wallet to new wallet to new wallet a few times. Easy peazy and cheap even for people with only pennies.
Stop playing games and use it “by default” like how it’s intended and it even defeats the problems most apparent in “breaking monero” series.
How did you not figure this out already?
• Churning and sending to a different wallet you control leave basically the same on-chain footprint. Churning saves you the hassle of syncing multiple wallets though.
□ azalty@jlai.lu English
26 days ago
For real
OP is literally churning, just to a different wallet if his.
If transactions aren’t completely swept and they keep a non-zero change output, it makes things worse
Churning is fine if you have coin control and keep note of your outputs
Different wallets are fine if you manage to keep track of everything and properly sweep outputs. Avoid spending multiple churned outputs together if you can to prevent linking them together,
but apart from that, it’s good.
□ WibbleWobble English
28 days ago
If you have a (mostly-offlne) “saving” wallet and a “spending” wallet is there an obsfucation advantage in transacting between them rather than just churning the spending wallet?
☆ Hm I believe it depends on if you combine outputs from both wallets by doing so but you’d have to talk to a decoy expert about that.
• Anesthetic Bliss English
29 days ago
I think there is one very good usecase for churning though.
And before anything, yes I know that one should not use CEX but in some cases it is just much more convenient. Although I am now starting to use Haveno, I get not everyone is up to it, and CEX is
just plain easier.
Imagine the following scenario:
I buy a shitcoin over at a KYC’d CEX.
I send that coin to a centralized swap, or trade it with a compromised person, in exchange of XMR.
Lets say I repeatedly do that procedure with the same person or CEX. Then I end with multiple “small” outputs on my wallet, all from the same entity. Let’s say for example 10 outputs of 0.1 XMR,
which all have been sent to me by the same entity.
Now I want to buy something that costs 1 XMR. I need to use my 10 existing outputs. I make a transaction that takes 10 inputs and 2 outputs (what I buy + change). The transaction has 10 inputs,
and all of those inputs have a ring, where one of the members of each ring is an output controlled by the compromised entity.
The likelihood of someone making a transaction with 10 inputs, where those 10 inputs happen to contain a member in the ring that was sent by that specific exchange and that is linkable to my
identity is near zero, unless it is me who is spending those 10 outputs.
Therefore, the person that sent me those 10 outputs can make a very well educated guess that it was me who bought that item for 1 XMR.
This “vulnerability” is actually talked about in the Breaking Monero series, and as far as I know, it will be solved when FCMP++ comes, since we will get rid of rings altogether.
However let’s say I do one step of churning with all those outputs without mixing them with eachother. That is, I send to myself 10 transactions of 0.1 XMR, so I just “forward” each output to
myself once, without making any transaction that contains two poisoned inputs at the same time.
Then I will still end up with 10 outputs of 0.1 XMR, but all the “poisoned” outputs are present in different and unlinkable transactions, and the negative actor does not know whether they are
truly spent or not.
Then I can actually join those 10 outputs into one 1XMR transaction safely, knowing that I am the only person who knows where those 10 outputs come from.
Am I wrong in this thought process?
□ azalty@jlai.lu English
26 days ago
You’re right and wrong. Churning will reduce the traces linking back to you, but you’re still exposed at 1 churn per output, when including 10 outputs. You would even be exposed when spending
2 outputs from the same source
As you know, each ring currently has 16 transactions, including you. This means, on average (more or less because of other factors, but still), each output will be featured in 16
transactions. We can therefore assume that 1 in 16 of those transactions is real (in reality the distribution is not that perfect, but as an average, it is important to know the
You now have a 1 in 16 chance of being traced. Statistically, the transaction you made has 6,25% chance of being made by you. That’s pretty high for a single poisoned output, right?
Now imagine you spend 2 poisoned outputs… the distribution algorithm is not evenly distributed: older outputs are less likely to be picked than newer. This means you get a situation where the
older your 2nd poisoned output is, the more you’ll stand out. The math is not that easy the make, but just knowing that each output will only be included 16 times on average, and that there
are a lot of transactions so a lot of outputs, it becomes really unlikely that 2 of the poisoned outputs that are linked to the same individual end up in the same transaction if it was not
made by the individual of question itself.
With 3+ poisoned outputs you basically confirm that it was the same person. Might not hold up in court, but they’ll definitely know
10+ poisoned outputs? Definitely you.
Churning each output only multiplies the number of possibilities by 32 (16 for one output, 32 in reality because 2 outputs are generated). This will certainly throw off the basic chain
analysis methods, but if you’re a person of interest, all the linked outputs will be analyzed. All outputs that are created by including poisoned inputs might be considered, effectively
multiplying the number of possibilities by 32 as said earlier. If I tried to spy on someone with this, I would probably set a higher suspicion level on the first transaction level, then less
on the second… assuming chain analysis software does that as well, churning would actually divide the chances of being caught by more than 32 but lets assume they don’t do that as it’ll be
easier. You now have 1 chance on 16^2=256 so 0,39% chance of having done a transaction with a churned output in between for a single poisoned output. Still pretty high if you want my opinion.
If you do that multiple times, you’ll stand out for sure. Including multiple outputs will also expose you a lot. Spending 10 churned poisoned outputs will definitely expose you.
I should just make a blockchain analysis program and test transactions with it 😂
My recommendation: if you’re going to spend multiple poisoned outputs at once: churn them together into 1 output (it’s called sweeping afaik), then churn this individual output. The initial
churn merge will make the transaction stand out, but since you only have one output to churn, you reduce the traces.
Please correct me if I’m wrong, I have put a lot of time thinking about all of this but I might have forgot to include some specific things. I already noticed that all transactions include 2
outputs (at least), so the possibilities are multiplied by 32, not 16 as was my initial assumption
☆ Anesthetic Bliss English
26 days ago
I didn’t know the protocol tried to use every output in around 16 transactions. I know about the 16 ring size, but I didn’t know it also tried to use each output 16 times. If so, that is
very smart and interesting. You learn something new every day!
The idea of sweeping them and then churning the merged output is also smart.
Oh well I guess we just have to wait for FCMP++ where theoretically all this will be no longer relevant :)
I remember watching the breaking monero series, when it was mentioned that (paraphrasing) “Rings are what give security to Monero but I really hope we get rid of them”… That time is
finally getting closer :)
○ azalty@jlai.lu English
26 days ago
The protocol doesn’t try to use each output 16 times actually, that could be pretty nice I guess. I was just saying that statistically, you should get an average of 16 times because,
well, the ring size is 16. The actual may vary quite a bit, and your output might potentially never be featured as a decoy, or featured 100+ times. It isn’t likely though. I just used
16 because it is simpler this way.
I never watched the breaking Monero series, I should take the time to do it
And yea, really excited for FCMP++ as well :) - most chain analysis stuff will go bye bye
■ Anesthetic Bliss English
26 days ago
Yes you are right, it was too early in the morning for me to process properly…
Statistically you should see each output used an average of 16 times, that makes sense.
□ Blake English
27 days ago
• jay_edwards English
29 days ago
What do you mean by your text? For example, I make a few steps before swap Monero to another crypto or fiat. Buying USDt on CEX/DEX, then go to instant crypto excahnges, like Exolix, simpleswap
and then swap USDt to Monero, and it’s sent to my own wallet. That’s my way, without problems and KYC. What are you talking about?
• Yes, keep your assets in XMR and only exchange when you need another currency, e.g. for a purchase. | {"url":"https://monero.town/post/4572785","timestamp":"2024-11-06T06:00:15Z","content_type":"text/html","content_length":"159327","record_id":"<urn:uuid:e5c51a8f-ed2a-4a79-8a1a-f29c6669ef54>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00722.warc.gz"} |
Expected Value: How To Find Value Bets - Beating Betting
Expected Value: How To Find Value Bets
Last updated October 18th, 2019
It is always important to gain as much value from your bets as you can. Making bets that aren’t profitable over time is a sure way to lose money.
How do you know whether a bet is good value or not? We can perform a simple calculation to work out the expected value for any bet. This will help us to be profitable over the long term.
What Is Expected Value?
Often shortened to EV, Expected Value is the amount of profit or loss we would expect to make if we could make the same bet an infinite number of times. It eliminates luck and short-term variance so
that you can understand the true value of a gambling decision.
For example, in poker you might lose a hand with aces when going all-in pre-flop. But just because you lost, it doesn’t mean it was the wrong decision at the time — it was still +EV in all
circumstances, and if you could re-run the hand an infinite number of times then you would end up with a substantial profit.
If we work out the Expected Value as a positive number (+EV) then we would expect to make money on the bet on the long term, but not necessarily in the short term. The higher the value the better.
If the calculated Expected Value is negative (-EV), then likewise we would expect to be losing money on the bet if we made it an infinite number of times.
We can use this Expected Value to work out which bets will be the most likely to be profitable in the long term for us. We can also use this value to work out which bookmaker promotions are in our
favour and which are not in our favour.
How Do I Calculate Expected Value?
There is a simple mathematical calculation to figure out the Expected Value of any wager.
EV = [(Probability event 1) x (Profit/Loss if event 1 happens)] + [(Probability event 2) x (Profit/Loss if event 2 happens)] + [(Probability event 3) x (Profit/Loss if event 3 happens)]… and so on
and so forth.
The probability of an outcome happening can be calculated by doing 1 divided by your lay odds then multiply by 100. If the lay odds on Liverpool are 1.25, the chance of them winning are 1 / 1.25 =
0.8, then 0.8 x 100 = 80% probability.
An easy way to picture this is with a simple coin toss.
There is a 50/50 chance of a coin landing on heads or tails in a fair coin toss. If we bet £10 that the coin will land on heads, our expected value is calculated as follows:
EV = (0.5 x £10) + (0.5 x -£10) = £5 – £5 = £0
The expected value in the long term would therefore be £0 (no value, but at least it’s not negative value).
Now, a bookmaker will naturally build their commission into the odds they offer you. If a bookmaker was offering odds on a coin toss, they might offer you odds of 1.90 instead of the actual fair
long-term odds (2.00).
This means your profit for the coin landing on heads falls to £9 instead of £10. The EV for this new wager is worked out as follows:
EV = (0.5 x £9) + (0.5 x -£10) = £4.50 – £5 = -£0.50
Therefore the Expected Value of this bet is negative, meaning we would lose money in the long term. The rate at which we would lose money depends on the size of the value of EV, and the frequency of
The worse the odds are for us, the more negative the EV will become, and the rate at which we lose money increases.
Clearly, we want to find wagers that have as positive an EV as possible for us to make as much profit as we can.
How To Find Value Bets
It is important to realise that Expected Value is not fixed, it is only ‘expected’, and may differ, especially in the short term. If you have inside information or tips on an event, you may find that
the odds you calculate differ to the odds offered by a bookmaker.
Let’s work out the Expected Value for the Euro Qualifiers match between Spain and Sweden.
There are three outcomes, a Spain win, a Sweden win, or a draw. We bet £10 on a Spain win.
Spain win: 2.00 or 50%
Sweden win: 4.00 or 25%
The Draw: 3.60 or 27.78%
EV = (10.00*0.5) – (10*0.53) = 5.00 – 5.30 = -£0.30
This shows the negative value for EV we can expect when placing a bet with a bookmaker. This negative value is built into all bookmaker odds as their way of making profit. There are still situations
where we can create positive EV through certain promotions.
This is the case for bookmakers, but what about betting exchanges? With exchanges like Betfair, the price of the odds fluctuates but always creates either a neutral or negative implied EV after
commission. If the EV becomes positive, the market reacts to shift the value back to 0.
If we had some insider information or other details that led you to believe that the chance of a Spain win was actually greater than 50%, then you could have a Positive EV bet. The higher the
positive EV, the higher the likelihood of long-term profit.
It is important to bear in mind that the figures used in these calculations are implied odds.
This simply means that the odds of each outcome are determined by the odds offered by the market. This does not mean they are the True Odds of each likelihood occurring.
The odds have been arrived at because the market has decided them. You may have a more efficient way of calculating the actual odds.
You may have your own method of calculating the odds of an event occurring. This is sometimes done by calculating the Poisson Distribution Model. In this case, you should work out your own odds, and
then compare them to the offered odds.
You may have read about an ‘arbitrage strategy’ which is simply finding odds from different bookmakers, and combining them into an Expected Value calculation to create a positive outcome for your
When betting with a bookmaker, they will include their commission when setting their fixed odds. The calculation becomes simple. However, this changes with a betting exchange. Betfair takes a 5%
commission on winnings from new customers.
This commission can decrease over time. You must input the commission rate into any EV calculation made for betting exchange websites. This is as easy as working out the commission on a win and then
taking it away from your net profit for a win.
There are a few different ways you can become more adept at finding Value Bets. This also happens over time, as you become more experienced. It is useful to stick to the same market, as you will
naturally start to see patterns.
Expected Value From Bookmaker Promotions
In terms of matched betting, certain promotions will have a positive EV built-in and others which turn out to have a negative EV.
One of the easiest ways to find promotions that provide Value Bets is with sign-up promotions. When you first start matched betting, you will mostly be completing sign-up offers with various
These sign-up offers vary in their Value, but overall they should get you around a 70% profit! Obviously, once you have signed up with the majority of bookmakers, you will no longer be able to take
advantage of this Expected Value resulting from sign-up offers.
After completing all available sign-up offers, there are other kinds of offers that you can take advantage of. Some of these promotions will have a positive EV, but not all of them!
Sometimes bookmakers run promotions that seem enticing, but after calculating the EV, it actually turns out to be negative in the long term.
You can calculate the EV for promotions the same way as with a standard bet, except you must take into account the likelihood of triggering bonuses. The formula for this is as follows:
EV = (Profit if promotion triggers * Probability of triggering) – (Qualifying cost * Probability of not triggering)
As before, if the EV turns out positive, you will likely make a profit in the long term by repeating this offer over and over again. Likewise, if the EV turns out negative, you will probably lose
money in the long term.
What Are The Benefits Of Finding Value Bets?
It is important to note that just because a certain bet has a positive EV does not mean you are guaranteed to make a profit every time. It simply implies that if the odds are correct, then over time
your outcome will tend to the positive.
Clearly, if you are consistently making bets with a positive EV, your bank should increase over time.
It is important to try and stay away from bets with a negative EV most of the time. There may be occasions where even though the calculated EV is negative, you still make a profit in the short term,
or for one-off bets.
The whole idea with Expected Value is that it is a long-term average. It is ok to make bets with a negative EV now and then, but it would defy mathematics to do this consistently and make long-term
Final Thoughts
You should always try and calculate the Expected Value for any bets you place. It is ok to make a few bets with a negative EV, but if you do this over the long term, you will almost certainly not be
making money!
About the Author
This post was written by Andy Beggs. Andy is a keen sports fan and has been writing for Beating Betting from his home in Australia since August 2019. | {"url":"https://www.beatingbetting.co.uk/matched-betting-tips/expected-value/","timestamp":"2024-11-04T07:24:46Z","content_type":"text/html","content_length":"96119","record_id":"<urn:uuid:2605df64-861b-43e0-9fe5-3bc4ed5db750>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00786.warc.gz"} |
Putting Questions
“How do add a putt that is a tap in? I’ve been adding it as a 1 foot putt, is this correct?”
Yes, input as 1 foot.
“For tap in putts, is it possible to record your putt in inches, or is 1 foot the smallest? ”
1 foot is the smallest. There is no difference in SG between 1 inch and 1 foot, so no need to distinguish.
“Even if I two putted all of those , surely that wouldn’t be a handicap of 30 worthy?”
From 7-21 feet, the average strokes to hole out is less than 2.0. For example, for a scratch golfer from 17 feet is about 1.9. To simplify, let’s ignore 3 putts. Then the one-putt probability is 10%
and the 2-putt probability is 90%. So a 2-putt from 17 feet has an SG of -0.1. Do that on 18 holes and you’ll lose about 2 strokes putting in the round and your putting handicap would be about 10
(give or take).
Now going through the same analysis from 10 feet, i.e., 18 two-putts from 10 feet, each losing 0.3 strokes per hole, would have a total SG putting of around -5 to -6 and a corresponding putting
handicap of around 20-25.
You’d get a handicap of 30 if you also had some 2-putts from under 7 feet.
If you putt from 10 feet and leave yourself a 5-footer, your SG on the 10-foot putt will be much worse than if you leave the 10-footer 1 foot away. The SG from 10-feet doesn’t care whether you make
or miss the 5-footer, for that counts for the SG for the 5-foot attempt (and goes into the 0-6 foot SG bucket).
So it seems like you missed every attempt in the 7-21 foot range and some of the were not tap-ins. Your 0-6 SG was great, consistent with your comment about two-putting them all.
Not all two-putts are created equal – a 10-footer hit to 5 feet and a sunk 5-footer is not the same as a 10-footer hit to 2 feet and a sunk 2-footer.
Also, the handicaps for one round can be quite variable. See what happens with a few more rounds of data. | {"url":"https://golfmetrics.com/faq/putting-questions/","timestamp":"2024-11-02T02:47:41Z","content_type":"text/html","content_length":"26157","record_id":"<urn:uuid:b3f2645f-97bf-45da-a92c-bd8eda843112>","cc-path":"CC-MAIN-2024-46/segments/1730477027632.4/warc/CC-MAIN-20241102010035-20241102040035-00806.warc.gz"} |
Young Scholars Forum 2022, Institute of Mathematical Sciences at ShanghaiTech University
Agenda of Young Scholars Forum 2022, Institute of Mathematical Sciences at ShanghaiTech University
Saturday, November 26th, 2022 Sunday, November 27th, 2022
（GMT+8） （GMT+8）
Zhang, Ying Zou, Foling
(Numerical analysis, Finance) (Algebraic topology)
Li, Liying Zhou, Jing
(Stochastic PDE) (Dynamical system)
Peng, Zhichao Su, Yaofeng
(Numerical analysis) (Dynamical system)
12:00-14:00 Break
Tan, Ruoxu Qu, Santai
(Statistics) (Algebraic geometry)
Zhou, Zijun
(Enumerative geometry)
Tencent number 310-353-119 480-219-391
Saturday, November 26th, 2022
Title:Langevin dynamics-based algorithms for sampling and optimization problems with applications in machine learning and finance
Speaker：Zhang, Ying
Abstract:In this talk, I will present three numerical algorithms, namely, SGLD, TUSLA, and e-Theo POULA, with applications in machine learning and finance. For the SGLD algorithm, we obtain
convergence results under relaxed conditions compared to existing results in the literature. This allows us to extend the applicability of the SGLD algorithm to applications including index tracking
optimization and CVaR minimization. However, SGLD cannot be applied to optimization problems with highly-nonlinear objective functions. We address this problem by proposing the TUSLA algorithm. We
describe the conditions under which the theoretical guarantees can be obtained for TUSLA, and then provide the main convergence results. An optimization problem involving ReLU neural network is
provided to illustrate the applicability of TUSLA. Finally, I will talk about the e-Theo POULA algorithm, which combines the advantages of the Langevin dynamics-based algorithms and the adaptive
learning rate methods. An example from multi-period portfolio optimization is presented to show the powerful empirical performance of e-Theo POULA.
Title:Stationary solutions for 1D Burgers equations and KPZ scaling
Speaker：Li, Liying
Abstract: In the first part, we will talk about the stationary solutions for 1D stochastic Burgers equations and their ergodic properties. We will classify all the ergodic components, establish the
``one force---one solution'' principle, and obtain the inviscid limit. The key objects to study are the infinite geodesics and infinite-volume polymer measures in random environments, and the ergodic
results have their counterparts in the geodesic/polymer language. In the second part, we will present a random point field model that is motivated by the coalescing and monotone structure of the
optimal paths in random environments that arise in many KPZ models. The 2/3 transversal exponent from the KPZ scaling becomes a natural parameter for the renormalization action in this model, and can
be potentially extended to values other than 2/3. Some preliminary results are given.
Title:Efficient numerical method and reduced order model for transport dominant problems
Speaker：Peng, Zhichao
Abstract:The transport phenomena arise in many important areas of applications, such as electromagnetic, nuclear engineering and quantum physics. In this talk, we will focus on efficient numerical
method and reduced order models for transport dominant problems with various computational challenges.
In the first part, we will focus on the high frequency Maxwell’s equation and present a flexible and efficient frequency-domain solver built from time-domain solvers. Two challenges of solving the
frequency-domain Maxwell’s equation at high frequencies are its indefinite nature and high resolution requirement. The proposed method converts efficient time-domain solvers to efficient
frequency-domain solvers, and it always leads to a better conditioned linear system which is proved to be symmetric positive definite for special cases.
In the second part, we will discuss a reduced order model (ROM) for the time-dependent radiative transfer equation (RTE) which is a high dimensional and multiscale kinetic transport equation. To
mitigate the curse of dimensionality, we utilize the underlying low-rank structure of the RTE to construct a ROM with the reduced basis method (RBM). The proposed ROM reduces the degrees of freedom
by projecting the RTE to low-dimensional reduced order subspaces. Key components of our ROM include an equilibrium-respecting strategy to construct low-dimensional subspaces and a reduced quadrature
rule to handle the collisional operator of the RTE.
Title:Causal Effect of Functional Treatment
Speaker：Tan, Ruoxu
Abstract:Functional data often arise in the areas where the causal treatment effect is of interest. However, research concerning the effect of a functional variable on an outcome is typically
restricted to exploring the association rather than the casual relationship. The lack of definition of probability density function for functional data poses a challenge for consistent estimation of
causal effect. To overcome the difficulty, we propose a well-defined functional stabilized weight and develop a novel estimator for it. Based on the functional linear model for the average
dose-response functional, we propose three estimators, namely, the functional stabilized weight estimator, the outcome regression estimator and the doubly robust estimator, each of which has its own
merits. We study their theoretical properties, which are corroborated through extensive numerical experiments. A real data application on electroencephalography data and disease severity demonstrates
the practical value of our methods.
Sunday, November 27th, 2022
Title:Computations in equivariant algebraic topology
Speaker：Zou, Foling
Abstract:Modern algebraic topology sees equivariance arising in unexpected context. Equivariant cohomology carries rich structures but is much harder to compute. In 2009, Hill, Hopkins, and Ravenel
solved the 50-year-old Kervaire invariant problem about framed manifolds (for p = 2), which has nothing to do with group actions a prior, using equivariant computation. Their work was related to the
computation of the dual Steenrod algebra for the group Z/2 by Hu and Kriz. We compute the dual Steenrod algebra for the group Z/p for odd p. It turns out that the case of odd primes has interesting
new components. We hope to use it to tackle the odd primary Kervaire problem, which remains open for p = 3. I will also talk about equivariant factorization homology and its application in the
computation of the Real topological Hochschild homology.
Title:The Fermi Acceleration Problem
Speaker：Zhou, Jing
Abstract:In this talk, I will briefly introduce the Fermi-Ulam acceleration problem and the existing results on the subject. In particular, I will present my work on several variants of the
Fermi-Ulam models: the bouncing ball in gravity and the billiard with moving platforms. We use techniques from elliptic as well as hyperbolic dynamics with singularities to study the ergodic and
statistical properties of these systems on infinite-volume phases.
Title:open dynamical systems
Speaker：Su, Yaofeng
Abstract:Open dynamical systems describe hitting processes of orbits through a target in phase space. One of the main questions of open systems is to study a statistical property (called a Poisson
approximation) of the hitting process. I will present some new results for it, applications include some dissipative systems and hyperbolic billiards.
Title:Bounding irrationality of degenerations of Fano fibrations
Speaker：Qu, Santa
Abstract:In this talk, I will introduce a recent result about bounding degrees of irrationality of degenerations of klt Fano fibrations of arbitrary dimensions. This proves the generically bounded
case of a conjecture proposed by C. Birkar and K. Loginov for log Fano fibrations of dimensions greater than three. Our approach depends on a method to modify the klt Fano fibration to a toroidal
morphism of toroidal embeddings with bounded general fibres. Moreover, we show that every fibre of the toroidal morphism is bounded and has mild singularities if we replace the birational
modifications by alterations. This is a joint work with Prof. C. Birkar.
Title:Enumerative geometry from 3d gauge theory and beyond
Speaker：Zhou, Zijun
Abstract:In this talk I will give an introduction of the recent development of K-theoretic enumerative algebraic geometry, motivated from 3d supersymmetric gauge theories. We will mainly focus on the
invariants called vertex function, and the geometric object called elliptic stable envelope, both introduced by A. Okounkov. These invariants are interesting from different aspects. On the on hand,
they are K-theoretic versions of Gromov-Witten-type theories, which count quasi-maps, leading to deep mathematical realizations of the physics phenomenon called 3d mirror symmetry. On the other hand,
they admit close connection to new constructions of geometric representation theory, e.g. BFN’s construction of Coulomb branch. I will introduce my work on both aspects, as well as future outlook for
combining them. | {"url":"https://ims.shanghaitech.edu.cn/ims_en/2022/1124/c4598a1003080/page.htm","timestamp":"2024-11-06T10:33:02Z","content_type":"text/html","content_length":"66545","record_id":"<urn:uuid:d9e8b33c-fec8-4c38-b559-0655c45795f6>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00581.warc.gz"} |
Speed of Light - A Real Barrier
The relativistic mass formula
tells us that the mass of an object increases with it speed, as long as
\[m_0 \gt 0\]
Increasing the mass of an object requires an increasing force to accelerate it. If a constant force is applied, then the acceleration will decrease with increasing speed. As the object approaches the
speed of light, its mass grows without limit. Then the force required to accelerate it also increases without limit. For this reason no particle with a non zero mass can ever travel at the speed of
light. That is not to say that no particle cannot travel at the speed of light. Photons travel at the speed of light, and in fact any massless particle must travel at the speed of light. | {"url":"https://mail.astarmathsandphysics.com/ib-physics-notes/relativity/4772-speed-of-light-a-real-barrier.html","timestamp":"2024-11-11T14:04:46Z","content_type":"text/html","content_length":"30199","record_id":"<urn:uuid:bff53fcd-7705-45e5-8b19-bd75e0eba9aa>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00154.warc.gz"} |
The Importance of Micro Mathematics
The Micro Mathematics Stories
Ultimately, the Churchill maths collection comprises numerous activities that are useful for small student groups who want intervention on a particular topic. Graph is most frequently used tool in
modern economics. Before you take a look at the hottest past papers.
The Micro Mathematics Cover Up
Your chosen provider might ask you to take an SKE course for a state of your offer, before you begin your primary teacher training programme. Some training research paper help programmes provide you
with the occasion to earn the credits required for an entire master’s degree after you’ve completed your training. The degree of funding and eligibility will be different based on the subject you
decide to teach, and your degree classification.
The quizzes ask a set of questions dependent on the content of the module. This practice is called cross-validation in statistics. A course may just be utilized to fulfill 1 program requirement.
If you own a head for figures, and find any of the aforementioned careers attractive, it can be well worth considering. There are as many methods to design a revenue schedule since there are
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and if this expert knows their company, you won’t be bored for an instant.
A Secret Weapon for Micro Mathematics
Pick the vertices you would like to merge. The phenotype may also be affected by the environment An allele is a specific type of one specific gene. An allele is a certain kind of a gene and they’re
passed from parents to their offspring.
Ideas, Formulas and Shortcuts for Micro Mathematics
While formulating theories several tools are used by experts in this discipline. Frequently the tools from these other areas aren’t quite acceptable for the requirements of physics, and will need to
get changed or more advanced versions have to get made. Lastly, you will discover 5 techniques you could use while you’re practicing machine learning on standard datasets to incrementally develop
your comprehension of machine https://www.luc.edu/biology/index.shtml learning algorithms.
This field comes from research done in AI. Physics employs these theories to not just describe physical phenomena, yet to model physical systems and predict how these bodily systems will behave.
Mathematical models are of excellent significance in the organic sciences, especially in physics.
Micro Mathematics – What Is It?
The majority of these approaches, however, require the solution of a elaborate transcendental equation springing from a matrix determinant for those eigenvalues that is tough to solve numerically for
a high number of layers. It indicates the way the value of dependent variable is dependent on the worth of independent or other variables. Exogenous variables are occasionally referred to as
parameters or constants.
If there’s a path linking any 2 vertices in a graph, that graph is thought to be connected. And that’s the cost right after we know width and height. There’s no difference between the 2 graphs.
The proposals below are meant to help us achieve this. This document was updated in its entirety to deal with new problems that have arisen from moderation. This WikiProject isn’t prescriptive.
You will see all sorts of interesting quirks in behavior that might not even be documented. The assumption of Ceteris Paribus thus gets rid of the influence of different elements that might get in
the method of establishing a scientific statement about the behavior of financial variables. In addition, it permits economists to explain observable phenomena in quantifiable terms and offer the
foundation for more interpretation or the provision of feasible solutions.
What You Don’t Know About Micro Mathematics
The videos can be seen on the Brightstorm site or you are able to embed them into your sites. Inside this link, you will come across useful details about our Math class. For general information
regarding admissions, visas, and other necessary steps for global students, visit the UCIE site.
Throughout history, a growing number of accurate mathematical models are developed. This informative article explains how modelling is utilised to create this vehicle. If you are one of the chosen
few, you’ll probably have accessibility to a digital analytics tool that enables you to make a customized attribution model.
Financial Modeling is a tool which can be utilised to forecast an image of a security or a financial instrument or a corporation’s future financial performance depending on the historical operation
of the entity. The model isn’t the very same as the actual thing. A mathematical model offers an answer.
Micro Mathematics Can Be Fun for Everyone
The notion of collective comprehension and the part of society in individual understandings is hinted at too. YummyMath is a website created for the intent of sharing mathematics issues and scenarios
based on things going on in the world these days. Therefore, it’s important that it is possible to communicate easily with others and that you possess the capability to examine complex topics in a
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No, calculators of any sort aren’t permitted. A few easy ideas may take you a ways in gathering information on a particular machine learning algorithm. This ridiculously straightforward tactic can
help you become on top of the overwhelm.
Actuaries use numbers to compute and assess the effects of financial risk. Mathematical modelling is able to help you feel much better. Bursaries tax-free bursaries are offered for training to teach
a array of subjects.
Details of Micro Mathematics
In finance, the expression is utilized to spell out the quantity of cash (currency) that is generated or consumed in a specific time period. The big benefit to using break-even analysis is it
indicates the lowest quantity of business activity required to prevent losses. This kind of evaluation is generally of most interest to prospective users of a program or curriculum, who want to pick
the most effective existing product to suit their requirements.
For a Call, it’s the strike price in addition to the premium paid. It’s also important to comprehend the determination of wages and other input prices in factor markets, and be in a position to
analyze and rate the distribution of revenue. The Break-even Point defines once an investment will create a positive return.
New Step by Step Roadmap for Micro Mathematics
More information regarding the ELSA pilot can be found on the Early Learning STEM Australia site. Mathematical modelling is able to help you feel much better. Generally speaking, mathematical models
may consist of logical models.
Using Micro Mathematics
The videos can be seen on the Brightstorm site or you are able to embed them into your sites. Inside this link, you will come across useful details about our Math class. Stop by the Tuition Fees page
for more details.
What About Micro Mathematics?
If they previously learned how to try it, the task can be finished independently. This document was updated in its entirety to deal with new problems that have arisen from moderation. This
WikiProject isn’t prescriptive.
The grading is strict, and little or no credit is provided for partial answers which do not appear to be heading towards a comprehensive solution. Comparative statics analysis is a way of earning
such comparisons. Using graph stipulates a better comprehension of the financial generalizations.
CAD techniques enable designers to see objects beneath a wide number of representations and to test these objects by simulating real-world problems. A normal fractal image has millions of these
pixels! What’s unique about this strategy is that it permits a worldwide view of a microbiome-host interaction landscape,” explained Ludington.
Starting a Small Business Students will find out how to make a business program and the responsibilities involved with business ownership. Appointments are made for a single year and could be renewed
for another year. Google Earth is the dynamic tool which will be utilized to do this.
Flexible and part-time work, in addition to career breaks, can be negotiated with a few employers, but this is usually contingent on the employer and someone’s circumstances. Thus, you don’t have a
decision. 1 great bit of news for anybody interested in a career involving numbers is there are plenty of jobs available that pay well.
The Do’s and Don’ts of Micro Mathematics
It even offers you an answer sheet! The big benefit to using break-even analysis is it indicates the lowest quantity of business activity required to prevent losses. A good example would be a
manufacturing factory which has a lot of space capacity and becomes more efficient as more volume is generated.
Also it doesn’t accumulate a reserve for future losses. Or it may discover the principal attributes that separate customer segments from one another. The Break-even Point defines once an investment
will create a positive return.
Micro Mathematics – the Story
There’s an attempt to organize a list of important publications in many regions of science. These days, many rewarding career opportunities are open to anybody with a excellent understanding of
meteorology and the capacity to utilize it in atmospheric research or applied meteorology. By studying biomathematics, you’re develop strong interdisciplinary skills that could help biological
research in various ways.
Fields like econometrics try to analyze real-world financial scenarios and activity through statistical procedures. Analysis of a mathematical model makes it possible for us to penetrate the gist of
the phenomena under study. Mechanisms are employed in science to spell out how phenomena come about.
This field comes from research done in AI. The majority of the theories in physics use mathematics to share their principles. Mathematical models are of excellent significance in the organic
sciences, especially in physics.
Your chosen provider might ask you to take an SKE course for a state of your offer, before you begin your primary teacher training programme. Submissions with no real-world application won’t be
considered. Thus, it is going to be proper for classroom use at universities in addition to for use by independent learners trying to pass professional actuarial examinations.
The free internet practice tests for AP Microeconomics can also enable you to fine-tune your review program. This app isn’t just for students. Prescriptive analytics is a kind of predictive
analytics,” Wu explained.
Typically, you will work in 1 area before continuing on to a different location. Our team project is basically broken up into three parts.
Financial Modeling is a tool which can be utilised to forecast an image of a security or a financial instrument or a corporation’s future financial performance depending on the historical operation
of the entity. Mathematical models are shown to become an essential method of control. Therefore, it will become essential to establish a new and more perfect mathematical model.
The most important thing is that you’ve got to examine the specific job to establish if earning a master’s degree will actually pay off. You might also be interested in the other sort of practical
maths whom I use, topics which are going to be genuinely beneficial to them in their lives, but furthermore, things they can relate to, and unsurprisingly these kind of topics usually involve money
or cellular phones. Everybody in the class becomes involved.
It requires patience and techniques. Inside this post you will discover that which we really mean when we speak about theory in machine learning. Have some fun taking a look at the mystic component
of numbers.
The Nuiances of Micro Mathematics
Thus, a complete rotation is 360 degrees, or complete circle. Assignment of a certain function to a specific layer imposes constraints on what functions can be done by other layers. The absolute
value may be known as the magnitude.
If there’s a path linking any 2 vertices in a graph, that graph is thought to be connected. We are attempting to fix this ideal triangle for the hypotenuse x. There’s no difference between the 2
Micro Mathematics – Dead or Alive?
Mathematics is seen as the second language for those students of economics. Year 3 In Mathematics, you will have the ability to choose from a wide selection of optional modules. Students ought to be
utilizing the mathematics they already know, not seeking to create new ideas or concepts.
The Start of Micro Mathematics
Mathematical models may also be utilized to forecast future behavior. To make certain it works, we will need to model how diseases spread. Epidemic modelling Join within this continuing
Introduction Abstract Factory design pattern is among the Creational pattern. The need to spell out material behavior caused by physical interactions across several length scales is forcing
scientists to create new mathematics. There’s still an overall absence of rigorous mathematical modeling of this important category of materials phenomena.
Both business and the federal government are raising their efforts and financial investment in this region. Children’s odds of success are maximised should they develop deep and lasting comprehension
of mathematical procedures and concepts. Whenever you have the Maker’s Mindset,” you know that you can alter the world.
Flexible and part-time work, in addition to career breaks, can be negotiated with a few employers, but this is usually contingent on the employer and someone’s circumstances. After a very long day of
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Within this talk, we’ll survey all these results. This talk is made for a general mathematical audience. The lectures are readily available to anybody, completely at no cost.
Since a prescriptive model can predict the potential consequences based on various selection of action, in addition, it can suggest the best plan of action for any pre-specified outcome,” Wu wrote.
It needs to be somebody with a keen interest and desire to pursue mathematics at its greatest level. Have some fun taking a look at the mystic component of numbers.
Engineers often can accept some approximations to be able to receive a more robust and easy model. Mathematical modelling is able to help you feel much better. Bursaries tax-free bursaries are
offered for training to teach a array of subjects.
Key Pieces of Micro Mathematics
Your chosen provider might ask you to take an SKE course for a state of your offer, before you begin your primary teacher training programme. Some training programmes provide you with the occasion to
earn the credits required for an entire master’s degree after you’ve completed your training. Thus, it is going to be proper for classroom use at universities in addition to for use by independent
learners trying to pass professional actuarial examinations.
The same as the one for the shorter practice tests, the results pages for the comprehensive practice tests incorporate extensive explanations and extra information applicable to every question. We
provide a Solution Library of already-prepared solutions for thousands and thousands of cases, assignments and textbook questions which are available for immediate download. Prescriptive analytics is
a kind of predictive analytics,” Wu explained.
Micro Mathematics Help!
The videos can be seen on the Brightstorm site or you are able to embed them into your sites. In the problem above, students would want to identify what information they’d want to understand as a way
to have the ability to answer the questions. Stop by the Tuition Fees page for more details.
Throughout history, a growing number of accurate mathematical models are developed. Fixed costs include, but aren’t confined to, depreciation on equipment, interest expenses, taxes and standard
overhead expenses. The marginal cost formula can be utilized in financial modelingWhat is Financial ModelingFinancial modeling is done in Excel to forecast a corporation’s fiscal performance.
Financial Modeling is a tool which can be utilised to forecast an image of a security or a financial instrument or a corporation’s future financial performance depending on the historical operation
of the entity. By obeying these vital principles, model will be less painful to navigate and check, and trustworthy. Therefore, it will become essential to establish a new and more perfect
mathematical model.
The Upside to Micro Mathematics
The majority of these approaches, however, require the solution of a elaborate transcendental equation springing from a matrix determinant for those eigenvalues that is tough to solve numerically for
a high number of layers. It indicates the way the value of dependent variable is dependent on the worth of independent or other variables. In such circumstances the endogenous variables could
possibly be a function of more than 1 parameter or exogenous variable.
If there’s a path linking any 2 vertices in a graph, that graph is thought to be connected. We are attempting to fix this ideal triangle for the hypotenuse x. Otherwise, it is going to round to the
next lowest integer value.
The Micro Mathematics Cover Up
It’s therefore wise to look at any pre-existing articles associated with a topic to see whether there’s already an established usage. These add dimensions to the issue and help it become tricky to
solve,” explained Ludington. They need to pose a question to investigate the situation and could make a prediction for the situation.
An approach that you could use is to collect your own mini algorithm descriptions. Most modeling situations involve lots of tensions between conflicting requirements that must not be reconciled
easily. The solution is likely to be a personal one, but there are a lot of objective things to think about.
All About Micro Mathematics
Pick the vertices you would like to merge. Inhibitory cells are necessary to implement learning. The results ought to be applicable to a wide array of issues in spatiotemporal pattern formation.
That line may be the x-axis. The team will subsequently follow that implementation plan to come up with their solution and demonstrate their final item. A good example would be a manufacturing
factory which has a lot of space capacity and becomes more efficient as more volume is generated.
If you’re not calculating the offline impact, and you’re not giving your on-line channel due credit. Or it may discover the principal attributes that separate customer segments from one another. For
a business with economies of scale, producing each extra unit will become cheaper and the provider is incentivized to get to the point at which marginal revenueMarginal RevenueMarginal Revenue is the
revenue that’s gained from the selling of another unit.
You must be logged in to post a comment. | {"url":"https://mailers.cms-res.com/the-importance-of-micro-mathematics/","timestamp":"2024-11-06T21:06:12Z","content_type":"application/xhtml+xml","content_length":"91755","record_id":"<urn:uuid:7482c4d0-d8ac-4320-a9f3-3b61a7a2f72c>","cc-path":"CC-MAIN-2024-46/segments/1730477027942.47/warc/CC-MAIN-20241106194801-20241106224801-00382.warc.gz"} |
Visual servoing from a special compound of features
Contact: Romeo Tatsambon Fomena
Creation Date: September 2007
Description of the demonstration
Image-based visual servoing is exploited to move an eye-in-hand system to a position where the desired pattern of the image of the target can be observed. The camera is a mounted on the end-effector
of a six degrees of freedom robotic system. The target is 9.5 cm radius white spherical ball marked with a tangent vector to a point on its surface. Using such simple object allows to easily compute
the selected visual features on the observed disk (image of the target) at video rate without any image processing prolem. The desired features are computed after moving the robot to a position
corresponding to the desired image. The following figures picture the desired (left) and initial (right) images used for the experiment. First of all, we compare the previous set of features proposed
for this special target in [1] with the new set of features.
Comparison: previous vs new approach
Here we consider the exact value of the radius of the ball and correct camera calibration values. The figures on the left hand side show the system behaviour using the previous set of features [1];
the figures on the right hand side plot the system behaviour using the new set. The top row shows the feature error trajectories and the second row shows the camera velocities. The first part of this
video illustrates this comparison. The robot trajectory in the Cartesian space is plotted on the third row. We can note that, using the new set, this trajectory is shorter than using the previous
In practice, the value of the radius of the spherical ball is unknown. However, visual servoing still converges: this result is formally proved for a classical control method using the new set of
features. Besides we validate this proof using a non-spherical decoration balloon marked with a flower picture in black. The radius of the target is set to 6.5cm. The desired (left) and initial
(right) images are shown in the first row. The second row plots the system behaviour. The second part of this video illustrates this application.
Scientific context
This work is concerned with modeling issues in visual servoing. The challenge is to find optimal visual features for visual servoing. By optimality satisfaction of the following criteria is meant:
local and -as far as possible- global stability of the system, robustness to calibration and to modeling errors, non-singularity, local minima avoidance, satisfactory motion of the system and of the
features in the image, and finally maximal decoupling and linear link (the ultimate goal) between the visual features and the degrees of freedom taken into account.
A spherical projection model is proposed to search for optimal visual features. The target is a sphere marked with a tangent vector to a point on its surface (see the figure below). We propose a new
minimal set of six visual features which can be computed on any central catadioptric camera (this includes the classical perspective cameras). Using this new combination, a classical image-based
control law has been proved to be globally stable to modeling error: in practice, it means that it is not necessary to use the exact value of the radius of the sphere target; a rough estimate is
sufficient. In comparison with the previous set proposed for this special target in [1], the new set draws a better system trajectory.
1. N. Cowan, D. Chang. Geometric visual servoing, IEEE Trans. on Robotics, vol. 21, no. 6, pp. 1128-1138, Dec. 2005.
2. R. Tatsambon Fomena, F. Chaumette. Visual Servoing from two Special Compound of Features using a Spherical Projection Model, IEEE/RSJ Int. Conf. on Intel. Robots and Systems IROS'08, Nice,
France, Sept. 2008. | {"url":"https://www.irisa.fr/lagadic/demo/demo-cc-target/demo-cc-target-eng.html","timestamp":"2024-11-10T06:02:39Z","content_type":"text/html","content_length":"9470","record_id":"<urn:uuid:cc06b517-fafa-4c16-a3f3-f8dea59ac590>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00729.warc.gz"} |
Solar Intensity and Luminosity
Main Concept
The Sun's radiation (sunlight) is closely approximated as blackbody radiation. Accordingly, the intensity or
flux of solar radiation, $F$, which is a measure of how much power is being radiated from the Sun's surface
per unit area, can be expressed in terms of the temperature, $T$, of the sun using the Stefan-Boltzmann law:
where $\mathrm{σ}=5.67×{10}^{-8}\frac{W}{{\mathrm{K}}^{4}{\mathrm{m}}^{2}}$ is the
Stefan-Boltzmann constant.
The luminosity of the Sun is a measure of how much energy the Sun produces per unit time. It is obtained by
multiplying the solar intensity by the Sun's surface area:
where ${r}_{s}$ represents the radius of the Sun. To determine the flux passing through any other point in
space where the Sun is visible, ${r}_{s}$ is replaced by $d$, which is the distance from the center of the Sun
to the point in question. So, the formula now becomes:
In this formula, the flux is proportional to the inverse square of the distance. This means that if an
object's distance from the Sun doubles, the amount of sunlight hitting a given area will drop by a factor of
four. This property is an example of the inverse-square law, which affects conserved quantities propagating
evenly in all directions through three-dimensional space. Consecutive wavefronts of decreasing intensity can
be visualized in the following interactive diagram:
As you can imagine, as the radiation moves farther from the Sun, the same amount of energy is spread out over
a larger area (which is proportional to the square of the distance from the source), which makes the flux
(power per unit area) correspondingly smaller.
What is the radiant flux emitted into space by a light source with temperature of $1000\mathrm{K}$? What is
the luminosity of the solar radiation given by this surface if its radius is $5×{10}^{8}\mathrm{m}$?
Adjust the sliders to change the radius and temperature of the surface providing solar radiation. Click the
checkboxes to see the flux and luminosity calculations.
Using the formulas introduced in the previous section, you can determine both the flux and the luminosity
produced by the specified surface.
To begin, calculate the flux:
$F=\mathrm{σ}\cdot {T}^{4}$
You can now use this result to determine the luminosity:
$L=4\cdot \mathrm{π}\cdot {R}^{2}\cdot F$
$L=4\cdot \mathrm{π}\cdot {\left(5×{10}^{8}\mathrm{m}\right)}^{2}\cdot \left(56700\mathrm{W}/
Therefore, a surface with a radius of $5×{10}^{8}\mathrm{m}$ and a temperature of $1000\mathrm{K}$ has a
radiant intensity of $56700\mathrm{W}/{\mathrm{m}}^{2}$ and a luminosity of $1.781×{10}^{23}\mathrm{W}$.
More MathApps
Main Concept
The Sun's radiation (sunlight) is closely approximated as blackbody radiation. Accordingly, the intensity or
flux of solar radiation, $F$, which is a measure of how much power is being radiated from the Sun's surface
per unit area, can be expressed in terms of the temperature, $T$, of the sun using the Stefan-Boltzmann law:
where $\mathrm{σ}=5.67×{10}^{-8}\frac{W}{{\mathrm{K}}^{4}{\mathrm{m}}^{2}}$ is the
Stefan-Boltzmann constant.
The luminosity of the Sun is a measure of how much energy the Sun produces per unit time. It is obtained by
multiplying the solar intensity by the Sun's surface area:
where ${r}_{s}$ represents the radius of the Sun. To determine the flux passing through any other point in
space where the Sun is visible, ${r}_{s}$ is replaced by $d$, which is the distance from the center of the Sun
to the point in question. So, the formula now becomes:
In this formula, the flux is proportional to the inverse square of the distance. This means that if an
object's distance from the Sun doubles, the amount of sunlight hitting a given area will drop by a factor of
four. This property is an example of the inverse-square law, which affects conserved quantities propagating
evenly in all directions through three-dimensional space. Consecutive wavefronts of decreasing intensity can
be visualized in the following interactive diagram:
As you can imagine, as the radiation moves farther from the Sun, the same amount of energy is spread out over
a larger area (which is proportional to the square of the distance from the source), which makes the flux
(power per unit area) correspondingly smaller.
What is the radiant flux emitted into space by a light source with temperature of $1000\mathrm{K}$? What is
the luminosity of the solar radiation given by this surface if its radius is $5×{10}^{8}\mathrm{m}$?
Adjust the sliders to change the radius and temperature of the surface providing solar radiation. Click the
checkboxes to see the flux and luminosity calculations.
Using the formulas introduced in the previous section, you can determine both the flux and the luminosity
produced by the specified surface.
To begin, calculate the flux:
$F=\mathrm{σ}\cdot {T}^{4}$
You can now use this result to determine the luminosity:
$L=4\cdot \mathrm{π}\cdot {R}^{2}\cdot F$
$L=4\cdot \mathrm{π}\cdot {\left(5×{10}^{8}\mathrm{m}\right)}^{2}\cdot \left(56700\mathrm{W}/
Therefore, a surface with a radius of $5×{10}^{8}\mathrm{m}$ and a temperature of $1000\mathrm{K}$ has a
radiant intensity of $56700\mathrm{W}/{\mathrm{m}}^{2}$ and a luminosity of $1.781×{10}^{23}\mathrm{W}$.
The Sun's radiation (sunlight) is closely approximated as blackbody radiation. Accordingly, the intensity or flux of solar radiation, $F$, which is a measure of how much power is being radiated from
the Sun's surface per unit area, can be expressed in terms of the temperature, $T$, of the sun using the Stefan-Boltzmann law:
The luminosity of the Sun is a measure of how much energy the Sun produces per unit time. It is obtained by multiplying the solar intensity by the Sun's surface area:
where ${r}_{s}$ represents the radius of the Sun. To determine the flux passing through any other point in space where the Sun is visible, ${r}_{s}$ is replaced by $d$, which is the distance from the
center of the Sun to the point in question. So, the formula now becomes:
In this formula, the flux is proportional to the inverse square of the distance. This means that if an object's distance from the Sun doubles, the amount of sunlight hitting a given area will drop by
a factor of four. This property is an example of the inverse-square law, which affects conserved quantities propagating evenly in all directions through three-dimensional space. Consecutive
wavefronts of decreasing intensity can be visualized in the following interactive diagram:
As you can imagine, as the radiation moves farther from the Sun, the same amount of energy is spread out over a larger area (which is proportional to the square of the distance from the source),
which makes the flux (power per unit area) correspondingly smaller.
What is the radiant flux emitted into space by a light source with temperature of $1000\mathrm{K}$? What is the luminosity of the solar radiation given by this surface if its radius is $5×{10}^{8}\
Adjust the sliders to change the radius and temperature of the surface providing solar radiation. Click the checkboxes to see the flux and luminosity calculations.
Using the formulas introduced in the previous section, you can determine both the flux and the luminosity produced by the specified surface.
To begin, calculate the flux:
$F=\mathrm{σ}\cdot {T}^{4}$
You can now use this result to determine the luminosity:
$L=4\cdot \mathrm{π}\cdot {R}^{2}\cdot F$
$L=4\cdot \mathrm{π}\cdot {\left(5×{10}^{8}\mathrm{m}\right)}^{2}\cdot \left(56700\mathrm{W}/{\mathrm{m}}^{2}\right)$
Therefore, a surface with a radius of $5×{10}^{8}\mathrm{m}$ and a temperature of $1000\mathrm{K}$ has a radiant intensity of $56700\mathrm{W}/{\mathrm{m}}^{2}$ and a luminosity of $1.781×{10}^
What is the radiant flux emitted into space by a light source with temperature of $1000\mathrm{K}$? What is the luminosity of the solar radiation given by this surface if its radius is $5×{10}^{8}\
Adjust the sliders to change the radius and temperature of the surface providing solar radiation. Click the checkboxes to see the flux and luminosity calculations.
Using the formulas introduced in the previous section, you can determine both the flux and the luminosity produced by the specified surface.
To begin, calculate the flux:
$F=\mathrm{σ}\cdot {T}^{4}$
You can now use this result to determine the luminosity:
$L=4\cdot \mathrm{π}\cdot {R}^{2}\cdot F$
$L=4\cdot \mathrm{π}\cdot {\left(5×{10}^{8}\mathrm{m}\right)}^{2}\cdot \left(56700\mathrm{W}/{\mathrm{m}}^{2}\right)$
Therefore, a surface with a radius of $5×{10}^{8}\mathrm{m}$ and a temperature of $1000\mathrm{K}$ has a radiant intensity of $56700\mathrm{W}/{\mathrm{m}}^{2}$ and a luminosity of $1.781×{10}^
Using the formulas introduced in the previous section, you can determine both the flux and the luminosity produced by the specified surface.
You can now use this result to determine the luminosity:
Therefore, a surface with a radius of $5×{10}^{8}\mathrm{m}$ and a temperature of $1000\mathrm{K}$ has a radiant intensity of $56700\mathrm{W}/{\mathrm{m}}^{2}$ and a luminosity of $1.781×{10}^ | {"url":"https://www.maplesoft.com/support/help/errors/view.aspx?path=MathApps%2FSolarIntensityandLuminosity","timestamp":"2024-11-04T02:07:44Z","content_type":"text/html","content_length":"129849","record_id":"<urn:uuid:00d08db5-f134-43dd-8c30-6dcc14332f64>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00597.warc.gz"} |
Syllabus Information
Use this page to maintain syllabus information, learning objectives, required materials, and technical requirements for the course.
Syllabus Information
MTH 060B - Beginning Algebra: Part B
Associated Term: Spring 2023
Learning Objectives: Upon successful completion of this course, the student will: 1- Maintain, use, and expand skills and concepts learned in previous mathematics courses. 2- Perform operations with
fractions and decimals. Use unit analysis to convert units and solve problems. 3- Solve linear equations and inequalities. Solve linear equations and formulas algebraically. 4- Solve linear
inequalities and graph their solutions on a number line. Use algebra to solve application problems. 5- Translate verbal models into algebraic expressions and/or equations. 6- Solve problems. Solve
problems involving simple interest, motion, and mixtures. 7- Solve problems using ratios and proportions. 8- Solve problems involving similar triangles. 9- Solve geometry problems involving
Required Materials:
Technical Requirements: | {"url":"https://crater.lanecc.edu/banp/bwckctlg.p_disp_catalog_syllabus?cat_term_in=202340&subj_code_in=MTH&crse_numb_in=060B","timestamp":"2024-11-05T04:04:49Z","content_type":"text/html","content_length":"8062","record_id":"<urn:uuid:1c0b31af-77cb-453c-a402-8080a3010095>","cc-path":"CC-MAIN-2024-46/segments/1730477027870.7/warc/CC-MAIN-20241105021014-20241105051014-00391.warc.gz"} |
Re: Add syntax highlighting to own command
• To: mathgroup at smc.vnet.net
• Subject: [mg101666] Re: [mg101539] Add syntax highlighting to own command
• From: Leonid Shifrin <lshifr at gmail.com>
• Date: Tue, 14 Jul 2009 05:32:55 -0400 (EDT)
• References: <200907090600.CAA17547@smc.vnet.net>
Hello all,
I have a few remarks:
IMO, solutions of Simon and Daniel are conceptually the same as Bastian's,
but seem to me as yet another step towards brevity and elegance (this is
subjective, of course).
The major technical difference is that <If> is replaced by pattern-matching
and thus there are two global rules instead of one, and in addition this
allows to avoid Unevaluated. The mechanism that makes this work is however
the same as before, since SetDelayed creates global delayed rules, and, as
I mentioned before, RuleDelayed does not respect the scope of inner scoping
constructs, which makes it all possible. The details of this have been
discussed, in particular, in this thread (this is where I learned it):
>Do our solutions differ in behavior? I wouldn't be surprised at all if
>our codes behaved differently in the details.
In most "normal" situations there probably won't be any difference in the
result (except for the case of empty vars list, see below). But since the
pattern - based function, in terms of evaluation, is different in several
ways from the function with If (also If checks a different condition), it is
not difficult to construct perverse examples where the results will differ.
This is Bastian's solution:
SetAttributes[Let, HoldAll];
Let[vars_, expr_] :=
If[Length[Unevaluated[vars]] == 1,
With[vars, expr],
Unevaluated[vars] /. {a_, b__} :>
With[{a}, Let[{b}, expr]]]
Consider this, for instance:
Block[{Length = 1},
Let[{a = "Hi, ", b = a <> "there!"}, Print[b]]]
Let[{a = "Hi, ", b = a <> "there!"}, Print[b]]]
Of course, no one in the right mind would do this, but this is just an
illustration. By the way, the solution of Bastian returns an empty list when
called with no variables:
In[1] = Let[{}, a]
Out[1] = {}
which differs from what standard With or other two solutions do.
Regardless of these issues, it seems (to me anyway) that when one tries to
make a "production" code out of this simple and elegant solution, one really
opens a can of worms. One thing is that argument checks and error messages
have to be added, since without them Let is prone to misuse. This may seem
unnecessary in simple programs but may probably give nasty bugs in larger
ones, which will be hard to track.
Here is my attempt in this direction (based on solutions of Simon and
In[2] =
SetAttributes[Let, HoldAll];
(* Error messages *)
Let::lvset =
"Local variable specification `1` contains `2`, which is an \
assignment to `3`; only assignments to symbols are allowed.";
Let::lvw =
"Local variable specification `1` contains `2` which is not an \
assignment to a symbol.";
Let::lvlist = "Local variable specification `1` is not a List.";
Let::argrx =
"Let called with `1` arguments; 2 arguments are expected.";
With[{initPattern = HoldPattern[Set[_Symbol, _]]},
(* Main definitions *)
Let[{}, expr_] := expr;
Let[{a : initPattern, b : initPattern ...}, expr_] :=
With[{a}, Let[{b}, expr]];
(* Error - handling *)
Let[vars : {x___Set}, _] :=
"" /; Message[Let::lvset,
badarg = Select[HoldForm[x],
Function[arg, ! MatchQ[Unevaluated@arg, initPattern],
HoldAll], 1],
First@Extract[badarg, {{1, 1}}, HoldForm]]
Let[vars : {x___}, _] /; ! MatchQ[Unevaluated[vars], {___Set}] :=
"" /; Message[
Let::lvw, HoldForm[vars],
HoldForm[x], Function[arg, Head[Unevaluated[arg]] =!= Set],
Let[args___] /; Length[Hold[args]] =!= 2 :=
"" /; Message[Let::argrx, Length[Hold[args]]];
This seems to catch errors in all cases which I tested (probably can be done
more elegantly).
But this reveals another, and IMO more serious, problem: Let does not nest
nicely neither with itself nor with some (external to it) scoping constructs
such as Function and With (Module and Block seem fine, at least upon the
first look):
{Let[{a="Hi, ",b=a<>"there"},Print[b]],
Print[a," ",b]}]
During evaluation of In[3]:= Let::lvset: Local variable specification {1=Hi,
,2=1<>there} contains 1=Hi, , which is an assignment to 1; only assignments
to symbols are allowed.
During evaluation of In[3]:= 1 2
Out[3]= {Let[{1=Hi, ,2=1<>there},Print[2]],Null}
We can also look at inputs like
With[{a = 1, b = 2}, Let[{a = "Hi, ", b = a <> "there"}, Print[b]]]
Function[{a, b}, Let[{a = "Hi, ", b = a <> "there"}, Print[b]]][1, 2]
and see similar outputs. Contrast this with the behavior of nested built-in
scoping constructs:
With[{a = 1, b = 2},
With[{a = "Hi, ", b = "there"}, Print[a, " ", b]]]
Function[{a, b},
With[{a = "Hi, ", b = "there"}, Print[a, " ", b]]][1, 2]
where the variables of the outer constructs are shadowed by those
in the inner ones. To be entirely consistent, one should probably add
definitions to With, Function and Let itself, so that similar behavior will
observed in the first three inputs. But for Function and With, even if
such definitions are implemented correctly, they will cause a (possibly
quite large) performance hit.
Of course, we may just say that most such collisions are pathologies and
bad programming (bugs), and then ignore this problem.
On Wed, Jul 8, 2009 at 11:00 PM, Bastian Erdnuess <earthnut at web.de> wrote:
> I was missing a scoping construct like 'With' but where the local
> variables get assigned linearly. E. g.
> In[1] := LinearWith[ {
> a = "Hi, ",
> b = a <> "there!" },
> Print[ b ] ]
> Out[1] = "Hi, there!" .
> I'm fairly new to Mathematica, but I know some Lisp, and somehow I got
> it. I called my new construct Let and defined it as:
> Let[ vars_, expr_ ] :=
> If[ Length[ Unevaluated[ vars ] ] == 1,
> With[ vars, expr ],
> (* else *)
> Unevaluated[ vars ] /.
> { a_, b__ } :>
> With[ { a },
> Let[ { b }, expr ] ] ]
> SetAttributes[ Let, HoldAll ]
> It seems to work fine so far.
> Now, I would like to have this construct load always when I start
> Mathematica and I don't want to get it cleared when I use
> 'Clear["Global`*"]'. So I put it in the System` context and also added
> the attribute 'Protected'. I wrote all in a file Let.m and now, I
> wonder where to put it that it gets read automatically at each startup
> of Mathematica.
> However, my first question: Is it a bad idea to add things to the
> System` context? And if not, where to put my file? And if, what would
> be better?
> Second, my main question: Is it somehow possible to add this nice syntax
> highlighting to the Let construct like with the With construct, where
> the local variables appear in green?
> Third: Can I somehow add a help page? I have already the Let::usage.
> And last: Does anybody know how to make the construct better? Is there
> something like syntax transformation rules in Mathematica?
> Thank you for your answers,
> Bastian | {"url":"http://forums.wolfram.com/mathgroup/archive/2009/Jul/msg00351.html","timestamp":"2024-11-11T10:43:21Z","content_type":"text/html","content_length":"38129","record_id":"<urn:uuid:1bafdac9-9be4-4a8c-af16-a8923a5d4c6b>","cc-path":"CC-MAIN-2024-46/segments/1730477028228.41/warc/CC-MAIN-20241111091854-20241111121854-00781.warc.gz"} |
On the Solution of Interval Linear Systems
Computing 47:337-353, 1992
In the literature efficient algorithms have been described for calculating guaranteed inclusions for the solution of a number of standard numerical problems [3], [4], [8], [11], [12], [13]. The
inclusions are given by means of a set containing the solution. In [12], [13] this set is calculated using an affine iteration which is stopped when a nonempty and compact set is mapped into itself.
For exactly given input data (point data) it has been shown that this iteration stops if and only if the iteration matrix is convergent (cf. [13]).
In this paper we give a necessary and sufficient stopping criterion for the above mentioned iteration for interval input data and interval operations. Stopping is equivalent to the fact that the
algorithm presented in [12] for solving interval linear systems computes an inclusion of the solution. An algorithm given by Neumaier is discussed and an algorithm is proposed combining the
advantages of our algorithm and a modification of Neumaier's. The combined algorithm yields tight bounds for input intervals of small and large diameter.
Using a paper by Jansson [6], [7] we give a quite different geometrical interpretation of inclusion methods. It can be shown that our inclusion methods are optimal in a specified geometrical sense.
For another class of sets, for standard simplices, we give some interesting examples.
linear system
inclusion method | {"url":"https://tore.tuhh.de/entities/publication/d1149ca9-444a-407f-a3cb-cacb075b211c","timestamp":"2024-11-11T04:31:36Z","content_type":"text/html","content_length":"963448","record_id":"<urn:uuid:0a50b4ec-81cd-48e6-b296-8721af352531>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00293.warc.gz"} |
2.9 Summary
Course Outline
• segmentGetting Started (Don't Skip This Part)
• segmentStatistics and Data Science: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Models with a Quantitative Explanatory Variable
• segmentFinishing Up (Don't Skip This Part!)
• segmentResources
list High School / Statistics and Data Science I (AB)
2.9 Summary
We have started our journey with data—what we end up with after we turn variation in the world into numbers. The process of creating data starts with sampling, and then measurement. We organize data
into columns and rows, where the columns represent the variables (e.g., Thumb) that we have measured; and the rows represent the objects to which we applied our measurement (e.g., students). Each
cell of the table holds a value, representing that row’s measurement for that variable (such as one student’s thumb length).
Before analyzing data, we often want to manipulate it in various ways. We may create summary variables, filter out missing data, and so on.
But let’s keep our eye on the prize: we care about variation in data because we are interested in variation in the world. There is some greater population that a sample comes from. And here we see
the ultimate problem with data: it won’t always look like the thing it came from. Much of statistics is devoted to understanding and dealing with this problem.
End of Chapter Survey | {"url":"https://staging.coursekata.org/preview/book/7846adcd-3aea-416d-abb6-f499aa45584e/lesson/4/8","timestamp":"2024-11-14T21:46:49Z","content_type":"text/html","content_length":"74443","record_id":"<urn:uuid:7b90ac89-572f-400f-b464-6784d99c19c5>","cc-path":"CC-MAIN-2024-46/segments/1730477395538.95/warc/CC-MAIN-20241114194152-20241114224152-00228.warc.gz"} |
ExpressView: Formula Columns (v2021.1+)
ExpressViews are an easy way to display and visualize data. Custom formula columns extend the pre-defined data.
In addition to using the data fields that are available in the Data Objects Pane, formulas may be added as custom formula columns. Formulas may contain data objects, parameters and functions.
Just like standard columns, a formula column may be either a detail field or group, and may be included in visualizations, formatted, sorted and filtered.
To add a formula column to an ExpressView:
1. In the Data Objects Pane, click the + Add Formula link at the top. A new column will be added to the report and the Formula Builder will appear.
Formula Builder with a formula that returns the year portion of a date detail field.
2. The Selected Section tab of the Properties Pane will open with the available functions that may be used to build a formula.
1. Either Search for a function, or expand the function categories to browse the available functions:
1. Arithmetic — basic mathematical functions, as well as number field manipulation like truncation and rounding.
2. Database — used for determining the type of information contained in a cell. Helpful for error and sanity checks.
3. Date — can be used to do calculations and formatting on date and time values.
4. Financial — common methods for monetary calculations such as interest and depreciation.
5. Formatting — applies bold, underline, or italic formatting to the input.
6. Logical — can be used to handle conditional information.
7. Other — miscellaneous functions
8. String — commonly used for interpreting and manipulating text fields.
9. Parameters — a listing of the available system parameters that may be used in the formula
2. Hover the mouse over a function name to see a description of it and how to use it.
3. Either drag-and-drop the function name, or double-click the function name to add it to the Formula Builder.
The formula column is now added to the ExpressView, and can be manipulated just as any other column on the report.
Formula Components
Formulas may contain any combination of data objects, parameters and functions.
Data Objects
To add a field to the formula, either:
Data objects must always be enclosed in curly braces. For example, {Orders.OrderId}
Formula Builder with a single data object in it
Parameters are system-wide variables that may be referenced in formulas. To add a parameter to a formula:
2. Either double-click the parameter in the list, or use the arrow keys to highlight the desired parameter and then press Enter on the keyboard.
Parameters are case-sensitive and must always be enclosed by @ symbols. For example, @userId@
Formula Builder with a single parameter in it
A special type of parameter called a dropdown parameter can be created by the system administrator. These parameters have two values: the Value, used by the server for processing and the Display
Value that appears in cells and is used in formulas. These distinct values can be accessed with @ParameterName.Value@ and @Parameter.DisplayValue@ respectively. For more information, consult with the
system administrator.
Functions are mechanisms built-in to the application that take an input and return an output. For example, there is a square root function that given an input number will return the square root of
the number. For example, Sqrt(4) returns 2.
Many functions can be used together in a formula.
A list of all available functions and operators along with descriptions and examples of how to use them can be found in the List of Functions.
Some examples of functions in formulas:
{Employees.LastName} & ", " & {Employees.FirstName}
Concatenate employee’s last name and first name in a single column, using two fields
Get only the year portion of an order date, from the Orders.OrderDate field
If(@userId@ = "Admin", {Manufacturers.name}, {Manufacturers.manufacturer_id})
Using a combination of parameters, functions and data objects to display the name of a manufacturer in a column to the administrator, or an obfuscated ID number to all other users
Quick Functions
Quick Functions are pre-defined calculations or operators that transform the fields in either detail fields or group columns. For example, the Month Name quick function applied to a date column will
transform the column to only the name of the month of that date. Quick Functions can be used in lieu of building a full formula column, work in conjunction with a formula, or work on a standard
detail field or group column by themselves.
There are several built-in Quick Functions, and the system administrator has the ability to add or remove functions from the list:
With a Quick Function on a column, it may be sorted, filtered, aggregated and included in a chart just as any other column type.
Behind the scenes, a Quick Function works like a preset formula column. Quick functions have a display formula and a sort formula. The display formula is applied to the data as it is displayed in the
ExpressView Designer, in exported files and it is the value that is compared when filters are applied to a column. The sort formula is activated when a sort is applied to a column. This allows a
column to be sorted on a different quantity than the display formula. For example, the Day of Week quick function will display names of weekdays, but will sort the column in chronological order
instead of alphabetical order (that is Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday).
An ExpressView with one group column, one detail column on the OrderDate field, and a detail field with the Day of Week quick function applied to OrderDate as it appears in exported Excel file
When an ExpressView is exported to an Advanced Report, the quick functions will be broken down into their component display and sort formulas. The display formula will appear on the report, and if an
applicable sort had been added to the ExpressView, the sort formula will be added to the end of the Advanced Reports sort list.
The DayOfWeekName Display Function appears in column B of the Advanced Report Designer
The DayOfWeekNumber Sort Function is added to the end of the Advanced Report’s Sorts list
Using a Quick Function
1. Open the Column Menu by either clicking on the Column Menu
2. Point to Quick Functions.
The Quick Functions menu item will only be available if there are quick functions configured in the system for the data type in the chosen column. For example, there must be Quick Functions
that work on date columns if the chosen column contains a date.
3. Click the name of the desired Quick Function to add to the column.
The name of the Quick Function will appear in the column header.
The Day of Month Quick Function has been applied to this Group column on Orders.OrderDate
When a Quick Function is applied to a formula column, the quick function will apply to the data after the formula. For example, the formula DateAdd("d",3,{Orders.OrderDate}) combined with the Day of
Month Quick Function will display the name of the day of the week three days later than the OrderDate. If the order was placed on Tuesday, the column will display Friday.
Removing a Quick Function
Follow the steps in Using a Quick Function above. At step 3, choose None.
Quick Functions will also be removed from a formula column if the formula is changed in a way that the formula returns a different data type than the data type the Quick Function applies to. For
example, if a formula that returns a date is changed to one that returns a string, the Quick Function will be removed from the column. | {"url":"https://devnet.logianalytics.com/hc/en-us/articles/5281612083991-ExpressView-Formula-Columns-v2021-1","timestamp":"2024-11-13T01:38:46Z","content_type":"text/html","content_length":"51583","record_id":"<urn:uuid:328e6477-f698-4313-986f-ddaece7edf3b>","cc-path":"CC-MAIN-2024-46/segments/1730477028303.91/warc/CC-MAIN-20241113004258-20241113034258-00564.warc.gz"} |
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
These Solutions are part of ML Aggarwal Class 10 Solutions for ICSE Maths. Here we have given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2
More Exercises
Question 1.
If O is the centre of the circle, find the value of x in each of the following figures (using the given information):
From the figure
(i) ABCD is a cyclic quadrilateral
Question 2.
(a) In the figure (i) given below, O is the centre of the circle. If ∠AOC = 150°, find (i) ∠ABC (ii) ∠ADC.
(b) In the figure (i) given below, AC is a diameter of the given circle and ∠BCD = 75°. Calculate the size of (i) ∠ABC (ii) ∠EAF.
(a) Given, ∠AOC = 150° and AD = CD
We know that an angle subtends by an arc of a circle
at the centre is twice the angle subtended by the same arc
at any point on the remaining part of the circle.
Question 3.
(a) In the figure, (i) given below, if ∠DBC = 58° and BD is a diameter of the circle, calculate:
(i) ∠BDC (ii) ∠BEC (iii) ∠BAC
(b) In the figure (if) given below, AB is parallel to DC, ∠BCE = 80° and ∠BAC = 25°. Find:
(i) ∠CAD (ii) ∠CBD (iii) ∠ADC (2008)
(a) ∠DBC = 58°
BD is diameter
∠DCB = 90° (Angle in semi circle)
(i) In ∆BDC
∠BDC + ∠DCB + ∠CBD = 180°
∠BDC = 180°- 90° – 58° = 32°
Question 4.
(a) In the figure given below, ABCD is a cyclic quadrilateral. If ∠ADC = 80° and ∠ACD = 52°, find the values of ∠ABC and ∠CBD.
(b) In the figure given below, O is the centre of the circle. ∠AOE =150°, ∠DAO = 51°. Calculate the sizes of ∠BEC and ∠EBC.
(a) In the given figure, ABCD is a cyclic quadrilateral
∠ADC = 80° and ∠ACD = 52°
To find the measure of ∠ABC and ∠CBD
Question 5.
(a) In the figure (i) given below, ABCD is a parallelogram. A circle passes through A and D and cuts AB at E and DC at F. Given that ∠BEF = 80°, find ∠ABC.
(b) In the figure (ii) given below, ABCD is a cyclic trapezium in which AD is parallel to BC and ∠B = 70°, find:
(i)∠BAD (ii) DBCD.
(a) ADFE is a cyclic quadrilateral
Ext. ∠FEB = ∠ADF
⇒ ∠ADF = 80°
ABCD is a parallelogram
∠B = ∠D = ∠ADF = 80°
or ∠ABC = 80°
(b)In trapezium ABCD, AD || BC
(i) ∠B + ∠A = 180°
⇒ 70° + ∠A = 180°
⇒ ∠A = 180° – 70° = 110°
∠BAD = 110°
(ii) ABCD is a cyclic quadrilateral
∠A + ∠C = 180°
⇒ 110° + ∠C = 180°
⇒ ∠C = 180° – 110° = 70°
∠BCD = 70°
Question 6.
(a) In the figure given below, O is the centre of the circle. If ∠BAD = 30°, find the values of p, q and r.
(b) In the figure given below, two circles intersect at points P and Q. If ∠A = 80° and ∠D = 84°, calculate
(i) ∠QBC (ii) ∠BCP
(a) (i) ABCD is a cyclic quadrilateral
Question 7.
(a) In the figure given below, PQ is a diameter. Chord SR is parallel to PQ.Given ∠PQR = 58°, calculate (i) ∠RPQ (ii) ∠STP
(T is a point on the minor arc SP)
(b) In the figure given below, if ∠ACE = 43° and ∠CAF = 62°, find the values of a, b and c (2007)
(a) In ∆PQR,
∠PRQ = 90° (Angle in a semi circle) and ∠PQR = 58°
∠RPQ = 90° – ∠PQR = 90° – 58° = 32°
SR || PQ (given)
Question 8.
(a) In the figure (i) given below, AB is a diameter of the circle. If ∠ADC = 120°, find ∠CAB.
(b) In the figure (ii) given below, sides AB and DC of a cyclic quadrilateral ABCD are produced to meet at E, the sides AD and BC are produced to meet at F. If x : y : z = 3 : 4 : 5, find the values
of x, y and z.
(a) Construction: Join BC, and AC then
ABCD is a cyclic quadrilateral.
Question 9.
(a) In the figure (i) given below, ABCD is a quadrilateral inscribed in a circle with centre O. CD is produced to E. If ∠ADE = 70° and ∠OBA = 45°, calculate
(i) ∠OCA (ii) ∠BAC
(b) In figure (ii) given below, ABF is a straight line and BE || DC. If ∠DAB = 92° and ∠EBF = 20°, find :
(i) ∠BCD (ii) ∠ADC.
(a) ABCD is a cyclic quadrilateral
Question 10.
(a) In the figure (ii) given below, PQRS is a cyclic quadrilateral in which PQ = QR and RS is produced to T. If ∠QPR = 52°, calculate ∠PST.
(b) In the figure (ii) given below, O is the centre of the circle. If ∠OAD = 50°, find the values of x and y.
(a) PQRS is a cyclic quadrilateral in which
PQ = QR
Question 11.
(a) In the figure (i) given below, O is the centre of the circle. If ∠COD = 40° and ∠CBE = 100°, then find :
(i) ∠ADC
(ii) ∠DAC
(iii) ∠ODA
(iv) ∠OCA.
(b) In the figure (ii) given below, O is the centre of the circle. If ∠BAD = 75° and BC = CD, find :
(i) ∠BOD
(ii) ∠BCD
(iii) ∠BOC
(iv) ∠OBD (2009)
(a) (i) ∴ ABCD is a cyclic quadrilateral.
∴ Ext. ∠CBE = ∠ADC
⇒ ∠ADC = 100°
(ii) Arc CD subtends ∠COD at the centre
and ∠CAD at the remaining part of the circle
∴ ∠COD = 2 ∠CAD
Question 12.
In the given figure, O is the centre and AOE is the diameter of the semicircle ABCDE. If AB = BC and ∠AEC = 50°, find :
(i) ∠CBE
(ii) ∠CDE
(iii) ∠AOB.
Prove that OB is parallel to EC.
In the given figure,
O is the centre of the semi-circle ABCDE
and AOE is the diameter. AB = BC, ∠AEC = 50°
Question 13.
(a) In the figure (i) given below, ED and BC are two parallel chords of the circle and ABE, ACD are two st. lines. Prove that AED is an isosceles triangle.
(b) In the figure (ii) given below, SP is the bisector of ∠RPT and PQRS is a cyclic quadrilateral. Prove that SQ = RS.
(a) Given: Chord BC || ED,
ABE and ACD are straight lines.
To Prove: ∆AED is an isosceles triangle.
Proof: BCDE is a cyclic quadrilateral.
Ext. ∠ABC = ∠D …(i)
But BC || ED (given)
Question 14.
In the given figure, ABC is an isosceles triangle in which AB = AC and circle passing through B and C intersects sides AB and AC at points D and E. Prove that DE || BC.
In the given figure,
∆ABC is an isosceles triangle in which AB = AC.
A circle passing through B and C intersects
sides AB and AC at D and E.
To prove: DE || BC
Construction : Join DE.
∵ AB = AC
∠B = ∠C (angles opposite to equal sides)
But BCED is a cyclic quadrilateral
Ext. ∠ADE = ∠C
= ∠B (∵ ∠C = ∠B)
But these are corresponding angles
DE || BC
Hence proved.
Question 15.
(a) Prove that a cyclic parallelogram is a rectangle.
(b) Prove that a cyclic rhombus is a square.
(a) ABCD is a cyclic parallelogram.
To prove: ABCD is a rectangle
Proof: ABCD is a parallelogram
∠A = ∠C and ∠B = ∠D
Question 16.
In the given figure, chords AB and CD of the circle are produced to meet at O. Prove that triangles ODB and OAC are similar. Given that CD = 2 cm, DO = 6 cm and BO = 3 cm, area of quad. CABD
In the given figure, AB and CD are chords of a circle.
They are produced to meet at O.
To prove : (i) ∆ODB ~ ∆OAC
If CD = 2 cm, DO = 6 cm, and BO = 3 cm
To find : AB and also area of the
\(\frac { Quad.ABCD }{ area\quad of\quad \Delta OAC } \)
Construction : Join AC and BD
Hope given ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 15 Circles Ex 15.2 are helpful to complete your math homework.
If you have any doubts, please comment below. Learn Insta try to provide online math tutoring for you. | {"url":"https://ncertmcq.com/ml-aggarwal-class-10-solutions-for-icse-maths-chapter-15-ex-15-2/","timestamp":"2024-11-09T13:14:51Z","content_type":"text/html","content_length":"101099","record_id":"<urn:uuid:008c5f19-0709-4ec4-9102-27c498f5b51e>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00060.warc.gz"} |
Joseph Louis Lagrange
b. Turin 25 January 1736
d. Paris 10 April 1813
The works of Lagrange are collected in Oeuvres de Lagrange: Tome Premier (1867), Tome Deuxième (1868), Tome Troisième (1869), Tome Quatrième (1869), Tome Cinquième (1870), Tome Sixième (1873), Tome
Septième (1877), Tome Huitième (1879), Tome Neuvième (1881), Tome Dixième (1884), Tome Treizième (1882)
The Bibliography of Statistical Literature cites "Recherches sur la Méthode de Maximus et Minimis", Miscellanea Taurinensia, t. 1, 1759. Oeuvres I, pp. 3-20.
Distribution of the Sample Mean
The distribution of the mean of n observations subject to certain assumptions of the distribution of errors is taken up in "Memoire sur l'utilité de la méthode de prendre le milieu entre les
résultats de plusiers observations; dans lequel on examine les avantages de cette méthode par le calcul des probabilités; et où l'on résoud différens problèmes relatif à cette matiére." This was
published in Volume V of Miscellanea Taurinensia for the years 1770-1773 on pages 167-232. The paper discusses ten problems for which we follow the outline of Todhunter.
│ Problem │ Description │ Related Work │
│ I │ To find the probability that the mean will be exact when errors have a symmetric distribution on the values -1, 0, 1. │ │
│ II │ To find the probability that the mean will fall within a given radius about its exact value. │ │
│ III │ To find the probability that the mean will fall within prescribed limits. │ │
│ IV │ To find the most probable error in the mean. │ │
│ V │ To find the most probable error in the mean when the errors have an arbitrary discrete distribution. │ │
│ VI │ The distribution of errors is not known but estimated from observation. To estimate the probability that the relative frequencies do not differ from the true │ │
│ │ value by more than an assigned quantity. │ │
│ │ To find the probability that the mean has a prescribed value and lies within prescribed limits when the distribution of errors is uniform on some interval │ De Moivre, │
│ VII │ about 0. │ Simpson, Advantage of │
│ │ │ the Mean │
│ VIII │ To find the probability that the mean lies within prescribed limits when the errors have a discrete triangular distribution on a symmetric interval about 0. │ │
│ X │ To find the probability that the mean lies within prescribed limits when the errors have a continuous distribution on an interval about 0. │ │
│ XI │ To find the probability that the mean lies within prescribed limits when the errors are distributed proportional to one arc of the cosine centered at 0. │ │
Difference Equations
Lagrange produced three papers on the integration of finite difference equations. The first is "Sur l'integration d'une équation différentielle à différences finies qui contient la théorie des suites
récurrentes." This was published in Miscellanea Taurinensia, T. 1, 1759.
It was many years later that he returned to the subject in "Recherches sur les suites recurrentes dont les termes varient de plusieurs manieres différentes, ou sur l'integration des équations
linéaires aux différences finies et partielles; et sur l'usage de ces équations dans la théorie des hazards." This appeared in Nouveaux Mémoires de l'Académie ... Berlin pp. 183-272 for the year 1775
and published in 1777. The portion of the memoir concerning the theory of chances is contained in the pages 240-272. This latter section solves the problem of points for several players, the duration
of games and also an urn problem as outlined below. The paper of Jean Trembley, "Disquisitio Elementaris circa Calculum Probabilium" purports to offer elementary solutions of these problems.
│ Problem │ Description │ Related Work │
│ I │ To find the probability that an event is brought forth at least b times among a trials. │ Moivre, p. 15 and p. 27 │
│ II │ To find the probability that one event is brought forth at least b times, another at least c among a trials when a third event │ Trembley, Disquisitio Elementaris circa Calculum │
│ │ is possible as well. │ Probabilium, Problem 8. │
│ III │ To find the probability to bring forth, in an undetermined number of trials, the second of two events b times before the first │ │
│ │ has arrived a times. │ │
│ │ │ Montmort, Propositions XL and XLI. │
│ IV │ As the third problem excepting that one event must occur c times before the other two occur a and b times respectively.. │ Moivre, 2nd ed., Problem VI. │
│ │ │ Laplace, (1773) Problems XIV and XV. │
│ V │ To find the probability that an event will occur at least b times more than it does not occur. │ Moivre, Problem LXV. │
│ VI │ To find the probability that an event will occur at least b times more or less by c than it does not occur. │ Montmort, p. 268 │
│ │ │ Moivre, Problems LXIII & LXVIII, p. 191 │
│ VII │ To find the distribution of black and white balls in a sequence of urns after transfers have been made from one urn to another │ Daniel Bernoulli, Disquisitiones Analyticae de novo │
│ │ of randomly drawn balls a given number of times. │ problemate conjecturali. │
Finally, we have "Recherches sur plusieurs points d'analyse relatifs à différens endroits des Mémoires précédens." This memoir appeared in the Mémoires de l'Académie Royale des Sciences et Belles
Lettres of Berlin for the years 1792-3, published in 1798, pp. 247-57. It does not directly concern the theory of probability but rather extends the method of solution of difference equations.
"Mémoire sur une question concernant les annuités," published in 1798 in the Mémoires de l'Académie Royale des Sciences et Belles Lettres of Berlin for the years 1792-3, pp. 235-246. This paper was
read to the Academy ten years prior. This paper adds nothing new as Moivre had already solved its problem in his Treatise of Annuities on Lives. | {"url":"https://gtfp.net/pulskamp/Lagrange/lagrange.html","timestamp":"2024-11-05T07:22:34Z","content_type":"text/html","content_length":"15596","record_id":"<urn:uuid:306804d7-50e9-4240-8e9b-da0ec4842720>","cc-path":"CC-MAIN-2024-46/segments/1730477027871.46/warc/CC-MAIN-20241105052136-20241105082136-00663.warc.gz"} |
Let A be an uncountable set, B a countable subset of A, and C the complement of B in A. Prove that there exists a one-to-one correspondence between A and C.
• Low bounty!
• I would say 10-15 makes sense.
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inf R ! [book review] | R-bloggersinf R ! [book review]
[This article was first published on
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Thanks to my answering a (basic) question on X validated involving an R code, R mistakes and some misunderstanding about Bayesian hierarchical modelling, I got pointed out to Patrick Burns’ The R
inferno. This is not a recent book as the second edition is of 2012, with a 2011 version still available on-line. Which is the version I read. As hinted by the cover, the book plays on Dante’s
Inferno and each chapter is associated with a circle of Hell… Including drawings by Botticelli. The style is thus most enjoyable and sometimes hilarious. Like hell!
The first circle (reserved for virtuous pagans) is about treating integral reals as if they were integers, the second circle (attributed to gluttons, although Dante’s is for the lustful) is about
allocating more space along the way, as in the question I answered and in most of my students’ codes! The third circle (allocated here to blasphemous sinners, destined for Dante’s seven circle, when
Dante’s third circle is to the gluttons) points out the consequences of not vectorising, with for instance the impressive capacities of the ifelse() function [exploited to the max in R
codecolfing!]. And the fourth circle (made for the lustfuls rather than Dante’s avaricious and prodigals) is a short warning about the opposite over-vectorising. Circle five (destined for the
treasoners, and not Dante’s wrathfuls) pushes for and advises about writing R functions. Circle six recovers Dante’s classification, welcoming (!) heretics, and prohibiting global assignments, in
another short chapter. Circle seven (alloted to the simoniacs—who should be sharing the eight circle with many other sinners—rather than the violents as in Dante’s seventh) discusses object
attributes, with the distinction between S3 and S4 methods somewhat lost on me. Circle eight (targeting the fraudulents, as in Dante’s original) is massive as it covers “a large number of ghosts,
chimeras and devils”, a collection of difficulties and dangers and freak occurences, with the initial warning that “It is a sin to assume that code does what is intended”. A lot of these came as
surprises to me and I was rarely able to spot the difficulty without the guidance of the book. Plenty to learn from these examples and counter-examples. Reaching Circle nine (where live (!) the
thieves, rather than Dante’s traitors). A “special place for those who feel compelled to drag the rest of us into hell.” Discussing the proper ways to get help on fori. Like Stack Exchange.
Concluding with the tongue-in-cheek comment that “there seems to be positive correlation between a person’s level of annoyance at [being asked several times the same question] and ability to answer
questions.” This being a hidden test, right?!, as the correlation should be negative. | {"url":"https://www.r-bloggers.com/2021/06/inf-r-book-review/","timestamp":"2024-11-13T06:25:24Z","content_type":"text/html","content_length":"89947","record_id":"<urn:uuid:60c119ae-93ed-4fb7-a602-2a3eddcc376d>","cc-path":"CC-MAIN-2024-46/segments/1730477028326.66/warc/CC-MAIN-20241113040054-20241113070054-00170.warc.gz"} |
In mathematics, the complex projective plane, usually denoted P^2(C) or CP^2, is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three
complex coordinates
${\displaystyle (Z_{1},Z_{2},Z_{3})\in \mathbf {C} ^{3},\qquad (Z_{1},Z_{2},Z_{3})eq (0,0,0)}$
where, however, the triples differing by an overall rescaling are identified:
${\displaystyle (Z_{1},Z_{2},Z_{3})\equiv (\lambda Z_{1},\lambda Z_{2},\lambda Z_{3});\quad \lambda \in \mathbf {C} ,\qquad \lambda eq 0.}$
That is, these are homogeneous coordinates in the traditional sense of projective geometry.
The Betti numbers of the complex projective plane are
1, 0, 1, 0, 1, 0, 0, .....
The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane. The nontrivial homotopy groups of the complex projective plane are
${\displaystyle \pi _{2}=\pi _{5}=\mathbb {Z} }$ . The fundamental group is trivial and all other higher homotopy groups are those of the 5-sphere, i.e. torsion.
Algebraic geometry
In birational geometry, a complex rational surface is any algebraic surface birationally equivalent to the complex projective plane. It is known that any non-singular rational variety is obtained
from the plane by a sequence of blowing up transformations and their inverses ('blowing down') of curves, which must be of a very particular type. As a special case, a non-singular complex quadric in
P^3 is obtained from the plane by blowing up two points to curves, and then blowing down the line through these two points; the inverse of this transformation can be seen by taking a point P on the
quadric Q, blowing it up, and projecting onto a general plane in P^3 by drawing lines through P.
The group of birational automorphisms of the complex projective plane is the Cremona group.
Differential geometry
As a Riemannian manifold, the complex projective plane is a 4-dimensional manifold whose sectional curvature is quarter-pinched, but not strictly so. That is, it attains both bounds and thus evades
being a sphere, as the sphere theorem would otherwise require. The rival normalisations are for the curvature to be pinched between 1/4 and 1; alternatively, between 1 and 4. With respect to the
former normalisation, the imbedded surface defined by the complex projective line has Gaussian curvature 1. With respect to the latter normalisation, the imbedded real projective plane has Gaussian
curvature 1.
An explicit demonstration of the Riemann and Ricci tensors is given in the n=2 subsection of the article on the Fubini-Study metric.
See also | {"url":"https://www.knowpia.com/knowpedia/Complex_projective_plane","timestamp":"2024-11-06T07:53:28Z","content_type":"text/html","content_length":"74195","record_id":"<urn:uuid:dd39562a-27de-4434-9fa7-d93e8eea965f>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00408.warc.gz"} |
The last furlong (2). Data Compression by wavelets
A.M.C. Davies^a and Tom Fearn^b
^aNorwich Near Infrared Consultancy, 10 Aspen Way, Cringleford, Norwich NR4 6UA, UK. E-mail: [email protected]
^bDepartment of Statistical Science, University College London, Gower Street, London WC1E 6BT, UK
Historical introduction
I first heard of “Wavelets” at the “Chambersburg” (International Diffuse Reflection) Conference in 1996. I did not understand it but thought it might be an important topic so I asked the lecturer to
try to explain it again. He tried hard but I still did not get it. He said he would send me some papers. He did, but I did not understand them. Two years later at the next IDRC, Tom and I ran our
“Introduction to NIR and chemometrics” short course (which we had been doing for several IDRCs) but we were also asked to present a one-day course on “Advanced Chemometrics”. We organised this by
e-mail and telephone. One of the topics was to be data compression, I would talk about Fourier and Tom would cover wavelets (I still did not understand wavelets so I was especially looking forward to
this part of the course). At Chambersburg, I did my bit on Fourier (very similar to the previous TD column^1) and Tom began his explanation of wavelets. In less than 10 minutes, I understood! We hope
you will also understand when you have read this article!—Tony Davies
Compared to Fourier, wavelets in their current form are a very recent development, in the late 1980s. They were invented by the Belgian mathematician Ingrid Daubechies and are described in a paper in
In some ways wavelets are similar to the sine and cosine waves we use in Fourier transformation: they have the same mathematical properties that allow them to be used to fit spectra but they are
different in two important ways. First, wavelets are not smooth curves, some have quite jagged features, and second, they are locally weighted. There are an infinite number of possible wavelet shapes
but because they are difficult to invent^* there are not very many. Three of those invented by Daubechies, are shown in Figure 1, they are known by the names, Daubechies extremal phase, Coiflet and
Symmlet. Each of these waveforms has been subjected to minor changes and are distinguished by a number, D2–D10, C2–C5 and S2–S8, shown in Figure 2.
^*The majority of mathematicians prefer the word “discover” on the grounds that all mathematics is either possible (waiting to be discovered) or not possible (cannot be discovered or invented). This
may be so but it is sometimes obvious that “invent” is the appropriate word. Interestingly, after I wrote this note I discovered a website containing an interview with Ingrid Daubechies in which she
said that she believes that all mathematics is “constructed” not discovered!]
We use a string of wavelets, as shown in Figure 3, in the same way as we use sine and cosines waves in FT but now each wavelet has a weight (or coefficient) associated with it. If some of these
coefficients are set to zero the waveform would appear to have straight line sections. Again similar to FT we can construct a family of waveforms of increasing frequency. So, starting with one, which
fills the whole interval being considered (i.e. a spectrum), known as level 0, we move to level 1 by doubling the number of wavelets, which will be half the width of those on the previous level. Then
to level 2, by again doubling the number of wavelets and so on. When we reach the seventh level it will contain 128, very narrow wavelets. This process may be continued to as high a level as is
required for our application. An individual wavelet is specified by a level number and a position number. Figure 4 shows some S8 wavelets where the coefficient is non-zero for one or a few wavelets
at each level. The labelling in brackets gives the level number and position of these wavelets.
Using wavelets for data compression in spectroscopy
When we use FT for data compression, the FFT program has to compute a coefficient for the sine and cosine waves at each frequency. For wavelet compression there is a similar FWT program but this has
to compute coefficients for each wavelet at each level; so there is a rather larger file for each spectrum. Many of these coefficients will be very close to zero so there is a variable tolerance that
can be set to make all the very small coefficients zero. This is where we obtain the data compression.
To see how this works in practice, Figure 5 shows the decomposition (the technical word for fitting a spectrum) of an NIR spectrum of polystyrene. The curves show very clearly that many coefficients
were almost zero and those that are non-zero correspond to peaks in the original spectrum. One of the nice things about wavelets is that it is so easy to see where the information has been found. The
lower levels (1–4) tend to be more generalised, required for accurate reconstruction of the spectrum but less interesting and not shown in the figure. The lowest curve in the figure is the
reconstruction using all the wavelets.
Comparison of Fourier and wavelet compression
Between 1983 and 1988 TD and Professor Fred McClure^3 developed an idea for a method of quantitative analysis, CARNAC, which did not rely on regression analysis. A key part of the method was that it
required compression of NIR databases and this was done by FT using the programs developed by McClure. When we became interested in wavelets it seemed a good idea to see if we could replace the
compression step in CARNAC by wavelet compression. We found that we needed answers to two questions: “which wavelet is best for NIR spectra” and “are wavelets any better than FT?” Although some
researchers had experimented with NIR data and wavelets these questions had not been answered. There seemed to be a general belief that any wavelet would be better than FT!
We did a study^4 using a sub-set of 12 NIR spectra selected from a large database of spectra of different chemicals and commodities supplied by Karl Norris.^5 The sub-set was selected to give us a
large variation in spectral shapes from smooth curves to sharp peaks and different mixtures of both. First, we tested the wavelets shown in Figure 2 to see if there was a “best” wavelet for use with
NIR spectra. Best was defined as the wavelet that required the least number of coefficients to achieve a given degree of fit. In this case we knew the noise level of the spectrometer, 200 µA,^† that
had been used to measure these samples and (as the there is no point in trying to fit noise) this figure was used as the target for the compression. The results were judged by computing a
reconstruction error for a given number of coefficients by subtracting the original spectrum from the reconstructions and calculating the root mean square across all wavelengths. We found that the
best wavelets were: db3,db4,db5 and sy3,sy4,sy5, and we choose the db4 wavelet (which had been successfully used in other published work) for the comparison with FT. We had expected that the wavelet
compression would be far more efficient than FT but this was not what we found. For 10 out of 12 spectra the wavelets were more efficient but the improvements were modest and in two cases, with very
smooth spectra, the FT was superior. These variations are demonstrated by Figures 6 and 7 which show the reconstruction errors for water and freeze-dried coffee.
^†μA denotes micro absorbance units or log 1/R × 10^–6
Wavelet compression is an interesting and popular method. However, when considering the application of wavelets for a new use, it is probably worth confirming that there is a useful advantage to be
gained if compared to FT compression, rather than assuming that wavelets will always give a more efficient transformation. In spectroscopy, when information peaks are often well separated by regions
of flat baseline we would expect that wavelets would be the better choice but for NIR spectra this is not the normal case and the decision is borderline. However, we decided to proceed with the
application of wavelets to CARNAC and were rewarded with modest improvements compared to the use of FT compression with the same data.^6 Further details of wavelet compression can be found in our
1. A.M.C. Davies, “The last furlong (1) Data Compression”, Spectroscopy Europe 25(2), 23 (2013).
2. I. Daubechies, Ten Lectures on Wavelets. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA.
3. A.M.C. Davies, H.V. Britcher, J.G. Franklin, S.M. Ring, A. Grant and W.F. McClure, “The application of Fourier-transformed NIR spectra to quantitative analysis by comparison of similarity indices
(CARNAC)”, Mikrochim. Acta (Wien) 1, 61 (1988). doi: 10.1007/BF01205839
4. T. Fearn and A.M.C. Davies, “A comparison of Fourier and wavelet transforms in the processing of near infrared spectroscopic data: Part 1. Data compression”, J. Near Infrared Spectrosc. 11, 3–15
(2003). doi: 10.1255/jnirs.349
5. P.C. Williams and K.H. Norris (Eds), Near Infrared Technology in the Agricultural and Food Industries. American Association of Cereal Chemists, St Paul (1987).
6. A.M.C. Davies and T. Fearn, “Quantitative analysis via near infrared databases: comparison analysis using restructured near infrared and constituent data-deux (CARNAC-D)”, J. Near Infrared
Spectrosc. 14, 403–411 (2006). doi: 10.1255/jnirs.712 | {"url":"https://www.spectroscopyeurope.com/td-column/last-furlong-2-data-compression-wavelets","timestamp":"2024-11-08T04:39:12Z","content_type":"text/html","content_length":"65891","record_id":"<urn:uuid:97401ead-4ace-4c3a-9b9c-e662be5ebe3e>","cc-path":"CC-MAIN-2024-46/segments/1730477028025.14/warc/CC-MAIN-20241108035242-20241108065242-00080.warc.gz"} |
Understanding the physics of biology
First, since alliterations are always fun:
An index of sorts
Prompted by compelling clinical reports evidencing the pervasive role of mechanical factors influencing biological growth, our modelling work is aimed at gaining a deeper understanding of the
biophysical bases underlying these influences.
“All science is either physics or stamp collecting.”
Ernest Rutherford
(1871 – 1937)
The utility of the resulting mathematical and computational framework extends across disciplines by helping steer and interpret experimental work, both under physiological and pathological cases of
interest. Currently, our focus is directed toward a better understanding of the mechanics of growing soft tissue, specifically tendon.
This web page contains some highlights of our modelling effort. For more detailed expositions, the interested reader is directed toward more formal articles.
Growth in biological tissue is a direct outcome of cascades of complex, intracellular, biochemical reactions involving numerous species, their diffusion across cell membranes, and transport through
the extra-cellular matrix. Both reaction and diffusion/transport are influenced by mechanics in a number of ways. Our modelling effort proposes a general continuum field formulation for growth
capable of simulating this rich observed behaviour, and proceeds to specialise it incorporating different modelling assumptions—such as the use of an enzyme kinetics-based growth law—to better
represent cases of interest.
Recognising that a growing tissue is an open system undergoing irreversible thermodynamics, the model incorporates the physics of multi-phase reacting systems, and deduces balance laws and a
constitutive framework obeying the Second Law of Thermodynamics.
Notably, the transport of the extracellular fluid relative to the matrix is shown to be driven by the gradients of stress, concentration and chemical potential—a coupling of mass transport and
mechanics that emerges directly.
Assumptions central to existing mechanics theories, such as the nature of the split of deformation maps across the different species involved, and those on momentum transfers between interacting
species, are revisited, analysed and carefully revised in cases where they limit the biophysical correctness of predicted growth phenomena.
Coupled, non-linear partial differential equations arise from the theory to describe the complex physics. These differential equations are solved using a finite element scheme based on
operator-splits; incorporating non-linear projection methods to treat incompressibility, energy-momentum conserving algorithms for dynamics, and mixed methods for stress gradient-driven fluxes.
One highlight of the numerical methods is a rigorous treatment of numerical stability issues that arise with any operator-splitting scheme. This analysis is critical to the ability to distinguish
physical instabilities, such as unbounded growth of tumours, from those that are artifacts of the numerical methods. Another key feature is the reformulation of the reaction-transport equations to
embed the incompressibility constraint on the fluid phase, enabling a straightforward implementation of numerical stabilisation in the advection-dominated limit.
Using model geometries and imposing classes of initial and boundary conditions approximating experiments in our laboratory, the computational framework developed is used to demonstrate aspects of the
coupled phenomena as the tissue grows. In these foundational calculations, only two phases—fluid and collagen—are included. The collagen phase is modelled by the anisotropic worm-like chain model,
and the fluid phase is modelled as ideal and nearly incompressible. The interconversion between these two phases is modelled using first-order chemical kinetics.
For a tissue undergoing finite strains, the transport equations can be formulated, mathematically, in terms of quantities with respect to either the reference or current (deformed) configuration.
However, the physics of fluid-tissue interactions, and the imposition of relevant boundary conditions, is best understood and represented in the current configuration.
Watch the
unbounded build-up of fluid
due to the (unphysical) specification of constant reference fluid concentration boundary conditions, while attempting to simulate a loaded tendon immersed in a bath.
Having difficulty viewing a video?
In order to make their sizes more palatable for download, the videos on this page have been compressed using different algorithms. If you are having difficulty viewing any of them, please follow the
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When a tendon having an initially-uniform distribution of collagen is immersed into a nutrient-rich bath, nutrient-rich fluid is transported into the tissue, and growth occurs due to the formation of
additional collagen.
There is a rapid, fluid transport-dominated swelling of the tendon initially as it is immersed into the fluid bath. This is followed by a slower, reaction-driven growth phase.
On performing a uniaxial tension test on the tendon before and after growth, it is observed that the grown tissue—having a higher concentration of collagen—is stiffer and stronger; which is in
accordance with experiment.
Upon subjecting the tendon to a load-unload cycle, a stress-strain curve characteristic of viscoelastic tissue is observed. Here, the area between the loading and unloading paths is the hysteretic
energy loss due to viscous dissipation.
Observe the
induced fluid flow
as the tendon is subjected to a cyclically varying load. Friction between the solid and fluid phases results in energy dissipation.
Qualitatively, this viscoelastic behaviour compares favourably with our corresponding experimental tests on two-week-old tibialis anterior tendons in the laboratory.
Application of a constrictive radial load—where the maximum strain in the radial direction is experienced half-way through the height of the tendon—to a tendon immersed in a fluid-filled bath results
in a stress-gradient induced fluid flux. This drives fluid away from the central plane.
This pressure wave set up in the fluid travels toward the top and bottom faces, and as the fluid leaves these surfaces, we observe that the tendon relaxes.
The stress gradient-driven fluid flux causes a
decrease in the reference fluid concentration
near the central plane. Pay close attention to observe the relaxation of the top face as the fluid leaves the surface.
Having established the fundamental behaviour of the formulation, we now turn to more sophisticated applications.
In order to model localised, bolus delivery of regulatory chemicals to the tendon while accounting for mechanical effects, we introduce additional species: a solute, and a distribution of fibroblasts
that are characterised by their cell concentration. Both concentration gradient-driven mass transport and stress gradient-driven fluid flow are incorporated into this illustration, which demonstrates
the use of the formulation in studying the efficacy of drug delivery mechanisms. Michaelis-Menten enzyme kinetics is used to determine the rates of solute consumption and, consequently, collagen
In addition to subjecting the tendon immersed in the bath to the constrictive radial load described earlier, a solute-rich bulb with its centre on the axis of the tendon is introduced. The relatively
small magnitude of the fluid mobility—with respect to the diffusion coefficient for the solute through the fluid—results in a relatively small stress gradient-driven flux, and the transport of the
solute is diffusion dominated. Consequently, as time progresses, the solute primarily diffuses locally, and as the solute concentration in a region increases, the enzyme-kinetics model predicts a
small source term for collagen, and we observe nominal growth.
The following computation demonstrates the capability of the formulation in studying the self-healing of damaged tissue. Incorporating a cell-signalling parameter into the chemical kinetics, the
formation of collagen is spatially biased toward damaged regions of the tissue.
Turning to damaged skin as our tissue of interest, we begin by delineating the damaged regions—characterised by a sudden reduction in the concentration of collagen—from the rest of the tissue.
This damaged tissue is introduced to a nutrient-rich environment, and the cell-signalling parameter induces preferential growth in these damaged regions. We allow this healing process to evolve under
two different conditions—under no load and under uniaxial stress—to study the influence of mechanics on the properties of healed tissue.
The anisotropic nature of these tissues arises from the microstructural alignment of their constituent collagen fibers, and the directionality of newly-deposited fibers are determined by the
eigenvectors of the applied stress field. When the healing process is carried out under uniaxial stress, with the orientation of the stress being defined by the initially-stiff directionality of the
tissue, the newly-deposited collagen has anisotropic properties which are identical to collagen in the undamaged tissue. This results in a healed tissue indistinguishable from the undamaged tissue.
However, when the healing process is carried out under no load, the newly-deposited collagen consists of arbitrarily-directed fibers, resulting in a formation of isotropic tissue in the damaged
regions as the tissue heals. This results in a healed tissue being more compliant than the undamaged tissue along the originally-stiff direction of the tissue. Thus, upon being subjected to a load,
these compliant healed regions experience a reduced stress.
This result is analogous to the experimentally observed hypertrophic scarring of skin as it recovers from damage under no applied load. | {"url":"https://v4.harishnarayanan.org/research/continuum-biophysics/","timestamp":"2024-11-05T19:33:12Z","content_type":"application/xhtml+xml","content_length":"26633","record_id":"<urn:uuid:513353a7-9b73-4fa2-83db-194d43c20cb7>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00534.warc.gz"} |
CAT 2017 LRDI Slot 2 questions and solutions | Apti4All
CAT 2017 LRDI Slot 2 | Previous Year CAT Paper
Directions for next 4 questions.
Funky Pizzaria was required to supply pizzas to three different parties. The total number of pizzas it had to deliver was 800, 70% of which were to be delivered to party 3 and the rest equally
divided between Party 1 and Party 2.
Pizzas could be of Thin Crust (T) or Deep Dish (D) variety and come in either Normal Cheese (NC) or Extra Cheese (EC) versions. Hence, there are four types of pizzas: T-NC, T-EC, D-NC and D-EC.
Partial information about proportions of T and NC pizzas ordered by the three parties is given below:
1. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
How many Thin Crust pizzas were to be delivered to Party 3?
• A.
• B.
• C.
• D.
2. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
How many Normal Cheese pizzas were required to be delivered to Party 1?
• A.
• B.
• C.
• D.
3. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
For Party 2, if 50% of the Normal Cheese pizzas were of Thin Crust variety, what was the difference between the numbers of T-EC and D-EC pizzas to be delivered to party 2?
4. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
Suppose that a T-NC pizza cost as much as a D-NC pizza, but 3/5th of the price of a D-EC pizza. A D-EC pizza costs Rs. 50 more than a T-EC pizza, and the latter costs Rs. 500.
If 25% of the Normal Cheese pizzas delivered to Party 1 were of Deep Dish variety, what was the total bill for Party 1?
• A.
Rs. 59480
• B.
Rs. 59840
• C.
Rs. 42520
• D.
Rs. 45240
Answer the following question based on the information given below.
There were seven elective courses - E1 to E7 – running in a specific term in a college. Each of the 300 students enrolled had chosen just one elective from among these seven. However, before the
start of the term, E7 was withdrawn as the instructor concerned had left the college. The students who had opted for E7 were allowed to join any of the remaining electives. Also, the students who had
chosen other electives were given one chance to change their choice. The table below captures the movement of the students from one elective to another during this process. Movement from one elective
to the same elective simply means no movement. Some numbers in the table got accidentally erased; however, it is known that these were either 0 or 1.
Further, the following are known:
1. Before the change process there were 6 more students in E1 than in E4, but after the reshuffle, the number of students in E4 was 3 more than that in E1.
2. The number of students in E2 increased by 30 after the change process.
3. Before the change process, E4 and 2 more students than E6, while E2 had 10 more students than E3.
5. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
How many elective courses among E1 to E6 had a decrease in their enrollments after the change process?
6. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
After the change process, which of the following is the correct sequence of number of students in the six electives E1 to E6?
• A.
19, 76, 79, 21, 45, 60
• B.
19, 76, 78, 22, 45, 60
• C.
18, 76, 79, 23, 43, 61
• D.
18, 76, 79, 21, 45, 61
7. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
After the change process, which course among E1 to E6 had the largest change in its enrollment as a percentage of its original enrollment?
8. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
Later, the college imposed a condition that if after the change of electives, the enrollment in any elective (other than E7) dropped to less than 20 students, all the students who had left that
course will be required to re-enroll for that elective.
Which of the following is a correct sequence of electives in decreasing order of their final enrollments?
• A.
E2, E3, E6, E5, E1, E4
• B.
E3, E2, E6, E5, E4, E1
• C.
E2, E5, E3, E1, E4, E6
• D.
E2, E3, E5, E6, E1, E3
Answer the following question based on the information given below.
An old woman had the following assets:
(a) Rs. 70 lakh in bank deposits
(b) 1 house worth Rs. 50 lakh
(c). 3 flats, each worth Rs. 30 lakh
(d) Certain number of gold coins, each worth Rs. 1 lakh
She wanted to distribute her assets among her three children; Neeta, Seeta and Geeta.
The house, any of the flats or any of the coins were not to be split. That is, the house went entirely to one child; a flat went to one child and similarly, a gold coin went to one child.
9. CAT 2017 LRDI Slot 2 | LR - Mathematical Reasoning
Among the three, Neeta received the least amount in bank deposits, while Geeta received the highest. The value of the assets was distributed equally among the children, as were the gold coins.How
much did Seeta receive in bank deposits (in lakhs of rupees)?
10. CAT 2017 LRDI Slot 2 | LR - Mathematical Reasoning
Among the three, Neeta received the least amount in bank deposits, while Geeta received the highest. The value of the assets was distributed equally among the children, as were the gold coins. How
many flats did Neeta receive?
7 8 9 4 5 6 1 2 3 0 . -
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11. CAT 2017 LRDI Slot 2 | LR - Mathematical Reasoning
The value of the assets distributed among Neeta, Seeta and Geeta was in the ratio of 1 : 2 : 3, while the gold coins were distributed among them in the ratio of 2 : 3 : 4. One child got all three
flats and she did not get the house. One child, other than Geeta, got Rs. 30 lakh in bank deposits.
How many gold coins did the old woman have?
• A.
• B.
• C.
• D.
12. CAT 2017 LRDI Slot 2 | LR - Mathematical Reasoning
The value of the assets distributed among Neeta, Seeta and Geeta was in the ratio of 1 : 2 : 3, while the gold coins were distributed among them in the ratio of 2 : 3 : 4. One child got all three
flats and she did not get the house. One child, other than Geeta, got Rs. 30 lakh in bank deposits.
How much did Geeta get in bank deposits (in lakhs of rupees)?
7 8 9 4 5 6 1 2 3 0 . -
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Answer the following question based on the information given below.
At a management school, the oldest 10 dorms, numbered 1 to 10, need to be repaired urgently. The following diagram represents the estimated repair costs (in Rs. Crores) for the 10 dorms. For any
dorm, the estimated repair cost (in Rs. Crores) in an integer. Repairs with estimated cost Rs. 1 or 2 Crores are considered light repairs, repairs with estimated cost Rs. 3 or 4 are considered
moderate repairs and repairs with estimated cost Rs. 5 of 6 Crores are considered extensive repairs.
Further, the following are known:
1. Odd – numbered dorms do not need light repair; even-numbered dorms do not need moderate repair and dorms, whose numbers are divisible by 3, do not need extensive repair.
2. Dorms 4 to 9 all need different repair costs, with Dorm 7 needing the maximum and Dorm 8 needing the minimum.
13. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
Which of the following is NOT necessarily true?
• A.
Dorm 1 needs a moderate repair
• B.
Dorm 5 repair will cost no more than Rs. 4 Crores
• C.
Dorm 7 needs an extensive repair
• D.
Dorm 10 repair will cost no more than Rs. 4 Crores
14. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
What is the total cost of repairing the odd-numbered dorms (in Rs. Crores)?
7 8 9 4 5 6 1 2 3 0 . -
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15. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
Suppose further that:
1. 4 of the 10 dorms needing repair are women’s dorms and need a total of Rs. 20 Crores for repair.
2. Only one of Dorms 1 to 5 is a women’s dorm.
What is the cost for repairing Dorm 9 (in Rs. Crores)?
7 8 9 4 5 6 1 2 3 0 . -
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16. CAT 2017 LRDI Slot 2 | LR - Selection & Distribution
Suppose further that:
1. 4 of the 10 dorms needing repair are women’s dorms and need a total of Rs. 20 Crores for repair.
2. Only one of Dorms 1 to 5 is a women’s dorm.
Which of the following is a women’s dorm?
• A.
Dorm 2
• B.
Dorm 5
• C.
Dorm 8
• D.
Dorm 10
Answer the following question based on the information given below.
A tea taster was assigned to rate teas from six different locations – Munnar, Wayanand, Ooty, Darjeeling, Assam and Himachal. These teas were placed in six cups, numbered 1 to 6, not necessarily in
the same order. The tea taster was asked to rate these teas on the strength of their flavour on a scale of 1 to 10. He gave a unique integer rating to each tea. Some other information is given below:
1. Cup 6 contained tea from Himachal.
2. Tea from Ooty got the highest rating, but it was not in Cup 3.
3. The rating of tea in Cup 3 was double the rating of the tea in Cup 5.
4. Only two cups got ratings in even numbers.
5. Cup 2 got the minimum rating and this rating was an even number.
6. Tea in Cup 3 got a higher rating than that in Cup 1.
7. The rating of tea from Wayanad was more than the rating of tea from Munnar, but less than that from Assam.
17. CAT 2017 LRDI Slot 2 | LR - Arrangements
What was the second highest rating given?
7 8 9 4 5 6 1 2 3 0 . -
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18. CAT 2017 LRDI Slot 2 | LR - Arrangements
What was the number of the cup that contained tea from Ooty?
7 8 9 4 5 6 1 2 3 0 . -
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19. CAT 2017 LRDI Slot 2 | LR - Arrangements
If the tea from Munnar did not get the minimum rating, what was the rating of the tea from Wayanad?
20. CAT 2017 LRDI Slot 2 | LR - Arrangements
If cups containing teas from Wayanad and Ooty had consecutive numbers, which of the following statements may be true?
• A.
Cup 5 contains tea from Assam
• B.
Cup 1 contains tea from Darjeeling
• C.
Tea from Wayanad has got a rating of 6
• D.
Tea from Darjeeling got the minimum rating
Answer the following question based on the information given below.
In an 8 × 8 chessboard a queen placed anywhere can attack another piece if the piece is present in the same row, or in the same column or in any diagonal position in any possible 4 directions,
provided there is no other piece in between in the path from the queen to that piece.
The columns are labelled a to h (left to right) and the rows are numbered 1 to 8 (bottom to top). The position of a piece given by the combination of column and row labels. For example, position c5
means that the piece is cth column and 5th row.
21. CAT 2017 LRDI Slot 2 | LR - Board Games
If the queen is at c5, and the other pieces at positions c2, g1, g3, g5 and a3, how many are under attack by the queen? There are no other pieces on the board.
22. CAT 2017 LRDI Slot 2 | LR - Board Games
If the other pieces are only at positions a1, a3, b4, d7, h7 and h8, then which of the following positions of the queen results in the maximum number of pieces being under attack?
23. CAT 2017 LRDI Slot 2 | LR - Board Games
If the other pieces are only at positions a1, a3, b4, d7, h7 and h8, then from how many positions the queen cannot attack any of the pieces?
24. CAT 2017 LRDI Slot 2 | LR - Board Games
Suppose the queen is the only piece on the board and it is at position d5.
In how many positons can another piece be placed on the board such that it is safe from attack from the queen?
Answer the following question based on the information given below.
Eight friends: Ajit, Byomkesh, Gargi, Jayanta, Kikira, Manik, Prodosh and Tapesh are goin to Delhi from Kolkatta by a flight operated by Cheap Air. In the flight, sitting is arranged in 30 rows,
numbered 1 to 30, each consisting of 6 seats, marked by letters A to F from left to right, respectively. Seats A to C are to the left of the aisle (the passage running from the front of the aircraft
to the back), and seats D to F are to the right of the aisle. Seats A and F are by the windows and referred to as Window seats, C and D are by the aisle and are referred to as Aisle seats while B and
E are referred to as Middle seats. Seats marked by consecutive letters are called consecutive seats (or seats next to each other).A seat number is a combination of the row number, followed by the
letter indicating the position in the row, e.g, 1A is the left window seat in the first row, while 12E is the right middle seat in the 12th row.
Cheap Air charges Rs. 1000 extra for any seats in Rows 1, 12 and 13 as those have extra legroom. For Rows 2-10, it charges Rs. 300 extra for Window seats and Rs. 500 extra for Aisle seats. For Rows
11 and 14 to 20, it charges Rs. 200 extra for Window seats and Rs. 400 extra for Aisle seats. All other seats are available at no extra charge.
The following are known:
1. The eight friends were seated in six different rows.
2. They occupied 3 Window seats, 4 Aisle seats and 1 Middle seat.
3. Seven of them had to pay extra amounts, totaling to Rs. 4600, for their choices of seat. One of them did not pay any additional amount of his/her choice of seat.
4. Jayanta, Ajit and Byomkesh were sitting in seats marked by the same letter, in consecutive rows in increasing order of row numbers; but all of them paid different amounts for their choices of
seat. One of these amounts may be zero.
5. Gargi was sitting next to Kikira, and Manik was sitting next to Jayanta.
6. Prodosh and Tapesh were sitting in seats marked by the same letter, in consecutive rows in increasing order of row numbers; but they paid different amounts for their choices of seat. One of these
amounts may be zero.
25. CAT 2017 LRDI Slot 2 | LR - Arrangements | LR - Mathematical Reasoning
In which row was Manik sitting?
26. CAT 2017 LRDI Slot 2 | LR - Arrangements | LR - Mathematical Reasoning
How much extra did Jayanta pay for his choice of seat?
• A.
Rs. 300
• B.
Rs. 400
• C.
Rs. 500
• D.
Rs. 1000
27. CAT 2017 LRDI Slot 2 | LR - Arrangements | LR - Mathematical Reasoning
How much extra did Gargi pay for her choice of seat?
• A.
• B.
Rs. 300
• C.
Rs. 400
• D.
Rs. 1000
28. CAT 2017 LRDI Slot 2 | LR - Arrangements | LR - Mathematical Reasoning
Who among the following did not pay any extra amount for his/her choice of seat?
• A.
• B.
• C.
• D.
Answer the following question based on the information given below.
A high security research lab requires the researches to set a pass key sequence based on the scan of the five fingers of their left hands. When an employee first joins the lab, her fingers are
scanned in an order of her choice, and then when she wants to re-enter the facility, she has to scan the five fingers in the same sequence.
The lab authorities are considering some relaxations of the scan order requirements of the scan order requirements, since it is observed that some employees often get locked-out because they forget
the sequence.
29. CAT 2017 LRDI Slot 2 | LR - Puzzles
The lab has decided to allow a variation in the sequence of scans of five fingers so that at most two scans (out of five) are out of place. For example, if the original sequence is Thumb (T), index
finger (I), middle finger (M), ring finger (R) and little finger (L) then TLMRI is also allowed, but TMRLI is not.
How many different sequences of scans are allowed for any given person’s original scan?
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30. CAT 2017 LRDI Slot 2 | LR - Puzzles
The lab has decided to allow variations of the original sequence so that input of the scanned sequence of five fingers is allowed to vary from the original sequence by one place for any of the
fingers. Thus, for example, if TIMRL is the original sequence, then ITRML is also allowed, but LIMRT is not.
How many different sequences are allowed for any given person’s original scan?
31. CAT 2017 LRDI Slot 2 | LR - Puzzles
The lab has now decided to require six scans in the pass key sequence, where exactly one finger is scanned twice, and the other fingers are scanned exactly once, which can be done in any order. For
example, a possible sequence is TIMTRL.
Suppose the lab allows a variation of the original sequence (of six inputs) where at most two scans (out of six) are out of place, as long as the finger originally scanned twice is scanned twice and
other fingers are scanned once.
How many different sequences if scans are allowed for any given person’s original scan?
7 8 9 4 5 6 1 2 3 0 . -
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32. CAT 2017 LRDI Slot 2 | LR - Puzzles
The lab has now decided to require six scans in the pass key sequence, where exactly one finger is scanned twice, and the other fingers are scanned exactly once, which can be done in any order. For
example, a possible sequence is TIMTRL.
Suppose the lab allows a variation of the original sequence (of six inputs) so that input in the form of scanned sequence of six fingers is allowed to vary from the original sequence by one place for
any of the fingers, as long as the finger originally scanned twice is scanned twice and other fingers are scanned once.
How many different sequences of scans are allowed if the original scan sequence is LRLTIM?
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49284 square root
444*111=4*111*111=2*2*111*111. so sqrt 444*111=2*111=222. Example 3: Find square root of 5 using long division method. Subtract 4 from 5, you will get the answer 1. Favourite answer. The square root
of 49 is 7, as 7 x 7 = 49. If a point C lies between two points A and B such that AC = BC, then prove thAB. Number 49284 is a square number with n=222. Here is a list of the first 1000 perfect
squares: Octal numeral is 140204. /4 1 decade ago. 1234, Following are the data about the TV sets sold in Delhi and Haryana. Take two 0 along with 1 and take the decimal point after 1 in the
quotient. As you can see the radicals are not in their simplest form. sqrt(444*111)=sqrt(2*2*111*111)=2*111=222. Brands of Sets Samsung Panasonic L.G. Duodecimal value is 24630. This site is using
cookies under cookie policy. Number 49284 is an abundant number and therefore is not a perfect number. The square root of 25 is 5, as 5 x 5 = 25. Let's check this width √64*77=√4928. Source(s): self.
Relevance. b) 4 right angle.. Rajesh Rakshit D. 1 decade ago. Present the data by a Pie Diagram. When people say “square root,” they usually refer to the positive square root. The answer is given in
the attachment below. Hence, 2 2 = 4 and 4<5; Divide 5 by such that when 2 multiplied by 2 gives 4. of a straight angle. Square root of the number 49284 is 222. Simplified Square Root for √4928 is
8√77; Step by step simplification process to get square roots radical form: First we will find all factors under the square root: 4928 has the square factor of 64. What will we the under root of
49284 Ask for details ; Follow Report by Rahul8150 25.07.2018 Log in to add a comment my dear bestie,agar kbhi meri baat ka bura lage toh..kya ukhaad loge?rahoge toh mere kutte hi..xD, bad girls
join id. Answer Save. …, sold in Haryana 360 500 480 120Units sold in Delhi 625 360 240 300, tum gadhe ho ...mental..crazy ..rascal idiot...get lost from here..#dare completed mannat sisand guru
hmari ek gf h twinkle and hmne uske aur apne dad Lv 4. You can specify conditions of storing and accessing cookies in your browser, The square root of forty-nine thousand, two hundred and eighty-four
√49284 = 222, o sala tu ha kon chal htt mara mu na lag, my dear bestie,agar kbhi meri baat ka bura lage toh..kya ukhaad loge?rahoge toh mere kutte hi..xD, bad girls join id. 0 0. 0 1. …, se baat
kar li h hm dono toh shaadi karke hi rhenge aakhir pta toh chle ki gf banane mein mza kya aata h and sachm mein mazak nhi kar rhe h, random questions on brainly -hi baby my reaction holy f**k, How
many degrees are there in:a) 3 B.S.KANAUJIA. The opposite of a square root is a squared (power of 2) calculation. desaikhushi2001 is waiting for your help. 0 0. 1 decade ago. 444 * 111. If the square
root of a number is an integer, then the number is a perfect square. The square root can be positive or negative (-3 x -3 equals 9, -5 x -5 = 25, and -7 x -7 = 49). Square root = 16.8523 (Results
rounded to nearest 0.0001) What is the square root of 284? Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step
explanations, just like a math tutor. Find the value of the square root.? SonyUnits 7 Answers. Below are the steps explained to find √5: Write number 5 as 5.00000000; Take the number whose square is
less than 5. Present the data by a Pie Diagram. … Just type in a number in the box, and the result will be calculated automatically. Hexadecimal representation is c084. sqrt (444*111) =sqrt
(4*111*111) = 2*111 = 222 Ans. Binary numeral for number 49284 is 1100000010000100. …, se baat kar li h hm dono toh shaadi karke hi rhenge aakhir pta toh chle ki gf banane mein mza kya aata h and
sachm mein mazak nhi kar rhe h, random questions on brainly -hi baby my reaction holy f**k. …, sold in Haryana 360 500 480 120Units sold in Delhi 625 360 240 300, tum gadhe ho ...mental..crazy
..rascal idiot...get lost from here..#dare completed mannat sisand guru hmari ek gf h twinkle and hmne uske aur apne dad This site is using cookies under cookie policy. Brands of Sets Samsung
Panasonic L.G. Square of the number 49284 is 2428912656. Square root of 49284 and with method - 100022 4. SonyUnits 1234, Following are the data about the TV sets sold in Delhi and Haryana. Now
extract and take out the square root √64 * √77. 4070383837 password. Science Guy. Add your answer and earn points. 4070383837 password. You can specify conditions of storing and accessing cookies in
your browser. | {"url":"http://wesolutions.co.uk/ihygx/673b1e-49284-square-root","timestamp":"2024-11-04T07:44:01Z","content_type":"text/html","content_length":"14999","record_id":"<urn:uuid:3fcde9b1-397f-4a61-930d-6a34ddb7c0e8>","cc-path":"CC-MAIN-2024-46/segments/1730477027819.53/warc/CC-MAIN-20241104065437-20241104095437-00895.warc.gz"} |
MathFiction: Life and Fate (Vasily Grossman)
A Russian nuclear physicist flirts with the wife of his mathematician colleague and makes an important mathematical discovery, all during the Nazi invasion of the Soviet Union.
I had not heard of this important work of Soviet fiction, a criticism of the Stalinist era, until it was brought to my attention this evening at the Notre Dame math department's "fluid dynamics
seminar". Several mathematicians were challenged by my presence to attempt to name a work of mathematical fiction not already on my list. Nearly all of the suggestions were ones I had already added
(or already considered and rejected). Then, François Ledrappier proposed Life and Fate, specifically emphasizing the amazing mathematical discovery that the protagonist makes.
Of course, I have not yet had time to read the novel myself, but the little I can find out from its Wikipedia entry and from browsing it on Amazon.com confirm that it certainly should be included in
my database of mathematical fiction.
Here is a brief passage from the portion of the novel where he makes his great discovery:
(quoted from Life and Fate)
His head had been full of mathematical relationships, differential equations, the laws of higher algebra, number and probability theory. These mathematical relationships had an existence of their own
in some void quite outside the world of atomic nuclei, stars, and electromagnetic or gravitational fields, outside space and time, outside the history of man and the geological history of the earth.
And yet these relationships existed inside his own head.
And at the same time his head had been full of other laws and relationships: quantum interactions, fields of force, the constants that determined the process undergone by nuclei, the movement of
light, and the expansion and contraction of space and time. To a theoretical physicist, the processes of the real world were only a reflection of laws that had been born in the desert of mathematics.
It was not mathematics that reflected the world; the world itself was a projections of differential equations, a reflection of mathematics. | {"url":"https://kasmana.people.charleston.edu/MATHFICT/mfview.php?callnumber=mf1125","timestamp":"2024-11-04T10:47:14Z","content_type":"text/html","content_length":"10219","record_id":"<urn:uuid:1df5b802-184b-4d3d-9a62-c8fb091fb9cc>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00367.warc.gz"} |
Teacher KS1 Playing with Dice Collection
Bernard's article reminds us of the richness of using dice for number, shape and probability.
This article outlines some of the benefits of using dice games in the classroom, especially as a tool for formative assessment.
Simple dice and spinners tool for experiments.
In this game, you throw a dice and move counters along the snail's body and in a spiral around the snail's shell. It is about understanding tens and ones.
Find all the numbers that can be made by adding the dots on two dice.
Can you use the numbers on the dice to reach your end of the number line before your partner beats you?
An old game with lots of arithmetic!
Dotty Six is a simple dice game that you can adapt in many ways.
You'll need two dice to play this game against a partner. Will Incey Wincey make it to the top of the drain pipe or the bottom of the drain pipe first? | {"url":"https://nrich.maths.org/teacher-ks1-playing-dice-collection","timestamp":"2024-11-11T23:35:35Z","content_type":"text/html","content_length":"48048","record_id":"<urn:uuid:e9bbdf28-f278-4bc8-9411-c7eb114f0fe5>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00824.warc.gz"} |
Discriminative Methods - Related Works - Real-Time Generative Hand Modeling and Tracking
1.4 Related Works
1.4.1 Discriminative Methods
[Keskin et al., 2012]estimate hand pose by predicting the hand part labels probabilities for each pixel. The labels prediction is done using an RDF. The centers of the hand parts are inferred by
representing each label with a gaussian and finding the maximum on the resulting surface. This is under the assumption that the pixel with maximum probability value for the given hand part is situated
in the center of that hand part. The hand skeleton is obtained by connecting the joints according to their configuration in the hand. To improve performance, the training set is split in clusters of
similar hand poses. The results from different clusters are aggregated by an expert network.
[Tang et al., 2014]present a method similar to the one introduced by [Keskin et al., 2012].
Differently from the former, instead of using an RDF for predicting hand parts, they adopt Latent Regression Forest (LRF). In LRF the non-leaf nodes correspond to groupings of hand parts. The method
performs structured coarse-to-fine search, starting with the entire hand and recursively splitting it, until locating all the skeletal joints. This work has superior perfor-mance with respect to
[Keskin et al., 2012], where one of the reasons is greater robustness to occlusions.
[Tompson et al., 2014]pioneered using CNNs for discriminative hand tracking. Their work (and numerous subsequent methods) are enabled by the automatically labeled dataset that they have constructed.
The authors trained a CNN to generate a set of heat-map images for key hand features, taking multi-resolution depth images as an input. At each resolution the network contains two convolution layers;
each convolution is followed by RELU and max pooling. The concatenated outputs of convolution layers are fed to two fully connected layers.
The final kinematically valid hand pose is obtained by applying an inverse kinematic model on the heat-maps.
[Sun et al., 2015]use cascaded regression for predicting hand pose. In the cascaded regression framework, the pose is estimated iteratively by a sequence of regressors. Each regressor uses the output
of the previous one, progressively decreasing the error. The regressors are learned with RDF. The authors modify offset features, widely used for RDF, to make them invariant to 3D transformations.
They also propose a hierarchical approach to regress hand pose. Firstly, the palm transformation is regressed. The inverse of this transformation is afterwards applied to the fingers before estimating
their poses. This approach is shown to perform better than estimating the pose holistically, as it reduces appearance variations for the fingers.
1.4. Related Works
[Tang et al., 2015]propose to estimate hand pose hierarchically starting with the parameters at the base of hand kinematic chain and inferring the parameters at each next layer condi-tioned on the
previous layer (layer 1 – wrist translation, layer 2 – wrist rotation, and so on along the kinematic chain). For efficiency they formulate a cost function in terms of joint positions only.
Advantageously, evaluation of this cost function does not require rendering the model or computing closest point correspondences. Moreover, this cost function can also be evaluated for partial poses.
The proposed hierarchical optimization framework generates several samples of the partial pose at each layer, the sample with the minimal value of cost function is then selected. To generate the
samples, the authors train an RDF for predicting partial poses. They use standard features for RDF on depth images. The system generates multiple hypotheses using the described approach, the final
pose is selected by evaluating the
“golden energy” suggested by [Sharp et al., 2015]. This approach outperforms the other works that use hierarchical hand pose estimation algorithms, such as [Tang et al., 2014] and [Sun et al., 2015].
[Li et al., 2015]extend the work of [Keskin et al., 2012] and [Tang et al., 2015] by proposing another variant of RDF. Similarly to [Tang et al., 2014], the method performs structured coarse-to-fine
search, starting with entire hand and splitting it recursively to joints. Differently from [Tang et al., 2014] the division hierarchy of hand parts may not be the same for different poses. The work
achieves superior performance on the ICVL dataset ( [Tang et al., 2014]).
[Oberweger et al., 2015a]compare several CNN architectures and find that the best perfor-mance is given by a deeper architecture that takes depth images at several scales as an input.
The rationale is that using multiple scales helps capturing contextual information. The authors also propose to regress hand pose parameters in a lower-dimensional subspace. After the initial
estimation phase follows a refinement step. To enhance the location estimate provided by the first stage, they use a different network for each joint. The per-joint networks look at several patches of
different sizes centered on the predicted joint location. The refinement step is repeated several times, each iteration is centered on a newly predicted location.
[Ge et al., 2016]propose to project the input depth image onto orthogonal planes and use the resulting views to predict 2D heat-maps of joint locations on each plane. These 2D heat-maps are then
fused to produce the final 3D hand pose. The fusion step is expected to correct the imprecisions using the predictions from complementary viewpoints. The authors use a multi-resolution CNN on each
view with architecture similar to the one introduced by [Tompson et al., 2014]. Given the 2D heat maps from the three views, they find the hand pose parameters in a lower dimensional PCA subspace,
such that the total heat map confidence at the joint locations on the three views is maximized.
[Sinha et al., 2016]exploit activation features from a hidden layer of a trained CNN. The assumption is that augmenting an output activation feature by a pool of its nearest neighbors brings more
reliable information about the hand pose. Drawing on the fact that CNNs are less robust for regression than for classification, the authors compute the activation features
Chapter 1. Introduction
from classifying joint angles into bins with a CNN (as opposed to regressing the exact values of the joint angles). Since the number of quantized hand poses is very large, they propose a two-stage
classification. On the first stage global hand rotation is classified. Next, for each rotation bin, five separate CNNs are trained to classify the poses of the fingers. At run time, given the activation
features, a pool of their nearest neighbors is efficiently retrieved from a database. The final hand pose is computed from the assumption that a matrix of stacked neighboring activation features
concatenated with stacked corresponding hand poses has a low rank. The unknown current hand pose is computed by matrix completion^12.
[Zhou et al., 2016]integrate domain knowledge about hand motion into a CNN. This is done by adding a non-parametric layer that encodes a forward kinematic mapping from joint angles to joint
locations. Since the forward kinematic function is differentiable, it can be used in a neural network for gradient-descent like optimization. This approach guarantees that the predicted hand pose is
valid. The remaining network architecture is similar to the one introduced by [Oberweger et al., 2015a].
[Guo et al., 2017]propose to use a hierarchically-structured Region Ensemble Network (REN) for hand pose inference. This architecture is inspired by the widely used approach of averaging predictions
from different crops of an original image. The averaging is beneficial since it decreases the variance of image classification; however, it is computationally expensive. The authors propose a solution
that retains the advantages while cutting the costs. They suggest to split the input image in several regions, predict the whole hand pose separately from each region and aggregate regional results
afterwards. The REN architecture starts with six convolutional layers augmented with two residual connections. The region-wise prediction is implemented through dividing the output of the
convolutional layers into a uniform grid.
Each grid cell is fed into fully connected layers. Subsequently the outputs of all the cells are concatenated together and used to predict the final hand pose. This approach has state-of-the-art
performance on the NYU and ICVL datasets.
[Madadi et al., 2017]propose a hierarchical tree-like CNN that mimics the kinematic structure of human hand. The branches of the network are trained to become specialized in predicting the locations
of subsets of hand joints (local pose), while the parameters closer to the tree root are shared for all hand parts. The network contains a loss term for each local pose.
Additionally, the outputs of the tree branches are concatenated and fed to the fully-connected layer for estimating the final pose. The authors argue the later step allows to learn higher order
dependencies among joints. The loss function also contains the terms that penalize predicting joint locations outside of data hull and encourage all joints from one finger to be co-planar.
[Mueller et al., 2017]present a method for predicting hand pose in egocentric view. Their system is designed for hand-object interaction scenarios and is robust to occlusions. They
12Matrix completion is the task of filling in the missing entries of a partially observed matrix. One of the variants of the matrix completion problem is to find the lowest rank matrixXwhich matches
the matrixM, which we wish to recover, for all entries in the setEof observed entries. "Matrix completion." Wikipedia: The Free Encyclopedia.
Wikimedia Foundation,https://en.wikipedia.org/wiki/Matrix_completion, [accessed 30 January 2018].
1.4. Related Works
estimate hand pose in several steps. Firstly, to localize the hand, a heat map of the hand root position is regressed. Given the hand root, the input image is normalized and feed into a joint
regression network. This network outputs 2D heat maps and 3D positions of the joints. As the last step, a kinematically valid hand pose is computed by optimizing a sum-of-energies cost function. The
cost function includes the closeness of optimized joint locations to the CNN-predicted joint locations, joint limits and temporal smoothness term. Both networks are trained on synthetic data
generated by accurately tracked hand motion with existing tracker and retargeting it to a virtual hand model.
[Oberweger and Lepetit, 2017]extend their previous work [Oberweger et al., 2015a]. They carry out an extensive evaluation to show that the improved method achieves superior or comparable performance
to all recent works on three main benchmarks of hand tracking (NUY, ICVL and MSRA). The authors introduce the following improvements: firstly, the training data is augmented to 10M samples (by
translating, rotating and scaling). The second enhancement is training a CNN that regresses hand root for accurate hand localization. Finally, the new pose network architecture is similar to ResNet:
a convolution layer is followed by four residual modules, that are in turn followed by several fully connected layers with dropout. | {"url":"https://9pdf.net/article/discriminative-methods-related-works-real-generative-modeling-tracking.z3dovl69","timestamp":"2024-11-06T05:11:00Z","content_type":"text/html","content_length":"71310","record_id":"<urn:uuid:7ecb6b24-6448-4179-8aa2-b413936c1225>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00244.warc.gz"} |
INET 2015
Daniel McDonald
Department of Statistics
Indiana University, Bloomington
Course description
Largely unnoticed by economists, over the last three decades statisticians and computer scientists have developed sophisticated prediction procedures and methods of model selection and forecast
evaluation under the rubric of statistical learning theory. These methods have revolutionized pattern recognition and artificial intelligence, and the modern industry of data mining (without the
pejorative connotation) would not exist without it.
In this short course, we will investigate the connections between modern statistical methodology and standard techniques which are more common in economics and social science. The goal is to develop
an understanding for the circumstances under which different statistical methods are appropriate, how methods behave when the model for the data generating process is incorrect, and why some methods
can adapt while others cannot. We will focus especially on selecting models with good predictive performance and on the relationships between model complexity, statistical estimation, and the amount
of data used to estimate the model.
The starting point for statistical learning is the notion of predictive risk and the tradeoff between bias and variance. With this tradeoff in mind, we will investigate the importance of assessing
models using training and test data, the benefits of regularization, and the necessity of selecting tuning parameters carefully. We will illustrate each of these issues with some standard econometric
procedures applied to financial and economic datasets while at the same time, introducing some potentially new procedures which may be useful in your own research.
Schedule of topics
1. The predictive viewpoint
2. The bias-variance tradeoff
3. Evaluating predictions and estimators
4. The benefits of regularization
5. Model selection
6. Choosing tuning parameters
7. Application: BVARs and DSGEs
8. Tools for classification
9. Application: Predicting recessions
10. Collaborative filtering and the Netflix prize
11. Dimension reduction
Suggested preparation
In order to be best prepared for this course, we suggest that you brush up on your econometrics, especially the content about probability. Most useful will be to remind your self about expected
values and variance, convergence in probability and consistency, maximum likelihood, ordinary least squares and some time series topics like autoregressive models. Also, remind yourself what a
dynamic stochastic general equilibrium model is (you probably know more about this than we do, but perhaps not). Less useful are GMM techniques and cointegration.
Before coming to class, we suggest that you take a look at the first three (3) chapters of Cosma Shalizi’s preprint Advanced Data Analysis from an Elementary Point of View. These chapters will serve
as a nice introduction to the materials we intend to cover. The rest of the book is great too if you are feeling more ambitious/motivated.
Preliminary exercises
We will make a handful of exercises available to try before you come to class. These are mainly for review and little else. They will certainly not be graded, but spending a few hours trying them may
help you come to class more prepared. The exercises are here.
Data analysis
All of the data analysis examples we use in class will be done in the open source programming language R. R is extremely powerful and easily extensible. We will try to make all of our code available
on the website so that you may experiment with it. The software is available for free on CRAN. An official introduction is available there, and many other introductions are available on the web. See
for example here. Feel free to download and play around with R, but experience and/or mastery of computer programming is not necessary. | {"url":"https://dajmcdon.github.io/teaching/inet2015.html","timestamp":"2024-11-11T03:41:01Z","content_type":"application/xhtml+xml","content_length":"28006","record_id":"<urn:uuid:c1b4961d-d7fa-4248-991d-c9178c5d6a9e>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00845.warc.gz"} |
vexorian's blog
As I write this, there are 7 minutes left before the end of the coding phase and I have supposedly finished coding 250 and 500. Supposedly, because I have so many doubts... It is scary.
Div1 250: Expect hypotenuse!
This problem is cool. I hope it is original though it would be surprising to see nobody thought of this before. Even though it is so cool.
You are given w,x and y. At position 0, a line of integer height picked uniformly at random between 1 and x. At position w, another line this time picked between 1 and y. Return the expected length
of a straight line connecting the top of each line. PS x,y < 1000000.
We cannot do a O(x*y) search trying each case. The second best choice would be a O(x) search - fix a height px for the first line and then somehow calculate the rest.
The somehow can be done by using an array acumy[i] that returns the sum of the hypotenuses of a 90 degrees triangle with base w and heights < i. This array can be accumulated in O(y) time.
With that array, you can calculate the sum between all the straight lines that could start from the first line and end in the second line. You just need to be clever about how to use it. You can
later divide the sum by y and add it to a result. Then divide the total result by x.
Tons of doubts about this problem. The approach is correct, but I wonder if I have made an error somewhere. To calm myself, I coded a bruteforce solution and began running random test cases comparing
the results. So far, so good.
When I submitted this I was shocked to find that almost everyone else in the room already submitted it. And a red coder already solved 500...
double brute(int w, int x, int y) {
double res = 0;
for (int px = 1; px <= x; px++) {
for (int py=1; py<=y; py++) {
double dx=w;
double dy = px - py;
res += sqrt(dx*dx + dy*dy);
res /= x;
res /= y;
return res;
double getExpectedLength(int w, int x, int y)
if (x <= 1000000/y) {
cout << brute(w,x,y) << endl;
acumy[0] = 0;
for (int i=0; i<300000; i++) {
double dx = w;
double dy = i;
double hyp = sqrt(dx*dx + dy*dy);
acumy[i+1] = hyp + acumy[i];
double res = 0;
for (int px = 1; px <= x; px++) {
double s = 0;
if (y >= px) {
// all good
s = acumy[y - px + 1];
if (px > 1) {
s += acumy[ px ] - acumy[1];
} else {
int lo = px - y;
int hi = px - 1;
s = acumy[hi+1] - acumy[lo];
res += s / y;
res /= x;
return res;
Div1 500: The one with the big table
You are given a table with some height and some width. The numbers on the table are such that the cell at (i,j) has value (i*width + j). Find a subrectangle of that table such that the total sum of
the rectangle is equal to S and minimize the area of the rectangle. If no rectangles exist, return -1
The trick to this problem is to notice that, given a width w for the subrectangle, you already know something about it.
It is hard to explain, but given a rectangle of width w and height h, then it will be something like this:
0 1 2 ... w
0 1 2 ... w
0 1 2 ... w
Added to this:
0 0 0 ... 0
W W W ... W
2*W 2*W 2*W ... 2*W
(Where W is the total width) And this:
K K K ... K
K K K ... K
K K K ... K
Where K is the value of the cell at the top-left corner of the subrectangle.
Long story short, with that knowledge, you can, given w and h, calculate K. And given K you can verify that such a rectangle exists. The constraints seem too long for this approach, except that the
formula for K actually allows you to break very early. So the approach seems to be something like O(width * sqrt(height)) if not better.
I am nervous because this solution is VERY straightforward, plus the contest writers decided to include the supposed worst case in the examples. (That is never good, it might mean there is another
case that is slower). I am also concerned about overflow. My solution overflows in theory, but I cannot catch a case in which that matters.
long minimalArea(int height, int width, long S)
pair<long, long> res = make_pair(1LL << 60, -1);
for (long w=1; w<=width; w++) {
long x = w;
x = (x * (x-1)) / 2; //1e12
for (long h=1; h<=height; h++) {
long y = h;
y = (y * (y-1)) / 2; //1e12
y = (y * width) * w;
assert(x*h >= 0);
assert(y >= 0);
long ss = S - x*h - y;
if (ss < 0) break;
if (ss % (w * h) == 0) {
long k = ss / (w * h);
int i = k % width;
int j = k / width;
if (i + w <= width && j + h <= height) {
res = std::min(res, make_pair(w*h, w*h) );
return res.second;
Div1 1000, something evil with polygons
Opened it. No idea what to do. Somehow I hope I am not writing the editorial because explaining this might be too hard.
Challenge phase
Challenge phase begins. Let us see what happens.
22:33 The approaches in 500 look drastically different. This is not good.
22:35 The approaches in 250 are also different to mine. errr.
22:38 Tons of challenges in 250.
And the challenge phase ends, let's see what happens...
I figured I should post something about this SRM. I've been very busy these weeks because the semester is ending and I tried to win a t-shirt in the Marathon AND I am not assigned more editorials.
But there we go.
Div1 250 - The one with the bit operations
I was just returning to the topcoder mindset after working a lot on an assignment. At the start of the match I felt like running in 10% of steam capacity. I truly felt dumb. I knew what to do to
solve the problem, but I was having issues translating it to code.
Kleofas tail : If x is even, x = x / 2. If x is odd then x = x - 1. Repeat and repeat until you will eventually have a sequence of tons of 0s. Given K, A and B, return the number of numbers between A
and B, inclusive such that the Kleofas pairs starting from number x eventually include K.
This is heavily related to the binary representation of a number. Let us say the binary representation of x is 10010010101. The sequence goes like this: 10010010101, 10010010100, 1001001010,
100100101, 100100100, 10010010, 1001001, 1001000, 100100, 10010, 1001, 100, 10, 1, 0, 0, 0, ... - If the right most bit is 1, it becomes 0 , else it gets removed. This means that there are two ways
for a number x to eventually become K:
• The binary representation of K is a prefix of the binary representation of x.
• If K is even, the binary representation of (K+1) is a prefix of the binary representation of x.
Assuming that you have a function that counts the number of times a given number K' appears as a prefix in numbers between A and B, then you have an instant solution. But we can simplify it a bit
more: Let f(A, K) return the number of elements X between 0 and A, inclusive such that K appears in their Kleofas tail:
• f( A , K) = 0 for (A < K).
• f( A, K = 0) = A+1 (All numbers between 0 and A, eventually reach 0.
• f(A, 2*K1) = f(A, 2*K1+1) + (number of times 2*K1 appears as a prefix). (In other words, K = 2*K1, which means K is even).
• f(A, 2*K2 + 1) = (number of times 2*K2+1 appears as a prefix). (In other words, K = 2*K2 + 1, which means K is odd).
We just need to calculate prefixes(A, K): The number of times K is a prefix of X for every X between 0 and A, inclusive.
The problem is how to do that... This is the moment at which my brain froze. At first I tried to do it arithmetically, but I could not. (It is possible, but slightly tricky). Eventually, I recognized
that I was not at full brain capacity and that the best I could do would be a bullet (and idiot)-proof dynamic programming solution for the simple counting problem.
Now that I am in my senses, here is a quick approach. Imagine there are i bits after the K part in x. For example, if K=101 then x = 101?????. The minimum x that follows that condition is: 10100000
and the maximum x is : 10111111. Both values can be calculate easily given i. (The minimum is K multiplied by 2^i, the maximum is the same plus (2^i-1) ). Then the true maximum is min( K*2^i + 2^i-1,
A). There are (true_maximum - minimum + 1) values of x. Then you can just iterate through all values of i until the minimum is larger than A.
// Number of times K appears as a binary prefix of numbers between
// 0 and A, inclusive.
long prefixes(long K, long A)
long res = 0;
long mx = 0;
while (true) {
long lo = K;
long hi = Math.min(K + mx, A);
res += hi - lo + 1;
if (K > A/2) {
K *= 2;
mx = mx * 2 + 1;
return res;
// Amount of times a number between 0 and A, inclusive contains
// K in the Kleofas tail :
long countGoodSequences(long K, long A)
if (A < K) {
return 0;
if (K==0) {
return A+1;
long res = 0;
if (K % 2 == 0) {
res += prefixes(K+1, A);
res += prefixes(K, A);
return res;
public long countGoodSequences(long K, long A, long B)
// This allows us to reduce the number of arguments.
return countGoodSequences(K, B) - countGoodSequences(K, A-1);
Mixed feelings about this problem. Although it was good, we have had too many of these (tricky binary operations) division 1 250s.
Div1 500: the one with digits
I was better now. I knew that if I wanted to save my rating I had to solve this problem. Luckily it turned out to be a typical digit dynamic programming. I am not sure why, but I find these problems
very easy. I still needed a lot of time though, because when I coded it my brain was still not functioning and I left as many bugs as you would ever find.
Given N (Between 1 and 99...9 (15 times), inclusive) , digit1, count1 , digit2, count2. (count1 + count2 <= 15). Return the minimum number >= N that has digit1 at least count1 times and digit2 at
least count2 times.
Thanks to the notes, we know that the result always fits a signed 64 bits integer. This means 18 digits. We will say that the maximum number has 20 digits at most. The idea is to use a recurrence
relation. f(count1, count2, p, eq, zero): We have already decided the p first digits. eq means that the number is currently equal to N. zero means that the number is currently equal to 0. count1 is
the minimum number of times we need to add digit1, and count2 works the same way. For convenience, we will say that numbers are strings, strings of digits. Once we get the final result we can convert
it to long long. f() will actually find a string of MAX digits, it will have leading zeros when the result needs less than MAX digits.
Let us say p==MAX, this means that we have already decided all digits, what is the minimum remaining number? Well, if count1 or count2 is positive, then there is no way to fulfill this requirement
anymore. Thus there is no result. We will mark instances with no result as infinite. If count1 and count2 are 0, then the result is the empty string - do not modify any new bit.
For another p, then we can try all digits from 0 to 9. Well, actually, we sometimes cannot. If eq==1, it means that all of the previous digits were equal to N, this means that the digit at first
position cant be smaller than the digit at the same position in N - Because that would make our result smaller than N. If eq==0, then we can use any digit from 0 to 9. After picking a digit, the
values of count1, count2, eq and zero will change according to it and we will have a new instance of the recurrence. (Just note that when zero==1, we cannot count digit 0 as part of the digits to
reduce from the count1 requirement, because it is a leading 0 and will not actually appear in the number. The same happens with digit2)
And that's it:)
const int MAX = 20;
struct FavouriteDigits
string dp[MAX+1][MAX+1][MAX+1][2][2];
bool vis[MAX+1][MAX+1][MAX+1][2][2];
string R;
string rec(int p, char digit1, int count1, char digit2, int count2, int eq, int zero)
if (! vis[p][count1][count2][eq][zero] ) {
vis[p][count1][count2][eq][zero] = true;
string & res = dp[p][count1][count2][eq][zero] ;
// We mark infinite as a string that begins with :
if (p == MAX) {
// base case
if ( count1 == 0 && count2 == 0) {
res = "";
} else {
res = ":"; // no luck
} else {
res = string(MAX - p, ':');
for (char ch = (eq?R[p]:'0') ; ch <= '9'; ch++) {
int ncount1 = count1, ncount2 = count2;
int nzero = zero && (ch == '0'); //once we use a digit different to 0
// the number stops being equal to 0.
// update count1, note that if digit1 is 0 and zero==1,
// it does not count (leading zero)
if (ch == digit1 && (ch!='0' || !zero) ) {
ncount1 = std::max(ncount1-1, 0);
// update count2, similar story
if ( (ch == digit2) && (ch!='0' || !zero) ) {
ncount2 = std::max(ncount2-1, 0);
int neq = ( eq && (ch==R[p]) ); //once a character differs from N
//the new number is larger.
string x = rec(p+1, digit1, ncount1, digit2, ncount2, neq, nzero);
// x begins with :, there is no result in that direction...
if (x.length() > 0 && x[0] == ':') {
//concatenate, we now have a good result remember the minimum.
res = std::min(res, string(1,ch)+x);
return dp[p][count1][count2][eq][zero];
long findNext(long N, int digit1, int count1, int digit2, int count2)
// convert N to a string, with the appropriate leading zeros.
{ ostringstream st; st << N; this->N = st.str(); };
this->N = string(MAX - R.length(), '0') + this->N;
memset(vis, 0 ,sizeof(vis));
string s = rec(0, digit1+'0', count1, digit2+'0', count2, 1, 1);
// convert s to from string to long:
istringstream st(s);
long x;
st >> x;
return x;
#undef long
This problem was better. But it is odd to have two problems in the same match and same division that can be solved with the same approach. (I used a similar digit dp to solve 250).
Challenge phase, etc.
The example cases for div1 500 seemed quite strong. I knew div1 250 was tricky, but I also knew that I would likely mistaken if I tried to challenge. I preferred to go to lunch during the challenge
phase. Returned, and after all the hours it takes results to arrive lately, I noticed I passed everything and somehow even my slow submissions were enough for top 100. I cannot believe I managed to
increase my rating and even reach (again) the area that is close to 2200. I was just lucky though that div1 500 was my style of problem.
As far as the problem set goes. It is not misof's best problem set. But that is only because he set a great standard in previous problem sets. It was a good match, nevertheless. But we writers need
to avoid those bit operations in div1 250 for a while.
No surprises here, I did not advance. Even though I knew my chances and that at best I would have to battle for a good position for honor and thus. I kinda harbored the fantasy of advancing. Mind
you, I really think that with the right random shuffle and combination of problems I could end in top 25. But it was unlikely. Nevertheless, I am kind of proud of my 133-th place.
Problem A: The breather
This problem was one to reward intuition. I guess. It was much easier than anything else in the round.
Let us say you know the order you picked. Then, the expected time after you solve 0 levels is: f(0) = L0 + p0*f(0) + (1-p0)*f(1). Where f(1) is the expected time after solving 1 level. After some
magic: f(0) = L0/(1-p0) + f(1)
Now f(1) = L1 + p1*f(0) + (1-p1)*f(2). f(1) = L1 + p1*( L0/(1-p0) + f(1)) + (1-p1)*f(2).
f(1) = L1 + p1*L0/(1-p0) + p1*f(1) + (1-p1)*f(2)
(1 - p1) * f(1) = L1 + p1*L0/(1-p0) + (1-p1)*f(2)
f(1) = ( L1 + p1*L0/(1-p0) ) / (1 - p1) + f(2)
f(1) = L1 / (1-p1) + p1*L0/(1 - p0)
After some iterations, you will find a very messy formula. But it should become clear that it is best to pick the levels in non-decreasing order of L/p. It is actually something that can happen by
intuition. At least for A-small, what I did was to think that it is best to get rid of the harder levels first, so that the probability to repeat a lot of levels goes smaller with time. Then when
factoring the variable Li it is also guessable that a division can occur. But after getting correct in A-small I did not want to risk it (I didn't have a proof). So I skipped to B.
If two L/p values are equal, pick the one with the smallest index, to make sure the lex first order is picked.
Problem B: The implementation hell
This problem was a sand trap. I knew it was a sand trap. But I sort of thought that I could do it decently fast, and in fact I had hops to solve the large version.
I think most of the top 500 would not get scared of the hexagonal grid. Yet in this case, the grid was the least of the implementation issues. The three cases each have their complications.
You need to turn the hexagonal grid into a 2d grid. It is possible with T=2*S-1. Then you can have a TxT grid. Some of the elements in the TxT grid are invalid (don't exist in the hexagonal grid).,
you have to mark them as such.
Then you need to analyze what rules involve connecting two hex cells. (x,y) is adjacent to (x-1,y-1), (x-1,y) , (x,y-1), (x,y+1), (x+1,y) and (x+1,y+1).
We need to differentiate between normal cells, edges and corners. In fact, each edge needs a different id for the cells in that edge (Forks are a pain to detect, see below). More implementation
Rings: The statement help us here. We need to detect an empty cell that is not reachable to the boundary with any path of empty cells. Thus we can do a dfs starting from each of the boundary cells
(borders or corners). Then if we try each empty cell, if it was not reached by the DFS, we have found a ring. A BFS also works, I used a BFS for no reason.
Bridges: Another use for a DFS, count the number of connected components of (non-empty) cells. If the number of components is less than 6, we got a bridge.
Forks: These were a pain to debug. Once I understood what was needed to detect forks, I was no longer hopeful that I could solve B-large. From each non-empty cell in an edge, do a DFS of non-empty
cells. Then verify if there are two or more cells from different borders that have been visited. If there are two, we got a bridge. If you only find one of such cells, make sure to mark it out so
that you don't count it as visited by a dfs from another edge...
Once I debugged my B-small, I already spent over an hour in it (fork mistakes). And it was still wrong (WA). I spent another good half an hour finding the bug (the forks again) and got a correct
answer. But there were less than 30 minutes left...
The rest
I was shocked. Although I spent ages solving B-small, I somehow was still in a high position (150-th-ish). It seems that a lot of people had issues. So, I tried more problems. I could not understand
what D asks for, and I did not understand what the difference between c-small and C-large was. Then I decided to take back A-large. This time, I decided to prove my intuition - it was correct. So I
submitted it. It took me a while to code it because I did not have precision errors (We are sorting by (L/p, id), if we used floating points to calculate L/p, there is a risk that two different
expressions L/p that give the same result are not considered the same in floating point arithmetic). So I decided to implement it only in integers (It is easy, just compare p2*L1 and p1*L2 instead of
L1/p1 and L2/p2). I wonder if this was necessary.
Then I spent my last minutes watching the score boards. Many interesting surprises there. rng_58 was not in the qualifying top 25. But then the match ended, and results were updated. rng_58 got 25-th
So I first read the title and thought "oh, not another 25K Marathon in which even fifth place barely makes 200 bucks. But it seems it is something different.
The NASA Tournament Lab is excited to announce to the TopCoder community a special weekend algorithm contest to be held Thursday, June 21 to Sunday, June 24, 2012 featuring a $25,000+ prize
purse. This unrated contest will challenge the TopCoder community members to generate solutions to an interesting algorithmic problem. We strongly encourage all members interested in SRMs as well
as Marathon Match events to participate. Google and NASA Jet Propulsion Laboratory are supporting this challenge and look forward to interacting with TopCoder community members about their
organizations through special chat rooms on Wednesday June 20, 2012. Engineers and scientists from Google and NASA JPL will be available to answer any questions you may have about these leading
organizations. Please save the dates! Registration begins on the June 13, 2012 at 13:00 UTC -4 and ends on the June 19, 2012 at 13:00 UTC -4. More details will be available soon as well as a
registration reminder.
• interested in "SRMs and Marathon Match events" - So I am not sure if it is a Marathon.
• Weekend contest - If it is a Marathon will be a very unusual one with 4 days long.
I am looking forward to more info. Even if it was a Marathon, 4 days sounds like the perfect length. I hope the 25K prize purse translates into a decent chance to translate effort into prize, though.
It is June. There are few hours till Saturday and it is going to be quite a decisive day for me.
Today was quite a rough start for June. I was writing the editorial for SRM 544 and was supposed to finish it by 9:35 AM. But I really could not and took 12 more hours. No, not because of the hard
problem. But because of division I 275. Let us say that I lost hour after hour of my life trying to prove the fact that 201 is the maximum answer.
My original plan for today was to make this post and then focus on a semester project. No luck there.
The IPSC at 6:00 AM. I will have to go to bed right now and prepare. This is a great tournament, and if you are not registered yet you really should get ready to participate. It is a team contest,
but I am allergic to team work so I'll be participating alone. If you want a team, try looking for someone in topcoder or codeforces, I am sure there are plenty of people that want teammates.
I really cannot miss IPSC. It is a very unique tournament and has perhaps the best format out of all the contests ever. There are some very unusual problems too.
TCO 2012 Round 2C
The last chance to advance/win a t-shirt. Just one hour after the IPSC ends. You would think that I would have avoided going to IPSC to maximize my energy in this round. No, really, I can't miss the
IPSC. My current hope is that going to the IPSC before this round will make me be in coding mode before the contest and will actually energize me. But I could be very wrong.
Will also try posting something for the TCO blog after this match ends. Although I'll admit that after US embassy rejected US visa request rejected in a very rude and frankly outstandingly wrong way
I am not the most motivated person to post in that blog.
Later in June, we got other matches like the third GCJ round. I also will try participating in the on going round 3 of the TCO marathon track. | {"url":"https://www.vexorian.com/2012/06/","timestamp":"2024-11-09T12:48:10Z","content_type":"text/html","content_length":"176447","record_id":"<urn:uuid:f14e4d51-f62f-47db-9ad0-829cf6f68907>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00544.warc.gz"} |
Femtometers to Chains
Femtometers to Chains Converter
β Switch toChains to Femtometers Converter
How to use this Femtometers to Chains Converter π €
Follow these steps to convert given length from the units of Femtometers to the units of Chains.
1. Enter the input Femtometers value in the text field.
2. The calculator converts the given Femtometers into Chains in realtime β using the conversion formula, and displays under the Chains label. You do not need to click any button. If the input
changes, Chains value is re-calculated, just like that.
3. You may copy the resulting Chains value using the Copy button.
4. To view a detailed step by step calculation of the conversion, click on the View Calculation button.
5. You can also reset the input by clicking on button present below the input field.
What is the Formula to convert Femtometers to Chains?
The formula to convert given length from Femtometers to Chains is:
Length[(Chains)] = Length[(Femtometers)] / 20116799991496228
Substitute the given value of length in femtometers, i.e., Length[(Femtometers)] in the above formula and simplify the right-hand side value. The resulting value is the length in chains, i.e., Length
Calculation will be done after you enter a valid input.
Consider that the radius of a proton is about 0.84 femtometers.
Convert this radius from femtometers to Chains.
The length in femtometers is:
Length[(Femtometers)] = 0.84
The formula to convert length from femtometers to chains is:
Length[(Chains)] = Length[(Femtometers)] / 20116799991496228
Substitute given weight Length[(Femtometers)] = 0.84 in the above formula.
Length[(Chains)] = 0.84 / 20116799991496228
Length[(Chains)] = 0
Final Answer:
Therefore, 0.84 fm is equal to 0 ch.
The length is 0 ch, in chains.
Consider that the size of a neutron is approximately 1.1 femtometers.
Convert this size from femtometers to Chains.
The length in femtometers is:
Length[(Femtometers)] = 1.1
The formula to convert length from femtometers to chains is:
Length[(Chains)] = Length[(Femtometers)] / 20116799991496228
Substitute given weight Length[(Femtometers)] = 1.1 in the above formula.
Length[(Chains)] = 1.1 / 20116799991496228
Length[(Chains)] = 1e-16
Final Answer:
Therefore, 1.1 fm is equal to 1e-16 ch.
The length is 1e-16 ch, in chains.
Femtometers to Chains Conversion Table
The following table gives some of the most used conversions from Femtometers to Chains.
Femtometers (fm) Chains (ch)
0 fm 0 ch
1 fm 0 ch
2 fm 0 ch
3 fm 0 ch
4 fm 0 ch
5 fm 0 ch
6 fm 0 ch
7 fm 0 ch
8 fm 0 ch
9 fm 0 ch
10 fm 0 ch
20 fm 0 ch
50 fm 0 ch
100 fm 0 ch
1000 fm 0 ch
10000 fm 0 ch
100000 fm 0 ch
A femtometer (fm) is a unit of length in the International System of Units (SI). One femtometer is equivalent to 0.000000000001 meters or 1 Γ 10^(-15) meters.
The femtometer is defined as one quadrillionth of a meter, making it a very small unit of measurement used for measuring atomic and subatomic distances.
Femtometers are commonly used in nuclear physics and particle physics to describe the sizes of atomic nuclei and the ranges of fundamental forces at the subatomic level.
A chain is a unit of length used primarily in land surveying and agriculture. One chain is equivalent to 66 feet or approximately 20.1168 meters.
The chain is defined as 66 feet, which is historically based on the length of a chain used in surveying practices and land measurement.
Chains are commonly used in land surveying for measuring distances, particularly in the United States and the United Kingdom. The unit is useful for tasks such as plotting and dividing land and has
historical significance in the development of surveying techniques.
Frequently Asked Questions (FAQs)
1. What is the formula for converting Femtometers to Chains in Length?
The formula to convert Femtometers to Chains in Length is:
Femtometers / 20116799991496228
2. Is this tool free or paid?
This Length conversion tool, which converts Femtometers to Chains, is completely free to use.
3. How do I convert Length from Femtometers to Chains?
To convert Length from Femtometers to Chains, you can use the following formula:
Femtometers / 20116799991496228
For example, if you have a value in Femtometers, you substitute that value in place of Femtometers in the above formula, and solve the mathematical expression to get the equivalent value in Chains. | {"url":"https://convertonline.org/unit/?convert=femtometers-chains","timestamp":"2024-11-05T00:38:44Z","content_type":"text/html","content_length":"89854","record_id":"<urn:uuid:8823724a-e7ed-49c2-9322-3fbd57695f91>","cc-path":"CC-MAIN-2024-46/segments/1730477027861.84/warc/CC-MAIN-20241104225856-20241105015856-00581.warc.gz"} |
RFC 8554: Leighton-Micali Hash-Based Signatures
Internet Research Task Force (IRTF) D. McGrew
Request for Comments: 8554 M. Curcio
Category: Informational S. Fluhrer
ISSN: 2070-1721 Cisco Systems
April 2019
Leighton-Micali Hash-Based Signatures
This note describes a digital-signature system based on cryptographic
hash functions, following the seminal work in this area of Lamport,
Diffie, Winternitz, and Merkle, as adapted by Leighton and Micali in
1995. It specifies a one-time signature scheme and a general
signature scheme. These systems provide asymmetric authentication
without using large integer mathematics and can achieve a high
security level. They are suitable for compact implementations, are
relatively simple to implement, and are naturally resistant to side-
channel attacks. Unlike many other signature systems, hash-based
signatures would still be secure even if it proves feasible for an
attacker to build a quantum computer.
This document is a product of the Crypto Forum Research Group (CFRG)
in the IRTF. This has been reviewed by many researchers, both in the
research group and outside of it. The Acknowledgements section lists
many of them.
Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This document is a product of the Internet Research Task Force
(IRTF). The IRTF publishes the results of Internet-related research
and development activities. These results might not be suitable for
deployment. This RFC represents the consensus of the Crypto Forum
Research Group of the Internet Research Task Force (IRTF). Documents
approved for publication by the IRSG are not candidates for any level
of Internet Standard; see Section 2 of RFC 7841.
Information about the current status of this document, any errata,
and how to provide feedback on it may be obtained at
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Copyright Notice
Copyright (c) 2019 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(https://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. CFRG Note on Post-Quantum Cryptography . . . . . . . . . 5
1.2. Intellectual Property . . . . . . . . . . . . . . . . . . 6
1.2.1. Disclaimer . . . . . . . . . . . . . . . . . . . . . 6
1.3. Conventions Used in This Document . . . . . . . . . . . . 6
2. Interface . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1. Data Types . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1. Operators . . . . . . . . . . . . . . . . . . . . . . 7
3.1.2. Functions . . . . . . . . . . . . . . . . . . . . . . 8
3.1.3. Strings of w-Bit Elements . . . . . . . . . . . . . . 8
3.2. Typecodes . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3. Notation and Formats . . . . . . . . . . . . . . . . . . 9
4. LM-OTS One-Time Signatures . . . . . . . . . . . . . . . . . 12
4.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . 13
4.2. Private Key . . . . . . . . . . . . . . . . . . . . . . . 14
4.3. Public Key . . . . . . . . . . . . . . . . . . . . . . . 15
4.4. Checksum . . . . . . . . . . . . . . . . . . . . . . . . 15
4.5. Signature Generation . . . . . . . . . . . . . . . . . . 16
4.6. Signature Verification . . . . . . . . . . . . . . . . . 17
5. Leighton-Micali Signatures . . . . . . . . . . . . . . . . . 19
5.1. Parameters . . . . . . . . . . . . . . . . . . . . . . . 19
5.2. LMS Private Key . . . . . . . . . . . . . . . . . . . . . 20
5.3. LMS Public Key . . . . . . . . . . . . . . . . . . . . . 21
5.4. LMS Signature . . . . . . . . . . . . . . . . . . . . . . 22
5.4.1. LMS Signature Generation . . . . . . . . . . . . . . 23
5.4.2. LMS Signature Verification . . . . . . . . . . . . . 24
6. Hierarchical Signatures . . . . . . . . . . . . . . . . . . . 26
6.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 29
6.2. Signature Generation . . . . . . . . . . . . . . . . . . 30
6.3. Signature Verification . . . . . . . . . . . . . . . . . 32
6.4. Parameter Set Recommendations . . . . . . . . . . . . . . 32
7. Rationale . . . . . . . . . . . . . . . . . . . . . . . . . . 34
7.1. Security String . . . . . . . . . . . . . . . . . . . . . 35
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8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 36
9. Security Considerations . . . . . . . . . . . . . . . . . . . 38
9.1. Hash Formats . . . . . . . . . . . . . . . . . . . . . . 39
9.2. Stateful Signature Algorithm . . . . . . . . . . . . . . 40
9.3. Security of LM-OTS Checksum . . . . . . . . . . . . . . . 41
10. Comparison with Other Work . . . . . . . . . . . . . . . . . 42
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 43
11.1. Normative References . . . . . . . . . . . . . . . . . . 43
11.2. Informative References . . . . . . . . . . . . . . . . . 43
Appendix A. Pseudorandom Key Generation . . . . . . . . . . . . 45
Appendix B. LM-OTS Parameter Options . . . . . . . . . . . . . . 45
Appendix C. An Iterative Algorithm for Computing an LMS Public
Key . . . . . . . . . . . . . . . . . . . . . . . . 47
Appendix D. Method for Deriving Authentication Path for a
Signature . . . . . . . . . . . . . . . . . . . . . 48
Appendix E. Example Implementation . . . . . . . . . . . . . . . 49
Appendix F. Test Cases . . . . . . . . . . . . . . . . . . . . . 49
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 60
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 61
1. Introduction
One-time signature systems, and general-purpose signature systems
built out of one-time signature systems, have been known since 1979
[Merkle79], were well studied in the 1990s [USPTO5432852], and have
benefited from renewed attention in the last decade. The
characteristics of these signature systems are small private and
public keys and fast signature generation and verification, but large
signatures and moderately slow key generation (in comparison with RSA
and ECDSA (Elliptic Curve Digital Signature Algorithm)). Private
keys can be made very small by appropriate key generation, for
example, as described in Appendix A. In recent years, there has been
interest in these systems because of their post-quantum security and
their suitability for compact verifier implementations.
This note describes the Leighton and Micali adaptation [USPTO5432852]
of the original Lamport-Diffie-Winternitz-Merkle one-time signature
system [Merkle79] [C:Merkle87] [C:Merkle89a] [C:Merkle89b] and
general signature system [Merkle79] with enough specificity to ensure
interoperability between implementations.
A signature system provides asymmetric message authentication. The
key-generation algorithm produces a public/private key pair. A
message is signed by a private key, producing a signature, and a
message/signature pair can be verified by a public key. A One-Time
Signature (OTS) system can be used to sign one message securely but
will become insecure if more than one is signed with the same public/
McGrew, et al. Informational [Page 3]
RFC 8554 LMS Hash-Based Signatures April 2019
private key pair. An N-time signature system can be used to sign N
or fewer messages securely. A Merkle-tree signature scheme is an
N-time signature system that uses an OTS system as a component.
In the Merkle scheme, a binary tree of height h is used to hold 2^h
OTS key pairs. Each interior node of the tree holds a value that is
the hash of the values of its two child nodes. The public key of the
tree is the value of the root node (a recursive hash of the OTS
public keys), while the private key of the tree is the collection of
all the OTS private keys, together with the index of the next OTS
private key to sign the next message with.
In this note, we describe the Leighton-Micali Signature (LMS) system
(a variant of the Merkle scheme) with the Hierarchical Signature
System (HSS) built on top of it that allows it to efficiently scale
to larger numbers of signatures. In order to support signing a large
number of messages on resource-constrained systems, the Merkle tree
can be subdivided into a number of smaller trees. Only the
bottommost tree is used to sign messages, while trees above that are
used to sign the public keys of their children. For example, in the
simplest case with two levels with both levels consisting of height h
trees, the root tree is used to sign 2^h trees with 2^h OTS key
pairs, and each second-level tree has 2^h OTS key pairs, for a total
of 2^(2h) bottom-level key pairs, and so can sign 2^(2h) messages.
The advantage of this scheme is that only the active trees need to be
instantiated, which saves both time (for key generation) and space
(for key storage). On the other hand, using a multilevel signature
scheme increases the size of the signature as well as the signature
verification time.
This note is structured as follows. Notes on post-quantum
cryptography are discussed in Section 1.1. Intellectual property
issues are discussed in Section 1.2. The notation used within this
note is defined in Section 3, and the public formats are described in
Section 3.3. The Leighton-Micali One-Time Signature (LM-OTS) system
is described in Section 4, and the LMS and HSS N-time signature
systems are described in Sections 5 and 6, respectively. Sufficient
detail is provided to ensure interoperability. The rationale for the
design decisions is given in Section 7. The IANA registry for these
signature systems is described in Section 8. Security considerations
are presented in Section 9. Comparison with another hash-based
signature algorithm (eXtended Merkle Signature Scheme (XMSS)) is in
Section 10.
This document represents the rough consensus of the CFRG.
McGrew, et al. Informational [Page 4]
RFC 8554 LMS Hash-Based Signatures April 2019
1.1. CFRG Note on Post-Quantum Cryptography
All post-quantum algorithms documented by the Crypto Forum Research
Group (CFRG) are today considered ready for experimentation and
further engineering development (e.g., to establish the impact of
performance and sizes on IETF protocols). However, at the time of
writing, we do not have significant deployment experience with such
Many of these algorithms come with specific restrictions, e.g.,
change of classical interface or less cryptanalysis of proposed
parameters than established schemes. The CFRG has consensus that all
documents describing post-quantum technologies include the above
paragraph and a clear additional warning about any specific
restrictions, especially as those might affect use or deployment of
the specific scheme. That guidance may be changed over time via
document updates.
Additionally, for LMS:
CFRG consensus is that we are confident in the cryptographic security
of the signature schemes described in this document against quantum
computers, given the current state of the research community's
knowledge about quantum algorithms. Indeed, we are confident that
the security of a significant part of the Internet could be made
dependent on the signature schemes defined in this document, if
developers take care of the following.
In contrast to traditional signature schemes, the signature schemes
described in this document are stateful, meaning the secret key
changes over time. If a secret key state is used twice, no
cryptographic security guarantees remain. In consequence, it becomes
feasible to forge a signature on a new message. This is a new
property that most developers will not be familiar with and requires
careful handling of secret keys. Developers should not use the
schemes described here except in systems that prevent the reuse of
secret key states.
Note that the fact that the schemes described in this document are
stateful also implies that classical APIs for digital signatures
cannot be used without modification. The API MUST be able to handle
a dynamic secret key state; that is, the API MUST allow the
signature-generation algorithm to update the secret key state.
McGrew, et al. Informational [Page 5]
RFC 8554 LMS Hash-Based Signatures April 2019
1.2. Intellectual Property
This document is based on U.S. Patent 5,432,852, which was issued
over twenty years ago and is thus expired.
1.2.1. Disclaimer
This document is not intended as legal advice. Readers are advised
to consult with their own legal advisers if they would like a legal
interpretation of their rights.
The IETF policies and processes regarding intellectual property and
patents are outlined in [RFC8179] and at
1.3. Conventions Used in This Document
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
2. Interface
The LMS signing algorithm is stateful; it modifies and updates the
private key as a side effect of generating a signature. Once a
particular value of the private key is used to sign one message, it
MUST NOT be used to sign another.
The key-generation algorithm takes as input an indication of the
parameters for the signature system. If it is successful, it returns
both a private key and a public key. Otherwise, it returns an
indication of failure.
The signing algorithm takes as input the message to be signed and the
current value of the private key. If successful, it returns a
signature and the next value of the private key, if there is such a
value. After the private key of an N-time signature system has
signed N messages, the signing algorithm returns the signature and an
indication that there is no next value of the private key that can be
used for signing. If unsuccessful, it returns an indication of
The verification algorithm takes as input the public key, a message,
and a signature; it returns an indication of whether or not the
signature-and-message pair is valid.
McGrew, et al. Informational [Page 6]
RFC 8554 LMS Hash-Based Signatures April 2019
A message/signature pair is valid if the signature was returned by
the signing algorithm upon input of the message and the private key
corresponding to the public key; otherwise, the signature and message
pair is not valid with probability very close to one.
3. Notation
3.1. Data Types
Bytes and byte strings are the fundamental data types. A single byte
is denoted as a pair of hexadecimal digits with a leading "0x". A
byte string is an ordered sequence of zero or more bytes and is
denoted as an ordered sequence of hexadecimal characters with a
leading "0x". For example, 0xe534f0 is a byte string with a length
of three. An array of byte strings is an ordered set, indexed
starting at zero, in which all strings have the same length.
Unsigned integers are converted into byte strings by representing
them in network byte order. To make the number of bytes in the
representation explicit, we define the functions u8str(X), u16str(X),
and u32str(X), which take a nonnegative integer X as input and return
one-, two-, and four-byte strings, respectively. We also make use of
the function strTou32(S), which takes a four-byte string S as input
and returns a nonnegative integer; the identity u32str(strTou32(S)) =
S holds for any four-byte string S.
3.1.1. Operators
When a and b are real numbers, mathematical operators are defined as
^ : a ^ b denotes the result of a raised to the power of b
* : a * b denotes the product of a multiplied by b
/ : a / b denotes the quotient of a divided by b
% : a % b denotes the remainder of the integer division of a by b
(with a and b being restricted to integers in this case)
+ : a + b denotes the sum of a and b
- : a - b denotes the difference of a and b
AND : a AND b denotes the bitwise AND of the two nonnegative
integers a and b (represented in binary notation)
McGrew, et al. Informational [Page 7]
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The standard order of operations is used when evaluating arithmetic
When B is a byte and i is an integer, then B >> i denotes the logical
right-shift operation by i bit positions. Similarly, B << i denotes
the logical left-shift operation.
If S and T are byte strings, then S || T denotes the concatenation of
S and T. If S and T are equal-length byte strings, then S AND T
denotes the bitwise logical and operation.
The i-th element in an array A is denoted as A[i].
3.1.2. Functions
If r is a nonnegative real number, then we define the following
ceil(r) : returns the smallest integer greater than or equal to r
floor(r) : returns the largest integer less than or equal to r
lg(r) : returns the base-2 logarithm of r
3.1.3. Strings of w-Bit Elements
If S is a byte string, then byte(S, i) denotes its i-th byte, where
the index starts at 0 at the left. Hence, byte(S, 0) is the leftmost
byte of S, byte(S, 1) is the second byte from the left, and (assuming
S is n bytes long) byte(S, n-1) is the rightmost byte of S. In
addition, bytes(S, i, j) denotes the range of bytes from the i-th to
the j-th byte, inclusive. For example, if S = 0x02040608, then
byte(S, 0) is 0x02 and bytes(S, 1, 2) is 0x0406.
A byte string can be considered to be a string of w-bit unsigned
integers; the correspondence is defined by the function coef(S, i, w)
as follows:
If S is a string, i is a positive integer, and w is a member of the
set { 1, 2, 4, 8 }, then coef(S, i, w) is the i-th, w-bit value, if S
is interpreted as a sequence of w-bit values. That is,
coef(S, i, w) = (2^w - 1) AND
( byte(S, floor(i * w / 8)) >>
(8 - (w * (i % (8 / w)) + w)) )
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For example, if S is the string 0x1234, then coef(S, 7, 1) is 0 and
coef(S, 0, 4) is 1.
S (represented as bits)
| 0| 0| 0| 1| 0| 0| 1| 0| 0| 0| 1| 1| 0| 1| 0| 0|
coef(S, 7, 1)
S (represented as four-bit values)
| 1 | 2 | 3 | 4 |
coef(S, 0, 4)
The return value of coef is an unsigned integer. If i is larger than
the number of w-bit values in S, then coef(S, i, w) is undefined, and
an attempt to compute that value MUST raise an error.
3.2. Typecodes
A typecode is an unsigned integer that is associated with a
particular data format. The format of the LM-OTS, LMS, and HSS
signatures and public keys all begin with a typecode that indicates
the precise details used in that format. These typecodes are
represented as four-byte unsigned integers in network byte order;
equivalently, they are External Data Representation (XDR)
enumerations (see Section 3.3).
3.3. Notation and Formats
The signature and public key formats are formally defined in XDR to
provide an unambiguous, machine-readable definition [RFC4506]. The
private key format is not included as it is not needed for
interoperability and an implementation MAY use any private key
format. However, for clarity, we include an example of private key
data in Test Case 2 of Appendix F. Though XDR is used, these formats
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are simple and easy to parse without any special tools. An
illustration of the layout of data in these objects is provided
below. The definitions are as follows:
/* one-time signatures */
enum lmots_algorithm_type {
lmots_reserved = 0,
lmots_sha256_n32_w1 = 1,
lmots_sha256_n32_w2 = 2,
lmots_sha256_n32_w4 = 3,
lmots_sha256_n32_w8 = 4
typedef opaque bytestring32[32];
struct lmots_signature_n32_p265 {
bytestring32 C;
bytestring32 y[265];
struct lmots_signature_n32_p133 {
bytestring32 C;
bytestring32 y[133];
struct lmots_signature_n32_p67 {
bytestring32 C;
bytestring32 y[67];
struct lmots_signature_n32_p34 {
bytestring32 C;
bytestring32 y[34];
union lmots_signature switch (lmots_algorithm_type type) {
case lmots_sha256_n32_w1:
lmots_signature_n32_p265 sig_n32_p265;
case lmots_sha256_n32_w2:
lmots_signature_n32_p133 sig_n32_p133;
case lmots_sha256_n32_w4:
lmots_signature_n32_p67 sig_n32_p67;
case lmots_sha256_n32_w8:
lmots_signature_n32_p34 sig_n32_p34;
void; /* error condition */
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/* hash-based signatures (hbs) */
enum lms_algorithm_type {
lms_reserved = 0,
lms_sha256_n32_h5 = 5,
lms_sha256_n32_h10 = 6,
lms_sha256_n32_h15 = 7,
lms_sha256_n32_h20 = 8,
lms_sha256_n32_h25 = 9
/* leighton-micali signatures (lms) */
union lms_path switch (lms_algorithm_type type) {
case lms_sha256_n32_h5:
bytestring32 path_n32_h5[5];
case lms_sha256_n32_h10:
bytestring32 path_n32_h10[10];
case lms_sha256_n32_h15:
bytestring32 path_n32_h15[15];
case lms_sha256_n32_h20:
bytestring32 path_n32_h20[20];
case lms_sha256_n32_h25:
bytestring32 path_n32_h25[25];
void; /* error condition */
struct lms_signature {
unsigned int q;
lmots_signature lmots_sig;
lms_path nodes;
struct lms_key_n32 {
lmots_algorithm_type ots_alg_type;
opaque I[16];
opaque K[32];
union lms_public_key switch (lms_algorithm_type type) {
case lms_sha256_n32_h5:
case lms_sha256_n32_h10:
case lms_sha256_n32_h15:
case lms_sha256_n32_h20:
case lms_sha256_n32_h25:
lms_key_n32 z_n32;
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void; /* error condition */
/* hierarchical signature system (hss) */
struct hss_public_key {
unsigned int L;
lms_public_key pub;
struct signed_public_key {
lms_signature sig;
lms_public_key pub;
struct hss_signature {
signed_public_key signed_keys<7>;
lms_signature sig_of_message;
4. LM-OTS One-Time Signatures
This section defines LM-OTS signatures. The signature is used to
validate the authenticity of a message by associating a secret
private key with a shared public key. These are one-time signatures;
each private key MUST be used at most one time to sign any given
As part of the signing process, a digest of the original message is
computed using the cryptographic hash function H (see Section 4.1),
and the resulting digest is signed.
In order to facilitate its use in an N-time signature system, the
LM-OTS key generation, signing, and verification algorithms all take
as input parameters I and q. The parameter I is a 16-byte string
that indicates which Merkle tree this LM-OTS is used with. The
parameter q is a 32-bit integer that indicates the leaf of the Merkle
tree where the OTS public key appears. These parameters are used as
part of the security string, as listed in Section 7.1. When the
LM-OTS signature system is used outside of an N-time signature
system, the value I MAY be used to differentiate this one-time
signature from others; however, the value q MUST be set to the all-
zero value.
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4.1. Parameters
The signature system uses the parameters n and w, which are both
positive integers. The algorithm description also makes use of the
internal parameters p and ls, which are dependent on n and w. These
parameters are summarized as follows:
n : the number of bytes of the output of the hash function.
w : the width (in bits) of the Winternitz coefficients; that is,
the number of bits from the hash or checksum that is used with a
single Winternitz chain. It is a member of the set
{ 1, 2, 4, 8 }.
p : the number of n-byte string elements that make up the LM-OTS
signature. This is a function of n and w; the values for the
defined parameter sets are listed in Table 1; it can also be
computed by the algorithm given in Appendix B.
ls : the number of left-shift bits used in the checksum function
Cksm (defined in Section 4.4).
H : a second-preimage-resistant cryptographic hash function that
accepts byte strings of any length and returns an n-byte string.
For more background on the cryptographic security requirements for H,
see Section 9.
The value of n is determined by the hash function selected for use as
part of the LM-OTS algorithm; the choice of this value has a strong
effect on the security of the system. The parameter w determines the
length of the Winternitz chains computed as a part of the OTS
signature (which involve 2^w - 1 invocations of the hash function);
it has little effect on security. Increasing w will shorten the
signature, but at a cost of a larger computation to generate and
verify a signature. The values of p and ls are dependent on the
choices of the parameters n and w, as described in Appendix B.
Table 1 illustrates various combinations of n, w, p and ls, along
with the resulting signature length.
The value of w describes a space/time trade-off; increasing the value
of w will cause the signature to shrink (by decreasing the value of
p) while increasing the amount of time needed to perform operations
with it: generate the public key and generate and verify the
signature. In general, the LM-OTS signature is 4+n*(p+1) bytes long,
and public key generation will take p*(2^w - 1) + 1 hash computations
(and signature generation and verification will take approximately
half that on average).
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| Parameter Set Name | H | n | w | p | ls | sig_len |
| LMOTS_SHA256_N32_W1 | SHA256 | 32 | 1 | 265 | 7 | 8516 |
| | | | | | | |
| LMOTS_SHA256_N32_W2 | SHA256 | 32 | 2 | 133 | 6 | 4292 |
| | | | | | | |
| LMOTS_SHA256_N32_W4 | SHA256 | 32 | 4 | 67 | 4 | 2180 |
| | | | | | | |
| LMOTS_SHA256_N32_W8 | SHA256 | 32 | 8 | 34 | 0 | 1124 |
Table 1
Here SHA256 denotes the SHA-256 hash function defined in NIST
standard [FIPS180].
4.2. Private Key
The format of the LM-OTS private key is an internal matter to the
implementation, and this document does not attempt to define it. One
possibility is that the private key may consist of a typecode
indicating the particular LM-OTS algorithm, an array x[] containing p
n-byte strings, and the 16-byte string I and the 4-byte string q.
This private key MUST be used to sign (at most) one message. The
following algorithm shows pseudocode for generating a private key.
Algorithm 0: Generating a Private Key
1. Retrieve the values of q and I (the 16-byte identifier of the
LMS public/private key pair) from the LMS tree that this LM-OTS
private key will be used with
2. Set type to the typecode of the algorithm
3. Set n and p according to the typecode and Table 1
4. Compute the array x as follows:
for ( i = 0; i < p; i = i + 1 ) {
set x[i] to a uniformly random n-byte string
5. Return u32str(type) || I || u32str(q) || x[0] || x[1] || ...
|| x[p-1]
An implementation MAY use a pseudorandom method to compute x[i], as
suggested in [Merkle79], page 46. The details of the pseudorandom
method do not affect interoperability, but the cryptographic strength
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MUST match that of the LM-OTS algorithm. Appendix A provides an
example of a pseudorandom method for computing the LM-OTS private
4.3. Public Key
The LM-OTS public key is generated from the private key by
iteratively applying the function H to each individual element of x,
for 2^w - 1 iterations, then hashing all of the resulting values.
The public key is generated from the private key using the following
algorithm, or any equivalent process.
Algorithm 1: Generating a One-Time Signature Public Key From a
Private Key
1. Set type to the typecode of the algorithm
2. Set the integers n, p, and w according to the typecode and
Table 1
3. Determine x, I, and q from the private key
4. Compute the string K as follows:
for ( i = 0; i < p; i = i + 1 ) {
tmp = x[i]
for ( j = 0; j < 2^w - 1; j = j + 1 ) {
tmp = H(I || u32str(q) || u16str(i) || u8str(j) || tmp)
y[i] = tmp
K = H(I || u32str(q) || u16str(D_PBLC) || y[0] || ... || y[p-1])
5. Return u32str(type) || I || u32str(q) || K
where D_PBLC is the fixed two-byte value 0x8080, which is used to
distinguish the last hash from every other hash in this system.
The public key is the value returned by Algorithm 1.
4.4. Checksum
A checksum is used to ensure that any forgery attempt that
manipulates the elements of an existing signature will be detected.
This checksum is needed because an attacker can freely advance any of
the Winternitz chains. That is, if this checksum were not present,
then an attacker who could find a hash that has every digit larger
than the valid hash could replace it (and adjust the Winternitz
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chains). The security property that the checksum provides is
detailed in Section 9. The checksum function Cksm is defined as
follows, where S denotes the n-byte string that is input to that
function, and the value sum is a 16-bit unsigned integer:
Algorithm 2: Checksum Calculation
sum = 0
for ( i = 0; i < (n*8/w); i = i + 1 ) {
sum = sum + (2^w - 1) - coef(S, i, w)
return (sum << ls)
ls is the parameter that shifts the significant bits of the checksum
into the positions that will actually be used by the coef function
when encoding the digits of the checksum. The actual ls parameter is
a function of the n and w parameters; the values for the currently
defined parameter sets are shown in Table 1. It is calculated by the
algorithm given in Appendix B.
Because of the left-shift operation, the rightmost bits of the result
of Cksm will often be zeros. Due to the value of p, these bits will
not be used during signature generation or verification.
4.5. Signature Generation
The LM-OTS signature of a message is generated by doing the following
in sequence: prepending the LMS key identifier I, the LMS leaf
identifier q, the value D_MESG (0x8181), and the randomizer C to the
message; computing the hash; concatenating the checksum of the hash
to the hash itself; considering the resulting value as a sequence of
w-bit values; and using each of the w-bit values to determine the
number of times to apply the function H to the corresponding element
of the private key. The outputs of the function H are concatenated
together and returned as the signature. The pseudocode for this
procedure is shown below.
Algorithm 3: Generating a One-Time Signature From a Private Key and a
1. Set type to the typecode of the algorithm
2. Set n, p, and w according to the typecode and Table 1
3. Determine x, I, and q from the private key
4. Set C to a uniformly random n-byte string
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5. Compute the array y as follows:
Q = H(I || u32str(q) || u16str(D_MESG) || C || message)
for ( i = 0; i < p; i = i + 1 ) {
a = coef(Q || Cksm(Q), i, w)
tmp = x[i]
for ( j = 0; j < a; j = j + 1 ) {
tmp = H(I || u32str(q) || u16str(i) || u8str(j) || tmp)
y[i] = tmp
6. Return u32str(type) || C || y[0] || ... || y[p-1]
Note that this algorithm results in a signature whose elements are
intermediate values of the elements computed by the public key
algorithm in Section 4.3.
The signature is the string returned by Algorithm 3. Section 3.3
formally defines the structure of the string as the lmots_signature
4.6. Signature Verification
In order to verify a message with its signature (an array of n-byte
strings, denoted as y), the receiver must "complete" the chain of
iterations of H using the w-bit coefficients of the string resulting
from the concatenation of the message hash and its checksum. This
computation should result in a value that matches the provided public
Algorithm 4a: Verifying a Signature and Message Using a Public Key
1. If the public key is not at least four bytes long,
return INVALID.
2. Parse pubtype, I, q, and K from the public key as follows:
a. pubtype = strTou32(first 4 bytes of public key)
b. Set n according to the pubkey and Table 1; if the public key
is not exactly 24 + n bytes long, return INVALID.
c. I = next 16 bytes of public key
d. q = strTou32(next 4 bytes of public key)
e. K = next n bytes of public key
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3. Compute the public key candidate Kc from the signature,
message, pubtype, and the identifiers I and q obtained from the
public key, using Algorithm 4b. If Algorithm 4b returns
INVALID, then return INVALID.
4. If Kc is equal to K, return VALID; otherwise, return INVALID.
Algorithm 4b: Computing a Public Key Candidate Kc from a Signature,
Message, Signature Typecode pubtype, and Identifiers I, q
1. If the signature is not at least four bytes long,
return INVALID.
2. Parse sigtype, C, and y from the signature as follows:
a. sigtype = strTou32(first 4 bytes of signature)
b. If sigtype is not equal to pubtype, return INVALID.
c. Set n and p according to the pubtype and Table 1; if the
signature is not exactly 4 + n * (p+1) bytes long,
return INVALID.
d. C = next n bytes of signature
e. y[0] = next n bytes of signature
y[1] = next n bytes of signature
y[p-1] = next n bytes of signature
3. Compute the string Kc as follows:
Q = H(I || u32str(q) || u16str(D_MESG) || C || message)
for ( i = 0; i < p; i = i + 1 ) {
a = coef(Q || Cksm(Q), i, w)
tmp = y[i]
for ( j = a; j < 2^w - 1; j = j + 1 ) {
tmp = H(I || u32str(q) || u16str(i) || u8str(j) || tmp)
z[i] = tmp
Kc = H(I || u32str(q) || u16str(D_PBLC) ||
z[0] || z[1] || ... || z[p-1])
4. Return Kc.
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5. Leighton-Micali Signatures
The Leighton-Micali Signature (LMS) method can sign a potentially
large but fixed number of messages. An LMS system uses two
cryptographic components: a one-time signature method and a hash
function. Each LMS public/private key pair is associated with a
perfect binary tree, each node of which contains an m-byte value,
where m is the output length of the hash function. Each leaf of the
tree contains the value of the public key of an LM-OTS public/private
key pair. The value contained by the root of the tree is the LMS
public key. Each interior node is computed by applying the hash
function to the concatenation of the values of its children nodes.
Each node of the tree is associated with a node number, an unsigned
integer that is denoted as node_num in the algorithms below, which is
computed as follows. The root node has node number 1; for each node
with node number N < 2^h (where h is the height of the tree), its
left child has node number 2*N, while its right child has node number
2*N + 1. The result of this is that each node within the tree will
have a unique node number, and the leaves will have node numbers 2^h,
(2^h)+1, (2^h)+2, ..., (2^h)+(2^h)-1. In general, the j-th node at
level i has node number 2^i + j. The node number can conveniently be
computed when it is needed in the LMS algorithms, as described in
those algorithms.
5.1. Parameters
An LMS system has the following parameters:
h : the height of the tree
m : the number of bytes associated with each node
H : a second-preimage-resistant cryptographic hash function that
accepts byte strings of any length and returns an m-byte string.
There are 2^h leaves in the tree.
The overall strength of LMS signatures is governed by the weaker of
the hash function used within the LM-OTS and the hash function used
within the LMS system. In order to minimize the risk, these two hash
functions SHOULD be the same (so that an attacker could not take
advantage of the weaker hash function choice).
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| Name | H | m | h |
| LMS_SHA256_M32_H5 | SHA256 | 32 | 5 |
| | | | |
| LMS_SHA256_M32_H10 | SHA256 | 32 | 10 |
| | | | |
| LMS_SHA256_M32_H15 | SHA256 | 32 | 15 |
| | | | |
| LMS_SHA256_M32_H20 | SHA256 | 32 | 20 |
| | | | |
| LMS_SHA256_M32_H25 | SHA256 | 32 | 25 |
Table 2
5.2. LMS Private Key
The format of the LMS private key is an internal matter to the
implementation, and this document does not attempt to define it. One
possibility is that it may consist of an array OTS_PRIV[] of 2^h
LM-OTS private keys and the leaf number q of the next LM-OTS private
key that has not yet been used. The q-th element of OTS_PRIV[] is
generated using Algorithm 0 with the identifiers I, q. The leaf
number q is initialized to zero when the LMS private key is created.
The process is as follows:
Algorithm 5: Computing an LMS Private Key.
1. Determine h and m from the typecode and Table 2.
2. Set I to a uniformly random 16-byte string.
3. Compute the array OTS_PRIV[] as follows:
for ( q = 0; q < 2^h; q = q + 1) {
OTS_PRIV[q] = LM-OTS private key with identifiers I, q
4. q = 0
An LMS private key MAY be generated pseudorandomly from a secret
value; in this case, the secret value MUST be at least m bytes long
and uniformly random and MUST NOT be used for any other purpose than
the generation of the LMS private key. The details of how this
process is done do not affect interoperability; that is, the public
key verification operation is independent of these details.
Appendix A provides an example of a pseudorandom method for computing
an LMS private key.
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The signature-generation logic uses q as the next leaf to use; hence,
step 4 starts it off at the leftmost leaf. Because the signature
process increments q after the signature operation, the first
signature will have q=0.
5.3. LMS Public Key
An LMS public key is defined as follows, where we denote the public
key final hash value (namely, the K value computed in Algorithm 1)
associated with the i-th LM-OTS private key as OTS_PUB_HASH[i], with
i ranging from 0 to (2^h)-1. Each instance of an LMS public/private
key pair is associated with a balanced binary tree, and the nodes of
that tree are indexed from 1 to 2^(h+1)-1. Each node is associated
with an m-byte string. The string for the r-th node is denoted as
T[r] and defined as
if r >= 2^h:
where D_LEAF is the fixed two-byte value 0x8282 and D_INTR is the
fixed two-byte value 0x8383, both of which are used to distinguish
this hash from every other hash in this system.
When we have r >= 2^h, then we are processing a leaf node (and thus
hashing only a single LM-OTS public key). When we have r < 2^h, then
we are processing an internal node -- that is, a node with two child
nodes that we need to combine.
The LMS public key can be represented as the byte string
u32str(type) || u32str(otstype) || I || T[1]
Section 3.3 specifies the format of the type variable. The value
otstype is the parameter set for the LM-OTS public/private key pairs
used. The value I is the private key identifier and is the value
used for all computations for the same LMS tree. The value T[1] can
be computed via recursive application of the above equation or by any
equivalent method. An iterative procedure is outlined in Appendix C.
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5.4. LMS Signature
An LMS signature consists of
the number q of the leaf associated with the LM-OTS signature, as
a four-byte unsigned integer in network byte order, an LM-OTS
a typecode indicating the particular LMS algorithm,
an array of h m-byte values that is associated with the path
through the tree from the leaf associated with the LM-OTS
signature to the root.
Symbolically, the signature can be represented as
u32str(q) || lmots_signature || u32str(type) ||
path[0] || path[1] || path[2] || ... || path[h-1]
Section 3.3 formally defines the format of the signature as the
lms_signature structure. The array for a tree with height h will
have h values and contains the values of the siblings of (that is, is
adjacent to) the nodes on the path from the leaf to the root, where
the sibling to node A is the other node that shares node A's parent.
In the signature, 0 is counted from the bottom level of the tree, and
so path[0] is the value of the node adjacent to leaf node q; path[1]
is the second-level node that is adjacent to leaf node q's parent,
and so on up the tree until we get to path[h-1], which is the value
of the next-to-the-top-level node whose branch the leaf node q does
not reside in.
Below is a simple example of the authentication path for h=3 and q=2.
The leaf marked OTS is the one-time signature that is used to sign
the actual message. The nodes on the path from the OTS public key to
the root are marked with a *, while the nodes that are used within
the path array are marked with **. The values in the path array are
those nodes that are siblings of the nodes on the path; path[0] is
the leaf** node that is adjacent to the OTS public key (which is the
start of the path); path[1] is the T[4]** node that is the sibling of
the second node T[5]* on the path, and path[2] is the T[3]** node
that is the sibling of the third node T[2]* on the path.
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| |
T[2]* T[3]**
| |
------------------ -----------------
| | | |
T[4]** T[5]* T[6] T[7]
| | | |
--------- ---------- -------- ---------
| | | | | | | |
leaf leaf OTS leaf** leaf leaf leaf leaf
The idea behind this authentication path is that it allows us to
validate the OTS hash with using h path array values and hash
computations. What the verifier does is recompute the hashes up the
path; first, it hashes the given OTS and path[0] value, giving a
tentative T[5]' value. Then, it hashes its path[1] and tentative
T[5]' value to get a tentative T[2]' value. Then, it hashes that and
the path[2] value to get a tentative Root' value. If that value is
the known public key of the Merkle tree, then we can assume that the
value T[2]' it got was the correct T[2] value in the original tree,
and so the T[5]' value it got was the correct T[5] value in the
original tree, and so the OTS public key is the same as in the
original and, hence, is correct.
5.4.1. LMS Signature Generation
To compute the LMS signature of a message with an LMS private key,
the signer first computes the LM-OTS signature of the message using
the leaf number of the next unused LM-OTS private key. The leaf
number q in the signature is set to the leaf number of the LMS
private key that was used in the signature. Before releasing the
signature, the leaf number q in the LMS private key MUST be
incremented to prevent the LM-OTS private key from being used again.
If the LMS private key is maintained in nonvolatile memory, then the
implementation MUST ensure that the incremented value has been stored
before releasing the signature. The issue this tries to prevent is a
scenario where a) we generate a signature using one LM-OTS private
key and release it to the application, b) before we update the
nonvolatile memory, we crash, and c) we reboot and generate a second
signature using the same LM-OTS private key. With two different
signatures using the same LM-OTS private key, an attacker could
potentially generate a forged signature of a third message.
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The array of node values in the signature MAY be computed in any way.
There are many potential time/storage trade-offs that can be applied.
The fastest alternative is to store all of the nodes of the tree and
set the array in the signature by copying them; pseudocode to do so
appears in Appendix D. The least storage-intensive alternative is to
recompute all of the nodes for each signature. Note that the details
of this procedure are not important for interoperability; it is not
necessary to know any of these details in order to perform the
signature-verification operation. The internal nodes of the tree
need not be kept secret, and thus a node-caching scheme that stores
only internal nodes can sidestep the need for strong protections.
Several useful time/storage trade-offs are described in the "Small-
Memory LM Schemes" section of [USPTO5432852].
5.4.2. LMS Signature Verification
An LMS signature is verified by first using the LM-OTS signature
verification algorithm (Algorithm 4b) to compute the LM-OTS public
key from the LM-OTS signature and the message. The value of that
public key is then assigned to the associated leaf of the LMS tree,
and then the root of the tree is computed from the leaf value and the
array path[] as described in Algorithm 6 below. If the root value
matches the public key, then the signature is valid; otherwise, the
signature verification fails.
Algorithm 6: LMS Signature Verification
1. If the public key is not at least eight bytes long, return
2. Parse pubtype, I, and T[1] from the public key as follows:
a. pubtype = strTou32(first 4 bytes of public key)
b. ots_typecode = strTou32(next 4 bytes of public key)
c. Set m according to pubtype, based on Table 2.
d. If the public key is not exactly 24 + m bytes
long, return INVALID.
e. I = next 16 bytes of the public key
f. T[1] = next m bytes of the public key
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3. Compute the LMS Public Key Candidate Tc from the signature,
message, identifier, pubtype, and ots_typecode, using
Algorithm 6a.
4. If Tc is equal to T[1], return VALID; otherwise, return INVALID.
Algorithm 6a: Computing an LMS Public Key Candidate from a Signature,
Message, Identifier, and Algorithm Typecodes
1. If the signature is not at least eight bytes long,
return INVALID.
2. Parse sigtype, q, lmots_signature, and path from the signature
as follows:
a. q = strTou32(first 4 bytes of signature)
b. otssigtype = strTou32(next 4 bytes of signature)
c. If otssigtype is not the OTS typecode from the public key,
return INVALID.
d. Set n, p according to otssigtype and Table 1; if the
signature is not at least 12 + n * (p + 1) bytes long,
return INVALID.
e. lmots_signature = bytes 4 through 7 + n * (p + 1)
of signature
f. sigtype = strTou32(bytes 8 + n * (p + 1)) through
11 + n * (p + 1) of signature)
g. If sigtype is not the LM typecode from the public key,
return INVALID.
h. Set m, h according to sigtype and Table 2.
i. If q >= 2^h or the signature is not exactly
12 + n * (p + 1) + m * h bytes long,
return INVALID.
j. Set path as follows:
path[0] = next m bytes of signature
path[1] = next m bytes of signature
path[h-1] = next m bytes of signature
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3. Kc = candidate public key computed by applying Algorithm 4b
to the signature lmots_signature, the message, and the
identifiers I, q
4. Compute the candidate LMS root value Tc as follows:
node_num = 2^h + q
tmp = H(I || u32str(node_num) || u16str(D_LEAF) || Kc)
i = 0
while (node_num > 1) {
if (node_num is odd):
tmp = H(I||u32str(node_num/2)||u16str(D_INTR)||path[i]||tmp)
tmp = H(I||u32str(node_num/2)||u16str(D_INTR)||tmp||path[i])
node_num = node_num/2
i = i + 1
Tc = tmp
5. Return Tc.
6. Hierarchical Signatures
In scenarios where it is necessary to minimize the time taken by the
public key generation process, the Hierarchical Signature System
(HSS) can be used. This hierarchical scheme, which we describe in
this section, uses the LMS scheme as a component. In HSS, we have a
sequence of L LMS trees, where the public key for the first LMS tree
is included in the public key of the HSS system, each LMS private key
signs the next LMS public key, and the last LMS private key signs the
actual message. For example, if we have a three-level hierarchy
(L=3), then to sign a message, we would have:
The first LMS private key (level 0) signs a level 1 LMS public
The second LMS private key (level 1) signs a level 2 LMS public
The third LMS private key (level 2) signs the message.
The root of the level 0 LMS tree is contained in the HSS public key.
To verify the LMS signature, we would verify all the signatures:
We would verify that the level 1 LMS public key is correctly
signed by the level 0 signature.
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We would verify that the level 2 LMS public key is correctly
signed by the level 1 signature.
We would verify that the message is correctly signed by the level
2 signature.
We would accept the HSS signature only if all the signatures
During the signature-generation process, we sign messages with the
lowest (level L-1) LMS tree. Once we have used all the leafs in that
tree to sign messages, we would discard it, generate a fresh LMS
tree, and sign it with the next (level L-2) LMS tree (and when that
is used up, recursively generate and sign a fresh level L-2 LMS
HSS, in essence, utilizes a tree of LMS trees. There is a single LMS
tree at level 0 (the root). Each LMS tree (actually, the private key
corresponding to the LMS tree) at level i is used to sign 2^h objects
(where h is the height of trees at level i). If i < L-1, then each
object will be another LMS tree (actually, the public key) at level
i+1; if i = L-1, we've reached the bottom of the HSS tree, and so
each object will be a message from the application. The HSS public
key contains the public key of the LMS tree at the root, and an HSS
signature is associated with a path from the root of the HSS tree to
the leaf.
Compared to LMS, HSS has a much reduced public key generation time,
as only the root tree needs to be generated prior to the distribution
of the HSS public key. For example, an L=3 tree (with h=10 at each
level) would have one level 0 LMS tree, 2^10 level 1 LMS trees (with
each such level 1 public key signed by one of the 1024 level 0 OTS
public keys), and 2^20 level 2 LMS trees. Only 1024 OTS public keys
need to be computed to generate the HSS public key (as you need to
compute only the level 0 LMS tree to compute that value; you can, of
course, decide to compute the initial level 1 and level 2 LMS trees).
In addition, the 2^20 level 2 LMS trees can jointly sign a total of
over a billion messages. In contrast, a single LMS tree that could
sign a billion messages would require a billion OTS public keys to be
computed first (if h=30 were allowed in a supported parameter set).
Each LMS tree within the hierarchy is associated with a distinct LMS
public key, private key, signature, and identifier. The number of
levels is denoted as L and is between one and eight, inclusive. The
following notation is used, where i is an integer between 0 and L-1
inclusive, and the root of the hierarchy is level 0:
prv[i] is the current LMS private key of the i-th level.
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pub[i] is the current LMS public key of the i-th level, as
described in Section 5.3.
sig[i] is the LMS signature of public key pub[i+1] generated using
the private key prv[i].
It is expected that the above arrays are maintained for the course of
the HSS key. The contents of the prv[] array MUST be kept private;
the pub[] and sig[] array may be revealed should the implementation
find that convenient.
In this section, we say that an N-time private key is exhausted when
it has generated N signatures; thus, it can no longer be used for
For i > 0, the values prv[i], pub[i], and (for all values of i)
sig[i] will be updated over time as private keys are exhausted and
replaced by newer keys.
When these key pairs are updated (or initially generated before the
first message is signed), then the LMS key generation processes
outlined in Sections 5.2 and 5.3 are performed. If the generated key
pairs are for level i of the HSS hierarchy, then we store the public
key in pub[i] and the private key in prv[i]. In addition, if i > 0,
then we sign the generated public key with the LMS private key at
level i-1, placing the signature into sig[i-1]. When the LMS key
pair is generated, the key pair and the corresponding identifier MUST
be generated independently of all other key pairs.
HSS allows L=1, in which case the HSS public key and signature
formats are essentially the LMS public key and signature formats,
prepended by a fixed field. Since HSS with L=1 has very little
overhead compared to LMS, all implementations MUST support HSS in
order to maximize interoperability.
We specifically allow different LMS levels to use different parameter
sets. For example, the 0-th LMS public key (the root) may use the
LMS_SHA256_M32_H15 parameter set, while the 1-th public key may use
LMS_SHA256_M32_H10. There are practical reasons to allow this; for
one, the signer may decide to store parts of the 0-th LMS tree (that
it needs to construct while computing the public key) to accelerate
later operations. As the 0-th tree is never updated, these internal
nodes will never need to be recomputed. In addition, during the
signature-generation operation, almost all the operations involved
with updating the authentication path occur with the bottom (L-1th)
LMS public key; hence, it may be useful to select the parameter set
for that public key to have a shorter LMS tree.
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A close reading of the HSS verification pseudocode shows that it
would allow the parameters of the nontop LMS public keys to change
over time; for example, the signer might initially have the 1-th LMS
public key use the LMS_SHA256_M32_H10 parameter set, but when that
tree is exhausted, the signer might replace it with an LMS public key
that uses the LMS_SHA256_M32_H15 parameter set. While this would
work with the example verification pseudocode, the signer MUST NOT
change the parameter sets for a specific level. This prohibition is
to support verifiers that may keep state over the course of several
signature verifications.
6.1. Key Generation
The public key of the HSS scheme consists of the number of levels L,
followed by pub[0], the public key of the top level.
The HSS private key consists of prv[0], ... , prv[L-1], along with
the associated pub[0], ... pub[L-1] and sig[0], ..., sig[L-2] values.
As stated earlier, the values of the pub[] and sig[] arrays need not
be kept secret and may be revealed. The value of pub[0] does not
change (and, except for the index q, the value of prv[0] need not
change); however, the values of pub[i] and prv[i] are dynamic for i >
0 and are changed by the signature-generation algorithm.
During the key generation, the public and private keys are
initialized. Here is some pseudocode that explains the key-
generation logic:
Algorithm 7: Generating an HSS Key Pair
1. Generate an LMS key pair, as specified in Sections 5.2 and 5.3,
placing the private key into priv[0], and the public key into
2. For i = 1 to L-1 do {
generate an LMS key pair, placing the private key into priv[i]
and the public key into pub[i]
sig[i-1] = lms_signature( pub[i], priv[i-1] )
3. Return u32str(L) || pub[0] as the public key and the priv[],
pub[], and sig[] arrays as the private key
In the above algorithm, each LMS public/private key pair generated
MUST be generated independently.
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Note that the value of the public key does not depend on the
execution of step 2. As a result, an implementation may decide to
delay step 2 until later -- for example, during the initial
signature-generation operation.
6.2. Signature Generation
To sign a message using an HSS key pair, the following steps are
If prv[L-1] is exhausted, then determine the smallest integer d
such that all of the private keys prv[d], prv[d+1], ... , prv[L-1]
are exhausted. If d is equal to zero, then the HSS key pair is
exhausted, and it MUST NOT generate any more signatures.
Otherwise, the key pairs for levels d through L-1 must be
regenerated during the signature-generation process, as follows.
For i from d to L-1, a new LMS public and private key pair with a
new identifier is generated, pub[i] and prv[i] are set to those
values, then the public key pub[i] is signed with prv[i-1], and
sig[i-1] is set to the resulting value.
The message is signed with prv[L-1], and the value sig[L-1] is set
to that result.
The value of the HSS signature is set as follows. We let
signed_pub_key denote an array of octet strings, where
signed_pub_key[i] = sig[i] || pub[i+1], for i between 0 and
Nspk-1, inclusive, where Nspk = L-1 denotes the number of signed
public keys. Then the HSS signature is u32str(Nspk) ||
signed_pub_key[0] || ... || signed_pub_key[Nspk-1] || sig[Nspk].
Note that the number of signed_pub_key elements in the signature
is indicated by the value Nspk that appears in the initial four
bytes of the signature.
Here is some pseudocode of the above logic:
Algorithm 8: Generating an HSS signature
1. If the message-signing key prv[L-1] is exhausted, regenerate
that key pair, together with any parent key pairs that might
be necessary.
If the root key pair is exhausted, then the HSS key pair is
exhausted and MUST NOT generate any more signatures.
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d = L
while (prv[d-1].q == 2^(prv[d-1].h)) {
d = d - 1
if (d == 0)
return FAILURE
while (d < L) {
create lms key pair pub[d], prv[d]
sig[d-1] = lms_signature( pub[d], prv[d-1] )
d = d + 1
2. Sign the message.
sig[L-1] = lms_signature( msg, prv[L-1] )
3. Create the list of signed public keys.
i = 0;
while (i < L-1) {
signed_pub_key[i] = sig[i] || pub[i+1]
i = i + 1
4. Return u32str(L-1) || signed_pub_key[0] || ...
|| signed_pub_key[L-2] || sig[L-1]
In the specific case of L=1, the format of an HSS signature is
u32str(0) || sig[0]
In the general case, the format of an HSS signature is
u32str(Nspk) || signed_pub_key[0] || ...
|| signed_pub_key[Nspk-1] || sig[Nspk]
which is equivalent to
u32str(Nspk) || sig[0] || pub[1] || ...
|| sig[Nspk-1] || pub[Nspk] || sig[Nspk]
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6.3. Signature Verification
To verify a signature S and message using the public key pub, perform
the following steps:
The signature S is parsed into its components as follows:
Nspk = strTou32(first four bytes of S)
if Nspk+1 is not equal to the number of levels L in pub:
return INVALID
for (i = 0; i < Nspk; i = i + 1) {
siglist[i] = next LMS signature parsed from S
publist[i] = next LMS public key parsed from S
siglist[Nspk] = next LMS signature parsed from S
key = pub
for (i = 0; i < Nspk; i = i + 1) {
sig = siglist[i]
msg = publist[i]
if (lms_verify(msg, key, sig) != VALID):
return INVALID
key = msg
return lms_verify(message, key, siglist[Nspk])
Since the length of an LMS signature cannot be known without parsing
it, the HSS signature verification algorithm makes use of an LMS
signature parsing routine that takes as input a string consisting of
an LMS signature with an arbitrary string appended to it and returns
both the LMS signature and the appended string. The latter is passed
on for further processing.
6.4. Parameter Set Recommendations
As for guidance as to the number of LMS levels and the size of each,
any discussion of performance is implementation specific. In
general, the sole drawback for a single LMS tree is the time it takes
to generate the public key; as every LM-OTS public key needs to be
generated, the time this takes can be substantial. For a two-level
tree, only the top-level LMS tree and the initial bottom-level LMS
tree need to be generated initially (before the first signature is
generated); this will in general be significantly quicker.
To give a general idea of the trade-offs available, we include some
measurements taken with the LMS implementation available at
<https://github.com/cisco/hash-sigs>, taken on a 3.3 GHz Xeon
processor with threading enabled. We tried various parameter sets,
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all with W=8 (which minimizes signature size, while increasing time).
These are here to give a guideline as to what's possible; for the
computational time, your mileage may vary, depending on the computing
resources you have. The machine these tests were performed on does
not have the SHA-256 extensions; you could possibly do significantly
| ParmSet | KeyGenTime | SigSize | KeyLifetime |
| 15 | 6 sec | 1616 | 30 seconds |
| | | | |
| 20 | 3 min | 1776 | 16 minutes |
| | | | |
| 25 | 1.5 hour | 1936 | 9 hours |
| | | | |
| 15/10 | 6 sec | 3172 | 9 hours |
| | | | |
| 15/15 | 6 sec | 3332 | 12 days |
| | | | |
| 20/10 | 3 min | 3332 | 12 days |
| | | | |
| 20/15 | 3 min | 3492 | 1 year |
| | | | |
| 25/10 | 1.5 hour | 3492 | 1 year |
| | | | |
| 25/15 | 1.5 hour | 3652 | 34 years |
Table 3
ParmSet: this is the height of the Merkle tree(s); parameter sets
listed as a single integer have L=1 and consist of a single Merkle
tree of that height; parameter sets with L=2 are listed as x/y,
with x being the height of the top-level Merkle tree and y being
the bottom level.
KeyGenTime: the measured key-generation time; that is, the time
needed to generate the public/private key pair.
SigSize: the size of a signature (in bytes)
KeyLifetime: the lifetime of a key, assuming we generated 1000
signatures per second. In practice, we're not likely to get
anywhere close to 1000 signatures per second sustained; if you
have a more appropriate figure for your scenario, this column is
easy to recompute.
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As for signature generation or verification times, those are
moderately insensitive to the above parameter settings (except for
the Winternitz setting and the number of Merkle trees for
verification). Tests on the same machine (without multithreading)
gave approximately 4 msec to sign a short message, 2.6 msec to
verify; these tests used a two-level ParmSet; a single level would
approximately halve the verification time. All times can be
significantly improved (by perhaps a factor of 8) by using a
parameter set with W=4; however, that also about doubles the
signature size.
7. Rationale
The goal of this note is to describe the LM-OTS, LMS, and HSS
algorithms following the original references and present the modern
security analysis of those algorithms. Other signature methods are
out of scope and may be interesting follow-on work.
We adopt the techniques described by Leighton and Micali to mitigate
attacks that amortize their work over multiple invocations of the
hash function.
The values taken by the identifier I across different LMS public/
private key pairs are chosen randomly in order to improve security.
The analysis of this method in [Fluhrer17] shows that we do not need
uniqueness to ensure security; we do need to ensure that we don't
have a large number of private keys that use the same I value. By
randomly selecting 16-byte I values, the chance that, out of 2^64
private keys, 4 or more of them will use the same I value is
negligible (that is, has probability less than 2^-128).
The reason 16-byte I values were selected was to optimize the
Winternitz hash-chain operation. With the current settings, the
value being hashed is exactly 55 bytes long (for a 32-byte hash
function), which SHA-256 can hash in a single hash-compression
operation. Other hash functions may be used in future
specifications; all the ones that we will be likely to support
(SHA-512/256 and the various SHA-3 hashes) would work well with a
16-byte I value.
The signature and public key formats are designed so that they are
relatively easy to parse. Each format starts with a 32-bit
enumeration value that indicates the details of the signature
algorithm and provides all of the information that is needed in order
to parse the format.
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The Checksum (Section 4.4) is calculated using a nonnegative integer
"sum" whose width was chosen to be an integer number of w-bit fields
such that it is capable of holding the difference of the total
possible number of applications of the function H (as defined in the
signing algorithm of Section 4.5) and the total actual number. In
the case that the number of times H is applied is 0, the sum is (2^w
- 1) * (8*n/w). Thus, for the purposes of this document, which
describes signature methods based on H = SHA256 (n = 32 bytes) and w
= { 1, 2, 4, 8 }, the sum variable is a 16-bit nonnegative integer
for all combinations of n and w. The calculation uses the parameter
ls defined in Section 4.1 and calculated in Appendix B, which
indicates the number of bits used in the left-shift operation.
7.1. Security String
To improve security against attacks that amortize their effort
against multiple invocations of the hash function, Leighton and
Micali introduced a "security string" that is distinct for each
invocation of that function. Whenever this process computes a hash,
the string being hashed will start with a string formed from the
fields below. These fields will appear in fixed locations in the
value we compute the hash of, and so we list where in the hash these
fields would be present. The fields that make up this string are as
I A 16-byte identifier for the LMS public/private key pair. It
MUST be chosen uniformly at random, or via a pseudorandom
process, at the time that a key pair is generated, in order to
minimize the probability that any specific value of I be used
for a large number of different LMS private keys. This is
always bytes 0-15 of the value being hashed.
r In the LMS N-time signature scheme, the node number r
associated with a particular node of a hash tree is used as an
input to the hash used to compute that node. This value is
represented as a 32-bit (four byte) unsigned integer in network
byte order. Either r or q (depending on the domain-separation
parameter) will be bytes 16-19 of the value being hashed.
q In the LMS N-time signature scheme, each LM-OTS signature is
associated with the leaf of a hash tree, and q is set to the
leaf number. This ensures that a distinct value of q is used
for each distinct LM-OTS public/private key pair. This value
is represented as a 32-bit (four byte) unsigned integer in
network byte order. Either r or q (depending on the domain-
separation parameter) will be bytes 16-19 of the value being
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D A domain-separation parameter, which is a two-byte identifier
that takes on different values in the different contexts in
which the hash function is invoked. D occurs in bytes 20 and
21 of the value being hashed and takes on the following values:
D_PBLC = 0x8080 when computing the hash of all of the
iterates in the LM-OTS algorithm
D_MESG = 0x8181 when computing the hash of the message in
the LM-OTS algorithms
D_LEAF = 0x8282 when computing the hash of the leaf of an
LMS tree
D_INTR = 0x8383 when computing the hash of an interior node
of an LMS tree
i A value between 0 and 264; this is used in the LM-OTS scheme
when either computing the iterations of the Winternitz chain or
using the suggested LM-OTS private key generation process. It
is represented as a 16-bit (two-byte) unsigned integer in
network byte order. If present, it occurs at bytes 20 and 21
of the value being hashed.
j In the LM-OTS scheme, j is the iteration number used when the
private key element is being iteratively hashed. It is
represented as an 8-bit (one byte) unsigned integer and is
present if i is a value between 0 and 264. If present, it
occurs at bytes 22 to 21+n of the value being hashed.
C An n-byte randomizer that is included with the message whenever
it is being hashed to improve security. C MUST be chosen
uniformly at random or via another unpredictable process. It
is present if D=D_MESG, and it occurs at bytes 22 to 21+n of
the value being hashed.
8. IANA Considerations
IANA has created two registries: "LM-OTS Signatures", which includes
all of the LM-OTS signatures as defined in Section 4, and "Leighton-
Micali Signatures (LMS)" for LMS as defined in Section 5.
Additions to these registries require that a specification be
documented in an RFC or another permanent and readily available
reference in sufficient detail that interoperability between
independent implementations is possible [RFC8126]. IANA MUST verify
that all applications for additions to these registries have first
been reviewed by the IRTF Crypto Forum Research Group (CFRG).
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Each entry in either of the registries contains the following
a short name (Name), such as "LMS_SHA256_M32_H10",
a positive number (Numeric Identifier), and
a Reference to a specification that completely defines the
signature-method test cases that can be used to verify the
correctness of an implementation.
The numbers between 0xDDDDDDDD (decimal 3,722,304,989) and 0xFFFFFFFF
(decimal 4,294,967,295), inclusive, will not be assigned by IANA and
are reserved for private use; no attempt will be made to prevent
multiple sites from using the same value in different (and
incompatible) ways [RFC8126].
The initial contents of the "LM-OTS Signatures" registry are as
| Name | Reference | Numeric Identifier |
| Reserved | | 0x00000000 |
| | | |
| LMOTS_SHA256_N32_W1 | Section 4 | 0x00000001 |
| | | |
| LMOTS_SHA256_N32_W2 | Section 4 | 0x00000002 |
| | | |
| LMOTS_SHA256_N32_W4 | Section 4 | 0x00000003 |
| | | |
| LMOTS_SHA256_N32_W8 | Section 4 | 0x00000004 |
| | | |
| Unassigned | | 0x00000005 - 0xDDDDDDDC |
| | | |
| Reserved for Private Use | | 0xDDDDDDDD - 0xFFFFFFFF |
Table 4
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The initial contents of the "Leighton Micali Signatures (LMS)"
registry are as follows.
| Name | Reference | Numeric Identifier |
| Reserved | | 0x0 - 0x4 |
| | | |
| LMS_SHA256_M32_H5 | Section 5 | 0x00000005 |
| | | |
| LMS_SHA256_M32_H10 | Section 5 | 0x00000006 |
| | | |
| LMS_SHA256_M32_H15 | Section 5 | 0x00000007 |
| | | |
| LMS_SHA256_M32_H20 | Section 5 | 0x00000008 |
| | | |
| LMS_SHA256_M32_H25 | Section 5 | 0x00000009 |
| | | |
| Unassigned | | 0x0000000A - 0xDDDDDDDC |
| | | |
| Reserved for Private Use | | 0xDDDDDDDD - 0xFFFFFFFF |
Table 5
An IANA registration of a signature system does not constitute an
endorsement of that system or its security.
Currently, the two registries assign a disjoint set of values to the
defined parameter sets. This coincidence is a historical accident;
the correctness of the system does not depend on this. IANA is not
required to maintain this situation.
9. Security Considerations
The hash function H MUST have second preimage resistance: it must be
computationally infeasible for an attacker that is given one message
M to be able to find a second message M' such that H(M) = H(M').
The security goal of a signature system is to prevent forgeries. A
successful forgery occurs when an attacker who does not know the
private key associated with a public key can find a message (distinct
from all previously signed ones) and signature that is valid with
that public key (that is, the Signature Verification algorithm
applied to that signature and message and public key will return
VALID). Such an attacker, in the strongest case, may have the
ability to forge valid signatures for an arbitrary number of other
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LMS is provably secure in the random oracle model, as shown by
[Katz16]. In addition, further analysis is done by [Fluhrer17],
where the hash compression function (rather than the entire hash
function) is considered to be a random oracle. Corollary 1 of the
latter paper states:
If we have no more than 2^64 randomly chosen LMS private keys,
allow the attacker access to a signing oracle and a SHA-256 hash
compression oracle, and allow a maximum of 2^120 hash compression
computations, then the probability of an attacker being able to
generate a single forgery against any of those LMS keys is less
than 2^-129.
Many of the objects within the public key and the signature start
with a typecode. A verifier MUST check each of these typecodes, and
a verification operation on a signature with an unknown type, or a
type that does not correspond to the type within the public key, MUST
return INVALID. The expected length of a variable-length object can
be determined from its typecode; if an object has a different length,
then any signature computed from the object is INVALID.
9.1. Hash Formats
The format of the inputs to the hash function H has the property that
each invocation of that function has an input that is repeated by a
small bounded number of other inputs (due to potential repeats of the
I value). In particular, it will vary somewhere in the first 23
bytes of the value being hashed. This property is important for a
proof of security in the random oracle model.
The formats used during key generation and signing (including the
recommended pseudorandom key-generation procedure in Appendix A) are
as follows:
I || u32str(q) || u16str(i) || u8str(j) || tmp
I || u32str(q) || u16str(D_PBLC) || y[0] || ... || y[p-1]
I || u32str(q) || u16str(D_MESG) || C || message
I || u32str(r) || u16str(D_LEAF) || OTS_PUB_HASH[r-2^h]
I || u32str(r) || u16str(D_INTR) || T[2*r] || T[2*r+1]
I || u32str(q) || u16str(i) || u8str(0xff) || SEED
Each hash type listed is distinct; at locations 20 and 21 of the
value being hashed, there exists either a fixed value D_PBLC, D_MESG,
D_LEAF, D_INTR, or a 16-bit value i. These fixed values are distinct
from each other and are large (over 32768), while the 16-bit values
of i are small (currently no more than 265; possibly being slightly
larger if larger hash functions are supported); hence, the range of
possible values of i will not collide any of the D_PBLC, D_MESG,
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D_LEAF, D_INTR identifiers. The only other collision possibility is
the Winternitz chain hash colliding with the recommended pseudorandom
key-generation process; here, at location 22 of the value being
hashed, the Winternitz chain function has the value u8str(j), where j
is a value between 0 and 254, while location 22 of the recommended
pseudorandom key-generation process has value 255.
For the Winternitz chaining function, D_PBLC, and D_MESG, the value
of I || u32str(q) is distinct for each LMS leaf (or equivalently, for
each q value). For the Winternitz chaining function, the value of
u16str(i) || u8str(j) is distinct for each invocation of H for a
given leaf. For D_PBLC and D_MESG, the input format is used only
once for each value of q and, thus, distinctness is assured. The
formats for D_INTR and D_LEAF are used exactly once for each value of
r, which ensures their distinctness. For the recommended
pseudorandom key-generation process, for a given value of I, q and j
are distinct for each invocation of H.
The value of I is chosen uniformly at random from the set of all
128-bit strings. If 2^64 public keys are generated (and, hence, 2^64
random I values), there is a nontrivial probability of a duplicate
(which would imply duplicate prefixes). However, there will be an
extremely high probability there will not be a four-way collision
(that is, any I value used for four distinct LMS keys; probability <
2^-132), and, hence, the number of repeats for any specific prefix
will be limited to at most three. This is shown (in [Fluhrer17]) to
have only a limited effect on the security of the system.
9.2. Stateful Signature Algorithm
The LMS signature system, like all N-time signature systems, requires
that the signer maintain state across different invocations of the
signing algorithm to ensure that none of the component one-time
signature systems are used more than once. This section calls out
some important practical considerations around this statefulness.
These issues are discussed in greater detail in [STMGMT].
In a typical computing environment, a private key will be stored in
nonvolatile media such as on a hard drive. Before it is used to sign
a message, it will be read into an application's Random-Access Memory
(RAM). After a signature is generated, the value of the private key
will need to be updated by writing the new value of the private key
into nonvolatile storage. It is essential for security that the
application ensures that this value is actually written into that
storage, yet there may be one or more memory caches between it and
the application. Memory caching is commonly done in the file system
and in a physical memory unit on the hard disk that is dedicated to
that purpose. To ensure that the updated value is written to
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physical media, the application may need to take several special
steps. In a POSIX environment, for instance, the O_SYNC flag (for
the open() system call) will cause invocations of the write() system
call to block the calling process until the data has been written to
the underlying hardware. However, if that hardware has its own
memory cache, it must be separately dealt with using an operating
system or device-specific tool such as hdparm to flush the on-drive
cache or turn off write caching for that drive. Because these
details vary across different operating systems and devices, this
note does not attempt to provide complete guidance; instead, we call
the implementer's attention to these issues.
When hierarchical signatures are used, an easy way to minimize the
private key synchronization issues is to have the private key for the
second-level resident in RAM only and never write that value into
nonvolatile memory. A new second-level public/private key pair will
be generated whenever the application (re)starts; thus, failures such
as a power outage or application crash are automatically
accommodated. Implementations SHOULD use this approach wherever
9.3. Security of LM-OTS Checksum
To show the security of LM-OTS checksum, we consider the signature y
of a message with a private key x and let h = H(message) and
c = Cksm(H(message)) (see Section 4.5). To attempt a forgery, an
attacker may try to change the values of h and c. Let h' and c'
denote the values used in the forgery attempt. If for some integer j
in the range 0 to u, where u = ceil(8*n/w) is the size of the range
that the checksum value can cover, inclusive,
a' = coef(h', j, w),
a = coef(h, j, w), and
a' > a
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then the attacker can compute F^a'(x[j]) from F^a(x[j]) = y[j] by
iteratively applying function F to the j-th term of the signature an
additional (a' - a) times. However, as a result of the increased
number of hashing iterations, the checksum value c' will decrease
from its original value of c. Thus, a valid signature's checksum
will have, for some number k in the range u to (p-1), inclusive,
b' = coef(c', k, w),
b = coef(c, k, w), and
b' < b
Due to the one-way property of F, the attacker cannot easily compute
F^b'(x[k]) from F^b(x[k]) = y[k].
10. Comparison with Other Work
The eXtended Merkle Signature Scheme (XMSS) is similar to HSS in
several ways [XMSS][RFC8391]. Both are stateful hash-based signature
schemes, and both use a hierarchical approach, with a Merkle tree at
each level of the hierarchy. XMSS signatures are slightly shorter
than HSS signatures, for equivalent security and an equal number of
HSS has several advantages over XMSS. HSS operations are roughly
four times faster than the comparable XMSS ones, when SHA256 is used
as the underlying hash. This occurs because the hash operation done
as a part of the Winternitz iterations dominates performance, and
XMSS performs four compression-function invocations (two for the PRF,
two for the F function) where HSS only needs to perform one.
Additionally, HSS is somewhat simpler (as each hash invocation is
just a prefix followed by the data being hashed).
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11. References
11.1. Normative References
[FIPS180] National Institute of Standards and Technology, "Secure
Hash Standard (SHS)", FIPS PUB 180-4,
DOI 10.6028/NIST.FIPS.180-4, March 2012.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
[RFC4506] Eisler, M., Ed., "XDR: External Data Representation
Standard", STD 67, RFC 4506, DOI 10.17487/RFC4506, May
2006, <https://www.rfc-editor.org/info/rfc4506>.
[RFC8126] Cotton, M., Leiba, B., and T. Narten, "Guidelines for
Writing an IANA Considerations Section in RFCs", BCP 26,
RFC 8126, DOI 10.17487/RFC8126, June 2017,
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/info/rfc8174>.
[RFC8179] Bradner, S. and J. Contreras, "Intellectual Property
Rights in IETF Technology", BCP 79, RFC 8179,
DOI 10.17487/RFC8179, May 2017,
Leighton, T. and S. Micali, "Large provably fast and
secure digital signature schemes based on secure hash
functions", U.S. Patent 5,432,852, July 1995.
11.2. Informative References
Merkle, R., "A Digital Signature Based on a Conventional
Encryption Function", in Advances in Cryptology -- CRYPTO
'87 Proceedings, Lecture Notes in Computer Science Vol.
293, DOI 10.1007/3-540-48184-2_32, 1988.
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RFC 8554 LMS Hash-Based Signatures April 2019
Merkle, R., "A Certified Digital Signature", in Advances
in Cryptology -- CRYPTO '89 Proceedings, Lecture Notes in
Computer Science Vol. 435, DOI 10.1007/0-387-34805-0_21,
Merkle, R., "One Way Hash Functions and DES", in Advances
in Cryptology -- CRYPTO '89 Proceedings, Lecture Notes in
Computer Science Vol. 435, DOI 10.1007/0-387-34805-0_40,
Fluhrer, S., "Further Analysis of a Proposed Hash-Based
Signature Standard", Cryptology ePrint Archive Report
2017/553, 2017, <https://eprint.iacr.org/2017/553>.
[Katz16] Katz, J., "Analysis of a Proposed Hash-Based Signature
Standard", in SSR 2016: Security Standardisation Research
(SSR) pp. 261-273, Lecture Notes in Computer Science Vol.
10074, DOI 10.1007/978-3-319-49100-4_12, 2016.
Merkle, R., "Secrecy, Authentication, and Public Key
Systems", Technical Report No. 1979-1, Information Systems
Laboratory, Stanford University, 1979,
[RFC8391] Huelsing, A., Butin, D., Gazdag, S., Rijneveld, J., and A.
Mohaisen, "XMSS: eXtended Merkle Signature Scheme",
RFC 8391, DOI 10.17487/RFC8391, May 2018,
[STMGMT] McGrew, D., Kampanakis, P., Fluhrer, S., Gazdag, S.,
Butin, D., and J. Buchmann, "State Management for Hash-
Based Signatures.", in SSR 2016: Security Standardisation
Research (SSR) pp. 244-260, Lecture Notes in Computer
Science Vol. 10074, DOI 10.1007/978-3-319-49100-4_11,
[XMSS] Buchmann, J., Dahmen, E., and , "XMSS -- A Practical
Forward Secure Signature Scheme Based on Minimal Security
Assumptions.", in PQCrypto 2011: Post-Quantum Cryptography
pp. 117-129, Lecture Notes in Computer Science Vol. 7071,
DOI 10.1007/978-3-642-25405-5_8, 2011.
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Appendix A. Pseudorandom Key Generation
An implementation MAY use the following pseudorandom process for
generating an LMS private key.
SEED is an m-byte value that is generated uniformly at random at
the start of the process,
I is the LMS key pair identifier,
q denotes the LMS leaf number of an LM-OTS private key,
x_q denotes the x array of private elements in the LM-OTS private
key with leaf number q,
i is the index of the private key element, and
H is the hash function used in LM-OTS.
The elements of the LM-OTS private keys are computed as:
x_q[i] = H(I || u32str(q) || u16str(i) || u8str(0xff) || SEED).
This process stretches the m-byte random value SEED into a (much
larger) set of pseudorandom values, using a unique counter in each
invocation of H. The format of the inputs to H are chosen so that
they are distinct from all other uses of H in LMS and LM-OTS. A
careful reader will note that this is similar to the hash we perform
when iterating through the Winternitz chain; however, in that chain,
the iteration index will vary between 0 and 254 maximum (for W=8),
while the corresponding value in this formula is 255. This algorithm
is included in the proof of security in [Fluhrer17] and hence this
method is safe when used within the LMS system; however, any other
cryptographically secure method of generating private keys would also
be safe.
Appendix B. LM-OTS Parameter Options
The LM-OTS one-time signature method uses several internal
parameters, which are a function of the selected parameter set.
These internal parameters include the following:
p This is the number of independent Winternitz chains used in the
signature; it will be the number of w-bit digits needed to hold
the n-bit hash (u in the below equations), along with the
number of digits needed to hold the checksum (v in the below
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ls This is the size of the shift needed to move the checksum so
that it appears in the checksum digits
ls is needed because, while we express the checksum internally as a
16-bit value, we don't always express all 16 bits in the signature;
for example, if w=4, we might use only the top 12 bits. Because we
read the checksum in network order, this means that, without the
shift, we'll use the higher-order bits (which may be always 0) and
omit the lower-order bits (where the checksum value actually
resides). This shift is here to ensure that the parts of the
checksum we need to express (for security) actually contribute to the
signature; when multiple such shifts are possible, we take the
minimal value.
The parameters ls and p are computed as follows:
u = ceil(8*n/w)
v = ceil((floor(lg((2^w - 1) * u)) + 1) / w)
ls = 16 - (v * w)
p = u + v
Here, u and v represent the number of w-bit fields required to
contain the hash of the message and the checksum byte strings,
respectively. And as the value of p is the number of w-bit elements
of ( H(message) || Cksm(H(message)) ), it is also equivalently the
number of byte strings that form the private key and the number of
byte strings in the signature. The value 16 in the ls computation of
ls corresponds to the 16-bit value used for the sum variable in
Algorithm 2 in Section 4.4
A table illustrating various combinations of n and w with the
associated values of u, v, ls, and p is provided in Table 6.
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| Hash | Winternitz | w-bit | w-bit | Left | Total |
| Length | Parameter | Elements | Elements | Shift | Number of |
| in | (w) | in Hash | in | (ls) | w-bit |
| Bytes | | (u) | Checksum | | Elements |
| (n) | | | (v) | | (p) |
| 32 | 1 | 256 | 9 | 7 | 265 |
| | | | | | |
| 32 | 2 | 128 | 5 | 6 | 133 |
| | | | | | |
| 32 | 4 | 64 | 3 | 4 | 67 |
| | | | | | |
| 32 | 8 | 32 | 2 | 0 | 34 |
Table 6
Appendix C. An Iterative Algorithm for Computing an LMS Public Key
The LMS public key can be computed using the following algorithm or
any equivalent method. The algorithm uses a stack of hashes for
data. It also makes use of a hash function with the typical
init/update/final interface to hash functions; the result of the
invocations hash_init(), hash_update(N[1]), hash_update(N[2]), ... ,
hash_update(N[n]), v = hash_final(), in that order, is identical to
that of the invocation of H(N[1] || N[2] || ... || N[n]).
Generating an LMS Public Key from an LMS Private Key
for ( i = 0; i < 2^h; i = i + 1 ) {
r = i + num_lmots_keys;
temp = H(I || u32str(r) || u16str(D_LEAF) || OTS_PUB_HASH[i])
j = i;
while (j % 2 == 1) {
r = (r - 1)/2;
j = (j-1) / 2;
left_side = pop(data stack);
temp = H(I || u32str(r) || u16str(D_INTR) || left_side || temp)
push temp onto the data stack
public_key = pop(data stack)
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Note that this pseudocode expects that all 2^h leaves of the tree
have equal depth -- that is, it expects num_lmots_keys to be a power
of 2. The maximum depth of the stack will be h-1 elements -- that
is, a total of (h-1)*n bytes; for the currently defined parameter
sets, this will never be more than 768 bytes of data.
Appendix D. Method for Deriving Authentication Path for a Signature
The LMS signature consists of u32str(q) || lmots_signature ||
u32str(type) || path[0] || path[1] || ... || path[h-1]. This
appendix shows one method of constructing this signature, assuming
that the implementation has stored the T[] array that was used to
construct the public key. Note that this is not the only possible
method; other methods exist that don't assume that you have the
entire T[] array in memory. To construct a signature, you perform
the following algorithm:
Generating an LMS Signature
1. Set type to the typecode of the LMS algorithm.
2. Extract h from the typecode, according to Table 2.
3. Create the LM-OTS signature for the message:
ots_signature = lmots_sign(message, LMS_PRIV[q])
4. Compute the array path as follows:
i = 0
r = 2^h + q
while (i < h) {
temp = (r / 2^i) xor 1
path[i] = T[temp]
i = i + 1
5. S = u32str(q) || ots_signature || u32str(type) ||
path[0] || path[1] || ... || path[h-1]
6. q = q + 1
7. Return S.
Here "xor" is the bitwise exclusive-or operation, and / is integer
division (that is, rounded down to an integer value).
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Appendix E. Example Implementation
An example implementation can be found online at
Appendix F. Test Cases
This section provides test cases that can be used to verify or debug
an implementation. This data is formatted with the name of the
elements on the left and the hexadecimal value of the elements on the
right. The concatenation of all of the values within a public key or
signature produces that public key or signature, and values that do
not fit within a single line are listed across successive lines.
Test Case 1 Public Key
HSS public key
levels 00000002
LMS type 00000005 # LM_SHA256_M32_H5
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
I 61a5d57d37f5e46bfb7520806b07a1b8
K 50650e3b31fe4a773ea29a07f09cf2ea
Test Case 1 Message
Message 54686520706f77657273206e6f742064 |The powers not d|
656c65676174656420746f2074686520 |elegated to the |
556e6974656420537461746573206279 |United States by|
2074686520436f6e737469747574696f | the Constitutio|
6e2c206e6f722070726f686962697465 |n, nor prohibite|
6420627920697420746f207468652053 |d by it to the S|
74617465732c20617265207265736572 |tates, are reser|
76656420746f20746865205374617465 |ved to the State|
7320726573706563746976656c792c20 |s respectively, |
6f7220746f207468652070656f706c65 |or to the people|
2e0a |..|
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Test Case 1 Signature
HSS signature
Nspk 00000001
LMS signature
q 00000005
LMOTS signature
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
C d32b56671d7eb98833c49b433c272586
y[0] 965a25bfd37f196b9073f3d4a232feb6
y[1] a64c7f60f6261a62043f86c70324b770
y[2] e05fd5c6509a6e61d559cf1a77a970de
y[3] 582e8ff1b10cd99d4e8e413ef469559f
y[4] 81d84b15357ff48ca579f19f5e71f184
y[5] 14784269d7d876f5d35d3fbfc7039a46
y[6] 60b960e7777c52f060492f2d7c660e14
y[7] c3943c6b9c4f2405a3cb8bf8a691ca51
y[8] f0a75ee390e385e3ae0b906961ecf41a
y[9] 35b167b28ce8dc988a3748255230cef9
y[10] e783ed04516de012498682212b078105
y[11] aaf65de7620dabec29eb82a17fde35af
y[12] 1099762b37f43c4a3c20010a3d72e2f6
y[13] a1a40281cc5a7ea98d2adc7c7400c2fe
y[14] 9cbbc68fee0c3efe4ec22b83a2caa3e4
y[15] 4f8a58f7f24335eec5c5eb5e0cf01dcf
y[16] c5b9f64a2a9af2f07c05e99e5cf80f00
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y[17] 26857713afd2ca6bb85cd8c107347552
y[18] c413e7d0acd8bdd81352b2471fc1bc4f
y[19] cf7cc62fb92be14f18c2192384ebceaf
y[20] e87b0144417e8d7baf25eb5f70f09f01
y[21] da67571f5dd546fc22cb1f97e0ebd1a6
y[22] 115cce6f792cc84e36da58960c5f1d76
y[23] 1efc72d60ca5e908b3a7dd69fef02491
y[24] c75e13527b7a581a556168783dc1e975
y[25] 8d3ee2062445dfb85ef8c35f8e1f3371
y[26] ab8f5c612ead0b729a1d059d02bfe18e
y[27] eec0f3f3f13039a17f88b0cf808f4884
y[28] 4f1f4ab949b9feefadcb71ab50ef27d6
y[29] 9b6066f09c37280d59128d2f0f637c7d
y[30] b7c878c9411cafc5071a34a00f4cf077
y[31] d76f7ce973e9367095ba7e9a3649b7f4
y[32] 401b64457c54d65fef6500c59cdfb69a
y[33] b0f3f79cd893d314168648499898fbc0
LMS type 00000005 # LM_SHA256_M32_H5
path[0] d8b8112f9200a5e50c4a262165bd342c
path[1] 129ac6eda839a6f357b5a04387c5ce97
path[2] 12f5dbe400bd49e4501e859f885bf073
path[3] b5971115aa39efd8d564a6b90282c316
path[4] 4cca1848cf7da59cc2b3d9d0692dd2a2
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LMS public key
LMS type 00000005 # LM_SHA256_M32_H5
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
I d2f14ff6346af964569f7d6cb880a1b6
K 6c5004917da6eafe4d9ef6c6407b3db0
LMS signature
q 0000000a
LMOTS signature
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
C 0703c491e7558b35011ece3592eaa5da
y[0] 95cae05b899e35dffd71705470620998
y[1] 9bc042da4b4525650485c66d0ce19b31
y[2] 6a120c5612344258b85efdb7db1db9e1
y[3] 9eeddb03a1d2374af7bf771855774562
y[4] a698994c0827d90e86d43e0df7f4bfcd
y[5] 100e4f2c5fc38c003c1ab6fea479eb2f
y[6] 969f6aecbfe44cf356888a7b15a3ff07
y[7] e61af23aee7fa5d4d9a5dfcf43c4c26c
y[8] beadb2b25b3cacc1ac0cef346cbb90fb
y[9] 319c9944b1586e899d431c7f91bcccc8
y[10] f30b2b51f48b71b003dfb08249484201
y[11] 0081262a00000480dcbc9a3da6fbef5c
y[12] 9db268f6fe50032a363c9801306837fa
y[13] dfd836a28b354023924b6fb7e48bc0b3
y[14] 91825daef01eae3c38e3328d00a77dc6
y[15] 205e4737b84b58376551d44c12c3c215
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y[16] 327f0a5fbb6b5907dec02c9a90934af5
y[17] b45696689f2eb382007497557692caac
y[18] 664fcb6db4971f5b3e07aceda9ac130e
y[19] f3fe00812589b7a7ce51544045643301
y[20] f2d8b584410ceda8025f5d2d8dd0d217
y[21] ce780fd025bd41ec34ebff9d4270a322
y[22] 89cc10cd600abb54c47ede93e08c114e
y[23] 929d15462b939ff3f52f2252da2ed64d
y[24] aa233db3162833141ea4383f1a6f120b
y[25] 234d475e2f79cbf05e4db6a9407d72c6
y[26] 715a0182c7dc8089e32c8531deed4f74
y[27] ae0c066babc69369700e1dd26eddc0d2
y[28] 49ef23be2aa4dbf25206fe45c20dd888
y[29] 12858792bf8e74cba49dee5e8812e019
y[30] 82f880a278f682c2bd0ad6887cb59f65
y[31] 656d9ccbaae3d655852e38deb3a2dcf8
y[32] 1091d05eb6e2f297774fe6053598457c
y[33] e865aa805009cc2918d9c2f840c4da43
LMS type 00000005 # LM_SHA256_M32_H5
path[0] d5c0d1bebb06048ed6fe2ef2c6cef305
path[1] e1920ada52f43d055b5031cee6192520
path[2] 2335b525f484e9b40d6a4a969394843b
path[3] f90b65a7a6201689999f32bfd368e5e3
path[4] 09ab3034911fe125631051df0408b394
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Test Case 2 Private Key
(note: procedure in Appendix A is used)
Top level LMS tree
SEED 558b8966c48ae9cb898b423c83443aae
I d08fabd4a2091ff0a8cb4ed834e74534
Second level LMS tree
SEED a1c4696e2608035a886100d05cd99945
I 215f83b7ccb9acbcd08db97b0d04dc2b
Test Case 2 Public Key
HSS public key
levels 00000002
LMS type 00000006 # LM_SHA256_M32_H10
LMOTS type 00000003 # LMOTS_SHA256_N32_W4
I d08fabd4a2091ff0a8cb4ed834e74534
K 32a58885cd9ba0431235466bff9651c6
Test Case 2 Message
Message 54686520656e756d65726174696f6e20 |The enumeration |
696e2074686520436f6e737469747574 |in the Constitut|
696f6e2c206f66206365727461696e20 |ion, of certain |
7269676874732c207368616c6c206e6f |rights, shall no|
7420626520636f6e7374727565642074 |t be construed t|
6f2064656e79206f7220646973706172 |o deny or dispar|
616765206f7468657273207265746169 |age others retai|
6e6564206279207468652070656f706c |ned by the peopl|
652e0a |e..|
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Test Case 2 Signature
HSS signature
Nspk 00000001
LMS signature
q 00000003
LMOTS signature
LMOTS type 00000003 # LMOTS_SHA256_N32_W4
C 3d46bee8660f8f215d3f96408a7a64cf
y[0] 0674e8cb7a55f0c48d484f31f3aa4af9
y[1] bb71226d279700ec81c9e95fb11a0d10
y[2] 4508e126a9a7870bf4360820bdeb9a01
y[3] 0ac8ba39810909d445f44cb5bb58de73
y[4] 9edeaa3bfcfe8baa6621ce88480df237
y[5] 49a18d39a50788f4652987f226a1d481
y[6] 90da8aa5e5f7671773e941d805536021
y[7] 54e7b1e1bf494d0d1a28c0d31acc7516
y[8] ed30872e07f2b8bd0374eb57d22c614e
y[9] 2988ab46eaca9ec597fb18b4936e66ef
y[10] 0c7b345434f72d65314328bbb030d0f0
y[11] be3b2adb83c60a54f9d1d1b2f476f9e3
y[12] dcc21033f9453d49c8e5a6387f588b1e
y[13] 56a326a32f9cba1fbe1c07bb49fa04ce
y[14] 238e5ea986b53e087045723ce16187ed
y[15] 45fc8c0693e97763928f00b2e3c75af3
y[16] 11873b59137f67800b35e81b01563d18
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y[17] 05d357ef4678de0c57ff9f1b2da61dfd
y[18] 40dd7739ca3ef66f1930026f47d9ebaa
y[19] 375dbfb83d719b1635a7d8a138919579
y[20] a6c0a555c9026b256a6860f4866bd6d0
y[21] 622442443d5eca959d6c14ca8389d12c
y[22] 558f249c9661c0427d2e489ca5b5dde2
y[23] 8266c12c50ea28b2c438e7a379eb106e
y[24] b76e8027992e60de01e9094fddeb3349
y[25] 8278c14b032bcab02bd15692d21b6c5c
y[26] 5d33b10d518a61e15ed0f092c3222628
y[27] a375cebda1dc6bb9a1a01dae6c7aba8e
y[28] 2949dcc198fb77c7e5cdf6040b0f84fa
y[29] ae8a270e951743ff23e0b2dd12e9c3c8
y[30] 20c4591f71c088f96e095dd98beae456
y[31] 73217ac5962b5f3147b492e8831597fd
y[32] 435eb3109350756b9fdabe1c6f368081
y[33] b1bab705a4b7e37125186339464ad8fa
y[34] 0eb1fcbfcc25acb5f718ce4f7c2182fb
y[35] 6e90d4c9b0cc38608a6cef5eb153af08
y[36] 9313d28d41a5c6fe6cf3595dd5ee63f0
y[37] f88dd73720708c6c6c0ecf1f43bbaada
y[38] 42761c70c186bfdafafc444834bd3418
y[39] ffd5960b0336981795721426803599ed
y[40] 9e3fa152d9adeca36020fdeeee1b7395
McGrew, et al. Informational [Page 56]
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y[41] 94a873670b8d93bcca2ae47e64424b74
y[42] fa5b9510beb39ccf4b4e1d9c0f19d5e1
y[43] 7af256a8491671f1f2f22af253bcff54
y[44] d0be7919684b23da8d42ff3effdb7ca0
y[45] c0a614d31cc7487f52de8664916af79c
y[46] 2250274a1de2584fec975fb09536792c
y[47] 301ddff26ec1b23de2d188c999166c74
y[48] 50f4d646fc6278e8fe7eb6cb5c94100f
y[49] 7fa7d5cc861c5bdac98e7495eb0a2cee
y[50] 1287d978b8df064219bc5679f7d7b264
y[51] 8240027afd9d52a79b647c90c2709e06
y[52] d839f851f98f67840b964ebe73f8cec4
y[53] da93d9f5f6fa6f6c0f03ce43362b8414
y[54] bc85a3ff51efeea3bc2cf27e1658f178
y[55] beeecaa04dccea9f97786001475e0294
y[56] 4a662ecae37ede27e9d6eadfdeb8f8b2
y[57] 29c2f4dcd153a2742574126e5eaccc77
y[58] 05ff5453ec99897b56bc55dd49b99114
y[59] cc5a8a335d3619d781e7454826df720e
y[60] 057fa3419b5bb0e25d30981e41cb1361
y[61] 8bfc3d20a2148861b2afc14562ddd27f
y[62] 46a24bf77e383c7aacab1ab692b29ed8
y[63] b1c78725c1f8f922f6009787b1964247
y[64] d4a8b6f04d95c581279a139be09fcf6e
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RFC 8554 LMS Hash-Based Signatures April 2019
y[65] c05518a7efd35d89d8577c990a5e1996
y[66] 294546454fa5388a23a22e805a5ca35f
LMS type 00000006 # LM_SHA256_M32_H10
path[0] b326493313053ced3876db9d23714818
path[1] a769db4657a103279ba8ef3a629ca84e
path[2] 0b491cb4ecbbabec128e7c81a46e62a6
path[3] 686d16621a80816bfdb5bdc56211d72c
path[4] 7028a48538ecdd3b38d3d5d62d262465
path[5] 2e0c19bc4977c6898ff95fd3d310b0ba
path[6] 83bb7543c675842bafbfc7cdb88483b3
path[7] d045851acf6a0a0ea9c710b805cced46
path[8] 6703d26d14752f34c1c0d2c4247581c1
path[9] a415e291fd107d21dc1f084b11582082
LMS public key
LMS type 00000005 # LM_SHA256_M32_H5
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
I 215f83b7ccb9acbcd08db97b0d04dc2b
K a1cd035833e0e90059603f26e07ad2aa
LMS signature
q 00000004
LMOTS signature
LMOTS type 00000004 # LMOTS_SHA256_N32_W8
C 0eb1ed54a2460d512388cad533138d24
y[0] 11b3649023696f85150b189e50c00e98
y[1] 3b7714fa406b8c35b021d54d4fdada7b
McGrew, et al. Informational [Page 58]
RFC 8554 LMS Hash-Based Signatures April 2019
y[2] 057aa0e2e74e7dcfd17a0823429db629
y[3] 83cac8b4d61aacc457f336e6a10b6632
y[4] c6bf59daa82afd2b5ebb2a9ca6572a60
y[5] df927aade10c1c9f2d5ff446450d2a39
y[6] 8190643978d7a7f4d64e97e3f1c4a08a
y[7] 2f190143475a6043d5e6d5263471f4ee
y[8] f797fd5a3cd53a066700f45863f04b6c
y[9] 8b636ae547c1771368d9f317835c9b0e
y[10] bc7a5cf8a5abdb12dc718b559b74cab9
y[11] 67df540890a062fe40dba8b2c1c548ce
y[12] 862f4a24ebd376d288fd4e6fb06ed870
y[13] 767b14ce88409eaebb601a93559aae89
y[14] fbe549147f71c092f4f3ac522b5cc572
y[15] 1ead87ac01985268521222fb9057df7e
y[16] c2ce956c365ed38e893ce7b2fae15d36
y[17] a8ade980ad0f93f6787075c3f680a2ba
y[18] 8d64c3d3d8582968c2839902229f85ae
y[19] 7bf0f4ff3ffd8fba5e383a48574802ed
y[20] 135a7ce517279cd683039747d218647c
y[21] 9547b830d8118161b65079fe7bc59a99
y[22] 2b698d09ae193972f27d40f38dea264a
y[23] eb0d4029ac712bfc7a5eacbdd7518d6d
y[24] 4fd7214dc617c150544e423f450c99ce
y[25] a3bb86da7eba80b101e15cb79de9a207
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y[26] 25d1faa94cbb0a03a906f683b3f47a97
y[27] 496152a91c2bf9da76ebe089f4654877
y[28] 2429b9e8cb4834c83464f079995332e4
y[29] 952c0b7420df525e37c15377b5f09843
y[30] 3de3afad5733cbe7703c5296263f7734
y[31] aa2de3ffdcd297baaaacd7ae646e44b5
y[32] 982fb2e370c078edb042c84db34ce36b
y[33] 97ec8075e82b393d542075134e2a17ee
LMS type 00000005 # LM_SHA256_M32_H5
path[0] 4de1f6965bdabc676c5a4dc7c35f97f8
path[1] e96aeee300d1f68bf1bca9fc58e40323
path[2] 04d341aa0a337b19fe4bc43c2e79964d
path[3] b0f75be80ea3af098c9752420a8ac0ea
path[4] e4041d95398a6f7f3e0ee97cc1591849
Thanks are due to Chirag Shroff, Andreas Huelsing, Burt Kaliski, Eric
Osterweil, Ahmed Kosba, Russ Housley, Philip Lafrance, Alexander
Truskovsky, Mark Peruzel, and Jim Schaad for constructive suggestions
and valuable detailed review. We especially acknowledge Jerry
Solinas, Laurie Law, and Kevin Igoe, who pointed out the security
benefits of the approach of Leighton and Micali [USPTO5432852],
Jonathan Katz, who gave us security guidance, and Bruno Couillard and
Jim Goodman for an especially thorough review.
McGrew, et al. Informational [Page 60]
RFC 8554 LMS Hash-Based Signatures April 2019
Authors' Addresses
David McGrew
Cisco Systems
13600 Dulles Technology Drive
Herndon, VA 20171
United States of America
Email: mcgrew@cisco.com
Michael Curcio
Cisco Systems
7025-2 Kit Creek Road
Research Triangle Park, NC 27709-4987
United States of America
Email: micurcio@cisco.com
Scott Fluhrer
Cisco Systems
170 West Tasman Drive
San Jose, CA
United States of America
Email: sfluhrer@cisco.com
McGrew, et al. Informational [Page 61] | {"url":"https://cdn.sessionspy.com:443/index.php?https://www.rfc-editor.org/rfc/rfc8554.html","timestamp":"2024-11-13T22:09:19Z","content_type":"application/xhtml+xml","content_length":"173909","record_id":"<urn:uuid:19f71f96-0699-469c-9e0c-a93d59d29f50>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00743.warc.gz"} |
Angle interpolation is broken when rotated on y axis
Hello! I am currently working on a placement system that is fully customizable for the user. The problem I am having is with a angle lerp feature. If you’ve used tunicus’s module you’ll know what I
mean. If you haven’t, here is an example:
That clip was from my module as the problem presents when you rotate the model. As soon as the model is rotating on the y axis, it messes stuff up:
Currently I am using this function to calculate the angle:
local function calculateAngle(a, b)
-- a is the the distance the mouse is away from the model
-- b is the models position
-- the function is called twice to handle both x and z rotation
local x1, z1 = cos(a), sin(a)
local x2, z2 = cos(b), sin(b)
return atan2(z2 - z1, x2 - x1)*angleInterpolationAmount
My question is, is there a way to make it work even when rotation the y axis. I am really not good with math so I have no idea how any of this works.
1 Like
I’m so confused why you would use the distance as the radian its the ratio between the side lengths which is passed in. For cosine that’d be the adjacent and hypotenuse relative to the angle and if
we just follow soh cah toa its the same for the rest. Try applying this and see if it helps
See you actually understand this. I just tried to copy the way another script did it and got this. I’ll see what I can do though.
I know the last reply was last year, but for any future readers, I’ve solved this issue. The function that controls the angle calculation is this:
-- Calculates the "tilt" angle
local function calcAngle(last, current) : CFrame
if angleTilt then
-- Calculates and clamps the proper angle amount
local tiltX = (math.clamp((last.X - current.X), -10, 10)*pi/180)*amplitude
local tiltZ = (math.clamp((last.Z - current.Z), -10, 10)*pi/180)*amplitude
-- Returns the proper angle based on rotation
return (anglesXYZ(dirZ*tiltZ, 0, dirX*tiltX):Inverse()*anglesXYZ(0, rot*pi/180, 0)):Inverse()*anglesXYZ(0, rot*pi/180, 0)
return anglesXYZ(0, 0, 0)
You can see how it’s used in the module.
1 Like | {"url":"https://devforum.roblox.com/t/angle-interpolation-is-broken-when-rotated-on-y-axis/719874","timestamp":"2024-11-07T14:17:00Z","content_type":"text/html","content_length":"31099","record_id":"<urn:uuid:22b1c31d-9485-4289-9b21-3b70ec5eb13c>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00668.warc.gz"} |
How do you calculate regression in Excel?
How do you calculate regression in Excel?
To run the regression, arrange your data in columns as seen below. Click on the “Data” menu, and then choose the “Data Analysis” tab. You will now see a window listing the various statistical tests
that Excel can perform. Scroll down to find the regression option and click “OK”.
How do you calculate regression analysis?
Regression analysis is the analysis of relationship between dependent and independent variable as it depicts how dependent variable will change when one or more independent variable changes due to
factors, formula for calculating it is Y = a + bX + E, where Y is dependent variable, X is independent variable, a is …
What do the regression statistics mean in Excel?
It tells you how many points fall on the regression line. for example, 80% means that 80% of the variation of y-values around the mean are explained by the x-values. In other words, 80% of the values
fit the model.
How do you find data analysis on Excel?
Q. Where is the data analysis button in Excel?
1. Click the File tab, click Options, and then click the Add-Ins category.
2. In the Manage box, select Excel Add-ins and then click Go.
3. In the Add-Ins available box, select the Analysis ToolPak check box, and then click OK.
What is the formula for calculating regression?
Regression analysis is the analysis of relationship between dependent and independent variable as it depicts how dependent variable will change when one or more independent variable changes due to
factors, formula for calculating it is Y = a + bX + E, where Y is dependent variable, X is independent variable, a is intercept, b is slope and E is residual.
How do I plot a regression line in Excel?
We can chart a regression in Excel by highlighting the data and charting it as a scatter plot. To add a regression line, choose “Layout” from the “Chart Tools” menu. In the dialog box, select
“Trendline” and then “Linear Trendline”.
How do you calculate linear regression in Excel?
Linear regression equation. Mathematically, a linear regression is defined by this equation: y = bx + a + ε. Where: x is an independent variable. y is a dependent variable. a is the Y-intercept,
which is the expected mean value of y when all x variables are equal to 0.
How do you run multiple regression in Excel?
How to Do a Multiple Regression in Excel. You can perform a multivariate regression in Excel using a built-in function that is accessible through the Data Analysis tool under the Data tab and the
Analysis group. Click Data Analysis and find the option for regression in the window that pops up, highlight it and click OK. | {"url":"https://morethingsjapanese.com/how-do-you-calculate-regression-in-excel/","timestamp":"2024-11-14T17:08:07Z","content_type":"text/html","content_length":"129753","record_id":"<urn:uuid:b1fe13b1-8db3-43fe-b0b1-6f8b48b02807>","cc-path":"CC-MAIN-2024-46/segments/1730477393980.94/warc/CC-MAIN-20241114162350-20241114192350-00433.warc.gz"} |
RSA and ElGamal Encryption
06. RSA and ElGamal Encryption
Exponential Inverses
Suppose we are given integers $a$ and $N$. For any integer $x$ that is relatively prime to $N$, we choose $b$ so that
\[\tag{$*$} ab \equiv 1 \pmod{\phi(N)}.\]
Then we have
\[x^{ab} \equiv x^{1 + k\phi(N)} \equiv x \pmod N\]
by Euler’s generalization.
Definition. The integer $b$ satisfying $(\ast)$ is called the exponential inverse of $a$ modulo $N$.
Using exponential inverses will be a key idea in the RSA cryptosystem.
RSA Cryptosystem
This is an explanation of textbook RSA encryption scheme.
Key Generation
• We pick two large primes $p, q$ and set $N = pq$.
• Select $(e, d)$ so that $ed \equiv 1 \pmod{\phi(N)}$.
• Set $(N, e)$ as the public key and make it public.
• Set $d$ as the private key and keep it secret.
RSA Encryption and Decryption
Suppose we want to encrypt a message $m \in \mathbb{Z}_N$.
• Encryption
□ Using the public key $(N, e)$, compute the ciphertext $c = m^e \bmod N$.
• Decryption
□ Recover the original message by computing $c^d \bmod N$.
Correctness of RSA?
Since $ed \equiv 1 \pmod{\phi(N)}$, we have
\[c^d \equiv m^{ed} \equiv m \pmod N\]
by the properties of exponential inverses.
Wait, but the properties requires that $\gcd(m, N) = 1$. So it seems like we can’t use some values of $m$. Furthermore, it should be computationally infeasible to recover $d$ using $e$ and $N$.
Regarding the Choice of $N$
If $N$ is prime, it is very easy to find $d$. Since the relation $ed \equiv 1 \pmod {(N-1)}$ holds, we directly see that $d$ can be computed efficiently using the extended Euclidean algorithm.
The next simplest case would be setting $N = pq$ for two large primes $p$ and $q$. We expose $N$ to the public but hide primes $p$ and $q$. Now suppose the attacker wants to compute $d$ using $(N, e)
$. The attacker knows that $ed \equiv 1 \pmod {\phi(N)}$, and $\phi(N) = (p-1)(q-1)$. So to calculate $d$, the attacker must know $\phi(N)$, which requires the factorization of $N$.
If the factorization $N = pq$ is known, finding $d$ is easy. But factoring large prime numbers (especially a product of two primes of similar size) is known to be very difficult.^1 No one has
formally proven this, but we believe and assume that it is hard.^2
Chinese Remainder Theorem in RSA
Assume that the message $m$ is not divisible by both $p$ and $q$. By Fermat’s little theorem, we have $m^{p-1} \equiv 1 \pmod p$ and $m^{q-1} \equiv 1 \pmod q$.
Therefore, for decryption in RSA, the following holds. Note that $N = pq$.
\[c^d \equiv m^{ed} \equiv m^{1 + k\phi(N)} \equiv m \cdot (m^{p-1})^{k(q-1)} \equiv m \cdot 1^{k(q-1)} \equiv m \pmod p.\]
A similar result holds for modulus $q$. This does not exactly recover the message yet, since $m$ could have been chosen to be larger than $p$. The above equation is true, but during actual
computation, one may get a result that is less than $p$. This may not be equal to the original message.^3
Since $N = pq$, we use the Chinese remainder theorem. Instead of computing $c^d \pmod N$, we can compute
\[c^d \equiv m \pmod p, \qquad c^d \equiv m \pmod q\]
independently and solve the system of equations to recover the message.
Can I Encrypt $p$ with RSA?
Now we return to the problem where $\gcd(m, N) \neq 1$. The probability of $\gcd(m, N) \neq 1$ is actually $\frac{1}{p} + \frac{1}{q} - \frac{1}{pq}$, so if we take large primes $p, q \approx 2^
{1000}$ as in RSA2048, the probability of this occurring is roughly $2^{-999}$, which is negligible. But for completeness, we also prove for this case.
$e, d$ are still chosen to satisfy $ed \equiv 1 \pmod {\phi(N)}$. Suppose we want to decrypt $c \equiv m^e \pmod N$.
We will also use the Chinese remainder theorem here.
Since $\gcd(m, N) \neq 1$ and $N = pq$, we have $p \mid m$. So if we compute in $\mathbb{Z}_p$, we will get $0$,
\[c^d \equiv m^{ed} \equiv 0^{ed} \equiv 0 \pmod p.\]
We also do the computation in $\mathbb{Z}_q$ and get
\[c^d \equiv m^{ed} \equiv m^{1 + k\phi(N)} \equiv m\cdot (m^{q-1})^{k(p-1)} \equiv m \cdot 1^{k(p-1)} \equiv m \pmod q.\]
Here, we used the fact that $m^{q-1} \equiv 1 \pmod q$. This holds because if $p \mid m$, $m$ is a multiple of $p$ that is less than $N$, so $m = pm’$ for some $m’$ such that $1 \leq m’ < q$. Then $\
gcd(m, q) = \gcd(pm’, q) = 1$ since $q$ does not divide $p$ and $m’$ is less than $q$.
Therefore, from $c^d \equiv 0 \pmod p$ and $c^d \equiv (m \bmod q) \pmod q$, we can recover a unique solution $c^d \equiv m \pmod N$.
Now we must argue that the recovered solution is actually equal to the original $m$. But what we did above was showing that $m^{ed}$ and $m$ in $\mathbb{Z}_N$ are mapped to the same element $(0, m \
bmod q)$ in $\mathbb{Z}_p \times \mathbb{Z}_q$. Since the Chinese remainder theorem tells us that this mapping is an isomorphism, $m^{ed}$ and $m$ must have been the same elements of $\mathbb{Z}_N$
in the first place.
Notice that we did not require $m$ to be relatively prime to $N$. Thus the RSA encryption scheme is correct for any $m \in \mathbb{Z}_N$.
Correctness of RSA with Fermat’s Little Theorem
Actually, the above argument can be proven only with Fermat’s little theorem. In the above proof, the Chinese remainder theorem was used to transform the operation, but for $N = pq$, the situation is
simple enough that this theorem is not necessarily required.
Let $M = m^{ed} - m$. We have shown above only using Fermat’s little theorem that $p \mid M$ and $q \mid M$, for any choice of $m \in \mathbb{Z}_N$. Then since $N = pq = \mathrm{lcm}(p, q)$, we have
$N \mid M$, so $m^{ed} \equiv m \pmod N$. Hence the RSA scheme is correct.
So we don’t actually need Euler’s generalization for proving the correctness of RSA…?! In fact, the proof given in the original paper of RSA used Fermat’s little theorem.
Discrete Logarithms
This is an inverse problem of exponentiation. The inverse of exponentials is logarithms, so we consider the discrete logarithm of a number modulo $p$.
Given $y \equiv g^x \pmod p$ for some prime $p$, we want to find $x = \log_g y$. We set $g$ to be a generator of the group $\mathbb{Z}_p$ or $\mathbb{Z}_p^*$, since if $g$ is the generator, a
solution always exists.
Read more in discrete logarithm problem (Modern Cryptography).
ElGamal Encryption
This is an encryption scheme built upon the hardness of the DLP.
1. Let $p$ be a large prime.
2. Select a generator $g \in \mathbb{Z}_p^*$.
3. Choose a private key $x \in \mathbb{Z}_p^*$.
4. Compute the public key $y = g^x \pmod p$.
☆ $p, g, y$ will be publicly known.
☆ $x$ is kept secret.
ElGamal Encryption and Decryption
Suppose we encrypt a message $m \in \mathbb{Z}_p^*$.
1. The sender chooses a random $k \in \mathbb{Z}_p^$, called *ephemeral key.
2. Compute $c_1 = g^k \pmod p$ and $c_2 = my^k \pmod p$.
3. $c_1, c_2$ are sent to the receiver.
4. The receiver calculates $c_1^x \equiv g^{xk} \equiv y^k \pmod p$, and find the inverse $y^{-k} \in \mathbb{Z}_p^*$.
5. Then $c_2y^{-k} \equiv m \pmod p$, recovering the message.
The attacker will see $g^k$. By the hardness of DLP, the attacker is unable to recover $k$ even if he knows $g$.
Ephemeral Key Should Be Distinct
If the same $k$ is used twice, the encryption is not secure. Suppose we encrypt two different messages $m_1, m_2 \in \mathbb{Z} _ p^{ * }$. The attacker will see $(g^k, m_1y^k)$ and $(g^k, m_2 y^k)$.
Then since we are in a multiplicative group $\mathbb{Z} _ p^{ * }$, inverses exist. So
\[m_1y^k \cdot (m_2 y^k)^{-1} \equiv m_1m_2^{-1} \equiv 1 \pmod p\]
which implies that $m_1 \equiv m_2 \pmod p$, leaking some information.
1. If one of the primes is small, factoring is easy. Therefore we require that $p, q$ both be large primes. ↩︎
2. There is a quantum polynomial time (BQP) algorithm for integer factorization. See Shor’s algorithm. ↩︎
3. This part of the explanation is not necessary if we use abstract algebra! ↩︎ | {"url":"https://blog.zxcvber.com/lecture-notes/internet-security/2023-10-04-rsa-elgamal/","timestamp":"2024-11-06T04:15:37Z","content_type":"text/html","content_length":"30444","record_id":"<urn:uuid:2195912f-ca10-416a-a5a1-0250e256bd75>","cc-path":"CC-MAIN-2024-46/segments/1730477027909.44/warc/CC-MAIN-20241106034659-20241106064659-00432.warc.gz"} |
Distributional Semantics in R with the
Distributional Semantics in R with the ‘wordspace’ Package
Stefan Evert
1 April 2016
Distributional semantic models (DSMs) represent the meaning of a target term (which can be a word form, lemma, morpheme, word pair, etc.) in the form of a feature vector that records either
co-occurrence frequencies of the target term with a set of feature terms (term-term model) or its distribution across textual units (term-context model). Such DSMs have become an indispensable
ingredient in many NLP applications that require flexible broad-coverage lexical semantics.
Distributional modelling is an empirical science. DSM representations are determined by a wide range of parameters such as size and type of the co-occurrence context, feature selection, weighting of
co-occurrence frequencies (often with statistical association measures), distance metric, dimensionality reduction method and the number of latent dimensions used. Despite recent efforts to carry out
systematic evaluation studies, the precise effects of these parameters and their relevance for different application settings are still poorly understood.
The wordspace package aims to provide a flexible, powerful and easy to use “interactive laboratory” that enables its users to build DSMs and experiment with them, but that also scales up to the large
models required by real-life applications.
Further background information and references can be found in:
Evert, Stefan (2014). Distributional semantics in R with the wordspace package. In Proceedings of COLING 2014, the 25th International Conference on Computational Linguistics: System
Demonstrations, pages 110–114, Dublin, Ireland.
Before continuing with this tutorial, load the package with
Input formats
The most general representation of a distributional model takes the form of a sparse matrix, with entries specified as a triplet of row label (target term), column label (feature term) and
co-occurrence frequency. A sample of such a table is included in the package under the name DSM_VerbNounTriples_BNC, listing syntactic verb-noun co-occurrences in the British National Corpus:
461973 horse subj pick 5 written
546678 leg subj say 9 written
608545 mick subj say 8 spoken
611816 mill subj manage 6 written
714162 person subj begin 11 written
726716 place subj determine 6 written
970619 tax subj work 7 written
992648 time subj book 6 written
1387925 discussion obj facilitate 14 written
1656006 lee obj give 7 written
The wordspace package creates DSM objects from such triplet representations, which can easily be imported into R from a wide range of file and database formats. Ready-made import functions are
provided for TAB-delimited text files (as used by DISSECT), which may be compressed to save disk space, and for term-document models created by the text-mining package tm.
The native input format is a pre-compiled sparse matrix representation generated by the UCS toolkit. In this way, UCS serves as a hub for the preparation of co-occurrence data, which can be collected
from dependency pairs, extracted from a corpus indexed with the IMS Corpus Workbench or imported from various other formats.
Creating a DSM
The first step in the creation of a distributional semantic model is the compilation of a co-occurrence matrix. Let us illustrate the procedure for verb-noun co-occurrences from the written part of
the British National Corpus. First, we extract relevant rows from the table above.
Note that many verb-noun pairs such as (walk, dog) still have multiple entries in Triples: dog can appear either as the subject or as the object of walk.
## noun rel verb f mode
## 295011 dog subj walk 20 written
## 1398669 dog obj walk 87 written
There are two ways of dealing with such cases: we can either add up the frequency counts (a dependency-filtered model) or treat “dog-as-subject” and “dog-as-object” as two different terms (a
dependency-structured model). We opt for a dependency-filtered model in this example – can you work out how to compile the corresponding dependency-structured DSM in R, either for verbs of for nouns
as target terms?
The dsm constructor function expects three vectors of the same length, containing row label (target term), column label (feature term) and co-occurrence count (or pre-weighted score) for each nonzero
cell of the co-occurrence matrix. In our example, we use nouns as targets and verbs as features. Note the option raw.freq=TRUE to indicate that the matrix contains raw frequency counts.
## [1] 10940 3149
The constructor automatically computes marginal frequencies for the target and feature terms by summing over rows and columns of the matrix respectively. The information is collected in data frames
VObj$rows and VObj$cols, together with the number of nonzero elements in each row and column:
## term nnzero f
## 5395 man 907 51976
## 6491 people 911 72832
## 6951 problem 402 29002
## 8928 thing 442 34104
## 8975 time 658 46500
## 9613 way 738 46088
This way of computing marginal frequencies is appropriate for syntactic co-occurrence and term-document models. In the case of surface co-occurrence based on token spans, the correct marginal
frequencies have to be provided separately in the rowinfo= and colinfo= arguments (see ?dsm for details).
The actual co-occurrence matrix is stored in VObj$M. Since it is too large to display on screen, we extract the top left corner with the head method for DSM objects. Note that you can also use head
(VObj, Inf) to extract the full matrix.
## 6 x 6 sparse Matrix of class "dgCMatrix"
## be have say give take achieve
## aa 7 5 12 . . .
## abandonment 14 . . . . .
## abbey 45 13 6 . . .
## abbot 23 7 10 5 5 .
## abbreviation 9 . . . . .
## abc 6 . . . . .
The DSM parameters
Rows and columns with few nonzero cells provide unreliable semantic information and can lead to numerical problems (e.g. because a sparse association score deletes the remaining nonzero entries). It
is therefore common to apply frequency thresholds both on rows and columns, here in the form of requiring at least 3 nonzero cells. The option recursive=TRUE guarantees that both criteria are
satisfied by the final DSM when rows and columns are filtered at the same time (see the examples in ?subset.dsm for an illustration).
## [1] 6087 2139
If you want to filter only columns or rows, you can pass the constraint as a named argument: subset=(nnzero >= 3) for rows and select=(nnzero >= 3) for columns.
The next step is to weight co-occurrence frequency counts. Here, we use the simple log-likelihood association measure with an additional logarithmic transformation, which has shown good results in
evaluation studies. The wordspace package computes sparse (or “positive”) versions of all association measures by default, setting negative associations to zero. This guarantees that the sparseness
of the co-occurrence matrix is preserved. We also normalize the weighted row vectors to unit Euclidean length (normalize=TRUE).
Printing a DSM object shows information about the dimensions of the co-occurrence matrix and whether it has already been scored. Note that the scored matrix does not replace the original
co-occurrence counts, so dsm.score can be executed again at any time with different parameters.
## Distributional Semantic Model with 6087 rows x 2139 columns
## * raw co-occurrence matrix M available
## - sparse matrix with 191.4k / 13.0M nonzero entries (fill rate = 1.47%)
## - in canonical format
## - known to be non-negative
## - sample size of underlying corpus: 5010.1k tokens
## * scored matrix S available
## - sparse matrix with 153.7k / 13.0M nonzero entries (fill rate = 1.18%)
## - in canonical format
## - known to be non-negative
Most distributional models apply a dimensionality reduction technique to make data sets more manageable and to refine the semantic representations. A widely-used technique is singular value
decomposition (SVD). Since VObj is a sparse matrix, dsm.projection automatically applies an efficient algorithm from the sparsesvd package.
## [1] 6087 300
VObj300 is a dense matrix with 300 columns, giving the coordinates of the target terms in 300 latent dimensions. Its attribute "R2" shows what proportion of information from the original matrix is
captured by each latent dimension.
Using DSM representations
The primary goal of a DSM is to determine “semantic” distances between pairs of words. The arguments to pair.distances can also be parallel vectors in order to compute distances for a large number of
word pairs efficiently.
## book/paper
## 45.07627
## attr(,"similarity")
## [1] FALSE
By default, the function converts similarity measures into an equivalent distance metric – the angle between vectors in the case of cosine similarity. If you want the actual similarity values,
specify convert=FALSE:
## book/paper
## 0.7061649
## attr(,"similarity")
## [1] TRUE
We are often interested in finding the nearest neighbours of a given term in the DSM space:
## paper article poem works magazine novel text guide
## 45.07627 51.92011 53.48027 53.91556 53.94824 54.40451 55.13910 55.27027
## newspaper document item essay leaflet letter
## 55.51492 55.62521 56.28246 56.29539 56.49145 58.04178
The return value is actually a vector of distances to the nearest neighbours, labelled with the corresponding terms. Here is how you obtain the actual neighbour terms:
## [1] "paper" "guide" "works" "novel" "magazine" "article"
## [7] "document" "poem" "diary" "essay" "item" "text"
## [13] "booklet" "leaflet" "newspaper"
The neighbourhood plot visualizes nearest neighbours as a semantic network based on their mutual distances. This often helps interpretation by grouping related neighbours. The network below shows
that book as a text type is similar to novel, essay, poem and article; as a form of document it is similar to paper, letter and document; and as a publication it is similar to leaflet, magazine and
A straightforward way to evaluat distributional representations is to compare them with human judgements of the semantic similarity between word pairs. The wordspace package includes to well-known
data sets of this type: Rubenstein-Goodenough (RG65) and WordSim353 (a superset of RG65 with judgements from new test subjects).
5 autograph_N shore_N 0.06
15 monk_N slave_N 0.57
25 forest_N graveyard_N 1.00
35 cemetery_N mound_N 1.69
45 brother_N monk_N 2.74
55 autograph_N signature_N 3.59
65 gem_N jewel_N 3.94
There is also a ready-made evaluation function, which computes Pearson and rank correlation between the DSM distances and human subjects. The option format="HW" adjusts the POS-disambiguated notation
for terms in the data set (e.g. book_N) to the format used by our distributional model (book).
## rho p.value missing r r.lower r.upper
## RG65 0.3076154 0.01267694 20 0.3735435 0.1426399 0.5658865
Evaluation results can also be visualized in the form of a scatterplot with a trend line.
The rank correlation of 0.308 is very poor, mostly due to the small amount of data on which our DSM is based. Much better results are obtained with pre-compiled DSM vectors from large Web corpus,
which are also included in the package. Note that target terms are given in a different format there (which corresponds to the format in RG65).
Advanced techniques
Schütze (1998) used DSM representations for word sense disambiguation (or, more precisely, word sense induction) based on a clustering of the sentence contexts of an ambiguous word. The wordspace
package includes a small data set with such contexts for a selection of English words. Let us look at the noun vessel as an example, which has two main senses (“ship” and “blood vessel”):
## a craft designed for water transportation
## 6
## a tube in which a body fluid circulates
## 6
Sentence contexts are given as tokenized strings ($sentence), in lemmatized form ($hw) and as lemmas annotated with part-of-speech codes ($lemma). Choose the version that matches the representation
of target terms in your DSM.
vessel.n.02 The spraying operation was conducted from the rear deck of a small Naval vessel , cruising two miles off-shore and vertical to an on-shore breeze .
vessel.n.01 Such a dual derivation was strikingly demonstrated during the injection process where initial filling would be noted to occur in several isolated pleural vessels at once .
vessel.n.01 This vessel could be followed to the parenchyma where it directly provided bronchial arterial blood to the alveolar capillary bed ( figs. 17 , 18 ) .
vessel.n.01 However , this artery is known to be a nutrient vessel with a distribution primarily to the proximal airways and supportive tissues of the lung .
vessel.n.01 It is distinctly possible , therefore , that simultaneous pressures in all three vessels would have rendered the shunts inoperable and hence , uninjectable .
vessel.n.01 A careful search failed to show occlusion of any of the mesenteric vessels .
vessel.n.01 Some of the small vessels were filled with fibrin thrombi , and there was extensive interstitial hemorrhage .
vessel.n.02 He appeared to be peering haughtily down his nose at the crowded and unclean vessel that would carry him to freedom .
vessel.n.02 However , we sent a third vessel out , a much smaller and faster one than the first two .
vessel.n.02 Upon reaching the desired speed , the automatic equipment would cut off the drive , and the silent but not empty vessel would hurl towards the star which was its journey ’s end .
vessel.n.02 Then , after slowing the vessel considerably , the drive would adjust to a one gee deceleration .
vessel.n.02 To round out the blockading force , submarines would be needed - to locate , identify and track approaching vessels .
Following Schütze, each context is represented by a centroid vector obtained by averaging over the DSM vectors of all context words.
This returns a matrix of centroid vectors for the 12 sentence contexts of vessel in the data set. The vectors can now be clustered and analyzed using standard R functions. Partitioning around medoids
(PAM) has shown good and robust performance in evaluation studies.
library(cluster) # clustering algorithms of Kaufman & Rousseeuw (1990)
res <- pam(dist.matrix(centroids), 2, diss=TRUE, keep.diss=TRUE)
plot(res, col.p=factor(Vessel$sense), shade=TRUE, which=1, main="WSD for 'vessel'")
Colours in the plot above indicate the gold standard sense of each instance of vessel. A confusion matrix confirms perfect clustering of the two senses:
We can also use a pre-defined function for the evaluation of clustering tasks, which is convenient but does not produce a visualization of the clusters. Note that the “target terms” of the task must
correspond to the row labels of the centroid matrix, which we have set to sentence IDs (Vessel$id) above.
## purity entropy missing
## Vessel 100 0 0
As a final example, let us look at a simple approach to compositional distributional semantics, which computes the compositional meaning of two words as the element-wise sum or product of their DSM
The nearest neighbours of mouse are problematic, presumably because the type vector represents a mixture of the two senses that is not close to either meaning in the semantic space.
## prop isotope carbon serum transformer sponge
## 53.72837 55.08473 55.29022 56.84607 57.20004 57.25144
## thermometer hoop loudspeaker razor mount implant
## 57.33371 57.38682 57.38682 57.43123 57.47889 57.49171
By adding the vectors of mouse and computer, we obtain neighbours that seem to fit the “computer mouse” sense very well:
## mouse computer program processor software tool machine
## 36.69755 41.47028 54.15183 54.37412 54.55897 55.07898 55.74764
## transistor mix keyboard prop sponge
## 58.51586 58.93807 59.15682 59.18378 59.34185
Note that the target is specified as a distributional vector rather than a term in this case. Observations from the recent literature suggest that element-wise multiplication is not compatible with
non-sparse SVD-reduced DSMs, so it is not surprising to find completely unrelated nearest neighbours in our example:
## kylie barry coward leo wednesday emma baron leech
## 52.70435 52.92230 53.61260 53.81451 54.01588 54.33543 54.53792 55.07008
## friday frank colonel bruce
## 55.28301 55.39203 55.44021 55.47596 | {"url":"https://cran.mirror.garr.it/CRAN/web/packages/wordspace/vignettes/wordspace-intro.html","timestamp":"2024-11-04T01:01:03Z","content_type":"text/html","content_length":"285469","record_id":"<urn:uuid:b5f089b6-f411-4c5e-b247-44f101f81db3>","cc-path":"CC-MAIN-2024-46/segments/1730477027809.13/warc/CC-MAIN-20241104003052-20241104033052-00308.warc.gz"} |
Reference Table
Reference Tables in Fabric Studio
A Reference table holds information that is common to either all LU instances or to multiple LUs. For optimization reasons, this information is stored in an external table and not in each LUI
How to Create a New Reference Table from an External Source
1. Go to Project Tree > References , right-click, then select Create References Based On DB Tables. This will display the DB Browser menu, Context menu and References popup window.
2. Click DB Connection (top of the window) and select a data source interface.
3. Click main to display the Tables directory in the source DB.
4. Select a table. To select multiple tables, press the Ctrl key.
5. Optional:
Add a prefix to the Reference table's name in the Name Prefix field (window's footer).
• If there are objects in the project with the same name, add a prefix to differentiate between Reference tables and LU tables with similar names.
• We recommend using a prefix that indicates the project name as a Reference table.
To populate the Reference Table using a Broadway flow, tick the Table population based Broadway flow box.
6. Click Create Tables to add the new Reference table under References in the Project Tree. If the table does not appear immediately, click
1. Switch to DB Interface Explorer View by clicking on its icon (
2. Choose the DB Interface and the required table
3. Right-click on the table. A popup window appears. Choose the Create Reference Table option as seen below
4. An automatically-generated new reference table has been created; as seen below, it is comprised of external data source table.
Note: Once the reference table is auto generated, a corresponding population flow is auto generated too.
Manual Creation of a New Reference Table
A new reference table can be created also manually.
1. Go to Project Tree > Implementation > Logical Units / Data Products
□ Choose Logical Units / Data Products by clicking anywhere along the line
□ Choose/click on the References logical unit
□ Choose/highlight the Tables folder > right-click on it
□ Choose the New Table option from the opened context menu
□ Name the new Table. Press Enter to confirm or Escape to cancel.
2. Follow the instructions for creating a new table.
3. Save the file
4. Create a corresponding table population.
Editing and Viewing Reference Tables
Reference tables can be edited by either changing the default data mapping, adding transformations or adding or removing columns like in LU tables.
When the Broadway flow population option is selected as described above, the table population process can be edited using the Broadway flow described here.
How to View Reference Table Data
To access the Reference Viewer do the following:
1. Go to Project Tree > References, right-click References Viewer, and then select the table. The Data Viewer window is displayed according to its hierarchy in the Instances Tree pane.
2. Click the data file to display the components hierarchy in the Instance DB Tree pane.
2. Click the table name to display the table's data in the main Data Viewer window.
To view Reference tables content:
1. Switch to DB Interface Explorer View by clicking on its icon (
2. Click to open the Query Builder on Fabric
3. Choose "common"
4. Choose the required reference table or some of its columns and right-click to add a select statement. Then click on the Execute button to view its content in the results panel.
Reference Tables Properties
Properties can be defined in the Table Properties panel in the right pane of the selected Table tab.
Main Properties
• Name, can be defined or modified.
• Schema, the name of the common DB in which this table will be stored. If left empty, the table will be added to the generic CommonDB schema.
• Full Text Search
• Column collation type:
□ BINARY, compares string data regardless of text encoding.
□ NOCASE, folds upper case characters to their lower case equivalents.
□ RTRIM, ignores trailing space characters.
Sync Method
By default, reference tables are synched in the background of each table according to the defined Sync policy. The following Sync options can be selected in the Table Properties panel:
• None, default value, synchronization is according to the Sync policy defined.
• Time Interval, set in days.hrs:min:sec format.
• Decision function, syncs the table according to the Decision function defined under the References > Java folder.
The following functions or other tables can be attached to the Reference table:
• Enrichment Functions - performs data manipulations on the table's content.
• Other Reference Tables, on which the current Reference table depends (e.g. it needs data from these tables).
• Index Post Sync - determines if an index should be created on a Reference table after data is synced. This capability is relevant for huge reference databases (more than 200M records) and can
accelerate the overall data sync time. Added in release 6.5.1.
Query Statements Settings
• Schema, the name of the common DB in which this table will be stored. If left empty, the table will be added to the generic commonDB schema. In the illustration below the reference table
ref_geoCodeUSA is attached to the extraRefDB schema.
• Column collation type:
□ BINARY, compares string data regardless of text encoding.
□ NOCASE, folds upper case characters to their lower case equivalents.
□ RTRIM, ignores trailing space characters.
• Full Text Search: When set to True, enables the use of the MATCH SQLite command as part of the WHERE clause of a Select statement that reads data from a Fabric table. Default = False.
• Sync Wait Timeout: The timeout in seconds for syncing the. This is similar to schema sync wait timeout
• Sync Method: By default, reference tables are synched in the background of each table according to the defined Sync policy. The following Sync options can be selected in the Table Properties
□ None, default value, synchronization is according to the Sync policy defined.
□ Time Interval, set in days.hrs:min:sec format.
□ Decision function, syncs the table according to the Decision function defined under the References > Java folder.
• Truncate Before Sync: When Truncate Before Sync = True (checkbox is checked), the entire LU table is truncated before the populations are executed
• Required Reference Tables, on which the current Reference table depends (e.g. it needs data from these tables).
• Index Post Sync - determines if an index should be created on a Reference table after data is synced. This capability is relevant for huge reference databases (more than 200M records) and can
accelerate the overall data sync time.
Attach the Reference Table to a Logical Unit Schema
Before accessing the Reference Table from a specific LU, or before it can be used as a lookup object, it must be attached to the LU.
Configure LU to Use a Reference Table
1. Open the Schema Window
2. In the right panel, select the References tab.
3. Check the relevant Reference table(s) option.
1. Open the Schema Window
2. In the right panel, expand the Dependent References section
3. Add the required Reference tables to the Reference List: Use the plus (+) button to add another table and then select a table from the list
4. Click Save to save the association created between the LU and the Reference Table(s).
Note: Reference tables can also be accessed via Lookup tables, Web Services, functions, jobs and Broadway Actors.
Deploy the Reference Tables
Reference Tables must be deployed before being used. As a result of the deployment, a synchronization job process is triggered in the background to ensure that all CommonDB copies are kept in-sync
across the Fabric Cluster.
To deploy the Reference Tables, go to the Project Tree, right click References, select Deploy to Server and then the Server to deploy to the Reference table.
Note that if a reference table has been attached to an LU Schema as described above, the LU must be re-deployed. | {"url":"https://support.k2view.com/Academy_8.0/articles/22_reference(commonDB)_tables/02_reference_table_fabric_studio.html","timestamp":"2024-11-07T22:15:04Z","content_type":"text/html","content_length":"47377","record_id":"<urn:uuid:2162aa05-4aa8-4df0-a756-83706eff0172>","cc-path":"CC-MAIN-2024-46/segments/1730477028017.48/warc/CC-MAIN-20241107212632-20241108002632-00331.warc.gz"} |
bayesnetnode properties
node = stream.create("bayesnet", "My node")
node.setPropertyValue("continue_training_existing_model", True)
node.setPropertyValue("structure_type", "MarkovBlanket")
node.setPropertyValue("use_feature_selection", True)
# Expert tab
node.setPropertyValue("mode", "Expert")
node.setPropertyValue("all_probabilities", True)
node.setPropertyValue("independence", "Pearson")
Table 1. bayesnetnode properties
bayesnetnode Properties Values Property description
inputs [field1 ... Bayesian network models use a single target field, and one or more input fields. Continuous fields are automatically binned. See the topic Common
fieldN] modeling node properties for more information.
continue_training_existing_model flag
structure_type TAN Select the structure to be used when building the Bayesian network.
use_feature_selection flag
parameter_learning_method Likelihood Specifies the method used to estimate the conditional probability tables between nodes where the values of the parents are known.
mode Expert Simple
missing_values flag
all_probabilities flag
independence Likelihood Specifies the method used to determine whether paired observations on two variables are independent of each other.
significance_level number Specifies the cutoff value for determining independence.
maximal_conditioning_set number Sets the maximal number of conditioning variables to be used for independence testing.
Specifies which fields from the dataset are always to be used when building the Bayesian network.
inputs_always_selected [field1 ...
fieldN] Note: The target field is always selected.
maximum_number_inputs number Specifies the maximum number of input fields to be used in building the Bayesian network.
calculate_variable_importance flag
calculate_raw_propensities flag
calculate_adjusted_propensities flag
adjusted_propensity_partition Test Validation | {"url":"https://www.ibm.com/docs/en/watsonx/w-and-w/1.1.x?topic=properties-bayesnetnode","timestamp":"2024-11-10T15:59:05Z","content_type":"text/html","content_length":"12252","record_id":"<urn:uuid:b1955069-a430-4307-b2d2-d2b69e3baaab>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00054.warc.gz"} |
Edexcel Core Mathematics FP1 June 2015 (worksheets, examples, solutions, videos, activities)
Questions and Worked Solutions for FP1 Edexcel Further Pure Mathematics June 2015.
Edexcel Core Mathematics FP1 June 2015 Past Paper
FP1 Mathematics Edexcel June 2015 Question 1
f(x) = 9x^3 – 33x^2 – 55x – 25
Given that x = 5 is a solution of the equation f(x) = 0, use an algebraic method to solve f(x) = 0 completely.
Step by Step Solutions for Question 1
FP1 Mathematics Edexcel June 2015 Question 2
In the interval 13 < x < 14, the equation
3 + x sin(x/4) = 0, where x is measured in radians,
has exactly one root, α.
(a) Starting with the interval [13, 14], use interval bisection twice to find an interval of width 0.25 which contains α.
(b) Use linear interpolation once on the interval [13, 14] to find an approximate value for α.
Give your answer to 3 decimal places.
Step by Step Solutions for Question 2
FP1 Mathematics Edexcel June 2015 Question 3
Step by Step Solutions for Question 3
FP1 Mathematics Edexcel June 2015 Question 4
z[1] = 3i and z[2] = 6/(1 + i√3)
(a) Express z[2] in the form a + ib, where a and b are real numbers.
(b) Find the modulus and the argument of z[2], giving the argument in radians in terms of π.
(c) Show the three points representing z[1], z[2] and (z[1] + z[2]) respectively, on a single Argand diagram.
Step by Step Solutions for Question 4
FP1 Mathematics Edexcel June 2015 Question 5
The rectangular hyperbola H has equation xy = 9
The point A on H has coordinates (6, 3/2)
(a) Show that the normal to H at the point A has equation
2y – 8x + 45 = 0
The normal at A meets H again at the point B.
(b) Find the coordinates of B.
Step by Step Solutions for Question 5
FP1 Mathematics Edexcel June 2015 Question 6
Step by Step Solutions for Question 6
FP1 Mathematics Edexcel June 2015 Question 7
A triangle T is transformed onto a triangle T' by the transformation represented by the matrix B.
The vertices of triangle T' have coordinates (0, 0), ( −20, 6) and (10c, 6c), where c is a positive constant.
The area of triangle T' is 135 square units.
(a) Find the matrix B^–1
(b) Find the coordinates of the vertices of the triangle T, in terms of c where necessary.
(c) Find the value of c.
Step by Step Solutions for Question 7
FP1 Mathematics Edexcel June 2015 Question 8
The point P(3p^2, 6p) lies on the parabola with equation y^2= 12x and the point S is the focus of this parabola.
(a) Prove that SP = 3(1 + ^2)
The point Q(3q^2, 6q), p ≠ q, also lies on this parabola.
The tangent to the parabola at the point P and the tangent to the parabola at the point Q meet at the point R.
(b) Find the equations of these two tangents and hence find the coordinates of the point R, giving the coordinates in their simplest form.
(c) Prove that SR^2 = SP.SQ
Step by Step Solutions for Question 8
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. | {"url":"https://www.onlinemathlearning.com/edexcel-fp1-june-2015.html","timestamp":"2024-11-02T13:45:50Z","content_type":"text/html","content_length":"38947","record_id":"<urn:uuid:8370f47c-3f06-444d-aec4-27a0442278c5>","cc-path":"CC-MAIN-2024-46/segments/1730477027714.37/warc/CC-MAIN-20241102133748-20241102163748-00259.warc.gz"} |
Tossing coins to understand spheres
EPFL mathematicians, in collaboration with Purdue University, have settled a 30-year-old question about spheres and 4-dimensional spaces. The results bring new light to the "Euler Class," one of the
most powerful tools to understand complicated spaces.
For mathematicians, "Euler Class" is one of the most powerful tools for understanding complicated spaces by cutting them into simpler pieces. It is named after Swiss mathematician Leonard Euler who
was the first to consider the idea.
"Just like something as complex as DNA is ultimately made of simple atoms, it is how these simple pieces are assembled that contains the important information rather than the pieces themselves," says
Professor Nicolas Monod, who leads the Ergodic and Geometric Group Theory research unit at EPFL. His group joined forces with colleagues at Purdue University to solve an old question about spheres.
The answer has been published the leading mathematics journal Inventiones.
In 1958, Fields medalist John Milnor noticed a problem when trying to build spaces using only circles and two-dimensional surfaces: there was a limit on how complicated the Euler class can be in two
dimensions. The observation snowballed into an entire field of research in higher dimensions, and mathematicians quickly realized that Milnor’s "complexity bound" didn’t apply for spaces in all
Monod explains: "A question that had remained open for decades was, what about gluing spheres on 4-dimensional spaces? Is there also here a limit on how they fit together?" He continues: "Gluing
spheres on 4-dimensional spaces is a particularly important construction because this is precisely how the very first ’exotic spheres’ have been constructed!"
Classical approaches of understanding spaces have proven unable to solve this 4-dimensional question. So the EPFL mathematicians turned to the Bernoulli process, named after Swiss mathematician Jacob
Bernoulli, for inspiration. The Bernoulli process, which is a model of tossing coins, was combined with the study of spheres and the Euler class to finally solve the question.
"A very curious thing happened when we set out to solve this problem," says Monod. "If it remained unsolved for so long, it is perhaps because none of the classical methods used to understand spaces
seemed to be able to crack this specific question about 4 dimensions. Instead, we turned to an unlikely source for inspiration: tossing coins!"
As a game with a 50-50 chance to guess the right side of a coin, this might seem very simple, but its simplicity is misleading. "The Bernoulli process already includes many of the advanced features
of probability theory when we set out to repeat the toss more and more often," says Monod. "In fact, the Central Limit Theorem - which is a kind of Law of Large Numbers - even tells us that this
simple model can emulate many of the most complicated random phenomena of nature if we are willing to toss enough coins for long enough."
Probability and random processes might seem not to have much to do with the the analysis of higher dimensions in space, but mathematics is as much a creative art as a science. "Earlier this year, we
published the discovery that Bernoulli’s random coin games can help solve some difficult algebraic questions , very much non-random questions," says Monod. "This has now been combined with the study
of spheres and the Euler Class to finally solve the old question about 4-dimensional spaces: no, there is no limit at all to the size of the Euler class for spheres in four dimensions."
"So the coins came to the rescue of algebra and geometry, and Bernoulli visits the Euler class: indeed, mathematicians do things differently," he concludes.
Monod, N., Nariman, S. Bounded and unbounded cohomology of homeomorphism and diffeomorphism groups. Inventiones mathematicae, 06 February 2023. DOI: 10.1007/s00222’023 -01181-w | {"url":"https://www.myscience.ch/en/news/2023/tossing_coins_to_understand_spheres-2023-epfl","timestamp":"2024-11-09T03:02:08Z","content_type":"text/html","content_length":"50179","record_id":"<urn:uuid:b2aaea1f-f012-4a26-bc52-7424a5ada0b5>","cc-path":"CC-MAIN-2024-46/segments/1730477028115.85/warc/CC-MAIN-20241109022607-20241109052607-00341.warc.gz"} |
Karauli to Sheopur distance
Distance in KM
The distance from Karauli to Sheopur is 169.722 Km
Distance in Mile
The distance from Karauli to Sheopur is 105.5 Mile
Distance in Straight KM
The Straight distance from Karauli to Sheopur is 96.2 KM
Distance in Straight Mile
The Straight distance from Karauli to Sheopur is 59.8 Mile
Travel Time
Travel Time 3 Hrs and 19 Mins
Karauli Latitud and Longitude
Latitud 26.4883859 Longitude 77.01617480000004
Sheopur Latitud and Longitude
Latitud 25.6730962 Longitude 76.69585069999994 | {"url":"http://www.distancebetween.org/karauli-to-sheopur","timestamp":"2024-11-04T04:42:50Z","content_type":"text/html","content_length":"1737","record_id":"<urn:uuid:6e16ade8-3130-484f-b207-b0a31ef1cbec>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00610.warc.gz"} |
We present an efficient spectral projected-gradient algorithm for optimization subject to a group one-norm constraint. Our approach is based on a novel linear-time algorithm for Euclidean projection
onto the one- and group one-norm constraints. Numerical experiments on large data sets suggest that the proposed method is substantially more efficient and scalable than existing methods. Citation …
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Branching proofs of infeasibility in low density subset sum problems
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On Theory of Compressive Sensing via L1-Minimization:
Compressive (or compressed) sensing (CS) is an emerging methodology in computational signal processing that has recently attracted intensive research activities. At present, the basic CS theory
includes recoverability and stability: the former quantifies the central fact that a sparse signal of length n can be exactly recovered from much less than n measurements via L1-minimization … Read
An Infeasible Interior-Point Algorithm with full-Newton Step for Linear Optimization
In this paper we present an infeasible interior-point algorithm for solving linear optimization problems. This algorithm is obtained by modifying the search direction in the algorithm [C. Roos, A
full-Newton step ${O}(n)$ infeasible interior-point algorithm for linear optimization, 16(4) 2006, 1110-1136.]. The analysis of our algorithm is much simpler than that of the Roos’s algorithm … Read
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After a brief introduction to Jordan algebras, we present a primal-dual interior-point algorithm for second-order conic optimization that uses full Nesterov-Todd-steps; no line searches are required.
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Closed-form solutions to static-arbitrage upper bounds on basket options
We provide a closed-form solution to the problem of computing the sharpest static-arbitrage upper bound on the price of a European basket option, given the prices of vanilla call options in the
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The submodular knapsack set is the discrete lower level set of a submodular function. The modular case reduces to the classical linear 0-1 knapsack set. One motivation for studying the submodular
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We address optimization of nonlinear functions of the form $f(Wx)$~, where $f:\R^d\rightarrow \R$ is a nonlinear function, $W$ is a $d\times n$ matrix, and feasible $x$ are in some large finite set $
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The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy
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10.7: Routh Stability - Ranges of Parameter Values that are Stable
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The stability of a process control system is extremely important to the overall control process. System stability serves as a key safety issue in most engineering processes. If a control system
becomes unstable, it can lead to unsafe conditions. For example, instability in reaction processes or reactors can lead to runaway reactions, resulting in negative economic and environmental
The absolute stability of a control process can be defined by its response to an external disturbance to the system. The system may be considered stable if it exists at a consistent state or setpoint
and returns to this state immediately after a system disturbance. In order to determine the stability of a system, one often must determine the eigenvalues of the matrix representing the system’s
governing set of differential equations. Unfortunately, sometimes the characteristic equation of the matrix (the polynomial representing its eigenvalues) can be difficult to solve; it may be too
large or contain unknown variables. In this situation, a method developed by British mathematician Edward Routh can yield the desired stability information without explicitly solving the equation.
Recall that in order to determine system stability one need only know the signs of the real components of the eigenvalues. Because of this, a method that can reveal the signs without actual
computation of the eigenvalues will often be adequate to determine system stability.
To quickly review, negative real eigenvalue components cause a system to return to a steady state point (where all partial derivatives equal zero) when it experiences disturbances. Positive real
components cause the system to move away from a stable point, and a zero real component indicates the system will not adjust after a disturbance. Imaginary components simply indicate oscillation with
a general trend in accordance with the real part. Using the method of Routh stability, one can determine the number of each type of root and thus see whether or not a system is stable. When unknown
variables exist in the equation, Routh stability can reveal the boundaries on these variables that keep the roots negative.
The Routh Array
The Routh array is a shortcut to determine the stability of the system. The number of positive (unstable) roots can be determined without factoring out any complex polynomial.
Generating the Array
The system in question must have a characteristic equation of a polynomial nature. as shown below:
\[P(S) = a_{n}S^n + a_{n-1}S^{n-1} + \cdots + a_{1}S+ a_0 \nonumber \]
In order to examine the roots, set P(S)=0, which will allow you to tell how many roots are in the left-hand plane, right hand plane, and on the j-omega axis. If the system involves trigonometric
functions it needs to be fit to a polynomial via a Taylor series expansion. One necessary condition for stability is that
This flow diagram shows the generation of a Routh array for an idealized case with m,n representing the location in the matrix.
The coefficients of the polynomial are placed into an array as seen below. The number of rows is one more than the order of the equation. The number of sign changes in the first column indicate the
number of positive roots for the equation.
Row 1 \[\\ \ a_{n } \ \ \qquad a_{n-2} \ \qquad a_{n-4} \qquad ... \nonumber \]
Row 2 \[\\ \ a_{n-1} \qquad a_{n-3} \qquad a_{n-5} \qquad ... \nonumber \]
Row 3 \[\\ \ b_1 \qquad \qquad b_2 \ \ \qquad b_3 \ \qquad \ ... \nonumber \]
Row 4 \[\\ \ c_1 \qquad \qquad c_2 \ \ \qquad c_3 \ \qquad \ ... \nonumber \]
\[\\ \ \vdots \qquad \quad \qquad \vdots \qquad \ \ \ \vdots \; \; \qquad \ \ \vdots \nonumber \]
Row 5 \[\\ \ p_1 \qquad \qquad p_2 \nonumber \]
Row 6 \[\\ \ q_1 \qquad \nonumber \]
Row 7 \[\\ \ v_1 \qquad \nonumber \]
In the array, the variables b1,b2,c1,c2,etc. are determined by calculating a determinant using elements from the previous two rows as shown below:
The general expression for any element x after the first two rows with index (m,n) is as follows:
Note, that if the Routh array starts with a zero, it may still be solved (assuming that all the other values in the row are not zero), by replacing the zero with a constant, and letting that constant
equal a very small positive number. Subsequent rows within that column that have this constant will be calculated based on the constant choosen.
Once the array is complete, apply the following theorems to determine stability:
1. If all of the values in the fist column of the Routh array are >0, then P(S) has all negative real roots and the system is stable.
2. If some of the values in the first column of the Routh array are <0, then the number of times the sign changes down the first column will = the number of positive real roots in the \(P(S)\)
3. If there is 1 pair of roots on the imaginary axis, both the same distance from the origin (meaning equidistant), then check to see if all the other roots are in the left hand plane. If so, then
the imaginary roots location may be found using AS^2 + B = 0, where A and B are the elements in the Routh array for the 2nd to last row.
To clarify even further, an example with real numbers is analyzed.
The following polynomial was generated from a sample system.
\[P(S) = 5S^3 -10S^2 + 7S+ 20 \nonumber \]
The preceding polynomial must be investigated in order to determine the stability of the system. This is done by generating a Routh array in the manner described above. The array as a result of this
polynomial is,
Row 1 \[\\ \ \ 5 \qquad 7 \nonumber \]
Row 2 \[\\ -10 \ \ \ \ 20 \nonumber \]
Row 3 \[\\ \ \ 17 \nonumber \]
Row 4 \[\\ \ \ 20 \nonumber \]
In the array shown above, the value found in the third row is calculated as follows.
Finding Stable Control Parameter Values
Often, for a unit operation, a PID parameter such as controller gain (K[c]), the integral time constant (T[i]), or the derivative time constant (T[d]) creates an additional variable in the
characteristic equation. This can be carried through the computations of the Routh array to indicate which values of the variable will provide stability to the system through by preventing positive
roots from occuring in the equation. For example, if a controller output is governed by the function:
\[ 10s^3 + 5s^2 + 8s + (T_d + 2) \nonumber \]
The stable values of T[d] can be found via a Routh array:
Row 1 \(\quad 10 \qquad\quad 8\)
Row 2 \(\quad 5 \ \qquad \ \ \ (T_d + 2)\)
Row 3 \( \ \ 4-2T_d\)
Row 4 \((T_d + 2)\)
We reveal \(-2 < T_d < 2 \qquad \) in order to keep the first column elements positive, so this is the stable range of values for this parameter.
If multiple parameters were in the equation, they would simply be solved for as a group, yielding constraints along the lines of "T[i] + K[c] > 2" etc, so any value chosen for one parameter would
give a different stable range for the other.
Special Cases
There are a few special cases that one should be aware of when using the Routh Test. These variances can arise during stability analysis of different control systems. When a special case is
encountered, the traditonal Routh stability solution methods are altered as presented below.
One of the coefficients in the characteristic equation equals zero
If the power of the 0*S^n is
Working Equation:
Row 1 \[\quad 2 \quad \quad \quad -24 \nonumber \]
Row 2 \[\quad \epsilon \quad \quad \quad 32 \nonumber \]
Row 3 \[\\frac{-24\epsilon-64}{\epsilon} \ \quad \qquad 0 \nonumber \]
Row 4 \[\\qquad 32 \nonumber \]
Since ε is positive we know that in the first column row 2 will be positive, row 4 will be positive, and row 3 will be negative. This means we will have a sign change from 2 to 3 and again from 3 to
4. Because of this, we know that two roots will have positive real components. If you actually factor out the equation you see that
One of the roots is zero
This case should be obvious simply from looking at the polynomial. The constant term will be missing, meaning the variable can be factored from every term. If you added an ε to the end as in case 1,
the last row would be ε and falsely indicate another sign change. Carry out Routh analysis with the last zero in place.
Row 1 \[\\ \ 1 \ \ \ -2 \nonumber \]
Row 2
Row 3
Row 4 \[\\ \ 0 \qquad \nonumber \]
As you can see in column one we have row 1 positive, row 2 and 3 negative, and row 4 zero. This is interpreted as one sign change, giving us one positive real root. Looking at this equation in
factored form,
we can see that indeed we have only one positive root equal which equals 2. The zero in the last row indicates an additional unstable root of zero. Alternatively, you may find it easier to just
factor out the variable and find the signs of the remaining eigenvalues. Just remember there is an extra root of zero.
Row Full of Zeros
When this happens you know you have either a pair of imaginary roots, or symmetric real roots. The row of zeros must be replaced. The following example illustrates this procedure.
Row 1 \[\\ \ 1 \qquad \qquad 10 \qquad \qquad 9 \nonumber \]
Row 2
Row 3 \[\\ \ 9 \qquad \qquad 9 \nonumber \]
Row 4 \[\\ \ 0 \qquad \qquad 0 \nonumber \]
Row 4 contains all zeros. To determine its replacement values, we first write an auxiliary polynomial A determined by the entries in Row 3 above.
Notice that the order decreases by 1 as we go down the table, but decreases by 2 as we go across.
We then take the derivative of this auxiliary polynomial.
The coefficients obtained after taking the derivative give us the values used to replace the zeros. From there, we can proceed the table calcuations normally. The new table is
Row 1 \[\\ \ 1 \qquad \qquad 10 \qquad \qquad 9 \nonumber \]
Row 2
Row 3 \[\\ \ 9 \qquad \qquad 9 \nonumber \]
Row 4 \[\\ \ 18 \qquad \qquad 0 \nonumber \]
Row 5 \[\\ \ 9 \qquad \qquad \nonumber \]
In fact, the purely imaginary or symmetric real roots of the original polynomial are the same as the roots of the auxiliary polynomial. Thus, we can find these roots.
Because we have two sign changes, we know the other two roots of the original polynomial are positive.
In fact, after factoring this polynomial, we obtain
Therefore, the roots are
Routh arrays are useful for classifying a system as stable or unstable based on the signs of its eigenvalues, and do not require complex computation. However, simply determing the stability is not
usually sufficient for the design of process control systems. It is important to develop the extent of stability as well as how close the system is to instability. Further stability analysis not
accounted for in the Routh analysis technique include finding the degree of stability, the steady state performance of the control system, and the transient response of the system to disturbances.
More involved techniques, such as those discussed in Eigenvalues and Eigenvectors, must be used to further characterize the system stability (with the exception of system polynomials resulting in
pure imaginary roots). Another limitation of the Routh method occurs when the polynomial in question becomes so large that Routh stability is too computationally time consuming (a personal judgment).
For this situation another method, such as a root locus plotmust be used.
Note that for defining stability, we will always start out with a polynomial. This polynomial arises from finding the eigenvalues of the linearized model. Thus we will never encounter other
functions, say exponenential functions or sin or cos functions in general for stability analysis in control theory.
Advantages Over Root Locus Plots
Routh stability evaluates the signs of the real parts of the roots of a polynomial without solving for the roots themselves. The system is stable if all real parts are negative. Therefore unlike root
locus plots, the actual eigenvalues do not need to be calculated for a Routh stability analysis. Furthermore, sometimes the system has too many unknowns to easily construct and interpret a root locus
plot (e.g. with two PID controllers there are the variables Kc1, Kc2, τi1, τi2, τd1, and τd2).
Assume the following polynomial describes the eigenvalues of a linearized model of your process. For this polynomial, complete a Routh array and determine the system's stability?
Since P(X) is a fourth-order polynomial, the Routh array contains five rows.
Row 1
Row 2
Row 3
Row 4
Row 5
Rows 1 and 2 correspond to the coefficients of the polynomial terms:
Rows 3, 4, and 5 contain the determinants using elements from the previous two rows.
The complete Routh array:
Row 1
Row 2
Row 3
Row 4
Row 5
Since all the values in the first column are positive, the equation P(x) has all negative roots and the system is stable.
Consider a system with the following characteristic equation:
\[20s^3+59s^2+46s+(4+K_c)=0 \nonumber \]
Using a P-only controller, find the range of controller gain that will yield a stable system.
Since the equation is a third-order polynomial, the Routh array has four rows. Routh Array:
Row 1 \[\\ \qquad \;20 \qquad \qquad \qquad 46 \qquad \nonumber \]
Row 2 \[\\ \qquad \;59 \qquad \qquad \qquad (4+K_c)\qquad \nonumber \]
Row 3 \[\\ \;46 - \frac{20}{59}(4+K_c) \quad \ \quad 0 \nonumber \]
Row 4 \[\\ \ \ \ \ \;(4+K_c) \qquad \qquad \ 0 \nonumber \]
For the system to be stable, all the elements of the first column of the Routh array have to be positive.
The first column will contain only positive elements if:
Acceptable stable range
Consider a system with the following characteristic equation:
\[s^5-3s^4+s^3+s^2+4=0 \nonumber \]
Determine the stability of this system.
One of the coefficients in the characteristic equation equals 0. We replace the zero with a quantity which would be positive (approach zero from the right-hand side) and continue with the analysis as
Working equation:
\[s^5-3s^4+s^3+s^2+ \epsilon s+4=0 \nonumber \]
Row 1 \[\\ \ \ 1 \qquad \qquad \qquad \qquad 1\qquad \qquad \epsilon \nonumber \]
Row 2
Row 3 \[\\ \frac{4}{3} \ \qquad \qquad \qquad \quad \epsilon + \frac {4}{3} \nonumber \]
Row 4 \[\\ \ 4+ \frac{9\epsilon}{4} \ \qquad \qquad \qquad 4 \nonumber \]
Row 5 \[\\ \epsilon + \frac {4}{3} - \frac{64}{27\epsilon+48} \qquad 0 \nonumber \]
Row 6 \[\\ 4 \nonumber \]
Since ε is positive, in the first column, there are two sign changes, from row 1 to row 2 and from row 2 to row 3. Thus, we know that the roots will have two positive real components. If you actually
factor out the equation you will see that,
Additional complication exists because at row 5, as ε goes to zero, the term also goes to zero, which means that for row 5, we are getting a row full of zeros. This means that we have a pair of
imaginary roots, and this situation can be solved using the equation,
In this case, the working equation is,
The imaginary roots are,
You are an engineer at an icecream factory. There is a storage vat with a cooling system that has a PI controller scheme. It has the following characteristic equation:
Your job is to determine the constraints on the values K[c] and T[i] such that the system is stable.
The goal is to make the matrix such that the first column has no sign changes. Since the first two entries in the first column are numbers and positive, therefore all other values in this column must
be positive.
Working equation:
Row 1 \[\\ 10 \qquad \qquad \qquad \qquad -K_c \qquad \qquad T_i+6 \nonumber \]
Row 2 \[\\ 3\qquad \qquad \qquad \qquad \qquad 3\qquad \nonumber \]
Row 3 \[\\ 10-K_c \ \qquad \qquad \qquad T_i+6 \nonumber \]
Row 4 \[\\ 3- \frac{3(T_i+6)}{(10-K_c)} \qquad \qquad 0 \nonumber \]
Row 5 \[\\ T_i+6 \nonumber \]
Thus the stable conditions for the constants are:
• Bequette, W.B. Process Control Modeling Design and Simulation., New Jersey: Prentice Hall, pp 170-178.
• Foecke, H.A. and Weinstein, A. "Complex roots via real roots and square roots using Routh's stability criterion." arxiv.org, January 5, 2007.
• Liptak, Bela G., Process Control and Optimization. Vol. II. New York: Taylor & Francis.
• Ogunnaike, Babatunde A.; Ray, W. Harmon. Process Dynamics, Modeling, and Control. New York Oxford: Oxford University Press, 1994.
• Authors: John D'Arcy, Matt Hagen, Adam Holewinski, and Alwin Ng
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Question #c440b | Socratic
Question #c440b
1 Answer
Your strategy here will be to use three conversion factors: one to take you from teaspoons to milliliters, one to take you from milliliters to milligrams, and one to take you from milligrams to grams
The problem tells you that
$\textcolor{p u r p \le}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\text{1 teaspoon " = " 5 mL}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
Use this to determine the volume of this drug that would be equivalent to $1.2$teaspoons
#1.2 color(red)(cancel(color(black)("teaspoons"))) * "5 mL"/(1color(red)(cancel(color(black)("teaspoon")))) = "6.0 mL"#
Next, the problem tells you that every $\text{5 mL}$ of this drug contains $\text{100. mg}$ of ibuprofen. This means that your sample will contain
#6.0 color(red)(cancel(color(black)("mL drug"))) * "100. mg ibuprofen"/(5color(red)(cancel(color(black)("mL drug")))) = "120 mg ibuprofen"#
Finally, use the fact that
$\textcolor{p u r p \le}{\overline{\underline{| \textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\text{1 g" = 10^3"mg}} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
to find the amount of ibuprofen in grams
#120 color(red)(cancel(color(black)("mg"))) * "1 g"/(10^3color(red)(cancel(color(black)("mg")))) = color(green)(bar(ul(|color(white)(a/a)color(black)("0.12 g")color(white)(a/a)|)))#
I'll leave the answer rounded to two sig figs.
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The Stacks project
Lemma 43.9.1. Let $X$ be a variety. Let $W \subset X$ be a subvariety of dimension $k + 1$. Let $f \in \mathbf{C}(W)^*$ be a nonzero rational function on $W$. Then $\text{div}_ W(f)$ is rationally
equivalent to zero on $X$. Conversely, these principal divisors generate the abelian group of cycles rationally equivalent to zero on $X$.
Comments (3)
Comment #6776 by Morten on
Why is the map $W'\rightarrow W$ generically finite? Could $W'$ not be a trivial family like $X\times \mathbb{P}^1$?
Comment #6777 by Johan on
Ha! Yes, this could happen. Of course, in that case $[W'_0]_k - [W'_\infty]_k$ is the zero cycle. I will fix this the next time I go through all the comments.
Comment #6933 by Johan on
Thanks and fixed here.
There are also:
• 2 comment(s) on Section 43.9: Rational equivalence and rational functions
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Math Worksheet
Classifying Triangles Worksheet 5-Pack - Math Worksheets Land - peirce cps k12 il
Name date triangles: worksheet 1 are the triangles equilateral, isosceles, or scalene? are the triangles acute, obtuse, or right? 1. 2. 3. 4. 5. 6. draw a triangle that fits the description. if the
triangle is not possible, tell why. 7 an acute...
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Writing down Variable Expressions Involving Two Operations
In this guide, the process of writing variable expressions that include two operations will be explained. By comprehending the use of two operations, composing variable expressions becomes
A step-by-step guide to Writing down Variable Expressions Involving Two Operations
Variables are considered letters whose values are unknown.
For instance, w is the variable in the expression: 2w\(+\)42
Coefficients are numerical values used along with a variable.
For instance, 2 is the coefficient in the expression 2w\(+\)42.
To identify the operations, you have to write down a mathematical expression and find the keywords.
To convert the description into an expression, you have to use the keywords to do one operation at a time.
Here is a step-by-step guide to writing down variable expressions involving two operations:
1. Identify the operations to be performed: Determine the two operations that need to be performed on the variables in the expression. For example, the operations could be addition and
multiplication, or subtraction and division.
2. Write down the expression using variables: Use variables to represent the values in the expression. For example, if you are given the expression “3 + 4x,” you would write “x + 4x.”
3. Use proper mathematical symbols for the operations: Use the appropriate mathematical symbols for the operations identified in step 1. For example, if the operations are addition and
multiplication, use the “+” symbol for addition and the “x” symbol for multiplication.
4. Simplify the expression: If possible, simplify the expression by combining like terms and using the distributive property.
5. Check your work: Make sure the expression makes sense and that the answer is in the correct form.
By following these steps, you will be able to write down variable expressions involving two operations.
Remember, the key is to clearly identify the operations to be performed, use variables to represent the values, and use proper mathematical symbols for the operations.
Writing down Variable Expressions Involving Two Operations – Example 1
Write an expression for this sequence of operations.
Multiply f by 9, then multiply w by the result.
Step 1: Find the keywords. “Multiply” is a keyword. Now write the expression.
Step 2: Change the first keyword: Multiply f by \(9→ f×9\)
Step 3: Change the second keyword: Multiply w by the result. w\(×\) (result)
Step 4: So, w\((f×9)\)
Writing down Variable Expressions Involving Two Operations – Example 2
Write an expression for this sequence of operations.
Divide 4 by x, then subtract 12 from the result.
Step 1: Find the keywords. Divide and subtract the keywords.
Step 2: Change the first keyword: Divide 4 by x→ \(\frac{4}{x}\)
Step 3: Change the second keyword: Subtract 12 from the result→ result \(-12\)
Step 4: So, \(\frac{4}{x}-12\)
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How can drivers optimize their Speed-to-Power for better performance? in context of speed to power
30 Aug 2024
Title: Optimizing Speed-to-Power: Strategies for Drivers to Enhance Performance
In the realm of high-performance driving, optimizing speed-to-power (S2P) is crucial for achieving exceptional acceleration and overall performance. This article delves into the concept of S2P, its
significance, and provides strategies for drivers to optimize their S2P ratio, thereby enhancing their driving experience.
Speed-to-Power (S2P) is a critical parameter in high-performance driving, measuring the relationship between an engine’s speed and power output. A higher S2P ratio indicates better acceleration and
overall performance. In this article, we will explore the concept of S2P, its importance, and provide guidelines for drivers to optimize their S2P ratio.
The Concept of Speed-to-Power:
S2P is calculated by dividing an engine’s power output (P) by its rotational speed (n):
S2P = P / n
In ASCII format:
S2P = P ÷ n
Significance of Speed-to-Power:
A higher S2P ratio indicates better acceleration, as the engine can produce more power at a given speed. This is particularly important in high-performance driving, where rapid acceleration and
deceleration are crucial.
Strategies for Optimizing Speed-to-Power:
1. Engine Tuning: Adjusting engine parameters such as compression ratio, camshaft profile, and fuel injection timing can optimize S2P.
2. Gear Selection: Selecting the correct gear for a given speed can maximize power output and minimize losses due to friction and heat.
3. Weight Reduction: Reducing vehicle weight through lightweight materials or optimized design can improve S2P by increasing power-to-weight ratio.
4. Aerodynamics: Optimizing aerodynamic features such as spoilers, splitters, and diffusers can reduce drag and improve S2P.
5. Driver Technique: Developing optimal driving techniques, including smooth acceleration and braking, can minimize energy losses and maximize S2P.
Optimizing speed-to-power is a critical aspect of high-performance driving. By understanding the concept of S2P and implementing strategies to optimize it, drivers can enhance their vehicle’s
acceleration and overall performance. This article has provided a comprehensive overview of the importance of S2P and offered practical guidelines for drivers to improve their S2P ratio.
• [1] “Speed-to-Power Optimization in High-Performance Vehicles” by J. Smith et al.
• [2] “The Effects of Engine Tuning on Speed-to-Power Ratio” by M. Johnson et al.
• [3] “Aerodynamic Optimization for Improved Speed-to-Power Performance” by T. Lee et al.
Note: The references provided are fictional and used only to demonstrate the format of an academic article.
Related articles for ‘speed to power ‘ :
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The Candlestick-RSI
Transforming the RSI Into a Candlestick Indicator Using Python
Candlestick charting does not have to be limited to price data. What if it can be used on technical indicators so that start testing some new pattern recognition techniques? In this article, the RSI
is looked at from the perspective of a simple candlestick chart where trading signals are derived. The basic idea is to start thinking about finding a robust strategy that uses the technique of
applying candlesticks to technical indicators in order to find either hidden patterns or trend reversal confirmation signals.
Knowledge must be accessible to everyone. This is why, from now on, a purchase of either one of my new books “Contrarian Trading Strategies in Python” or “Trend Following Strategies in Python” comes
with free PDF copies of my first three books (Therefore, purchasing one of the new books gets you 4 books in total). The two new books listed above feature a lot of advanced indicators and strategies
with a GitHub page. You can use the below link to purchase one of the two books (Please specify which one and make sure to include your e-mail in the note).
Pay Kaabar using PayPal.Me
Go to paypal.me/sofienkaabar and type in the amount. Since it’s PayPal, it’s easy and secure. Don’t have a PayPal…www.paypal.com
The Relative Strength Index
The RSI is without a doubt the most famous momentum indicator out there, and this is to be expected as it has many strengths especially in ranging markets. It is also bounded between 0 and 100 which
makes it easier to interpret. The RSI is notably used in many ways, among them:
• The extremes strategy: Where we search for overbought an oversold levels to initiate contrarian trades. The idea is that an overbought market has a high reading on the RSI signifying too much
bullish momentum and a possible correction or even a reversal may take place. Similarly, an oversold market has a low reading on the RSI signifying too much bearish momentum and a possible
correction or even a reversal may take place.
• The divergence strategy: When there is an established trend, the overall direction does not move linearly with regards to magnitude and force, meaning that when a market is rising, its momentum
strength is not stable. At the begining it is usually strong but by profit taking and lower convictions, the market starts to lose some strength, thus, continuing to rise but struggling to do so.
This is where we see a bearish divergence on the RSI. A divergence signifies the trend’s exhaustion and may signal a correction or even a reversal. Visually, a bearish divergence is when we see
the market making higher highs while the RSI is making lower highs. Similarly, a bullish divergence is when we see the market making lower lows while the RSI is making higher lows.
The fact that it is famous, contributes to the efficacy of the RSI. This is because the more traders and portfolio managers look at the indicator, the more people will react based on its signals and
this in turn can push market prices. Of course, we cannot prove this idea, but it is intuitive as one of the basis of Technical Analysis is that it is self-fulfilling.
The RSI is calculated using a rather simple way. We first start by taking price differences of one period. This means that we have to subtract every closing price from the one before it. Then, we
will calculate the smoothed average of the positive differences and divide it by the smoothed average of the negative differences. The last calculation gives us the Relative Strength which is then
used in the RSI formula to be transformed into a measure between 0 and 100.
To calculate the Relative Strength Index through the following function, we need an OHLC array (not a data frame). This means that we will be looking at an array of 4 columns. The function for the
Relative Strength Index is therefore:
def rsi(Data, lookback, close, where, width = 1, genre = 'Smoothed'):
# Adding a few columns
Data = adder(Data, 7)
# Calculating Differences
for i in range(len(Data)):
Data[i, where] = Data[i, close] - Data[i - width, close]
# Calculating the Up and Down absolute values
for i in range(len(Data)):
if Data[i, where] > 0:
Data[i, where + 1] = Data[i, where]
elif Data[i, where] < 0:
Data[i, where + 2] = abs(Data[i, where])
# Calculating the Smoothed Moving Average on Up and Down
absolute values
if genre == 'Smoothed':
lookback = (lookback * 2) - 1 # From exponential to smoothed
Data = ema(Data, 2, lookback, where + 1, where + 3)
Data = ema(Data, 2, lookback, where + 2, where + 4)
if genre == 'Simple':
Data = ma(Data, lookback, where + 1, where + 3)
Data = ma(Data, lookback, where + 2, where + 4)
# Calculating the Relative Strength
Data[:, where + 5] = Data[:, where + 3] / Data[:, where + 4]
# Calculate the Relative Strength Index
Data[:, where + 6] = (100 - (100 / (1 + Data[:, where + 5])))
# Cleaning
Data = deleter(Data, where, 6)
Data = jump(Data, lookback)
return Data
We need to define the primal manipulation functions first in order to use the RSI’s function on OHLC data arrays.
# The function to add a certain number of columns
def adder(Data, times):
for i in range(1, times + 1):
z = np.zeros((len(Data), 1), dtype = float)
Data = np.append(Data, z, axis = 1)
return Data
# The function to deleter a certain number of columns
def deleter(Data, index, times):
for i in range(1, times + 1):
Data = np.delete(Data, index, axis = 1)
return Data
# The function to delete a certain number of rows from the beginning
def jump(Data, jump):
Data = Data[jump:, ]
return Data
Candlestick Charting
Candlestick charts are among the most famous ways to analyze the time series visually. They contain more information than a simple line chart and have more visual interpretability than bar charts.
Many libraries in Python offer charting functions but being someone who suffers from malfunctioning import of libraries and functions alongside their fogginess, I have created my own simple function
that charts candlesticks manually with no exogenous help needed.
OHLC data is an abbreviation for Open, High, Low, and Close price. They are the four main ingredients for a timestamp. It is always better to have these four values together so that our analysis
reflects more the reality. Here is a table that summarizes the OHLC data of hypothetical security:
Our job now is to plot the data so that we can visually interpret what kind of trend is the price following. We will start with the basic line plot before we move on to candlestick plotting.
Note that you can download the data manually or using Python. In case you have an excel file that has OHLC only data starting from the first row and column, you can import it using the below code
import numpy as np
import pandas as pd
# Importing the Data
my_ohlc_data = pd.read_excel('my_ohlc_data.xlsx')
# Converting to Array
my_ohlc_data = np.array(my_ohlc_data)
Plotting basic line plots is extremely easy in Python and requires only one line of code. We have to make sure that we have imported a library called matplotlib and then we will call a function that
plots the data for us.
# Importing the necessary charting library
import matplotlib.pyplot as plt
# The syntax to plot a line chart
plt.plot(my_ohlc_data, color = 'black', label = 'EURUSD')
# The syntax to add the label created above
# The syntax to add a grid
Now that we have seen how to create normal line charts, it is time to take it to the next level with candlestick charts. The way to do this with no complications is to think about vertical lines.
Here is the intuition (followed by an application of the function below):
• Select a lookback period. This is the number of values you want to appear on the chart.
• Plot vertical lines for each row representing the highs and lows. For example, on OHLC data, we will use a matplotlib function called vlines which plots a vertical line on the chart using a
minimum (low) value and a maximum (high value).
• Make a color condition which states that if the closing price is greater than the opening price, then execute the selected block of code (which naturally contains the color green). Do this with
the color red (bearish candle) and the color black (Doji candle).
• Plot vertical lines using the conditions with the min and max values representing closing prices and opening prices. Make sure to make the line’s width extra big so that the body of the candle
appears sufficiently enough that the chart is deemed a candlestick chart.
def ohlc_plot(Data, window, name):
Chosen = Data[-window:, ]
for i in range(len(Chosen)):
plt.vlines(x = i, ymin = Chosen[i, 2], ymax = Chosen[i, 1], color = 'black', linewidth = 1)
if Chosen[i, 3] > Chosen[i, 0]:
color_chosen = 'green'
plt.vlines(x = i, ymin = Chosen[i, 0], ymax = Chosen[i, 3], color = color_chosen, linewidth = 4)
if Chosen[i, 3] < Chosen[i, 0]:
color_chosen = 'red'
plt.vlines(x = i, ymin = Chosen[i, 3], ymax = Chosen[i, 0], color = color_chosen, linewidth = 4)
if Chosen[i, 3] == Chosen[i, 0]:
color_chosen = 'black'
plt.vlines(x = i, ymin = Chosen[i, 3], ymax = Chosen[i, 0], color = color_chosen, linewidth = 4)
# Using the function
ohlc_plot(my_ohlc_data, 50, '')
The Candlestick-RSI
The Candlestick RSI will be a regular 13-period RSI applied to the open prices and the closing prices, thus forming a simplistic type of candles where we can start seeing the psychology of the move.
We can also add the high and low calculations so that we add them later into the signal. Therefore, the first step into creating the indicator is the following syntax:
lookback = 13
# Calculating a 13-period RSI on opening prices
my_data = rsi(my_data, lookback, 0, 5, genre = 'Smoothed')
# Calculating a 13-period RSI on high prices
my_data = rsi(my_data, lookback, 1, 6, genre = 'Smoothed')
# Calculating a 13-period RSI on low prices
my_data = rsi(my_data, lookback, 2, 7, genre = 'Smoothed')
# Calculating a 13-period RSI on closing prices
my_data = rsi(my_data, lookback, 3, 8, genre = 'Smoothed')
The below chart shows the candlestick RSI in the first panel with the EURUSD hourly values in the second panel.
The above chart is a bit unusual, as we are used to putting the market price in the first panel. This time, we have switched and put the Candlestick-RSI above. We are now ready to derive the trading
conditions based on a choice of candlestick patterns and the usual extremes strategy on the RSI.
The above chart shows an example on the USDCAD where barriers of 20 and 80 are more suited to the Candlestick RSI. Note that the word barriers refer to the oversold/overbought levels.
Applying the Pattern Recognition Strategy
By fetching the conditions from candlestick pattern recognition and the RSI’s contrarian methods, we can develop the following trading rules:
• Go long (Buy) whenever the current RSI (Based on closing prices) is greater than the current RSI (Based on opening prices) while simultaneously the current RSI (Based on low prices) is lower than
the lower barrier of 20 (or 30 in the case of certain currency pairs).
• Go short (Sell) whenever the current RSI (Based on closing prices) is lower than the current RSI (Based on opening prices) while simultaneously the current RSI (Based on high prices) is higher
than the upper barrier of 80 (or 70 in the case of certain currency pairs).
def signal(Data):
Data = adder(Data, 20)
for i in range(len(Data)):
if Data[i, 8] > Data[i, 5] and Data[i, 7] < lower_barrier and Data[i - 1, 9] == 0:
Data[i, 9] = 1
elif Data[i, 8] < Data[i, 5] and Data[i, 6] > upper_barrier and Data[i - 1, 10] == 0:
Data[i, 10] = -1
return Data
Evaluating the Signal Quality
Having had the signals, we now know when the algorithm would have placed its buy and sell orders, meaning, that we have an approximate replica of the past where can can control our decisions with no
hindsight bias. We have to simulate how the strategy would have done given our conditions.
This means that we need to calculate the returns and analyze the performance metrics. Let us see a neutral metric that can give us somewhat a clue on the predictability of the indicator or the
strategy. For this study, we will use the Signal Quality metric.
The signal quality is a metric that resembles a fixed holding period strategy. It is simply the reaction of the market after a specified time period following the signal. Generally, when trading, we
tend to use a variable period where we open the positions and close out when we get a signal on the other direction or when we get stopped out (either positively or negatively). Sometimes, we close
out at random time periods. Therefore, the signal quality is a very simple measure that assumes a fixed holding period and then checks the market level at that time point to compare it with the entry
level. In other words, it measures market timing by checking the reaction of the market. For the performance evaluation to be unbiased, we will use three signal periods:
• 3 closing bars: This signal period relates to quick market timing strategies based on immediate reactions. In other words, we will be measuring the difference between the closing price 3 periods
after the signal and the entry price at the trigger.
• 8 closing bars: This signal period relates to market timing strategies with some slight lag. In other words, we will be measuring the difference between the closing price 8 periods after the
signal and the entry price at the trigger.
• 21 closing bars: This signal period relates to more important reactions that can even signal a change in the trend. In other words, we will be measuring the difference between the closing price
21 periods after the signal and the entry price at the trigger.
# Choosing an example of 3 periods
period = 3
def signal_quality(Data, closing, buy, sell, period, where):
Data = adder(Data, 1)
for i in range(len(Data)):
if Data[i, buy] == 1:
Data[i + period, where] = Data[i + period, closing] - Data[i, closing]
if Data[i, sell] == -1:
Data[i + period, where] = Data[i, closing] - Data[i + period, closing]
return Data
# Using 3 Periods as a Window of signal Quality Check
my_data = signal_quality(my_data, 3, 6, 7, period, 8)
positives = my_data[my_data[:, 8] > 0]
negatives = my_data[my_data[:, 8] < 0]
# Calculating Signal Quality
signal_quality = len(positives) / (len(negatives) + len(positives))
print('Signal Quality = ', round(signal_quality * 100, 2), '%')
# Output for 3 periods USDCAD Hourly: 53.42%
# Output for 8 periods USDCAD Hourly: 50.75%
# Output for 21 periods USDCAD Hourly: 51.09%
A signal quality of 51.09% means that on 100 trades, we tend to see in 51 of the cases a higher price 21 periods after getting the signal. This shows that the pattern on hourly data of the USDCAD
since 2011 (~ 600 trades) is not very predictive. The same job must be done on other pairs and other assets to have a full idea on the quality of the pattern.
Unlimited tweaking and patterns can be tested on this interesting and promising field. We will do that in the coming articles in further details.
If you want to see how to create all sorts of algorithms yourself, feel free to check out Lumiwealth. From algorithmic trading to blockchain and machine learning, they have hands-on detailed courses
that I highly recommend.
Learn Algorithmic Trading with Python Lumiwealth
Learn how to create your own trading algorithms for stocks, options, crypto and more from the experts at Lumiwealth. Click to learn more
To sum up, what I am trying to do is to simply contribute to the world of objective technical analysis which is promoting more transparent techniques and strategies that need to be back-tested before
being implemented. This way, technical analysis will get rid of the bad reputation of being subjective and scientifically unfounded.
I recommend you always follow the the below steps whenever you come across a trading technique or strategy:
• Have a critical mindset and get rid of any emotions.
• Back-test it using real life simulation and conditions.
• If you find potential, try optimizing it and running a forward test.
• Always include transaction costs and any slippage simulation in your tests.
• Always include risk management and position sizing in your tests.
Finally, even after making sure of the above, stay careful and monitor the strategy because market dynamics may shift and make the strategy unprofitable. | {"url":"https://abouttrading.substack.com/p/the-candlestick-rsi-790","timestamp":"2024-11-09T12:27:42Z","content_type":"text/html","content_length":"262361","record_id":"<urn:uuid:c41511d7-3212-4b16-9fe4-d77643457b69>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00362.warc.gz"} |
How to Convert From SHGC to Shading Coefficients | Homesteady
How to Convert From SHGC to Shading Coefficients
Solar heat gain coefficient (SHGC) is the total percentage of solar energy at a window opening that is either absorbed and reflected into a building or directly transmitted through the window. In hot
climates, like Arizona, a low SHGC helps reduce energy costs.
In cold climates, however, a higher SHGC helps reduce energy costs. SHGC replaces a measurement called shading coefficient (SC). You may find references to shading coefficient in the ratings of older
windows or in older reference materials. The relationship between the two ratings systems is straightforward, and the coefficients can be converted with a simple formula.
1. Use the formula SHGC = SC*0.87 to convert between SHGC and SC. Enter the value that you have and solve algebraically to calculate the value that you don't have.
2. Multiply SHGC by 1.15 to convert the value from SHGC to SC. This conversion factor is the result of solving algebraically and is a simple, shorthand version of the formula. You can use this step
in place of algebraic conversion.
3. Multiply SC by 0.87 to convert from SC to SHGC.
The Drip Cap
• Solar heat gain coefficient (SHGC) is the total percentage of solar energy at a window opening that is either absorbed and reflected into a building or directly transmitted through the window.
• The relationship between the two ratings systems is straightforward, and the coefficients can be converted with a simple formula.
• Multiply SHGC by 1.15 to convert the value from SHGC to SC.
• Green Building: Project Planning and Cost Estimating; R.S. Means Co.
• Residential Windows: A Guide to New Technologies and Energy Performance; John Carmody
Writer Bio
Chance Woods has been a personal trainer since 2002, specializing in fitness and nutrition. She holds a Bachelor of Science in dietetics.
Photo Credits
• Jupiterimages/Creatas/Getty Images
• Jupiterimages/Creatas/Getty Images
More Articles | {"url":"https://homesteady.com/13401929/how-to-convert-from-shgc-to-shading-coefficients","timestamp":"2024-11-05T18:38:21Z","content_type":"text/html","content_length":"121277","record_id":"<urn:uuid:fdeb27e5-d591-4808-a83a-89683b89e1c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027889.1/warc/CC-MAIN-20241105180955-20241105210955-00822.warc.gz"} |
The influence of non-linearly variable normal load on the resonant vibration of shrouded blade
Introducing friction damping devices in blade structures is an effective measure to reduce the failure rate of aero-engine. The component motion perpendicular to contact surface leads the normal load
to vary during vibration, and phase difference of adjacent blades makes the variation non-linear. This article analyzed the influence of non-linearly variable normal load on damper’s
vibration-reducing effect, and deduced the formula of hysteresis curve. Using the model raised, this article calculated the vibratory response of a shrouded blade structure and compared the result
with experimental data. The conclusion is that macroslip model under the condition of non-linearly variable normal load is accurate and effective in terms of predicting the vibratory response of
shrouded blade structure.
1. Introduction
A large amount of damage in gas-turbine structures can be attributed to the high cycle fatigue caused by high stress during vibrating. Reducing the vibratory stress has always been an important topic
of blade structure design. It is an effective way to incorporate friction damping facilities into the blade structure during design phase, which normally includes under platform dampers and shrouds.
Macroslip model is widely adopted by researchers when studying vibration reduction problems. Griffin [1] came up with the macroslip model back in 1980, where he treated the damper as series
connection of a spring and a coulomb friction pair. Since then, a large number of researchers have did further research with regard of almost every aspect of macroslip model. Sinha [2] studied the
effects bought by the difference between maximum static friction force and sliding friction force. Griffin [3] studied mistuned blade-disk structure analyzing the influence of mistuning on damper’s
vibration reduction function. Chen [4] came up with the 3D contact model to simulate the three dimensional motion of contact point.
During the vibration of turbine blade with shrouds, the direction of blade vibration is not parallel to the contact surface, which leads to the variation of normal load on the contact surface.
Meanwhile, because of the phase difference between adjacent blades the change of normal load on the contact surface is normally non-linearly. This article firstly induces how the normal load change
during vibration, and then come up with the hysteresis curves describing the change of friction force on contact surface along with the displacement of contact point. Thus, the macroslip model under
the condition of non-linearly variable normal load is came forward. Using this very model this article analyzes the vibratory response characteristics of a shrouded blade and compares the simulation
results with the experimental data.
2. Macroslip model
Macroslip model is illustrated in Fig. 1 friction damper is simplified as the series connection of a spring and a coulomb friction pair. During vibration, the blade pulls or pushes the spring, but
the contact point won’t slip until the force of spring surpasses the maximum statistic friction force.
3. Non-linearly variable normal load
In shrouded blade structure, the vibration direction of blade is not parallel to shroud’s surface, which means the normal load on contact surface is changing during the process of vibration, as can
be seen in Fig. 2.
Fig. 2Vibration direction of shroud
Suppose the displacement of blade vibration is $s\left(t\right)$, which could be decomposed as horizontal component $x\left(t\right)$ which is parallel to contact surface and vertical component $y\
left(t\right)$ which is perpendicular to contact surface.
Under harmonic excitation, $x\left(t\right)=X\mathrm{c}\mathrm{o}\mathrm{s}\theta$. Suppose $\phi$ is the phase difference between the motion parallel to contact surface and the motion perpendicular
to contact surface, then:
$N={N}_{0}+{k}_{n}y\left(t\right)={N}_{0}+\gamma {k}_{d}X\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta +\phi \right),$
where $\gamma =\mathrm{c}\mathrm{o}\mathrm{t}\beta \left({k}_{n}/{k}_{d}\right)$, $\beta$ is the includes angle between blade chord line and shroud’s surface, and $\gamma$ is constant determined by
blade structure’s geometrical and material characteristics.
Thus, the non-linear relationship between normal load and vibration displacement can be expressed as:
$N\left(\theta \right)=\left\{\begin{array}{ll}{N}_{0}+\gamma {k}_{d}X\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta +\phi \right),& \theta \le \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm
{s}\left(-\frac{{N}_{0}}{\gamma {k}_{d}X}\right)-\phi ,\\ 0,& \theta \ge \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{s}\left(-\frac{{N}_{0}}{\gamma {k}_{d}X}\right)-\phi .\end{array}\
It is can be seen in Eq. (2), when $\phi =0$, the change of normal load along with displacement of contact point is linear; when $\phi e 0$, the change is non-linear.
4. Hysteresis curves
With the increase of amplitude, stick, slip and separation will occur on contact surface successively. One thing should be noticed is that when normal load changes non-linearly, separations in reload
and unload process don’t occur at same time because of the asymmetry of friction force’s upper and lower bounds. Therefore, with vibratory amplitude increasing gradually, contact surface will
experience four states, which are stick, slip but no separation, separation only in unload process, separation in both unload and reload process.
This article induces the formula describing how friction force changes along with vibratory displacement and draws the hysteresis curves accordingly.
Fig. 3 illustrates the hysteresis curves of different amplitudes when $\phi =\mathrm{}$45° ($\phi$ is phase difference as mentioned above), where from top to bottom, from left to right, amplitudes
increases. As can be seen, as amplitude getting larger, four corresponding states occur on contact surface: stick, slip but no separation, separation only in unload process, separation in both unload
and reload process.
Fig. 3Hysteresis curves of different amplitudes when φ= 45°
b) Slip but no liftoff ($\phi =$ 45°)
c) Liftoff during unloading ($\phi =$ 45°)
d) Liftoff during unloading and reloading ($\phi =$ 45°)
5. Calculate equivalent stiffness and equivalent damping using HBM method
After getting the hysteresis curves, the equivalent damping and equivalent stiffness is analyzed using HBM method.
Suppose the displacement of contact point changes harmonically: $x=X\mathrm{c}\mathrm{o}\mathrm{s}\theta$.
The damper’s friction force can be expressed using equivalent damping and equivalent stiffness as:
The damper’s friction force can also be expanded in Fourier series and truncated after its fundamental terms:
${f}_{n}=\frac{{a}_{1}}{X}X\mathrm{c}\mathrm{o}\mathrm{s}\theta +\frac{{b}_{1}}{\omega X}\omega X\mathrm{s}\mathrm{i}\mathrm{n}\theta =\frac{{a}_{1}}{X}x+\left(-\frac{{b}_{1}}{\omega X}\right)\
Compare Eq. (3) and Eq. (4), one can get the expressions of equivalent damping and equivalent stiffness:
${k}_{eq}=\frac{{a}_{1}}{X}=\frac{1}{\pi X}{\int }_{0}^{2\pi }f\left(X\mathrm{c}\mathrm{o}\mathrm{s}\theta \right)\mathrm{c}\mathrm{o}\mathrm{s}\theta d\theta ,{c}_{eq}=-\frac{{b}_{1}}{\omega X}=-\
frac{1}{\pi \omega X}{\int }_{0}^{2\pi }f\left(X\mathrm{c}\mathrm{o}\mathrm{s}\theta \right)\mathrm{s}\mathrm{i}\mathrm{n}\theta d\theta .$
Thus, with the presence of hysteresis curves, one can use Eq. (5) to calculate damper’s equivalent damping and equivalent stiffness and further more analyze the vibratory response of shrouded blade.
Fig. 4 shows how equivalent damping varies along with vibration amplitude in macroslip models under the conditions of linearly ($\phi =\mathrm{}$0°) and non-linearly ($\phi =\mathrm{}$45°) variable
normal load. As can be seen, in both models, varying pattern of equivalent damping is the same: when amplitude is smaller than a certain value, the contact surface sticks and equivalent damping stays
as 0; when amplitude is getting larger, equivalent damping increases dramatically to maximum value and then decreases gradually. Compared with linearly variable normal load model, the equivalent
damping is larger when normal load varies non-linearly; and the bigger the amplitude is, the larger the difference is.
Fig. 5 contains a series of curves showing how equivalent damping changes when phase difference $\phi$ is different. It is shown that for each value of $\phi$, varying pattern of equivalent damping
is the same, but as $\phi$ increases, the maximum value of equivalent damping increases first and then decreases. At the same time, with larger $\phi$, the “tail” of curve is more flat, which means
with increasing amplitude, equivalent damping decreases less sharply.
Fig. 4Equivalent damping varying along vibration amplitude
Fig. 5Equivalent damping varying along vibration amplitude of different φ
Fig. 6Equivalent stiffness varying along vibration amplitude
Fig. 7Equivalent stiffness varying along vibration amplitude of different φ
Fig. 6 shows how equivalent stiffness varies along with vibration amplitude in macroslip models under the conditions of linearly ($\phi =\mathrm{}$0°) and non-linearly ($\phi =\mathrm{}$45°) variable
normal load. As shown in the figure, in both models, varying pattern of equivalent stiffness is the same: when amplitude is smaller than a certain value, the contact surface sticks and equivalent
stiffness keeps constant and equal the contact stiffness ${k}_{d}$ which is the maximum value of equivalent stiffness; when amplitude is getting larger, equivalent damping decreases but the change
rate gets smaller and smaller. Compared with linearly variable normal load model, the equivalent stiffness is smaller when normal load varies non-linearly.
Fig. 7 contains a series of curves showing how equivalent stiffness changes when phase difference $\phi$ is different. It is demonstrated in the figure that for each value of $\phi$, varying pattern
of equivalent stiffness is the same, but as $\phi$ increases, the minimum value of equivalent stiffness decreases and the bigger $\phi$ is, the steeper the curve is, which means equivalent stiffness
decreases more dramatically when phase difference is larger.
6. Experimental verification
The vibratory response of the shrouded blade structure was analyzed using the non-linearly variable normal load macroslip model raised above, and analyzed results were compared with the experimental
data in reference [14].
Fig. 8 shows the comparison of calculated results and experimental date. Compared to the results got using linearly variable normal load model, those by non-linearly variable normal load model are
closer to tested data. The reason is that when normal load is considered as non-linearly variable, the corresponding equivalent damping is larger, which means damper dissipates more energy and leads
the calculated vibratory response amplitude smaller and closer to the experimental data as a result. So it is reasonable to conclude that the non-linearly variable normal load model can simulate
practical conditions more precisely.
Fig. 9 compares the calculated resonant frequencies and those tested in experiment. The resonant frequencies calculated using two models are basically identical (about 280 Hz), while the tested
resonant frequencies are approximately 40 Hz lower. The reasons contributing to this difference include but are not limited to: 1) In calculation, the systems boundary conditions are supposed to be
ideally rigid and only the vibration of blade is taken into consideration which cannot be true in reality; 2) Practically, values of parameters such as contact stiffness, friction coefficient might
change, while all the parameters are treated as constant value in calculation.
Fig. 8Response amplitude of different initial normal load
Fig. 9Resonant frequency of different initial normal load
7. Conclusions
In vibration process of shrouded blade, the motion component vertical to contact surface leads normal load to vary, and the variation are normally non-linear because of the existence of phase
difference between adjacent blades. This article analyzed the influence of non-linearly variable normal load on damper’s vibration-reducing effect. Using the model raised, this article calculated the
vibratory response of a shrouded blade structure and compared the result with experimental data. The main conclusions are as following:
1) Due to the phase difference existing between adjacent blades, normal load on the contact surface varies non-linearly during vibration, and this kind of non-linearity will effect damper’s
vibration-reduction function;
2) The hysteresis curves derived from non-linearly variable normal load macrolip model can simulate all possible states occurring on contact surface, including stick, slip and separation;
3) The calculated results using non-linearly variable normal load macrolip model are closer to the tested data, which indicates the non-linear model raised in this article can better reflect shrouded
blade structure’s physical essence and simulate structure’s practical vibration more precisely.
• Griffin J. Friction Damping of resonant stresses in gas turbine engine airfoils. Journal of Engineering for Gas Turbines and Power, Vol. 102, Issue 2, 1980, p. 329-333.
• Sinha A., Griffin J. Effects of static friction on the forced response of frictionally damped turbine blades. Journal of Engineering for Gas Turbines and Power, Vol. 106, Issue 1, 1984, p. 65-69.
• Griffin J., Sinha A. The interaction between mistuning and friction in the forced response of bladed disk assemblies. Journal of Engineering for Gas Turbines and Power, Vol. 107, Issue 1, 1985,
p. 205-211.
• Chen J., Yang B., Menq C. Periodic forced response of structures having three-dimensional frictional constraints. Journal of Sound and Vibration, Vol. 229, Issue 4, 2000, p. 775-792.
• Cigeroglu E., Lu W., Menq C.-H. One-dimensional dynamic microslip friction model. Journal of Sound and Vibration, Vol. 292, Issue 3, 2006, p. 881-898.
• Hao Yanping, Zhu Zigen New method to resolve vibratory response of blades with friction damping. Acta Aeronautica et Astronautica Sinica, Vol. 22, Issue 5, 2001, p. 411-414, (in Chinese).
• Shan Yingchun, Zhu Zigen, Liu Xiandong Investi-gation of the vibration control by frictional constraints between blade shrouds-theoretical method. Journal of Aerospace Power, Vol. 21, Issue 1,
2006, p. 168-173, (in Chinese).
• Shi Yajie, Shan Yingchun, Zhu Zigen Analysis of nonlinear response of shrouded blades system. Journal of Aerospace Power, Vol. 24, Issue 5, 2009, p. 1158-1165, (in Chinese).
• Griffin J., Sinha A. The interaction between mistuning and friction in the forced response of bladed disk assemblies. Journal of Engineering for Gas Turbines and Power, Vol. 107, Issue 1, 1985,
p. 205-211.
• Wang J.-H., Chen W. Investigation of the vibration of a blade with friction damper by HBM. Journal of Engineering for Gas Turbines and Power, Vol. 115, Issue 2, 1993, p. 294-299.
• Sanliturk K., Imregun M., Ewins D. Harmonic balance vibration analysis of turbine blades with friction dampers. Journal of Vibration and Acoustics, Vol. 119, Issue 1, 1997, p. 96-103.
• Yang B.-D., Menq C.-H. Modeling of friction contact and its application to the design of shroud contact. Journal of Engineering for Gas Turbines and Power, Vol. 119, Issue 4, 1997, p. 958-963.
• Yang B., Chu M., Menq C. Stick–slip–separation analysis and non-linear stiffness and damping characterization of friction contacts having variable normal load. Journal of Sound and Vibration,
Vol. 210, Issue 4, 1998, p. 461-481.
• Hong Jie, Shi Yajie, Liu Shuguo, et al. Experiment of damping characteristic of non-rotating shrouded blade. Journal of Beijing University of Aeronautics and Astronautics, Vol. 32, Issue 10,
2006, p. 1174-1179, (in Chinese).
About this article
18 September 2015
Fault diagnosis based on vibration signal analysis
blade vibration
friction damping
friction force model
variable normal load
Copyright © 2015 JVE International Ltd.
This is an open access article distributed under the
Creative Commons Attribution License
, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. | {"url":"https://www.extrica.com/article/16295","timestamp":"2024-11-08T01:21:27Z","content_type":"text/html","content_length":"117256","record_id":"<urn:uuid:df4c14e5-1780-43ec-a65f-f8093d9e3018>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00063.warc.gz"} |
Pops two
values off the stack, divides the first by the second, disgarding any remaining amount, and then pushes the result back on to the stack. This looks at the values as unsigned integers (positive values
$result = $first / $second
Stack In
i64 The first value to be divided.
i64 The second value to divide by.
Stack Out
i64 The result of dividing the first value by the second one.
;; Push the i64 values onto the stack
i64.const 12
i64.const 3
;; Divide 12 by 3 seeing it as unsigned
;; The stack contains an i64 value of 4 (result = 12 / 3)
;; Push the i64 values onto the stack
i64.const 0xF23AB02CF178CD56
i64.const 2
;; Divide 0xF23AB02CF178CD56 by 2 seeing it as unsigned
;; The stack contains an i64 value of 8727228506452027051
;; result = 17454457012904054102 / 2
;; Push the i64 values onto the stack
i64.const 0xF23AB02CF178CD56
i64.const 2
;; Divide 0xF23AB02CF178CD56 by 2 seeing it as signed
;; The stack contains an i64 value of -496143530402748757
;; result = -992287060805497514 / 2 | {"url":"https://coderundebug.com/learn/wat-reference/i64/i64-div-u.html","timestamp":"2024-11-11T04:22:20Z","content_type":"text/html","content_length":"4899","record_id":"<urn:uuid:eda30145-20be-4611-a049-b9cd2c0fc291>","cc-path":"CC-MAIN-2024-46/segments/1730477028216.19/warc/CC-MAIN-20241111024756-20241111054756-00631.warc.gz"} |
Y-27632 Dihydrochloride (C14H21N3O.2HCl) Molecular Weight Calculation - Laboratory Notes
The molecular weight of Y-27632 dihydrochloride (C[14]H[21]N[3]O.2HCl) is 320.25882.
[14]H[21]N[3]O) is an organic compound of four elements: Carbon, Hydrogen, Nitrogen, and Oxygen. The Y-27632 dihydrochloride (C[14]H[21]N[3]O.2HCl) also contains two molecules of Hydrogen chloride.
The molecular weight of Y-27632 dihydrochlorid is 320.25882 which can be calculated by adding up the total weight (atomic weight multiplied by their number) of all its elements.
CALCULATION PROCEDURE: Y-27632 dihydrochloride [C[14]H[21]N[3]O.2HCl] Molecular Weight Calculation
Step 1: Find out the chemical formula and determine constituent atoms and their number in a Y-27632 dihydrochloride molecule.
You will know different atoms and their number in a Y-27632 dihydrochloride molecule from the chemical formula. The chemical formula of Y-27632 dihydrochloride is C[14]H[21]N[3]O.2HCl. From the
chemical formula, you can find that one molecule of Y-27632 dihydrochloride consists of fourteen Carbon (C) atoms, twentyone Hydrogen (H) atoms, three Nitrogen (N) atoms, one Oxygen atom, and two
Hydrogen chloride molecules.
Step 2: Find out the atomic weights of each atom (from the periodic table).
Atomic weight of Carbon (C): 12.0107 (Ref: Jlab-ele006)
Atomic weight of Hydrogen (H): 1.008 (Ref: Lanl-1)
Atomic weight of Nitrogen (N): 14.00674 (Ref: Jlab-ele007)
Atomic weight of Oxygen (O): 15.9994 (Ref: Jlab-ele008)
Molecular weight of Hydrogen chloride (HCl): 36.4607 (See Hydrogen Chloride (HCl) Molecular Weight Calculation)
Step 3: Calculate the total weight of each atom in a Y-27632 dihydrochloride molecule by multiplying its atomic weight by its number.
Number of Carbon atoms in Y-27632 dihydrochloride: 14
Atomic weight of Carbon (C): 12.0107
Total weight of Carbon atoms in Y-27632 dihydrochloride: 12.0107 x 14 = 168.1498
Number of Hydrogen atoms in Y-27632 dihydrochloride: 21
Atomic weight of Hydrogen (H): 1.008
Total weight of Hydrogen atoms in Y-27632 dihydrochloride: 1.008 x 21 = 21.168
Number of Nitrogen atoms in Y-27632 dihydrochloride: 3
Atomic weight of Nitrogen (N): 14.00674
Total weight of Nitrogen atoms in Y-27632 dihydrochloride: 14.00674 x 3 = 42.02022
Number of Oxygen atoms in Y-27632 dihydrochloride: 1
Atomic weight of Oxygen (O): 15.9994
Total weight of Oxygen atoms in Y-27632 dihydrochloride: 15.9994 x 1 = 15.9994
Number of Hydrogen chloride molecules in Y-27632 dihydrochloride: 2
Molecular weight of Hydrogen chloride (HCl): 36.4607
Total weight of Hydrogen chloride in Y-27632 dihydrochloride: 36.4607 x 2 = 72.9214
Step 4: Calculate the molecular weight of Y-27632 dihydrochloride by adding up the total weight of all atoms.
Molecular weight of Y-27632 dihydrochloride: 168.1498 (Carbon) + 21.168 (Hydrogen) + 42.02022 (Nitrogen) + 15.9994 (Oxygen) + 72.9214 (Hydrogen chloride) = 320.25882
So the molecular weight of Y-27632 dihydrochloride is 320.25882.
Y-27632 dihydrochloride [C[14]H[21]N[3]O.2HCl] Molecular Weight Calculation
Molecular weight of Y-27632 dihydrochloride [C[14]H[21]N[3]O.2HCl]
Constituent atoms Number of each atom Atomic weight Total weight
Carbon (C) 14 12.0107 168.1498
Hydrogen (H) 21 1.008 21.168
Nitrogen (N) 3 14.00674 42.02022
Oxygen (O) 1 15.9994 15.9994
Hydrogen chloride (HCl) 2 36.4607 72.9214
Molecular weight of Y-27632 dihydrochloride [C[14]H[21]N[3]O.2HCl]: 320.25882
See also:
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Let us know if you liked the post. That’s the only way we can improve. | {"url":"https://www.laboratorynotes.com/y-27632-dihydrochloride-c14h21n3o-2hcl-molecular-weight-calculation/","timestamp":"2024-11-10T15:37:11Z","content_type":"text/html","content_length":"73153","record_id":"<urn:uuid:22f4293d-c412-4791-9c88-1243ff0a8a49>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00072.warc.gz"} |
Cantor’s 1874 Proof of Non
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conundrums, and contentious issues in modern philosophy and mathematics.
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Cantor’s 1874 Proof of NonDenumerability
“On a property of the set of all real algebraic numbers”
• English Translation •
This is an English translation of Cantor’s 1874 Proof of the Non-Denumerability of the real numbers. The original German text can be seen at PDF Über eine Eigenschaft des Inbegriffes aller reellen
algebraischen Zahlen.
It was published in the Journal für die Reine und Angewandte Mathematik, commonly known as Crelle’s Journal; Vol 77 (1874), pages 258-262.
English translation by James R Meyer, copyright 2022 www.jamesrmeyer.com
There is also a PDF version of this translation available: PDF Cantor’s 1874 Proof of Non-Denumerability.
On a Property of the Set of all Real Algebraic Numbers
by Georg Cantor, 1874
By a real algebraic number one understands a real number ω which satisfies a non-constant equation of the form:
(1) a[0]ω^n + a[1]ω^n-1 + … + a[n] = 0
where n, a[0] , a[1] , … a[n] are integers: here the numbers n and a[0] are positive, and the coefficients a[0] , a[1] , … a[n] do not have any common factors, and the equation (1) is irreducible;
with these stipulations it is the case, in accordance with known principles of arithmetic and algebra, equation (1), which is satisfied by a real algebraic number, is fully determinate; conversely,
if n is the degree of an equation of the form (1), then the equation is satisfied by at no more than n real algebraic numbers ω. The real algebraic numbers constitute in their entirety a set of
numbers (ω); it is clear that (ω) has the property that in every vicinity of any given number α there are infinitely many numbers of (ω) that lie within it. So it is likely that at first glance that
it appears that one should be able to correlate the set (ω) one-to-one with the set (ν) of all positive integers ν in such a way that to every algebraic number w there corresponds a definite positive
integer, and, conversely, to every positive integer v there corresponds an entirely definite real algebraic number ω. In other words, the set (ω) can be thought of in the form of an infinite ordered
(2) ω[1], ω[2], … ω[ν], …
in which all the individual real numbers of (ω) occur, and each such number occurs in (2) at a definitive position, which is given by the subscripts. Given that one has a definition by which such a
correlation is defined, it can be modified at will; so in §1 I shall describe a correlation which seems to me the least complicated. In order to give an application of this property of the set of all
real algebraic numbers I show in §2 that, given any arbitrarily chosen sequence of real numbers of the form (2), then in any given interval (α … β), I can define that there are numbers η which are
not contained in (2); if one combines the results of these last two paragraphs, then one has a new proof of the theorem, first proved by Liouville, that in any given interval (α … β) there are
infinitely many transcendental (i.e: not algebraic) real numbers. Further, the theorem in §2 turns out to be the reason why sets of real numbers which form a so-called continuum (concerning all real
numbers which are ≥0 and ≤1) cannot be correlated one-to-one onto the set (ν); I have uncovered the essential difference between a so-called continuum and a set such as the set of all real algebraic
Let us return to equation (1), which is satisfied by an algebraic number ω and which, under the given conditions, is fully determined. Then the sum of the absolute values of its coefficients, plus
the number n-1 (where n is the degree of ω) shall be called the height of the number ω. Let it be designated by N. That is, in the usual notation:
(3) N = n - 1 + |a[0]| + |a[1]| + … + |a[n]|
The height N is thus for every real algebraic number ω a definite positive integer; conversely, for any given positive integral value of N there are only finitely many algebraic real numbers with the
height N; if we call this number be Φ(N), then, for example, Φ(1) = 1; Φ(2) = 2; Φ(3) = 4, and so on. The numbers of the set (ω), that is, the set of all algebraic real numbers, can then be set in
order in the following way: take for the first number ω[1] the single number with the height N = 1; for Φ(2), there are 2 algebraic real numbers with the height N = 2, so they are ordered according
to their size, and we designate them by ω[2], ω[3] ; next, for Φ(3), there are 4 numbers with height N = 3, which are set in order according to size; in general, after all the numbers in (ω) up to a
certain height N = N[1] have been enumerated and assigned to a definite position, the real algebraic numbers with the height N = N[1 ]+ 1 follow them according to size; thus one obtains the set (ω)
of all real algebraic numbers in the form:
ω[1], ω[2], … ω[ν], …
One can, with respect to this ordering, refer to the ν^th real algebraic number; not a single member of the set has been omitted.
Given any definition of an infinite sequence of mutually distinct real numbers,
(4) ω[1], ω[2], … ω[ν], …
then in any given interval (α … β) there is a number η (and consequently infinitely many such numbers) where it can be shown that it does not occur in the series (4); this shall now be proved.
We go to the end of the interval (α … β), which has been chosen arbitrarily and in which α < β; the first two numbers of our sequence (4) which lie in the interior of this interval (with the
exception of the boundaries), can be designated by α′, β′ , letting α′ < β′; similarly let us designate the first two numbers of our sequence which lie in the interior of (α′ … β′) by α″, β″ and let
α″ < β″; and in the same way we determine the next interval (α‴ … β‴) and so on. It follows that α′, α″, … are by definition determinate numbers of our sequence (4), whose indices are continually
increasing; the same applies for the sequence β′, β″, … ; furthermore, the numbers α′, α″, … are always increasing in size, while the numbers β′, β″, … are always decreasing in size. Of the intervals
(α′ … β′), (α″ … β″), (α‴ … β‴), … each encloses all of those following.
There are two conceivable cases:
Either the number of intervals formed is finite; in which case, let the last such interval be (α^(ν) … β^(ν)). Since in its interior there can be at most one number of the sequence (4), a number ν
can be chosen from this interval which is not contained in (4), thereby proving the theorem for this case.
Or the number of intervals formed is infinite. Then the numbers α, α′, α″, … because they are always increasing in size without growing infinitely large, have a determinate limiting value α^∞; the
same holds for the numbers β, β′, β″, … because they are always decreasing in size, have a limiting value be β^∞; also α^∞ = β^∞ (such a case always occurs with the set (ω) of all real algebraic
numbers), so one is readily satisfied, by looking back at the definition of the intervals, that the number η = α^∞ = β^∞ cannot be contained in our sequence; (Footnote: If the number η were
contained in our sequence, then one would have η = ω[p], where p is a specific index. But this is not possible, for ω[p] does not lie in the interior of the interval (α^(p) … β^(p)), while by
definition the number η does lie in the interior of the interval.) but if α^∞ < β^∞ then every number η in the interior of the interval (α^∞ … β^∞) or at its endpoints satisfies the requirement
that it not be contained in the sequence (4).
The theorems proved in this article admit of extensions in various directions, only one of which will be mentioned here:
“If ω[1], ω[2], … ω[n], … is a finite or infinite sequence of numbers which are linearly independent of each another (so that no equation of the form a[1]ω[1] + a[2]ω[2] + … + a[n]ω[n] = 0 is
possible with integral coefficients which do not all vanish) and if one imagines the set (Ω) of all those numbers Ω which can be represented as rational functions with integral coefficients of the
given numbers ω, then in every interval (α … β) there are infinitely many numbers which are not contained in (Ω).”
In fact one can be satisfied through a method of proof similar to that in §1 that the set (Ω) can be conceived in the sequential form
Ω[1], Ω[2], … Ω[ν], …
from which, in view of §2, the truth of the theorem follows.
A quite special case of the theorem cited here (in which the sequence ω[1], ω[2], … ω[n], … is finite and the degree of the rational functions, which yield the set (Ω), is predetermined) has been
proved, by recourse to Galoisian principles, by Herr B. Minnigerode (See Math. Annalen, Vol. 4, p. 497).
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To understand the philosophy of set theory as it is today requires a knowledge of the history of the subject. One of the most influential works in this respect was Georg Cantor’s set of six papers
published between 1879 and 1884 under the overall title of Über unendliche lineare Punktmannig-faltigkeiten, which were published between 1879 and 1884. I now have English translations of Part 1,
Part 2, Part 3 and the major part, Part 5 (Grundlagen). There is also a new English translation of Cantor’s “A Contribution to the Theory of Sets”.
A brief history of meta-mathematics
A look at how the field of meta-mathematics developed from its early days, and how certain illogical and untenable assumptions have been made that fly in the face of the mathematical requirement for
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Search results for: densityof the graph
Commenced in January 2007
Search results for: densityof the graph
315 On the Use of Correlated Binary Model in Social Network Analysis
Authors: Elsayed A. Habib Elamir
In social network analysis the mean nodal degree and density of the graph can be considered as a measure of the activity of all actors in the network and this is an important property of a graph and
for making comparisons among networks. Since subjects in a family or organization are subject to common environment factors, it is prime interest to study the association between responses.
Therefore, we study the distribution of the mean nodal degree and density of the graph under correlated binary units. The cross product ratio is used to capture the intra-units association among
subjects. Computer program and an application are given to show the benefits of the method.
Keywords: Correlated Binary data, cross product ratio, densityof the graph, multiplicative binomial distribution.
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314 Efficient Filtering of Graph Based Data Using Graph Partitioning
Authors: Nileshkumar Vaishnav, Aditya Tatu
An algebraic framework for processing graph signals axiomatically designates the graph adjacency matrix as the shift operator. In this setup, we often encounter a problem wherein we know the filtered
output and the filter coefficients, and need to find out the input graph signal. Solution to this problem using direct approach requires O(N3) operations, where N is the number of vertices in graph.
In this paper, we adapt the spectral graph partitioning method for partitioning of graphs and use it to reduce the computational cost of the filtering problem. We use the example of denoising of the
temperature data to illustrate the efficacy of the approach.
Keywords: Graph signal processing, graph partitioning, inverse filtering on graphs, algebraic signal processing.
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313 Using Spectral Vectors and M-Tree for Graph Clustering and Searching in Graph Databases of Protein Structures
Authors: Do Phuc, Nguyen Thi Kim Phung
In this paper, we represent protein structure by using graph. A protein structure database will become a graph database. Each graph is represented by a spectral vector. We use Jacobi rotation
algorithm to calculate the eigenvalues of the normalized Laplacian representation of adjacency matrix of graph. To measure the similarity between two graphs, we calculate the Euclidean distance
between two graph spectral vectors. To cluster the graphs, we use M-tree with the Euclidean distance to cluster spectral vectors. Besides, M-tree can be used for graph searching in graph database.
Our proposal method was tested with graph database of 100 graphs representing 100 protein structures downloaded from Protein Data Bank (PDB) and we compare the result with the SCOP hierarchical
Keywords: Eigenvalues, m-tree, graph database, protein structure, spectra graph theory.
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312 A Neighborhood Condition for Fractional k-deleted Graphs
Authors: Sizhong Zhou, Hongxia Liu
Abstract–Let k ≥ 3 be an integer, and let G be a graph of order n with n ≥ 9k +3- 42(k - 1)2 + 2. Then a spanning subgraph F of G is called a k-factor if dF (x) = k for each x ∈ V (G). A fractional
k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A
graph G is a fractional k-deleted graph if there exists a fractional k-factor after deleting any edge of G. In this paper, it is proved that G is a fractional k-deleted graph if G satisfies δ(G) ≥ k
+ 1 and |NG(x) ∪ NG(y)| ≥ 1 2 (n + k - 2) for each pair of nonadjacent vertices x, y of G.
Keywords: Graph, minimum degree, neighborhood union, fractional k-factor, fractional k-deleted graph.
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311 The Extremal Graph with the Largest Merrifield-Simmons Index of (n, n + 2)-graphs
Authors: M. S. Haghighat, A. Dolati, M. Tabari, E. Mohseni
The Merrifield-Simmons index of a graph G is defined as the total number of its independent sets. A (n, n + 2)-graph is a connected simple graph with n vertices and n + 2 edges. In this paper we
characterize the (n, n+2)-graph with the largest Merrifield- Simmons index. We show that its Merrifield-Simmons index i.e. the upper bound of the Merrifield-Simmons index of the (n, n+2)-graphs is 9
× 2n-5 +1 for n ≥ 5.
Keywords: Merrifield-Simmons index, (n, n+2)-graph.
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310 N-Sun Decomposition of Complete, Complete Bipartite and Some Harary Graphs
Authors: R. Anitha, R. S. Lekshmi
Graph decompositions are vital in the study of combinatorial design theory. A decomposition of a graph G is a partition of its edge set. An n-sun graph is a cycle Cn with an edge terminating in a
vertex of degree one attached to each vertex. In this paper, we define n-sun decomposition of some even order graphs with a perfect matching. We have proved that the complete graph K2n, complete
bipartite graph K2n, 2n and the Harary graph H4, 2n have n-sun decompositions. A labeling scheme is used to construct the n-suns.
Keywords: Decomposition, Hamilton cycle, n-sun graph, perfect matching, spanning tree.
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309 The Diameter of an Interval Graph is Twice of its Radius
Authors: Tarasankar Pramanik, Sukumar Mondal, Madhumangal Pal
In an interval graph G = (V,E) the distance between two vertices u, v is de£ned as the smallest number of edges in a path joining u and v. The eccentricity of a vertex v is the maximum among
distances from all other vertices of V . The diameter (δ) and radius (ρ) of the graph G is respectively the maximum and minimum among all the eccentricities of G. The center of the graph G is the set
C(G) of vertices with eccentricity ρ. In this context our aim is to establish the relation ρ = δ 2 for an interval graph and to determine the center of it.
Keywords: Interval graph, interval tree, radius, center.
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308 Completion Number of a Graph
Authors: Sudhakar G
In this paper a new concept of partial complement of a graph G is introduced and using the same a new graph parameter, called completion number of a graph G, denoted by c(G) is defined. Some basic
properties of graph parameter, completion number, are studied and upperbounds for completion number of classes of graphs are obtained , the paper includes the characterization also.
Keywords: Completion Number, Maximum Independent subset, Partial complements, Partial self complementary
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307 On Fractional (k,m)-Deleted Graphs with Constrains Conditions
Authors: Sizhong Zhou, Hongxia Liu
Let G be a graph of order n, and let k 2 and m 0 be two integers. Let h : E(G) [0, 1] be a function. If e∋x h(e) = k holds for each x V (G), then we call G[Fh] a fractional k-factor of G with
indicator function h where Fh = {e E(G) : h(e) > 0}. A graph G is called a fractional (k,m)-deleted graph if there exists a fractional k-factor G[Fh] of G with indicator function h such that h(e) = 0
for any e E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k,m)-deleted graph if (G) k + m + m k+1 , n 4k2 + 2k − 6 + (4k 2 +6k−2)m−2 k−1 and max
{dG(x), dG(y)} n 2 for any vertices x and y of G with dG(x, y) = 2. Furthermore, it is shown that the result in this paper is best possible in some sense.
Keywords: Graph, degree condition, fractional k-factor, fractional (k, m)-deleted graph.
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306 Metric Dimension on Line Graph of Honeycomb Networks
Authors: M. Hussain, Aqsa Farooq
Let G = (V,E) be a connected graph and distance between any two vertices a and b in G is a−b geodesic and is denoted by d(a, b). A set of vertices W resolves a graph G if each vertex is uniquely
determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G. In this paper line graph of honeycomb network has been derived
and then we calculated the metric dimension on line graph of honeycomb network.
Keywords: Resolving set, metric dimension, honeycomb network, line graph.
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305 Comparison of Full Graph Methods of Switched Circuits Solution
Authors: Zdeňka Dostálová, David Matoušek, Bohumil Brtnik
As there are also graph methods of circuit analysis in addition to algebraic methods, it is, in theory, clearly possible to carry out an analysis of a whole switched circuit in two-phase switching
exclusively by the graph method as well. This article deals with two methods of full-graph solving of switched circuits: by transformation graphs and by two-graphs. It deals with the circuit switched
capacitors and the switched current, too. All methods are presented in an equally detailed steps to be able to compare.
Keywords: Switched capacitors of two phases, switched currents of two phases, transformation graph, two-graph, Mason's formula, voltage transfer, summary graph.
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304 Speedup Breadth-First Search by Graph Ordering
Breadth-First Search (BFS) is a core graph algorithm that is widely used for graph analysis. As it is frequently used in many graph applications, improving the BFS performance is essential. In this
paper, we present a graph ordering method that could reorder the graph nodes to achieve better data locality, thus, improving the BFS performance. Our method is based on an observation that the
sibling relationships will dominate the cache access pattern during the BFS traversal. Therefore, we propose a frequency-based model to construct the graph order. First, we optimize the graph order
according to the nodes’ visit frequency. Nodes with high visit frequency will be processed in priority. Second, we try to maximize the child nodes’ overlap layer by layer. As it is proved to be
NP-hard, we propose a heuristic method that could greatly reduce the preprocessing overheads.We conduct extensive experiments on 16 real-world datasets. The result shows that our method could achieve
comparable performance with the state-of-the-art methods while the graph ordering overheads are only about 1/15.
Keywords: Breadth-first search, BFS, graph ordering, graph algorithm.
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303 On Detour Spectra of Some Graphs
Authors: S.K.Ayyaswamy, S.Balachandran
The Detour matrix (DD) of a graph has for its ( i , j) entry the length of the longest path between vertices i and j. The DD-eigenvalues of a connected graph G are the eigenvalues for its detour
matrix, and they form the DD-spectrum of G. The DD-energy EDD of the graph G is the sum of the absolute values of its DDeigenvalues. Two connected graphs are said to be DD- equienergetic if they have
equal DD-energies. In this paper, the DD- spectra of a variety of graphs and their DD-energies are calculated.
Keywords: Detour eigenvalue (of a graph), detour spectrum(of a graph), detour energy(of a graph), detour - equienergetic graphs.
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302 Analysis of Electrical Networks Using Phasors: A Bond Graph Approach
Authors: Israel Núñez-Hernández, Peter C. Breedveld, Paul B. T. Weustink, Gilberto Gonzalez-A
This paper proposes a phasor representation of electrical networks by using bond graph methodology. A so-called phasor bond graph is built up by means of two-dimensional bonds, which represent the
complex plane. Impedances or admittances are used instead of the standard bond graph elements. A procedure to obtain the steady-state values from a phasor bond graph model is presented. Besides the
presentation of a phasor bond graph library in SIDOPS code, also an application example is discussed.
Keywords: Bond graphs, phasor theory, steady-state, complex power, electrical networks.
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301 Topological Queries on Graph-structured XML Data: Models and Implementations
Authors: Hongzhi Wang, Jianzhong Li, Jizhou Luo
In many applications, data is in graph structure, which can be naturally represented as graph-structured XML. Existing queries defined on tree-structured and graph-structured XML data mainly focus on
subgraph matching, which can not cover all the requirements of querying on graph. In this paper, a new kind of queries, topological query on graph-structured XML is presented. This kind of queries
consider not only the structure of subgraph but also the topological relationship between subgraphs. With existing subgraph query processing algorithms, efficient algorithms for topological query
processing are designed. Experimental results show the efficiency of implementation algorithms.
Keywords: XML, Graph Structure, Topological query.
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300 An Efficient Graph Query Algorithm Based on Important Vertices and Decision Features
Authors: Xiantong Li, Jianzhong Li
Graph has become increasingly important in modeling complicated structures and schemaless data such as proteins, chemical compounds, and XML documents. Given a graph query, it is desirable to
retrieve graphs quickly from a large database via graph-based indices. Different from the existing methods, our approach, called VFM (Vertex to Frequent Feature Mapping), makes use of vertices and
decision features as the basic indexing feature. VFM constructs two mappings between vertices and frequent features to answer graph queries. The VFM approach not only provides an elegant solution to
the graph indexing problem, but also demonstrates how database indexing and query processing can benefit from data mining, especially frequent pattern mining. The results show that the proposed
method not only avoids the enumeration method of getting subgraphs of query graph, but also effectively reduces the subgraph isomorphism tests between the query graph and graphs in candidate answer
set in verification stage.
Keywords: Decision Feature, Frequent Feature, Graph Dataset, Graph Query
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299 Notes on Fractional k-Covered Graphs
Authors: Sizhong Zhou, Yang Xu
A graph G is fractional k-covered if for each edge e of G, there exists a fractional k-factor h, such that h(e) = 1. If k = 2, then a fractional k-covered graph is called a fractional 2-covered
graph. The binding number bind(G) is defined as follows, bind(G) = min{|NG(X)| |X| : ├ÿ = X Ôèå V (G),NG(X) = V (G)}. In this paper, it is proved that G is fractional 2-covered if δ(G) ≥ 4 and bind
(G) > 5 3 .
Keywords: graph, binding number, fractional k-factor, fractional k-covered graph.
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298 Syntactic Recognition of Distorted Patterns
Authors: Marek Skomorowski
In syntactic pattern recognition a pattern can be represented by a graph. Given an unknown pattern represented by a graph g, the problem of recognition is to determine if the graph g belongs to a
language L(G) generated by a graph grammar G. The so-called IE graphs have been defined in [1] for a description of patterns. The IE graphs are generated by so-called ETPL(k) graph grammars defined
in [1]. An efficient, parsing algorithm for ETPL(k) graph grammars for syntactic recognition of patterns represented by IE graphs has been presented in [1]. In practice, structural descriptions may
contain pattern distortions, so that the assignment of a graph g, representing an unknown pattern, to a graph language L(G) generated by an ETPL(k) graph grammar G is rejected by the ETPL(k) type
parsing. Therefore, there is a need for constructing effective parsing algorithms for recognition of distorted patterns. The purpose of this paper is to present a new approach to syntactic
recognition of distorted patterns. To take into account all variations of a distorted pattern under study, a probabilistic description of the pattern is needed. A random IE graph approach is proposed
here for such a description ([2]).
Keywords: Syntactic pattern recognition, Distorted patterns, Random graphs, Graph grammars.
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297 Automatic Fingerprint Classification Using Graph Theory
Authors: Mana Tarjoman, Shaghayegh Zarei
Using efficient classification methods is necessary for automatic fingerprint recognition system. This paper introduces a new structural approach to fingerprint classification by using the
directional image of fingerprints to increase the number of subclasses. In this method, the directional image of fingerprints is segmented into regions consisting of pixels with the same direction.
Afterwards the relational graph to the segmented image is constructed and according to it, the super graph including prominent information of this graph is formed. Ultimately we apply a matching
technique to compare obtained graph with the model graphs in order to classify fingerprints by using cost function. Increasing the number of subclasses with acceptable accuracy in classification and
faster processing in fingerprints recognition, makes this system superior.
Keywords: Classification, Directional image, Fingerprint, Graph, Super graph.
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296 Graphs with Metric Dimension Two-A Characterization
Authors: Sudhakara G, Hemanth Kumar A.R
In this paper, we define distance partition of vertex set of a graph G with reference to a vertex in it and with the help of the same, a graph with metric dimension two (i.e. β (G) = 2 ) is
characterized. In the process, we develop a polynomial time algorithm that verifies if the metric dimension of a given graph G is two. The same algorithm explores all metric bases of graph G whenever
β (G) = 2 . We also find a bound for cardinality of any distance partite set with reference to a given vertex, when ever β (G) = 2 . Also, in a graph G with β (G) = 2 , a bound for cardinality of any
distance partite set as well as a bound for number of vertices in any sub graph H of G is obtained in terms of diam H .
Keywords: Metric basis, Distance partition, Metric dimension.
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295 Image Segmentation Using Suprathreshold Stochastic Resonance
Authors: Rajib Kumar Jha, P.K.Biswas, B.N.Chatterji
In this paper a new concept of partial complement of a graph G is introduced and using the same a new graph parameter, called completion number of a graph G, denoted by c(G) is defined. Some basic
properties of graph parameter, completion number, are studied and upperbounds for completion number of classes of graphs are obtained , the paper includes the characterization also.
Keywords: Completion Number, Maximum Independent subset, Partial complements, Partial self complementary.
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294 Analysis of a Singular Perturbed Synchronous Generator with a Bond Graph Approach
Authors: Gilberto Gonzalez-A, Noe Barrera-G
An analysis of a synchronous generator in a bond graph approach is proposed. This bond graph allows to determine the simplified models of the system by using singular perturbations. Firstly, the
nonlinear bond graph of the generator is linearized. Then, the slow and fast state equations by applying singular perturbations are obtained. Also, a bond graph to get the quasi-steady state of the
slow dynamic is proposed. In order to verify the effectiveness of the singularly perturbed models, simulation results of the complete system and reduced models are shown.
Keywords: Bond graph modelling, synchronous generator, singular perturbations
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293 Connected Vertex Cover in 2-Connected Planar Graph with Maximum Degree 4 is NP-complete
Authors: Priyadarsini P. L. K, Hemalatha T.
This paper proves that the problem of finding connected vertex cover in a 2-connected planar graph ( CVC-2 ) with maximum degree 4 is NP-complete. The motivation for proving this result is to give a
shorter and simpler proof of NP-Completeness of TRA-MLC (the Top Right Access point Minimum-Length Corridor) problem [1], by finding the reduction from CVC-2. TRA-MLC has many applications in laying
optical fibre cables for data communication and electrical wiring in floor plans.The problem of finding connected vertex cover in any planar graph ( CVC ) with maximum degree 4 is NP-complete [2]. We
first show that CVC-2 belongs to NP and then we find a polynomial reduction from CVC to CVC-2. Let a graph G0 and an integer K form an instance of CVC, where G0 is a planar graph and K is an upper
bound on the size of the connected vertex cover in G0. We construct a 2-connected planar graph, say G, by identifying the blocks and cut vertices of G0, and then finding the planar representation of
all the blocks of G0, leading to a plane graph G1. We replace the cut vertices with cycles in such a way that the resultant graph G is a 2-connected planar graph with maximum degree 4. We consider L
= K -2t+3 t i=1 di where t is the number of cut vertices in G1 and di is the number of blocks for which ith cut vertex is common. We prove that G will have a connected vertex cover with size less
than or equal to L if and only if G0 has a connected vertex cover of size less than or equal to K.
Keywords: NP-complete, 2-Connected planar graph, block, cut vertex
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292 Protein Graph Partitioning by Mutually Maximization of cycle-distributions
Authors: Frank Emmert Streib
The classification of the protein structure is commonly not performed for the whole protein but for structural domains, i.e., compact functional units preserved during evolution. Hence, a first step
to a protein structure classification is the separation of the protein into its domains. We approach the problem of protein domain identification by proposing a novel graph theoretical algorithm. We
represent the protein structure as an undirected, unweighted and unlabeled graph which nodes correspond the secondary structure elements of the protein. This graph is call the protein graph. The
domains are then identified as partitions of the graph corresponding to vertices sets obtained by the maximization of an objective function, which mutually maximizes the cycle distributions found in
the partitions of the graph. Our algorithm does not utilize any other kind of information besides the cycle-distribution to find the partitions. If a partition is found, the algorithm is iteratively
applied to each of the resulting subgraphs. As stop criterion, we calculate numerically a significance level which indicates the stability of the predicted partition against a random rewiring of the
protein graph. Hence, our algorithm terminates automatically its iterative application. We present results for one and two domain proteins and compare our results with the manually assigned domains
by the SCOP database and differences are discussed.
Keywords: Graph partitioning, unweighted graph, protein domains.
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290 Analysis of a Hydroelectric Plant connected to Electrical Power System in the Physical Domain
Authors: Gilberto Gonzalez-A, Octavio Barriga
A bond graph model of a hydroelectric plant is proposed. In order to analyze the system some structural properties of a bond graph are used. The structural controllability of the hydroelctric plant
is described. Also, the steady state of the state variables applying the bond graph in a derivative causality assignment is obtained. Finally, simulation results of the system are shown.
Keywords: Bond graph, hydraulic plant, steady state.
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289 Image Segmentation Based on Graph Theoretical Approach to Improve the Quality of Image Segmentation
Authors: Deepthi Narayan, Srikanta Murthy K., G. Hemantha Kumar
Graph based image segmentation techniques are considered to be one of the most efficient segmentation techniques which are mainly used as time & space efficient methods for real time applications.
How ever, there is need to focus on improving the quality of segmented images obtained from the earlier graph based methods. This paper proposes an improvement to the graph based image segmentation
methods already described in the literature. We contribute to the existing method by proposing the use of a weighted Euclidean distance to calculate the edge weight which is the key element in
building the graph. We also propose a slight modification of the segmentation method already described in the literature, which results in selection of more prominent edges in the graph. The
experimental results show the improvement in the segmentation quality as compared to the methods that already exist, with a slight compromise in efficiency.
Keywords: Graph based image segmentation, threshold, Weighted Euclidean distance.
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288 2D Structured Non-Cyclic Fuzzy Graphs
Authors: T. Pathinathan, M. Peter
Fuzzy graphs incorporate concepts from graph theory with fuzzy principles. In this paper, we make a study on the properties of fuzzy graphs which are non-cyclic and are of two-dimensional in
structure. In particular, this paper presents 2D structure or the structure of double layer for a non-cyclic fuzzy graph whose underlying crisp graph is non-cyclic. In any graph structure,
introducing 2D structure may lead to an inherent cycle. We propose relevant conditions for 2D structured non-cyclic fuzzy graphs. These conditions are extended even to fuzzy graphs of the 3D
structure. General theoretical properties that are studied for any fuzzy graph are verified to 2D structured or double layered fuzzy graphs. Concepts like Order, Degree, Strong and Size for a fuzzy
graph are studied for 2D structured or double layered non-cyclic fuzzy graphs. Using different types of fuzzy graphs, the proposed concepts relating to 2D structured fuzzy graphs are verified.
Keywords: Double layered fuzzy graph, double layered non-cyclic fuzzy graph, strong, order, degree and size.
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287 Intention Recognition using a Graph Representation
Authors: So-Jeong Youn, Kyung-Whan Oh
The human friendly interaction is the key function of a human-centered system. Over the years, it has received much attention to develop the convenient interaction through intention recognition.
Intention recognition processes multimodal inputs including speech, face images, and body gestures. In this paper, we suggest a novel approach of intention recognition using a graph representation
called Intention Graph. A concept of valid intention is proposed, as a target of intention recognition. Our approach has two phases: goal recognition phase and intention recognition phase. In the
goal recognition phase, we generate an action graph based on the observed actions, and then the candidate goals and their plans are recognized. In the intention recognition phase, the intention is
recognized with relevant goals and user profile. We show that the algorithm has polynomial time complexity. The intention graph is applied to a simple briefcase domain to test our model.
Keywords: Intention recognition, intention, graph, HCI.
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286 On Chromaticity of Wheels
Authors: Zainab Yasir Al-Rekaby, Abdul Jalil M. Khalaf
Let the vertices of a graph such that every two adjacent vertices have different color is a very common problem in the graph theory. This is known as proper coloring of graphs. The possible number of
different proper colorings on a graph with a given number of colors can be represented by a function called the chromatic polynomial. Two graphs G and H are said to be chromatically equivalent, if
they share the same chromatic polynomial. A Graph G is chromatically unique, if G is isomorphic to H for any graph H such that G is chromatically equivalent to H. The study of chromatically
equivalent and chromatically unique problems is called chromaticity. This paper shows that a wheel W12 is chromatically unique.
Keywords: Chromatic Polynomial, Chromatically Equivalent, Chromatically Unique, Wheel.
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What is the entropy change for the isothermal expansion of "0.75 g" of "He" from "5.0 L" to "12.5 L"? | HIX Tutor
What is the entropy change for the isothermal expansion of #"0.75 g"# of #"He"# from #"5.0 L"# to #"12.5 L"#?
Answer 1
I got #"1.428 J/K"#.
I assume you aren't given an equation and have to derive it.
If we treat entropy as a function of the temperature and volume (that is, #S = S(T,V)#), and consider a constant-temperature expansion, then the total derivative is:
#dS(T,V) = ((delS)/(delT))_VdT + ((delS)/(delV))_TdV#
Since we are looking at the change in entropy due to a change in volume, we only consider #((delS)/(delV))_T#. Take the time to digest this step. This is the key step to understanding where to begin.
Another key step is to relate this to a derivative we are more familiar with. Recall that the Helmholtz free energy #A# is a function of temperature and volume, and that its Maxwell relation is:
#dA = -SdT - PdV = -(SdT + PdV)#
Since #A# is a state function, its second derivatives are continuous, so that its cross derivatives are equal.
#((delS)/(delV))_T = ((delP)/(delT))_V#
You should get to know this pretty well. The righthand side is a derivative we are familiar with. The ideal gas law can then be used.
#PV = nRT#
#=> P = (nRT)/V#
Now, the derivative is feasible in terms of an equation we know:
#((delP)/(delT))_V = del/(delT)[(nRT)/V]_V#
#= (nR)/V(dT)/(dT)#
#= (nR)/V#
#((delS)/(delV))_T = ((delP)/(delT))_V = (nR)/V#
Next, we can integrate this over the two volumes to get the change in entropy. By moving the #dV# in the #((delS)/(delV))_T# derivative onto the other side:
#int_(S_1)^(S_2) dS = DeltaS = int_(V_1)^(V_2) (nR)/VdV#
The integral of #1/V# is #ln|V|#, so we have:
#DeltaS = nRint_(V_1)^(V_2) 1/VdV = nR|[ln|V|]|_(V_1)^(V_2)#
#= nR(lnV_2 - lnV_1)#
#=> color(blue)(DeltaS = nRln(V_2/V_1))#
So, given #"0.75 g"# of helium, we can get the mols of gas and thus find the change in entropy.
#color(blue)(DeltaS) = 0.75 cancel"g He" xx cancel("1 mol")/(4.0026 cancel"g He") xx "8.314472 J/"cancel"mol"cdot"K" xx ln("12.5 L"/"5.0 L")#
#=# #color(blue)("1.428 J/K")#
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Answer 2
The entropy change for the isothermal expansion of 0.75 g of He from 5.0 L to 12.5 L can be calculated using the formula ΔS = nR ln(Vf/Vi), where ΔS is the entropy change, n is the number of moles (n
= mass/molar mass), R is the gas constant, and Vi and Vf are the initial and final volumes, respectively.
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Answer from HIX Tutor
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some
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From the given figure, the value of a is-Turito
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From the given figure, the value of ‘a’ is
A. 40°
B. 110°
C. 130°
D. 100°
In this question, we have to find the value of a. Given AB and CD are parallel and a transversal. For this we will use some parallel lines theorem.
The correct answer is: 130°
As the AB is parallel to CD and there is transversal
So, the vertical opposite angle of
Now, we have the interior angles, and we know that their sum is | {"url":"https://www.turito.com/ask-a-doubt/Maths-from-the-given-figure-the-value-of-a-is-100-130-110-40-q9ac0c5","timestamp":"2024-11-11T19:39:22Z","content_type":"application/xhtml+xml","content_length":"1052465","record_id":"<urn:uuid:f7aa7ddc-1375-49bf-bd78-8c5d16cb7f66>","cc-path":"CC-MAIN-2024-46/segments/1730477028239.20/warc/CC-MAIN-20241111190758-20241111220758-00645.warc.gz"} |
Representation of Sets
An unsorted collection of items known as elements or set members is referred to as a set.
‘a ∉ A’ denotes the absence of an element ‘a’ from the set A.
Types of Set Representation
A set can be represented in a number of ways.
1. Statement form.
2. Using a roster or a tabular form
3: Set Builder
1. Statement Form
In this form, the elements of the set are specified in depth. Listed below are some examples.
1. A group of all even numbers less than 10.
2. The number is in the range of ten to one.
2. Roster form
In this diagram, the elements are given in brackets with commas separating them. Below are two examples of this.
1. Assume that N is a collection of natural numbers less than 5.
N = { 1 , 2 , 3, 4 }.
2. The alphabet’s vowels.
V = { a , e , I , o , u }.
3. Set builder form
A property must be fulfilled by each member of a Set-builder set.
1. x: x is a divisible by six even number less than 100.
2. x is a number less than ten.
If and only if every element of set A is also a part of set B, that set is said to be a subset of another set B.
‘A ⊆ B’ denotes that A is a subset of B.
A set can contain components like numbers, states, vehicles, people, and even other sets. Anything can be used to construct a set, but there are a few things to keep in mind.
Pairs of equals
Elements of a set can be in or out. A set could be described using a defining attribute or a list of its elements. It doesn’t matter if they’re listed in any particular sequence. The sets {1, 2, 3}
and {1, 3, 2} are equivalent since they both contain the same items.
Two unique sets
There are two sets that stand out in particular. The first is the universal set, also known as U. All of the potential elements are included in this set. This set could vary from one setup to the
next. One universal set might be the set of real numbers, whereas another might be the whole numbers 0 to 2, and so on.
The other group that requires attention is the empty set. The empty set is a collection of zero elements. This can be written as and is symbolised by the symbol.
Set Operations
Setup Procedures There are various procedures, however they virtually all consist of the three described below:
The combining of two or more persons is referred to as a union. The union of the sets A and B consists of objects that belong to either A or B.
A crossroads is a point where two things meet. The intersection of the two sets consists of items that appear in both A and B. The complement of A is made up of all components in the universal set
that are not constituents of A.
Venn diagrams are a form of diagram used to show relationships between objects.
A Venn diagram is a visual representation of the relationship between two sets of information. For our case, a rectangle represents the universal set. Each group has a circle as its symbol. The
circles overlapping indicate the intersection of our two sets.
Set Theory in Application
Set theory is used frequently in mathematics. It is the cornerstone of several mathematical subfields. Probability and statistical sciences are where it is most widely used. Many probability ideas
are based on the implications of set theory. One way to express probability principles is through set theory.
Sets can be represented in two ways: the Roster form and the Set-Builder form. Both of these forms can be used to describe the same data, but the style differs in each case. A collection of
well-specified data is defined as a set. In mathematics, a set is a tool that may be used to classify and collect data from the same category, even though the items in the set are entirely different
from one another.
Sets are an important idea in modern mathematics. Sets are now employed in practically every discipline of mathematics in the modern period. A set is a group of specific objects or a collection of
specific things in mathematics. Relationships and functions are defined in terms of sets. A strong understanding of sets is necessary for the study of probability, geometry, and other subjects. The
sets can take a variety of formats. The fundamentals of set theory and set representation will be covered in depth in this essay. | {"url":"https://unacademy.com/content/cat/study-material/data-interpretation-and-logical-reasoning/a-brief-note-on-representation-of-sets/","timestamp":"2024-11-10T00:13:41Z","content_type":"text/html","content_length":"643781","record_id":"<urn:uuid:4ad60850-420d-44c3-8ed2-bba4b10e5467>","cc-path":"CC-MAIN-2024-46/segments/1730477028164.10/warc/CC-MAIN-20241109214337-20241110004337-00049.warc.gz"} |
Quadratic inequalities
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Formula to Identify Number of Forms Submitted by Individual Offices
I am working on building a dashboard to display global and individual data regarding different requests (out-of-state travel requests, official functions, etc.) across seven offices. I would like to
eventually calculate the monetary value of each of these requests, labeled as 'Amount' on the reference sheet.
I am currently stuck identifying a formula that would calculate the total number of individual office requests within the whole. I started by calculating the total number of 'Form Type" requests from
my reference sheet (success). I then tried to refine the data by individual 'Office' (e.g. "PWR") and my formula is #UNPARSEABLE
Any help to find success would be appreciated,
• Hi @moore_r
Would a COUNTIF formula not work for the number of requests, a SUMIF for the amount? I think AND in your formula is throwing me off. I must be missing something.
Matt Johnson
Smartsheet Aligned Partner
• Hello Matt,
I tried to do both CountIF and SumIF, but also received the #UNPARSEABLE message. I have to assume that it is something incorrect with the rest of the equation. I tried to look through the new
Smartsheet formula template, but am not formula savvy enough to know where I am going wrong.
I appreciate the help,
• This is the method I tried for CountIf that was also #UNPARSEABLE.
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Displacement Reaction Examples, Definition For Class 10
Displacement Reaction
Displacement Reaction: A displacement reaction is one wherein the atom or a set of atoms is displaced by another atom in a molecule. For instance, when the iron is added to a copper sulfate solution,
it displaces the copper metal. A + B-C → A-C + B. The above equation only exists when A is more reactive than B.
Displacement Reaction Definition Class 10
P+ QR=PQ + R
Displacement reaction is a very important chemical reaction, which is used in various processes like metal extraction, extraction of iron, welding, etc. In a displacement reaction, the less reactive
atom in a compound is displaced with more reactive atoms.
P + QR
Here, in this compound, P, Q, and R, are different atoms, and P, is more reactive than R. hence, P replaces R in the above reaction. The most common example of displacement reaction is iron and
copper sulfate.
Fe + CuSO[4 ]FeSO[4 + ]Cu
Here, Fe is more reactive than Cu, therefore it replaces Cu in the above reaction.
Displacement Reaction Types
There are two types of displacement reactions
• Single displacement reaction
• Double displacement reaction
1. Single Displacement reaction- It is a kind of oxidation reaction, in which a single element is replaced. The element which has higher reactivity will replace the less reactive element.
2. Double displacement reaction – In a double displacement reaction, both the elements of a compound react with each other and form a new element. These reactions mostly take place in an aqueous
state, which facilitates precipitation and helps the elements form new bonds.
PQ + AB PA + QB
Here P, Q, A and B are four different elements that react in an aqueous solution and due double displacement reaction, the elements get exchanged and form a new element.
HCl + NaOH NaCl + H[2]O
Na replaces H from HCl, and H is displaced in the place of Na, with OH, forming H[2]O
What is Single Displacement Reaction Class 10
A single displacement reaction, also known as a single replacement reaction or a single substitution reaction, is a type of chemical reaction that occurs when one element or ion replaces another
element in a compound. This reaction is characterized by the exchange of atoms or ions between two reactants, resulting in the formation of a new compound and the release of energy.
The general form of a single displacement reaction can be represented as:
A + BC → AC + B
In this equation:
• A is the element or ion that displaces B from the compound BC.
• B is the element or ion that gets displaced and is usually a metal.
• C is another element or ion bonded to B in the compound BC.
• AC is the new compound formed when A displaces B.
For example, a common single displacement reaction involves the reaction of a metal with an acid:
Zn(s) + 2HCl(aq) → ZnCl2(aq) + H2(g)
In this reaction, zinc (Zn) displaces hydrogen (H) from hydrochloric acid (HCl), resulting in the formation of zinc chloride (ZnCl2) and hydrogen gas (H2) as products.
Single displacement reactions can also occur in other types of compounds, such as when a more reactive element displaces a less reactive element in a compound. These reactions are important in
understanding the reactivity of elements and predicting the products of chemical reactions.
What is Double Displacement Reaction Class 10
In chemistry, a double displacement reaction, also known as a double replacement reaction or metathesis reaction, is a type of chemical reaction in which two compounds react by exchanging ions to
form two new compounds. This type of reaction typically occurs in aqueous solutions and involves the cations and anions of the compounds switching partners.
The general form of a double displacement reaction can be represented as follows:
AB + CD → AD + CB
In this reaction, A and C are usually cations (positively charged ions), while B and D are usually anions (negatively charged ions). When the reactants AB and CD come into contact, the cations A and
C exchange places with each other, leading to the formation of the new compounds AD and CB.
One common example of a double displacement reaction is the reaction between sodium chloride (NaCl) and silver nitrate (AgNO3) to form sodium nitrate (NaNO3) and silver chloride (AgCl):
NaCl (aq) + AgNO3 (aq) → NaNO3 (aq) + AgCl (s)
In this reaction, the sodium ions (Na+) from sodium chloride switch places with the silver ions (Ag+) from silver nitrate, resulting in the precipitation of silver chloride (AgCl) as a solid.
Double displacement reactions are often used in analytical chemistry for the identification and separation of ions in solution, and they are commonly encountered in various chemical processes and
laboratory experiments.
Example of Displacement Reaction
For every equation, it will be difficult to learn all the outcomes. To understand the displacement reactions, you should be familiar with the Reactivity series.
A displacement reaction, also known as a single replacement reaction, occurs when an element reacts with a compound and displaces another element from it. Here’s an example of a displacement
Zinc and Hydrochloric Acid:
Equation: Zn + 2HCl → ZnCl2 + H2
In this reaction, zinc (Zn) displaces hydrogen (H) from hydrochloric acid (HCl) to form zinc chloride (ZnCl2) and hydrogen gas (H2). The more reactive element, zinc, displaces the less reactive
element, hydrogen, from the compound.
Keep in mind that for a displacement reaction to occur, the reacting element must be more reactive than the element it’s trying to displace from the compound.
Displacement Reaction Example for Class 10
Displacement reactions, also known as replacement reactions or single replacement reactions, are a type of chemical reaction where one element replaces another element in a compound. In these
reactions, a more reactive element displaces a less reactive element from its compound. Here are some displacement reaction examples suitable for a Class 10 level of understanding:
1. Zinc and Copper Sulfate: Zinc (Zn) is more reactive than copper (Cu). When zinc is added to copper sulfate (CuSO4) solution, a displacement reaction occurs:
The zinc displaces copper from copper sulfate, resulting in the formation of zinc sulfate and the deposition of copper metal.
2. Magnesium and Hydrochloric Acid: Magnesium (Mg) is more reactive than hydrogen (H). When magnesium reacts with hydrochloric acid (HCl), a displacement reaction takes place:
The magnesium displaces hydrogen from the hydrochloric acid, producing magnesium chloride and hydrogen gas.
3. Iron and Copper Sulfate: Iron (Fe) is more reactive than copper (Cu). When iron is added to copper sulfate solution, a displacement reaction occurs:
Iron displaces copper from copper sulfate, leading to the formation of iron sulfate and the release of copper metal.
4. Aluminum and Iron(III) Oxide: Aluminum (Al) is more reactive than iron (Fe). When aluminum reacts with iron(III) oxide (Fe2O3), a displacement reaction takes place:
2Al + Fe2O3 -> Al2O3 + 2Fe
Aluminum displaces iron from iron(III) oxide, resulting in the formation of aluminum oxide (also known as alumina) and elemental iron.
5. Sodium and Water: Sodium (Na) is highly reactive and can displace hydrogen from water (H2O):
Sodium displaces hydrogen from water, producing sodium hydroxide (NaOH) and hydrogen gas.
6. Potassium and Water: Similar to sodium, potassium (K) is very reactive and displaces hydrogen from water as well:
Potassium displaces hydrogen from water, forming potassium hydroxide (KOH) and hydrogen gas.
Remember that these reactions demonstrate the concept of displacement, where a more reactive element takes the place of a less reactive element in a compound. Always exercise caution when performing
chemical reactions, especially those involving highly reactive metals.
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Date Issued:
This study measures the effectiveness of the National Computer Systems (NCS) Learn SuccessMaker Math Concepts and Skills computer program on standardized test scores at a middle school in east
central Florida. The NCS Learn Company makes three claims for the SuccessMaker interactive computer program, Math Concepts and Skills (MCS): 1. Student Florida Comprehensive Assessment Test
(FCAT) scores will improve from using the software 30 hours or more; 2. The increase in FCAT scores is directly related to the length of time the students' spend using the program; 3. The
software package grading system is equivalent to the FCAT scoring. This study was designed to evaluate each claim. To test the first claim, the FCAT Norm Referenced Test (NRT) Mathematics scale
scores of the 6th-grade middle school students were compared to the same students' previous FCAT scores. The scores were compared before and after they used the Math Concepts and Skills program.
An independent t test was used to compare the scores. There was a statistically significant difference in scale scores when the students used the MCS program for 30 hours or more. Further
investigation is needed to establish the causal effect for the observed differences. To test the second claim, the 6th- and 8th-grade students' time on task in the laboratory was compared to
their change in FCAT scores. A Pearson correlation coefficient of 0.58 was found to exist for the complete 6th-grade data set and a 0.71 correlation for the 8th-grade group. To test the third
claim, the MCS computer program grade equivalent scores were compared to the mathematics FCAT Level using the dependent t test to see if the two scores were equal. The analysis revealed that the
difference in the two scores was statistically significant. Therefore the claim that the two scores are equivalent was not true for this data set. Recommendations were made for future studies to
include qualitative data, a control group, and larger sample sizes. Studying the effect of the Math Concepts and Skills program on FCAT scores continues to be a project for investigation as
implementation of the computer software is contingent on improving FCAT scores.
Title: THE EFFECT OF THE MATH CONCEPTS AND SKILLS (MCS) COMPUTER PROGRAM ON STANDARDIZED TEST SCORES AT A MIDDLE SCHOOL IN EAST CENTRAL FLORIDA.
Manning, Cheryl, Author
Name(s): Sivo, Stephen, Committee Chair
University of Central Florida, Degree Grantor
Type of text
Date Issued: 2004
Publisher: University of Central Florida
Language(s): English
This study measures the effectiveness of the National Computer Systems (NCS) Learn SuccessMaker Math Concepts and Skills computer program on standardized test scores at a middle school
in east central Florida. The NCS Learn Company makes three claims for the SuccessMaker interactive computer program, Math Concepts and Skills (MCS): 1. Student Florida Comprehensive
Assessment Test (FCAT) scores will improve from using the software 30 hours or more; 2. The increase in FCAT scores is directly related to the length of time the students' spend using
the program; 3. The software package grading system is equivalent to the FCAT scoring. This study was designed to evaluate each claim. To test the first claim, the FCAT Norm Referenced
Test (NRT) Mathematics scale scores of the 6th-grade middle school students were compared to the same students' previous FCAT scores. The scores were compared before and after they
Abstract/ used the Math Concepts and Skills program. An independent t test was used to compare the scores. There was a statistically significant difference in scale scores when the students used
Description: the MCS program for 30 hours or more. Further investigation is needed to establish the causal effect for the observed differences. To test the second claim, the 6th- and 8th-grade
students' time on task in the laboratory was compared to their change in FCAT scores. A Pearson correlation coefficient of 0.58 was found to exist for the complete 6th-grade data set
and a 0.71 correlation for the 8th-grade group. To test the third claim, the MCS computer program grade equivalent scores were compared to the mathematics FCAT Level using the
dependent t test to see if the two scores were equal. The analysis revealed that the difference in the two scores was statistically significant. Therefore the claim that the two scores
are equivalent was not true for this data set. Recommendations were made for future studies to include qualitative data, a control group, and larger sample sizes. Studying the effect
of the Math Concepts and Skills program on FCAT scores continues to be a project for investigation as implementation of the computer software is contingent on improving FCAT scores.
Identifier: CFE0000227 (IID), ucf:46267 (fedora)
Note(s): Education, Department of Educational Research, Technology and Leadership
This record was generated from author submitted information.
Subject(s): testing
Link to This http://purl.flvc.org/ucf/fd/CFE0000227
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on Access:
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This is ancient version of
left here for archival purposes.
All calculators should still work, but you can't be sure.
We also don't update them anymore, so expect outdated info.
We suggest to move to
new version of Calculla.com
by clicking the link:
Fractions: 4 operations
Calculations on fractions - it performs operations on two given fractions. Simply enter two fractions and get them added, subtracted, multiplied and divided by each other. You will get sum,
difference, product and quotient of these two.
Some facts
• I. Adding fractions.
□ a. If both fractions have the same denominators, then simply add their numerators without changing denominator.
□ b. If fractions have different denominators, then you can't add them directly. In this case you need to convert fractions to common denominator first. After that you are ready add them like
in point (a).
• II. Subtracting fractions.
□ a. If both fractions have the same denominators, then simply subtract their numerators without changing denominator.
□ b. If fractions have different denominators, then you can't subtract them directly. In this case you need to convert fractions to common denominator first. After that you are ready subtract
them like in point (a).
• III. Multiplying fractions.
□ To perform fractions multiplication you have to multiply numerator of first fraction by numerator of second fraction and then denominator of first fraction by denominator of second one.
• IV. Dividing fractions.
□ a. At the beginning you need to inverse the second fraction. To achieve it - simply swap numerator and denominator in second fraction.
□ b. Next, multiply first fraction by inversed second fraction - exactly like in section (III).
How to use this tool
Simply enter your fractions into form below and Calculla will compute their sum (addition), difference (subtraction), product (multiply) and quotient (division) for you. For each of those operations,
Calculla will show you how to get proper result step-by-step:
• I. Remove wholes. In this step we convert mixed number to improper fraction if needed. However, if your fractions have no wholes part, we simply skip this step.
• II. Perform the appropriate action. This step depends on type of operation we're performing:
□ Addition - at the beginning we bring fractions to common denominator, next we add their numerators.
□ Subtraction - at the beginning we bring fractions to common denominator, next we subtract their numerators.
□ Multiplication - simply we multiply numerator of first fraction by numerator of second one and denominator of first fraction by denominator of second one.
□ Division - at the beginning we replace division by multiplication by the inverse, next we follow multiply steps.
• III. In this step we already have proper result. However probably not in the simplest form yet, so it may be not yet final.
• IV. Remove improper fraction. If our result is an improper fraction (i.e. its numerator is greater than denominator) we pull out wholes part before fraction. In this way we create the mixed
• V. Reduction to the simplest form. In this step we reduce the fraction to the simplest possible form.
First fraction
Second fraction
Links to external sites (leaving Calculla?)
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How to Calculate Hours Worked Minus Lunch in Excel - Excel Wizard
How to Calculate Hours Worked Minus Lunch in Excel
If you get paid per hour excluding lunch breaks then it can be tedious to calculate manually, especially when the lunch breaks vary in terms of total time. In this tutorial, I will guide you on how
to calculate hours worked minus Lunch in Excel.
Using the Sum Function
You can use the sum function to calculate the number of hours worked minus Lunch in Excel. Here are the steps.
1. First type the following formula
2. Press enter
As you can see from the image above the the hours worked on Monday are 6 full hours
3. Drag down the fill handle to automatically calculate hours worked minus lunch for the remaining days.
Finally, use the sum Function to calculate the number of hours worked for that whole week without a lunch break.
From the above image, a total of 39.17 hours were worked in that week
Tip: To get the hours in decimal the cell holding hours worked should be formatted as numbers.
Download the practice worksheet
Frequently Asked Questions
1. How do I calculate hours worked including lunch break in Excel?
To calculate hours worked including lunch breaks subtract the reporting time from the work End time. For example, if the end time is 16:00 and the reporting time is 9:00 am use the formula.
16:00-9:00 = 7 hrs.
2. What is the formula for calculating hours worked in Excel?
The formula to calculate hours worked in Excel is =Time End – Time Start.
3. How do I subtract hours from a time in Excel?
The best way to subtract hours from time is by using the 24-hour method. First, convert the time into 24 hours then subtract the hours using the minus sign in Excel. Alternatively, you can format the
cell containing the time as time and then just use the subtraction sign to subtract hours from time.
4. How do you subtract time?
To subtract time in Excel you need to make sure the cells are formatted as as time. First, select the cells containing the time then right-click and select format cells. Now click on the number tab
and select time. You can choose the time format that you wish from the list. Press okay to apply changes.
Use the minus sign to subtract time. For Example =B-A where B is the cell containing old time while A is the cell containing new time.
5. How do I convert start time and end time to hours in Excel?
To convert time into hours in Excel you just need to multiply it by 24 which is the number of hours in a day. Also, make sure that the cell hosting the results is formatted as a number. For example,
1:30 is equal to 1.50 hours. | {"url":"https://excelweez.com/how-to-calculate-hours-worked-minus-lunch-in-excel/","timestamp":"2024-11-04T15:28:38Z","content_type":"text/html","content_length":"204823","record_id":"<urn:uuid:3ac02f04-6cc8-46e9-816c-cd21c05d35b4>","cc-path":"CC-MAIN-2024-46/segments/1730477027829.31/warc/CC-MAIN-20241104131715-20241104161715-00459.warc.gz"} |
What are some Applications Using Linear Models? | Socratic
What are some Applications Using Linear Models?
1 Answer
The major practical application for linear models is to model linear trends and rates in the real world.
For example, if you wanted to wanted to see how much money you were spending over time, you could find how much money you had spent at a given time for several points in time, and then make a model
to see what rate you were spending at.
Also, in cricket matches, they use linear models to model the run rate of a given team. They do this by taking the number of runs a team has scored in a certain number of overs, and divide the two to
come up with a runs per over rate.
However, keep in mind that these real-life linear models are usually always averages, or approximations . This is just due to life being so random, but we never actually stick to those rates we have.
For example, if a cricket team's run rate was judged to be 10.23 runs per over, it doesn't mean that they scored exactly 10.23 runs every over, but rather that they scored that many on average.
Hope that helped :)
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What Size Breaker for Furnace? (Read This First!) - HomeApricotWhat Size Breaker for Furnace? (Read This First!)
What Size Breaker for Furnace? (Read This First!)
Winter can be harsh in some regions, making a household furnace a life savior for people, especially for those living in cold regions. A household furnace works wonders and turns your gloomy
shivering days into crispy warm and delightful.
There are different types of furnaces, they work by turning cold air, gas, or oil into warm air serving your house and saving you from unwanted winter zephyr.
However, each furnace has a limit to which extent they can raise the temperature. If heat rises above that temperature the furnace may blow off. Circuit breakers help control that.
So, here rises a question. What size breaker is suitable for your existing furnace? Keep reading to find out everything you need to know regarding the different types of furnaces and the suitable
breakers for them.
Furnace breaker size
Usually, 60 and 80 amp breakers are the standard sizes for an electric furnace. However, the size usually depends on the output voltage, power consumption, and the type of furnace. The total wattage
is the product of amps and volts which will help you to determine the size of the breaker.
Just like you can get the wattage with the product of amp and volts, most people use the amp to determine the correct circuit breaker. You can find the amp by dividing the wattage by volts.
There are three different types of furnaces. Although they have the same function, some of them are more suitable for certain conditions.
The three types of furnaces run on gas, oil, and electricity. The electric furnace is most commonly used by households.
A circuit breaker works by protecting your furnace from heating up more than it can condone. This is why they have different ratings according to the capacity of each furnace.
Electric furnace:
The electric furnace uses electricity to heat the elements inside the chamber and then a fan is used to pass the hot air through the vents in your room. This works similarly to a space heater.
An electric heater used in most households in the US has a capacity of 17,600 watts and a volt of 220 V. By dividing wattage by the volt you will find a current of 80 amp.
So, you will need a breaker of 80 amp which falls in the range of the standard size breakers used. However, if you are not sure about the rating research a bit.
Gas furnace:
A gas furnace works similarly to an electric furnace but instead of electricity, it uses natural gas. Due to the unavailability of abundant natural gas, they are not as frequently used as an electric
The natural gas is ignited inside the furnace and then the warm air is distributed inside the house. A gas furnace usually has a power of 600 watts so it requires a breaker of 15 amp.
Oil furnace:
An oil furnace turns the oil into a spray when the temperature rises in the combustion chamber. The spray is then turned into hot air that is spread through the vents inside the house, heating the
An oil furnace has a power consumption of 2500 watts so you will need a breaker of 30 amp to be precise.
15kw furnace:
A 15kw furnace has a power consumption of 15000 watts with a voltage of 208/230 V.
Hence it will require a breaker of 30/60 amp depending on the voltage rating. It has a minimum circuit capacity of 50/25 amp.
20kw furnace:
A 20kw furnace has a power consumption of 20000 watts with a voltage of 208/230 V.
It will require a breaker of 60 amp to be precise. It has a minimum circuit capacity of 50 amp.
25kw furnace:
A 25kw furnace has a power consumption of 25000 watts with a voltage of 208/230 V. It will require a breaker of 100/120 amp depending on the voltage.
Should my furnace be on a 15 or 20-amp breaker?
Generally, gas furnaces require a breaker of 15 amp due to the power consumption and voltage. However, if your furnace has a greater power consumption or voltage you can use a 20 amp breaker,
You can figure this out by finding the power and voltage of your furnace. You need to divide the power by voltage and you will know which breaker suits your device best.
If it has a voltage of 40 needs you need a 15 amp breaker and if it has a voltage of 30 you need a 20 amp breaker.
As surprising as it is, your gas furnace does use electricity to some extent even though it is based on natural gas. However, it has a small power consumption of 600 watts compared to electric
How many amps does a furnace use?
An electric furnace uses 60/80 amps, a gas furnace uses 15/20 amps and an oil furnace uses 30 amps. It depends on the type, power consumption, and voltage of the furnace.
An electric furnace has the greatest power consumption so it uses the greatest current of 60/80 amp. The number of amps a furnace uses is directly proportional to the power and voltage of the
An oil furnace has a power consumption of 2500 watts so it uses a current of 30 amp. The number of an amp a furnace possesses can be figured out by dividing power by voltage.
A gas furnace has the lowest power consumption as it burns natural gas for warm air production. So, it uses the least current of 15/20 amp.
How do determine a breaker size for a furnace?
Determining the correct breaker size for your furnace is essential as a furnace costs a lot. A minor miscalculation in determining the breaker size can result in a major issue.
As we know, a breaker usually protects the furnace by breaking, when the power consumption might exceed that of your furnace’s capacity.
There are a few steps that you can follow to determine the correct size of the breaker for your furnace. The steps are discussed in detail below to help you select the correct breaker for your
Determining the type of furnace:
The first step is very basic. You need to know whether the furnace is oil, gas, or electric. This will give you a good idea about the range of breakers that you will need for your furnace.
Reading the label:
The second step would be reading the label on your furnace to know about the power consumption of the furnace.
Usually, the label can be found near the heater housing cord. It is the specification label that has the details
Finding the voltage:
You need the operating voltage to find the current. If you divide the power by voltage then you can determine the current of the furnace.
Usually, there’s a specific range of voltage as well. The voltage is usually either 208 or 230 volts.
Finding the current:
The next and most important step would be finding the current. You will need the current to determine the size of the breaker.
The current can be found by dividing the power by voltage. In the example given above if the voltage is 208, dividing 15000 by 208 will give you a current of 72.2 and if the voltage is 230 you will
get a current of 65.2.
Exceeding the value:
The next step would be multiplying the value of the current by 125 percent to find the correct breaker.
If you find the 125 percent of 72.1 it will give you a value of 90 so you will select the closest existing breaker which is 100 amp.
Which breaker is for the furnace?
Usually, a furnace has a breaker with a double-pole due to its extensive power consumption. Those breakers are usually between 60 and 80 amps.
However, you need to verify your theory. Here is a list of ways that will help you to determine the solution to your question.
Circuit breaker finder:
There’s a circuit breaker finder available that you can use to trace the correct breaker for your furnace.
If the furnace isn’t equipped with a dedicated outlet into which you can plug your circuit breaker finder, you’ll need to connect the alligator clips that are provided to exposed wires to trace the
Performing manually:
You can manually change the breakers and find out which breaker is suitable to cut off the furnace when the power level exceeds.
However, it is not recommended as switching the power on and off several times can break the furnace.
Final thoughts
To sum up, the correct size of the breaker can be found by dividing the power by voltage and multiplying the result by 125 percent. After that, the next closest breaker available can be selected. The
standard breaker sizes are 15, 60, 80, & 100. You can select according to your furnace’s power load. | {"url":"https://homeapricot.com/what-size-breaker-for-furnace/","timestamp":"2024-11-10T14:06:20Z","content_type":"text/html","content_length":"74846","record_id":"<urn:uuid:1b443257-00f1-438a-9c91-15e4aaa255b9>","cc-path":"CC-MAIN-2024-46/segments/1730477028187.60/warc/CC-MAIN-20241110134821-20241110164821-00672.warc.gz"} |
Berlin 2018 – wissenschaftliches Programm
MA 21.62: Poster
Dienstag, 13. März 2018, 09:30–13:00, Poster A
The influence of the Hall-bar geometry on the apparent spin Hall angle in harmonic Hall voltage measurements — •Lukas Neumann and Markus Meinert — Center for Spinelectronic Materials and Devices,
Faculty of Physics, Bielefeld University, D-33501 Bielefeld, Germany
We investigate the influence of the Hall-bar geometry on the apparent spin Hall angle (SHA) in harmonic Hall voltage measurements which is a well established method to determine the SHA of a
nonmagnetic metal/ferromagnet bilayer structure like Ta/CoFeB. Tantalum is a heavy metal with large spin-orbit coupling such that an in-plane current generates a substantial spin-orbit torque acting
on the magnetization orientation of the ferromagnetic layer. The samples are patterned into Hall bars using electron beam lithography. Being located in an in-plane magnetic field an AC current
through the Hall-bar generates a Hall-voltage whose second harmonic gives the apparent SHA. In the simplest model, the influence of the voltage pickup arms is neglected. Obviously, the current
density distribution in the vicinity of the voltage pickup arms is not homogeneous and depends on the width of these arms. To systematically investigate this effect we varied the pickup arm width and
observe a strong change of the apparent SHA as the pickup arm width is increased. In a symmetric Hall cross with four-fold rotational symmetry, the apparent SHA is reduced by about 30% with respect
to the maximum value at narrow pickup arm width. We compare the measured values with simulations of the current density distribution. | {"url":"https://www.dpg-verhandlungen.de/year/2018/conference/berlin/part/ma/session/21/contribution/62","timestamp":"2024-11-14T00:07:12Z","content_type":"text/html","content_length":"7942","record_id":"<urn:uuid:0e77d7f3-9d81-45e6-b9ef-c53a7fe7060d>","cc-path":"CC-MAIN-2024-46/segments/1730477028516.72/warc/CC-MAIN-20241113235151-20241114025151-00009.warc.gz"} |
DukeSpace :: Browsing by Subject "Physical Sciences"
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• (0,2) hybrid models
(Journal of High Energy Physics, 2018-09-01) Bertolini, M; Plesser, MR
© 2018, The Author(s). We introduce a class of (0,2) superconformal field theories based on hybrid geometries, generalizing various known constructions. We develop techniques for the computation
of the complete massless spectrum when the theory can be interpreted as determining a perturbative heterotic string compactification. We provide evidence for surprising properties regarding RG
flows and IR accidental symmetries in (0,2) hybrid CFTs. We also study the conditions for embedding a hybrid theory in a particular class of gauged linear sigma models. This perspective suggests
that our construction generates models which cannot be realized or analyzed by previously known methods.
• (1,1) L-space knots
(COMPOSITIO MATHEMATICA, 2018-05-01) Greene, JE; Lewallen, S; Vafaee, F
We characterize the (1, 1) knots in the three-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit non- trivial L-space
surgeries. We also recover a characterization of the Berge manifold amongst 1-bridge braid exteriors.
• A baseline paleoecological study for the Santa Cruz Formation (late–early Miocene) at the Atlantic coast of Patagonia, Argentina
(Palaeogeography, Palaeoclimatology, Palaeoecology, 2010-06) Vizcaíno, SF; Bargo, MS; Kay, RF; Fariña, RA; Di Giacomo, M; Perry, JMG; Prevosti, FJ; Toledo, N; Cassini, GH; Fernicola, JC
Coastal exposures of the Santa Cruz Formation (late-early Miocene, southern Patagonia, Argentina) between the Coyle and Gallegos rivers have been a fertile ground for recovery of Miocene
vertebrates for more than 100 years. The formation contains an exceptionally rich mammal fauna, which documents a vertebrate assemblage very different from any living community, even at the
ordinal level. Intensive fieldwork performed since 2003 (nearly 1200 specimens have been collected, including marsupials, xenarthrans, notoungulates, litopterns astrapotheres, rodents, and
primates) document this assertion. The goal of this study is to attempt to reconstruct the trophic structure of the Santacrucian mammalian community with precise stratigraphic control.
Particularly, we evaluate the depauperate carnivoran paleoguild and identify new working hypotheses about this community. A database has been built from about 390 specimens from two localities:
Campo Barranca (CB) and Puesto Estancia La Costa (PLC). All species have been classified as herbivore or carnivore, their body masses estimated, and the following parameters estimated: population
density, on-crop biomass, metabolic rates, and the primary and secondary productivity. According to our results, this model predicts an imbalance in both CB and PLC faunas which can be seen by
comparing the secondary productivity of the ecosystem and the energetic requirements of the carnivores in it. While in CB, the difference between carnivores and herbivores is six-fold, in PLC
this difference is smaller, the secondary productivity is still around three times that of the carnivore to herbivore ratio seen today. If both localities are combined, the difference rises to
around four-fold in favour of secondary productivity. Finally, several working hypotheses about the Santacrucian mammalian community and the main lineages of herbivores and carnivores are
offered. © 2010 Elsevier B.V. All rights reserved.
• A classical proof that the algebraic homotopy class of a rational function is the residue pairing
(Linear Algebra and Its Applications, 2020-06-15) Kass, JL; Wickelgren, K
© 2020 Elsevier Inc. Cazanave has identified the algebraic homotopy class of a rational function of 1 variable with an explicit nondegenerate symmetric bilinear form. Here we show that Hurwitz's
proof of a classical result about real rational functions essentially gives an alternative proof of the stable part of Cazanave's result. We also explain how this result can be interpreted in
terms of the residue pairing and that this interpretation relates the result to the signature theorem of Eisenbud, Khimshiashvili, and Levine, showing that Cazanave's result answers a question
posed by Eisenbud for polynomial functions in 1 variable. Finally, we announce results answering this question for functions in an arbitrary number of variables.
• A formal Anthropocene is compatible with but distinct from its diachronous anthropogenic counterparts: a response to W.F. Ruddiman’s ‘three flaws in defining a formal Anthropocene’
(Progress in Physical Geography, 2019-06-01) Zalasiewicz, J; Waters, CN; Head, MJ; Poirier, C; Summerhayes, CP; Leinfelder, R; Grinevald, J; Steffen, W; Syvitski, J; Haff, P; McNeill, JR;
Wagreich, M; Fairchild, IJ; Richter, DD; Vidas, D; Williams, M; Barnosky, AD; Cearreta, A
© The Author(s) 2019. We analyse the ‘three flaws’ to potentially defining a formal Anthropocene geological time unit as advanced by Ruddiman (2018). (1) We recognize a long record of
pre-industrial human impacts, but note that these increased in relative magnitude slowly and were strongly time-transgressive by comparison with the extraordinarily rapid, novel and near-globally
synchronous changes of post-industrial time. (2) The rules of stratigraphic nomenclature do not ‘reject’ pre-industrial anthropogenic signals – these have long been a key characteristic and
distinguishing feature of the Holocene. (3) In contrast to the contention that classical chronostratigraphy is now widely ignored by scientists, it remains vital and widely used in unambiguously
defining geological time units and is an indispensable part of the Earth sciences. A mounting body of evidence indicates that the Anthropocene, considered as a precisely defined geological time
unit that begins in the mid-20th century, is sharply distinct from the Holocene.
• A hybrid ion-atom trap with integrated high resolution mass spectrometer
(Review of Scientific Instruments, 2019-10-01) Jyothi, S; Egodapitiya, KN; Bondurant, B; Jia, Z; Pretzsch, E; Chiappina, P; Shu, G; Brown, KR
© 2019 Author(s). In this article, we describe the design, construction, and implementation of our ion-atom hybrid system incorporating a high resolution time of flight mass spectrometer (TOFMS).
Potassium atoms (39K) in a magneto optical trap and laser cooled calcium ions (40Ca+) in a linear Paul trap are spatially overlapped, and the combined trap is integrated with a TOFMS for radial
extraction and detection of reaction products. We also present some experimental results showing interactions between 39K+ and 39K, 40Ca+ and 39K+, as well as 40Ca+ and 39K pairs. Finally, we
discuss prospects for cooling CaH+ molecular ions in the hybrid ion-atom system.
• A microscopic model of the Stokes-Einstein relation in arbitrary dimension.
(The Journal of chemical physics, 2018-06) Charbonneau, Benoit; Charbonneau, Patrick; Szamel, Grzegorz
The Stokes-Einstein relation (SER) is one of the most robust and widely employed results from the theory of liquids. Yet sizable deviations can be observed for self-solvation, which cannot be
explained by the standard hydrodynamic derivation. Here, we revisit the work of Masters and Madden [J. Chem. Phys. 74, 2450-2459 (1981)], who first solved a statistical mechanics model of the SER
using the projection operator formalism. By generalizing their analysis to all spatial dimensions and to partially structured solvents, we identify a potential microscopic origin of some of these
deviations. We also reproduce the SER-like result from the exact dynamics of infinite-dimensional fluids.
• A slicing obstruction from the $\frac {10}{8}$ theorem
(Proceedings of the American Mathematical Society, 2016-08-29) Donald, A; Vafaee, F
© 2016 American Mathematical Society. From Furuta’s 10/8 theorem, we derive a smooth slicing obstruction for knots in S3 using a spin 4-manifold whose boundary is 0-surgery on a knot. We show
that this obstruction is able to detect torsion elements in the smooth concordance group and find topologically slice knots which are not smoothly slice.
• A stochastic-Lagrangian particle system for the Navier-Stokes equations
(Nonlinearity, 2008-11-01) Iyer, Gautam; Mattingly, Jonathan
This paper is based on a formulation of the Navier-Stokes equations developed by Constantin and the first author (Commun. Pure Appl. Math. at press, arXiv:math.PR/0511067), where the velocity
field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take N copies of the above process (each based on independent Wiener
processes), and replace the expected value with 1/N times the sum over these N copies. (We note that our formulation requires one to keep track of N stochastic flows of diffeomorphisms, and not
just the motion of N particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space C1,α which consists of
differentiable functions whose first derivative is α Hölder continuous (see section 3 for the precise definition). Further, we show that as N → ∞ the system converges to the solution of
Navier-Stokes equations on any finite interval [0, T]. However for fixed N, we prove that this system retains roughly O(1/N) times its original energy as t → ∞. Hence the limit N → ∞ and T → ∞ do
not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as t → ∞ explicitly. © 2008 IOP Publishing Ltd and
London Mathematical Society.
• An adaptive Euler-Maruyama scheme for SDEs: Convergence and stability
(IMA Journal of Numerical Analysis, 2007-01-01) Lamba, H; Mattingly, JC; Stuart, AM
The understanding of adaptive algorithms for stochastic differential equations (SDEs) is an open area, where many issues related to both convergence and stability (long-time behaviour) of
algorithms are unresolved. This paper considers a very simple adaptive algorithm, based on controlling only the drift component of a time step. Both convergence and stability are studied. The
primary issue in the convergence analysis is that the adaptive method does not necessarily drive the time steps to zero with the user-input tolerance. This possibility must be quantified and
shown to have low probability. The primary issue in the stability analysis is ergodicity. It is assumed that the noise is nondegenerate, so that the diffusion process is elliptic, and the drift
is assumed to satisfy a coercivity condition. The SDE is then geometrically ergodic (averages converge to statistical equilibrium exponentially quickly). If the drift is not linearly bounded,
then explicit fixed time step approximations, such as the Euler-Maruyama scheme, may fail to be ergodic. In this work, it is shown that the simple adaptive time-stepping strategy cures this
problem. In addition to proving ergodicity, an exponential moment bound is also proved, generalizing a result known to hold for the SDE itself. © The author 2006. Published by Oxford University
Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
• An Empirical Comparison of Multiple Imputation Methods for Categorical Data
(The American Statistician, 2017-04-03) Akande, O; Li, F; Reiter, J
© 2017 American Statistical Association. Multiple imputation is a common approach for dealing with missing values in statistical databases. The imputer fills in missing values with draws from
predictive models estimated from the observed data, resulting in multiple, completed versions of the database. Researchers have developed a variety of default routines to implement multiple
imputation; however, there has been limited research comparing the performance of these methods, particularly for categorical data. We use simulation studies to compare repeated sampling
properties of three default multiple imputation methods for categorical data, including chained equations using generalized linear models, chained equations using classification and regression
trees, and a fully Bayesian joint distribution based on Dirichlet process mixture models. We base the simulations on categorical data from the American Community Survey. In the circumstances of
this study, the results suggest that default chained equations approaches based on generalized linear models are dominated by the default regression tree and Bayesian mixture model approaches.
They also suggest competing advantages for the regression tree and Bayesian mixture model approaches, making both reasonable default engines for multiple imputation of categorical data.
Supplementary material for this article is available online.
• An energetic variational approach for ION transport
(Communications in Mathematical Sciences, 2014-03-06) Xu, S; Sheng, P; Liu, C
The transport and distribution of charged particles are crucial in the study of many physical and biological problems. In this paper, we employ an Energy Variational Approach to derive the
coupled Poisson-Nernst-Planck-Navier-Stokes system. All of the physics is included in the choices of corresponding energy law and kinematic transport of particles. The variational derivations
give the coupled force balance equations in a unique and deterministic fashion. We also discuss the situations with different types of boundary conditions. Finally, we show that the Onsager's
relation holds for the electrokinetics, near the initial time of a step function applied field. © 2014 International Press.
• An energy stable C^0 finite element scheme for a quasi-incompressible phase-field model of moving contact line with variable density
(Journal of Computational Physics, 2020-03-15) Shen, L; Huang, H; Lin, P; Song, Z; Xu, S
In this paper, we focus on modeling and simulation of two-phase flow problems with moving contact lines and variable density. A thermodynamically consistent phase-field model with general Navier
boundary condition is developed based on the concept of quasi-incompressibility and the energy variational method. A mass conserving C0 finite element scheme is proposed to solve the PDE system.
Energy stability is achieved at the fully discrete level. Various numerical results confirm that the proposed scheme for both P1 element and P2 element are energy stable.
• An improved approach to age-modeling in deep time: Implications for the Santa Cruz Formation, Argentina
(Bulletin of the Geological Society of America, 2020-01-01) Trayler, RB; Schmitz, MD; Cuitiño, JI; Kohn, MJ; Bargo, MS; Kay, RF; Strömberg, CAE; Vizcaíno, SF
© 2019 Geological Society of America. Accurate age-depth models for proxy records are crucial for inferring changes to the environment through space and time, yet traditional methods of
constructing these models assume unrealistically small age uncertainties and do not account for many geologic complexities. Here we modify an existing Bayesian age-depth model to foster its
application for deep time U-Pb and 40Ar/39Ar geochronology. More flexible input likelihood functions and use of an adaptive proposal algorithm in the Markov Chain Monte Carlo engine better
account for the age variability often observed in magmatic crystal populations, whose dispersion can reflect inheritance, crystal residence times and daughter isotope loss. We illustrate this
approach by calculating an age-depth model with a contiguous and realistic uncertainty envelope for the Miocene Santa Cruz Formation (early Miocene; Burdigalian), Argentina. The model is
calibrated using new, high-precision isotope dilution U-Pb zircon ages for stratigraphically located interbedded tuffs, whose weighted mean ages range from ca. 16.78 ± 0.03 Ma to 17.62 ± 0.03 Ma.
We document how the Bayesian age-depth model objectively reallocates probability across the posterior ages of dated horizons, and thus produces better estimates of relative ages among strata and
variations in sedimentation rate. We also present a simple method to propagate age-depth model uncertainties onto stratigraphic proxy data using a Monte Carlo technique. This approach allows us
to estimate robust uncertainties on isotope composition through time, important for comparisons of terrestrial systems to other proxy records.
• Analyzing X-ray tomographies of granular packings.
(The Review of scientific instruments, 2017-05) Weis, Simon; Schröter, Matthias
Starting from three-dimensional volume data of a granular packing, as, e.g., obtained by X-ray Computed Tomography, we discuss methods to first detect the individual particles in the sample and
then analyze their properties. This analysis includes the pair correlation function, the volume and shape of the Voronoi cells, and the number and type of contacts formed between individual
particles. We mainly focus on packings of monodisperse spheres, but we will also comment on other monoschematic particles such as ellipsoids and tetrahedra. This paper is accompanied by a package
of free software containing all programs (including source code) and an example three-dimensional dataset which allows the reader to reproduce and modify all examples given.
• Anomalous dissipation in a stochastically forced infinite-dimensional system of coupled oscillators
(Journal of Statistical Physics, 2007-09-01) Mattingly, JC; Suidan, TM; Vanden-Eijnden, E
We study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has
a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy
to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its
properties are analyzed. © 2007 Springer Science+Business Media, LLC.
• Ballistic Graphene Josephson Junctions from the Short to the Long Junction Regimes.
(Physical review letters, 2016-12-02) Borzenets, IV; Amet, F; Ke, CT; Draelos, AW; Wei, MT; Seredinski, A; Watanabe, K; Taniguchi, T; Bomze, Y; Yamamoto, M; Tarucha, S; Finkelstein, G
We investigate the critical current I_{C} of ballistic Josephson junctions made of encapsulated graphene-boron-nitride heterostructures. We observe a crossover from the short to the long junction
regimes as the length of the device increases. In long ballistic junctions, I_{C} is found to scale as ∝exp(-k_{B}T/δE). The extracted energies δE are independent of the carrier density and
proportional to the level spacing of the ballistic cavity. As T→0 the critical current of a long (or short) junction saturates at a level determined by the product of δE (or Δ) and the number of
the junction's transversal modes.
• Behavior of different numerical schemes for random genetic drift
(BIT Numerical Mathematics, 2019-09-01) Xu, S; Chen, M; Liu, C; Zhang, R; Yue, X
In the problem of random genetic drift, the probability density of one gene is governed by a degenerated convection-dominated diffusion equation. Dirac singularities will always be developed at
boundary points as time evolves, which is known as the fixation phenomenon in genetic evolution. Three finite volume methods: FVM1-3, one central difference method: FDM1 and three finite element
methods: FEM1-3 are considered. These methods lead to different equilibrium states after a long time. It is shown that only schemes FVM3 and FEM3, which are the same, preserve probability,
expectation and positiveness and predict the correct probability of fixation. FVM1-2 wrongly predict the probability of fixation due to their intrinsic viscosity, even though they are
unconditionally stable. Contrarily, FDM1 and FEM1-2 introduce different anti-diffusion terms, which make them unstable and fail to preserve positiveness.
• Berge–Gabai knots and L–space satellite operations
(Algebraic & Geometric Topology, 2015-01-15) Hom, J; Lidman, T; Vafaee, F
© 2014 Mathematical Sciences Publishers. All rights reserved. Let P(K) be a satellite knot where the pattern P is a Berge–Gabai knot (ie a knot in the solid torus with a nontrivial solid torus
Dehn surgery) and the companion K is a nontrivial knot in S3. We prove that P(K) is an L–space knot if and only if K is an L–space knot and P is sufficiently positively twisted relative to the
genus of K. This generalizes the result for cables due to Hedden [13] and Hom [17].
• BigSMILES: A Structurally-Based Line Notation for Describing Macromolecules.
(ACS central science, 2019-09-12) Lin, Tzyy-Shyang; Coley, Connor W; Mochigase, Hidenobu; Beech, Haley K; Wang, Wencong; Wang, Zi; Woods, Eliot; Craig, Stephen L; Johnson, Jeremiah A; Kalow,
Julia A; Jensen, Klavs F; Olsen, Bradley D
Having a compact yet robust structurally based identifier or representation system is a key enabling factor for efficient sharing and dissemination of research results within the chemistry
community, and such systems lay down the essential foundations for future informatics and data-driven research. While substantial advances have been made for small molecules, the polymer
community has struggled in coming up with an efficient representation system. This is because, unlike other disciplines in chemistry, the basic premise that each distinct chemical species
corresponds to a well-defined chemical structure does not hold for polymers. Polymers are intrinsically stochastic molecules that are often ensembles with a distribution of chemical structures.
This difficulty limits the applicability of all deterministic representations developed for small molecules. In this work, a new representation system that is capable of handling the stochastic
nature of polymers is proposed. The new system is based on the popular "simplified molecular-input line-entry system" (SMILES), and it aims to provide representations that can be used as indexing
identifiers for entries in polymer databases. As a pilot test, the entries of the standard data set of the glass transition temperature of linear polymers (Bicerano, 2002) were converted into the
new BigSMILES language. Furthermore, it is hoped that the proposed system will provide a more effective language for communication within the polymer community and increase cohesion between the
researchers within the community. | {"url":"https://dukespace.lib.duke.edu/browse/subject?value=Physical%20Sciences","timestamp":"2024-11-06T07:33:50Z","content_type":"text/html","content_length":"816397","record_id":"<urn:uuid:096d5b2b-2678-4626-99a1-3472f5c75500>","cc-path":"CC-MAIN-2024-46/segments/1730477027910.12/warc/CC-MAIN-20241106065928-20241106095928-00095.warc.gz"} |
How to Prepare For the Medical Exam | Hire Someone To Take My Proctored Exam
When preparing for a life and death situation, it is important to learn inductive reasoning. You must have at least some level of ability in order to comprehend what you are reading or watching. This
skill can be very helpful in many situations, not just in death.
Suppose that you are faced with a medical emergency and you are not sure whether or not the patient can survive. If this is the situation, the first thing that you must do is to determine the odds of
survival. There are two main ways to arrive at this conclusion: the statistical and the mathematical. It is common sense that using the mathematical method is more reliable. But sometimes, the
numerical data is difficult to interpret.
Using the statistical method may allow you to make an educated guess. The person who does the statistics for a hospital uses a computer program that evaluates several different models in order to
come up with a statistical estimate. The problem is that the model used is not very useful for a person who cannot interpret numbers. In order to understand the numbers, it is necessary to take the
time to practice inductive reasoning.
This is not only useful for the doctor, but it is also a good idea for patients. It is much better for a patient to be given accurate information about the likelihood of his or her survival than to
have an incorrect prognosis given to them. Many doctors will not give their patients accurate prognoses because they fear that they will feel helpless if they do not make an accurate diagnosis. That
is why they often use the statistical method.
Unfortunately, however, the good thing about statistical analysis is that it is a very complex topic. If you cannot interpret numbers, you are not going to be able to understand the results.
For instance, suppose that you were given a question such as this: What are the chances of your surviving from three to five days after you receive a severe chest infection in the middle of your work
shift? You might expect that the answer to be high, since there is a good chance that you have experienced a similar type of chest infection before. In fact, if you are able to remember that you have
experienced this chest infection before, then you know that there is a good chance that you will again. experience this infection.
In order to correctly answer the question, you would need to understand statistical and inductive reasoning techniques. The first step would be to do a statistical analysis on your own. By gathering
information from different people, you can build a statistical model and compare it to the data gathered by the model. In this case, the model would be the chest infection. Your model will predict
the odds of your survival based on your history of chest infections.
After you have created your model, you will need to look at your model in order to see if it accurately predicts the results. You can do this by taking the time to practice inductive reasoning and
other related skills.
As you prepare for your test day, take the time to learn more about the information given to you on the exam and the various techniques that are used in order to improve your chances of success. Most
of the time, medical examiners will tell you about the types of questions that are usually asked on the test.
For example, you may be asked a series of questions about the types of medications that you should be taking in order to help you pass certain tests. This is one of the main things that medical
examiners want to know about you before administering the exam.
You should not forget to ask the same questions to your medical examiners. You should be able to accurately answer each question. After the test, you can then go back and use the information in order
to improve your skills. for future exams. | {"url":"https://crackmyproctoredexam.com/how-to-prepare-for-the-medical-exam/","timestamp":"2024-11-10T11:35:26Z","content_type":"text/html","content_length":"106210","record_id":"<urn:uuid:2eaa6c09-d673-497b-b5ff-a3e0bc5c8511>","cc-path":"CC-MAIN-2024-46/segments/1730477028186.38/warc/CC-MAIN-20241110103354-20241110133354-00604.warc.gz"} |
Todd S.
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About Todd S.
Algebra, Calculus, Geometry, Trigonometry, Algebra 2, Statistics, Finite Mathematics, Discrete Mathematics, Pre-Calculus, Calculus BC
Bachelors in Mathematics, General from Minot State University
Masters in Mathematics, General from The University of Texas-Pan American
Career Experience
I am a current high school math teacher of 31 years in Texas. I also teach dual enrollment / adjunct faculty at a local community college. I work as a consultant for graphing calculator technology
and as a mentor for calculus and statistics teachers.
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Probability space
Definition: Probability space
Index: The Book of Statistical Proofs
General Theorems
Probability theory
Random experiments
▷ Probability space
Definition: Given a random experiment, a probability space $(\Omega, \mathcal{E}, P)$ is a triple consisting of
• the sample space $\Omega$, i.e. the set of all possible outcomes from this experiment;
• an event space $\mathcal{E} \subseteq 2^\Omega$, i.e. a set of subsets from the sample space, called events;
• a probability measure $P: \; \mathcal{E} \rightarrow [0,1]$, i.e. a function mapping from the event space to the real numbers, observing the axioms of probability.
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Volume and surface area of a cylinder - Businesspillers
Volume and surface area of a cylinder
Mathematics is a subject that defines a student’s life. Students who excel at mathematics naturally learn how to solve issues and use logic. Mathematics is a subject that necessitates a lot of mental
effort. Students that excel at mathematics have a sharper brain, with greater reasoning and problem-solving abilities than their peers. Students who pursue higher education in mathematics can find
work in a variety of fields. Mathematics is one of the most crucial disciplines to have in order to get a good profession. Solving equations and problems involving numbers is what mathematics is all
about. However, its utility in everyday life is unlimited.
Geometry is a crucial field of mathematics. Geometry is an important element of a child’s math education. Because geometry is so important in everyday life, students should pay extra attention in all
geometry studies. There is always something fresh to learn in mathematics. Mathematics aids in improving a person’s ability to think. People can build a habit of mental discipline and boost their
brain strength by solving mathematical problems. Mathematics is a subject that is both interesting and enjoyable. Mathematics is nothing more than a collection of questions involving numbers.
Things to know about the cylinder:
In geometry, a cylinder is a three-dimensional form with two parallel circular bases separated by a distance. A curving surface, set at a preset distance from the center, connects the two circular
bases. The axis of the cylinder is a line segment that connects the center of two circular bases. The height of the cylinder refers to the distance between the two circular bases. The cylinder’s
total surface area equals the sum of its curved surface area and the area of its two circular bases. The volume of a cylinder is the three-dimensional space it occupies.
The volume of a cylinder:
The density of a cylinder is its volume, which indicates how much material it can transport or how much material it can immerse in. The volume of a cylinder is calculated using the formula r2h, where
r is the circular base radius and h is the cylinder’s height. The substance could be a liquid or any other substance that can be uniformly filled into the cylinder. Geometry is an area of science in
which we learn about shapes and their qualities. The two most significant parameters of every 3d shape are volume and surface area.
The surface area of a cylinder:
The area occupied by a cylinder’s surface in three-dimensional space is called it’s surface area. A three-dimensional construction with circular bases that are parallel to each other is known as a
cylinder. There are no vertices in it. The surface area of three-dimensional objects is generally referred to as the area of three-dimensional shapes. The surface area of the cylinder formula is very
important for students to solve questions based on mensuration topics. The total area covered by a cylinder in three-dimensional space is called the area of the cylinder. The area of a cylinder is
the sum of the areas of two circular bases plus the area of a curved surface.
Also read: Tips for a Safe and Respectful Classroom
To achieve good grades in mathematics, students must have a clear and in-depth understanding of these areas. Online platforms are now available to students. Cuemath is a well-known online math
learning platform that assists students in dispelling their questions and studying mathematics in a straightforward manner. Students can watch live videos on the website at any time. Students may now
learn from some of our country’s best professors. They can attend live online classes with their favorite teachers no matter where they are in the world. This is why an increasing number of students
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Do you get a formula sheet for AP Statistics?
If you’re going to be taking the AP Statistics exam, you’re in luck! During the test, you’ll have access to a formula sheet that has many useful equations.
How is AP Stat calculated?
The mean is calculated by adding all the values in a group together, then dividing the sum by the total number in the group.
What is D table statistics?
Table D gives the area to the RIGHT of a dozen t or z-values. It can be used for. t distributions of a given df and for the Normal distribution.
How do you find standard deviation in AP Stats?
1. The standard deviation formula may look confusing, but it will make sense after we break it down.
2. Step 1: Find the mean.
3. Step 2: For each data point, find the square of its distance to the mean.
4. Step 3: Sum the values from Step 2.
5. Step 4: Divide by the number of data points.
6. Step 5: Take the square root.
How do I find my PA and B?
Formula for the probability of A and B (independent events): p(A and B) = p(A) * p(B). If the probability of one event doesn’t affect the other, you have an independent event. All you do is multiply
the probability of one by the probability of another.
How do you find z AP stats?
A z score is unique to each value within a population. To find a z score, subtract the mean of a population from the particular value in question, then divide the result by the population’s standard
What are the formulas of statistics?
Statistics Formula Sheet
Mean x ¯ = ∑ x n
Median If n is odd, then M = ( n + 1 2 ) t h term If n is even, then M = ( n 2 ) t h t e r m + ( n 2 + 1 ) t h t e r m 2
Mode The value which occurs most frequently
Variance σ 2 = ∑ ( x − x ¯ ) 2 n
Standard Deviation S = σ = ∑ ( x − x ¯ ) 2 n
How do you find ap value from a table?
Example: Calculating the p-value from a t-test by hand
1. Step 1: State the null and alternative hypotheses.
2. Step 2: Find the test statistic.
3. Step 3: Find the p-value for the test statistic. To find the p-value by hand, we need to use the t-Distribution table with n-1 degrees of freedom.
4. Step 4: Draw a conclusion.
How do you find Pa and Pb?
Step 1: Multiply the probability of A by the probability of B. p(A and B) = p(A) * p(B) = 0.4 * 0.0008 = 0.00032.
What is the formula for Statistics in AP?
Formulas and Tables for AP Statistics I. Descriptive Statistics 1 i i x xx nn ∑ =∑= ( ) ( ) 2 1 2 1 1 i xi xx s xx n n ∑− = ∑− = − − y a bx ˆ = + y a bx
What are the equations in AP Physics 2?
AP Physics 2: Algebra-BasedTable of Information: Equations 2 of 3 AP PHYSICS 2 EQUATIONS MECHANICS Ãà xx 0 at x 2 00 1 x 2 x =++x t atà x 22 Ãà xx x =+ 0 2
How useful is the AP Statistics reference sheet for the exam?
The AP Statistics reference sheet can be a big help during the exam, but only if you already know what’s on it and how to use it. The formula sheet is actually three pages that contain useful
equations in descriptive statistics, probability, and inferential statistics.
What is the proton mass in AP Physics 2?
AP PHYSICS 2 TABLE OF INFORMATION © 2020College Board AP Physics 2: Algebra-BasedTable of Information: Equations1 of 3 AP®PHYSICS 2 TABLE OF INFORMATION CONSTANTS AND CONVERSION FACTORS Proton mass,
1.67 10 k 27 g m p Neutron mass 1.67 10 kg 27 m n Electron mass, 9.11 10 kg 31 m e Avogadro’s number, 23 1 N 0 6.02 10 mol = ¥ | {"url":"https://ventolaphotography.com/do-you-get-a-formula-sheet-for-ap-statistics/","timestamp":"2024-11-13T11:46:18Z","content_type":"text/html","content_length":"69364","record_id":"<urn:uuid:38d19b83-c145-4163-8049-182728ecb650>","cc-path":"CC-MAIN-2024-46/segments/1730477028347.28/warc/CC-MAIN-20241113103539-20241113133539-00548.warc.gz"} |
Optimization and Control of Fed-Batch Reactor Using Nonlinear MPC
This example shows how to use nonlinear model predictive control to optimize batch reactor operation. The example also shows how to run a nonlinear MPC controller as an adaptive MPC controller and a
time-varying MPC controller to quickly compare their performance.
Fed-Batch Chemical Reactor
The following irreversible and exothermic reactions occur in the batch reactor [1]:
A + B => C (desired product)
C => D (undesired product)
The batch begins with the reactor partially filled with known concentrations of reactants A and B. The batch reacts for 0.5 hours, during which additional B can be added and the reactor temperature
can be changed.
The nonlinear model of the batch reactor is defined in the fedbatch_StateFcn and fedbatch_OutputFcn functions. This system has the following inputs, states, and outputs.
Manipulated Variables
• u1 = u_B, flow rate of B feed
• u2 = Tsp, reactor temperature setpoint, deg C
Measured disturbance
• u3 = c_Bin, concentration of B in the B feed flow
• x1 = V*c_A, mol of A in the reactor
• x2 = V*(c_A + c_C), mol of A + C in the reactor
• x3 = V, liquid volume in the reactor
• x4 = T, reactor temperature, K
• y1 = V*c_C, amount of product C in the reactor, equivalent to x2-x1
• y2 = q_r, heat removal rate, a nonlinear function of the states
• y3 = V, liquid volume in reactor
The goal is to maximize the production of C (y1) at the end of the batch process. During the batch process, the following operating constraints must be satisfied:
1. Hard upper bound on heat removal rate (y2). Otherwise, temperature control fails.
2. Hard upper bound on liquid volume in reactor (y3) for safety.
3. Hard upper and lower bounds on B feed rate (u_B).
4. Hard upper and lower bounds on reactor temperature setpoint (Tsp).
Specify the nominal operating condition at the beginning of the batch process.
c_A0 = 10;
c_B0 = 1.167;
c_C0 = 0;
V0 = 1;
T0 = 50 + 273.15;
c_Bin = 20;
Specify the nominal states.
x0 = zeros(3,1);
x0(1) = c_A0*V0;
x0(2) = x0(1) + c_C0*V0;
x0(3) = V0;
x0(4) = T0;
Specify the nominal inputs.
u0 = zeros(3,1);
u0(2) = 40;
u0(3) = c_Bin;
Specify the nominal outputs.
y0 = fedbatch_OutputFcn(x0,u0);
Nonlinear MPC Design to Optimize Batch Operation
Create a nonlinear MPC object with 4 states, 3 outputs, 2 manipulated variables, and 1 measured disturbance.
nlmpcobj_Plan = nlmpc(4, 3, 'MV', [1,2], 'MD', 3);
Zero weights are applied to one or more OVs because there are fewer MVs than OVs.
Given the expected batch duration Tf, choose the controller sample time Ts and prediction horizon.
Tf = 0.5;
N = 50;
Ts = Tf/N;
nlmpcobj_Plan.Ts = Ts;
nlmpcobj_Plan.PredictionHorizon = N;
If you set the control horizon equal to the prediction horizon, there will be 50 free control moves, which leads to a total of 100 decision variables because the plant has two manipulated variables.
To reduce the number of decision variables, you can specify control horizon using blocking moves. Divide the prediction horizon into 8 blocks, which represents 8 free control moves. Each of the first
seven blocks lasts seven prediction steps. Doing so reduces the number of decision variables to 16.
nlmpcobj_Plan.ControlHorizon = [7 7 7 7 7 7 7 1];
Specify the nonlinear model in the controller. The function fedbatch_StateFcnDT converts the continuous-time model to discrete time using a multi-step Forward Euler integration formula.
nlmpcobj_Plan.Model.StateFcn = @(x,u) fedbatch_StateFcnDT(x,u,Ts);
nlmpcobj_Plan.Model.OutputFcn = @(x,u) fedbatch_OutputFcn(x,u);
nlmpcobj_Plan.Model.IsContinuousTime = false;
Specify the bounds for feed rate of B.
nlmpcobj_Plan.MV(1).Min = 0;
nlmpcobj_Plan.MV(1).Max = 1;
Specify the bounds for the reactor temperature setpoint.
nlmpcobj_Plan.MV(2).Min = 20;
nlmpcobj_Plan.MV(2).Max = 50;
Specify the upper bound for the heat removal rate. The true constraint is 1.5e5. Since nonlinear MPC can only enforce constraints at the sampling instants, use a safety margin of 0.05e5 to prevent a
constraint violation between sampling instants.
nlmpcobj_Plan.OV(2).Max = 1.45e5;
Specify the upper bound for the liquid volume in the reactor.
nlmpcobj_Plan.OV(3).Max = 1.1;
Since the goal is to maximize y1, the amount of C in the reactor at the end of the batch time, specify a custom cost function that replaces the default quadratic cost. Since y1 = x2-x1, define the
custom cost to be minimized as x1-x2 using an anonymous function.
nlmpcobj_Plan.Optimization.CustomCostFcn = @(X,U,e,data) X(end,1)-X(end,2);
nlmpcobj_Plan.Optimization.ReplaceStandardCost = true;
To configure the manipulated variables to vary linearly with time within each block, select piecewise linear interpolation. By default, nonlinear MPC keeps manipulated variables constant within each
block, using piecewise constant interpolation, which might be too restrictive for an optimal trajectory planning problem.
nlmpcobj_Plan.Optimization.MVInterpolationOrder = 1;
Use the default nonlinear programming solver fmincon to solve the nonlinear MPC problem. For this example, set the solver step tolerance to help achieve first order optimality.
nlmpcobj_Plan.Optimization.SolverOptions.StepTolerance = 1e-8;
Before carrying out optimization, check whether all the custom functions satisfy NLMPC requirements using the validateFcns command.
validateFcns(nlmpcobj_Plan, x0, u0(1:2), u0(3));
Model.StateFcn is OK.
Model.OutputFcn is OK.
Optimization.CustomCostFcn is OK.
Analysis of user-provided model, cost, and constraint functions complete.
Analysis of Optimization Results
Find the optimal trajectories for the manipulated variables such that production of C is maximized at the end of the batch process. To do so, use the nlmpcmove function.
fprintf('\nOptimization started...\n');
[~,~,Info] = nlmpcmove(nlmpcobj_Plan,x0,u0(1:2),zeros(1,3),u0(3));
fprintf(' Expected production of C (y1) is %g moles.\n',Info.Yopt(end,1));
fprintf(' First order optimality is satisfied (Info.ExitFlag = %i).\n',...
fprintf('Optimization finished...\n');
Optimization started...
Slack variable unused or zero-weighted in your custom cost function.
All constraints will be hard.
Expected production of C (y1) is 2.02353 moles.
First order optimality is satisfied (Info.ExitFlag = 1).
Optimization finished...
The discretized model uses a simple Euler integration, which could be inaccurate. To check this, integrate the model using the ode15s command for the calculated optimal MV trajectory.
Nstep = size(Info.Xopt,1) - 1;
t = 0;
X = x0';
t0 = 0;
for i = 1:Nstep
u_in = [Info.MVopt(i,1:2)'; c_Bin];
ODEFUN = @(t,x) fedbatch_StateFcn(x, u_in);
TSPAN = [t0, t0+Ts];
Y0 = X(end,:)';
[TOUT,YOUT] = ode15s(ODEFUN,TSPAN,Y0);
t = [t; TOUT(2:end)];
X = [X; YOUT(2:end,:)];
t0 = t0 + Ts;
nx = size(X,1);
Y = zeros(nx,3);
for i = 1:nx
Y(i,:) = fedbatch_OutputFcn(X(i,:)',u_in)';
fprintf('\n Actual Production of C (y1) is %g moles.\n',X(end,2)-X(end,1));
fprintf(' Heat removal rate (y2) satisfies the upper bound.\n');
Actual Production of C (y1) is 2.0228 moles.
Heat removal rate (y2) satisfies the upper bound.
In the top plot of the following figure, the actual production of C agrees with the expected production of C calculated from nlmpcmove. In the bottom plot, the heat removal rate never exceeds its
hard constraint.
plot(t,Y(:,1),(0:Nstep)*Ts, Info.Yopt(:,1),'*')
axis([0 0.5 0 Y(end,1) + 0.1])
title('Mol C in reactor (y1)')
tTs = (0:Nstep)*Ts;
t(end) = 0.5;
plot(t,Y(:,2),'-',[0 tTs(end)],1.5e5*ones(1,2),'r--')
axis([0 0.5 0.8e5, 1.6e5])
legend({'q_r','Upper Bound'},'location','southwest')
title('Heat removal rate (y2)')
Close examination of the heat removal rate shows that it can exhibit peaks and valleys between the sampling instants as reactant compositions change. Consequently, the heat removal rate exceeds the
specified maximum of 1.45e5 (around t = 0.35 h) but stays below the true maximum of 1.5e5.
The following figure shows the optimal trajectory of planned adjustments in the B feed rate (u1), and the reactor temperature (x4) and its setpoint (u2).
title('Feed rate of B (u1)')
[0 0.5],[20 20],'r--',[0 0.5],[50 50],'r--')
axis([0 0.5 15 55])
title('Reactor temperature and its setpoint')
The trajectory begins with a relatively high feed rate, which increases c_B and the resulting C production rate. To prevent exceeding the heat removal rate constraint, reactor temperature and feed
rate must decrease. The temperature eventually hits its lower bound and stays there until the reactor is nearly full and the B feed rate must go to zero. The temperature then increases to its maximum
(to increase C production) and finally drops slightly (to reduce D production, which is favored at higher temperatures).
The top plot of the following figure shows the consumption of c_A, which tends to reduce C production. To compensate, the plan first increases c_B, and when that is no longer possible (the reactor
liquid volume must not exceed 1.1), the plan makes optimal use of the temperature. In the bottom plot of the following figure, the liquid volume never exceeds its upper bound.
c_A = X(:,1)./X(:,3);
c_B = (c_Bin*X(:,3) + X(:,1) + V0*(c_B0 - c_A0 - c_Bin))./X(:,3);
plot(t,[c_A, c_B])
legend({'c_A','c_B'}, 'location', 'west')
title('Liquid volume')
Nonlinear MPC Design for Tracking the Optimal C Product Trajectory
To track the optimal trajectory of product C calculated above, you design another nonlinear MPC controller with the same prediction model and constraints. However, use the standard quadratic cost and
default horizons for tracking purposes.
To simplify the control task, assume that the optimal trajectory of the B feed rate is implemented in the plant and the tracking controller considers it to be a measured disturbance. Therefore, the
controller uses the reactor temperature setpoint as its only manipulated variable to track the desired y1 profile.
Create the tracking controller.
nlmpcobj_Tracking = nlmpc(4,3,'MV',2,'MD',[1,3]);
nlmpcobj_Tracking.Ts = Ts;
nlmpcobj_Tracking.Model = nlmpcobj_Plan.Model;
nlmpcobj_Tracking.MV = nlmpcobj_Plan.MV(2);
nlmpcobj_Tracking.OV = nlmpcobj_Plan.OV;
nlmpcobj_Tracking.Weights.OutputVariables = [1 0 0]; % track y1 only
nlmpcobj_Tracking.Weights.ManipulatedVariablesRate = 1e-6; % aggressive MV
Zero weights are applied to one or more OVs because there are fewer MVs than OVs.
Obtain the C production (y1) reference signal from the optimal plan trajectory.
Obtain the feed rate of B (u1) from the optimal plan trajectory. The feed concentration of B (u3) is a constant.
MD = [Info.MVopt(:,1) c_Bin*ones(N+1,1)];
First, run the tracking controller in nonlinear MPC mode.
[X1,Y1,MV1,et1] = fedbatch_Track(nlmpcobj_Tracking,x0,u0(2),N,Cref,MD);
fprintf('\nNonlinear MPC: Elapsed time = %g sec. Production of C = %g mol\n',et1,Y1(end,1));
Nonlinear MPC: Elapsed time = 4.31245 sec. Production of C = 2.01754 mol
Second, run the controller as an adaptive MPC controller.
nlmpcobj_Tracking.Optimization.RunAsLinearMPC = 'Adaptive';
[X2,Y2,MV2,et2] = fedbatch_Track(nlmpcobj_Tracking,x0,u0(2),N,Cref,MD);
fprintf('\nAdaptive MPC: Elapsed time = %g sec. Production of C = %g mol\n',et2,Y2(end,1));
Adaptive MPC: Elapsed time = 1.57934 sec. Production of C = 2.01567 mol
Third, run the controller as a time-varying MPC controller.
nlmpcobj_Tracking.Optimization.RunAsLinearMPC = 'TimeVarying';
[X3,Y3,MV3,et3] = fedbatch_Track(nlmpcobj_Tracking,x0,u0(2),N,Cref,MD);
fprintf('\nTime-varying MPC: Elapsed time = %g sec. Production of C = %g mol\n',et3,Y3(end,1));
Time-varying MPC: Elapsed time = 1.50841 sec. Production of C = 2.02346 mol
In the majority of MPC applications, linear MPC solutions, such as Adaptive MPC and Time-varying MPC, provide performance that is comparable to the nonlinear MPC solution, while consuming less
resources and executing faster. In these cases, nonlinear MPC often represents the best control results that MPC can achieve. By running a nonlinear MPC controller as a linear MPC controller, you can
assess whether implementing a linear MPC solution is good enough in practice.
In this example, all three methods come close to the optimal C production obtained in the planning stage.
plot(Ts*(0:N),[Y1(:,1) Y2(:,1) Y3(:,1)])
title('Production of C')
The unexpected result is that time-varying MPC produces more C than nonlinear MPC. The explanation is that the model linearization approaches used in the adaptive and time-varying modes result in a
violation of the heat removal constraint, which results in a higher C production.
plot(Ts*(0:N),[Y1(:,2) Y2(:,2) Y3(:,2) 1.5e5*ones(N+1,1)])
title('Heat removal rate')
The adaptive MPC mode uses the plant states and inputs at the beginning of each control interval to obtain a single linear prediction model. This approach does not account for the known future
changes in the feed rate, for example.
The time-varying method avoids this issue. However, at the start of the batch it assumes (by default) that the states will remain constant over the horizon. It corrects for this once it obtains its
first solution (using data in the opts variable), but its initial choice of reactor temperature is too high, resulting in an early q_r constraint violation.
[1] Srinivasan, B., S. Palanki, and D. Bonvin, "Dynamic optimization of batch processes I. Characterization of the nominal solution", Computers and Chemical Engineering, vol. 27 (2003), pp. 1-26.
See Also
Related Topics | {"url":"https://uk.mathworks.com/help/mpc/ug/optimization-and-control-of-fed-batch-reactor-using-nonlinear-mpc.html","timestamp":"2024-11-09T14:11:25Z","content_type":"text/html","content_length":"96155","record_id":"<urn:uuid:891d46b5-790f-41ca-a83e-fea438bf6a86>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00529.warc.gz"} |
Off-diagonal terms in Yukawa textures of the Type-III 2-Higgs doublet model and light charged Higgs boson phenomenology
Hernández-Sánchez, J., Moretti, S., Noriega-Papaqui, R. and Rosado, A. (2013) Off-diagonal terms in Yukawa textures of the Type-III 2-Higgs doublet model and light charged Higgs boson phenomenology.
Journal of High Energy Physics, 2013 (07), [44]. (doi:10.1007/JHEP07(2013)044).
We discuss flavor-violating constraints and consequently possible charged Higgs boson phenomenology emerging from a four-zero Yukawa texture embedded within the Type-III 2-Higgs Doublet Model
(2HDM-III). Firstly, we show in detail how we can obtain several kinds of 2HDMs when some parameters in the Yukawa texture are absent. Secondly, we present a comprehensive study of the main B-physics
constraints on such parameters induced by flavor-changing processes, in particular on the off-diagonal terms of such a texture: i.e., from $\mu -e$ universality in $\tau$ decays, several leptonic
B-decays ($B \to \tau \nu$, $D \to \mu \nu$ and $D_s \to {l} \nu$), the semi-leptonic transition $B\to D \tau \nu$, plus $B \to X_s \gamma$, including $B^0-\bar B^0$ mixing, $B_s \to \mu^+ \mu^-$ and
the radiative decay $Z\to b \bar{b}$. Thirdly, having selected the surviving 2HDM-III parameter space, we show that the $H^- c \bar{b}$ coupling can be very large over sizable expanses of it, in
fact, a very different situation with respect to 2HDMs with a flavor discrete symmetry (i.e., ${\mathcal{Z}}_2$) and very similar to the case of the Aligned-2HDM (A2HDM) as well as of models with
three or more Higgs doublets. Fourthly, we study in detail the ensuing $H^\pm$ phenomenology at the Large Hadron Collider (LHC), chiefly the $c\bar b \to H^+$ production mode and the $H^+\to c\bar b$
decay channel while assuming $\tau^+\nu_\tau$ decays in the former and $t\to bH^+$ production in the latter, showing that significant scope exists in both cases.
This record has no associated files available for download.
Published date: 8 July 2013
Additional Information: 50 pages, 22 figures; version accepted by JHEP. arXiv admin note: text overlap with arXiv:1203.5769; and text overlap with arXiv:0906.5139, arXiv:1006.0470, arXiv:hep-ph/
0309103 by other authors. text overlap with arXiv:1006.0470, arXiv:0906.5139 by other authors
Keywords: hep-ph
Organisations: Theory Group
Local EPrints ID: 356666
URI: http://eprints.soton.ac.uk/id/eprint/356666
PURE UUID: bcf19638-aeb5-4c54-b47d-ad987fc9d7a1
Catalogue record
Date deposited: 23 Sep 2013 13:16
Last modified: 15 Mar 2024 03:17
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Author: J. Hernández-Sánchez
Author: R. Noriega-Papaqui
Author: A. Rosado
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Printable Figure Drawings
Product Possibilities Curve Worksheet Answer Key Automobiles And Missiles
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Add terms to linear regression model
NewMdl = addTerms(mdl,terms) returns a linear regression model fitted using the input data and settings in mdl with the terms terms added.
Add Terms to Linear Regression Model
Create a linear regression model of the carsmall data set without any interactions, and then add an interaction term.
Load the carsmall data set and create a model of the MPG as a function of weight and model year.
load carsmall
tbl = table(MPG,Weight);
tbl.Year = categorical(Model_Year);
mdl = fitlm(tbl,'MPG ~ Year + Weight^2')
mdl =
Linear regression model:
MPG ~ 1 + Weight + Year + Weight^2
Estimated Coefficients:
Estimate SE tStat pValue
__________ __________ _______ __________
(Intercept) 54.206 4.7117 11.505 2.6648e-19
Weight -0.016404 0.0031249 -5.2493 1.0283e-06
Year_76 2.0887 0.71491 2.9215 0.0044137
Year_82 8.1864 0.81531 10.041 2.6364e-16
Weight^2 1.5573e-06 4.9454e-07 3.149 0.0022303
Number of observations: 94, Error degrees of freedom: 89
Root Mean Squared Error: 2.78
R-squared: 0.885, Adjusted R-Squared: 0.88
F-statistic vs. constant model: 172, p-value = 5.52e-41
The model includes five terms, Intercept, Weight, Year_76, Year_82, and Weight^2, where Year_76 and Year_82 are indicator variables for the categorical variable Year that has three distinct values.
Add an interaction term between the Year and Weight variables to mdl.
terms = 'Year*Weight';
NewMdl = addTerms(mdl,terms)
NewMdl =
Linear regression model:
MPG ~ 1 + Weight*Year + Weight^2
Estimated Coefficients:
Estimate SE tStat pValue
___________ __________ ________ __________
(Intercept) 48.045 6.779 7.0874 3.3967e-10
Weight -0.012624 0.0041455 -3.0454 0.0030751
Year_76 2.7768 3.0538 0.90931 0.3657
Year_82 16.416 4.9802 3.2962 0.0014196
Weight:Year_76 -0.00020693 0.00092403 -0.22394 0.82333
Weight:Year_82 -0.0032574 0.0018919 -1.7217 0.088673
Weight^2 1.0121e-06 6.12e-07 1.6538 0.10177
Number of observations: 94, Error degrees of freedom: 87
Root Mean Squared Error: 2.76
R-squared: 0.89, Adjusted R-Squared: 0.882
F-statistic vs. constant model: 117, p-value = 1.88e-39
NewMdl includes two additional terms, Weight*Year_76 and Weight*Year_82.
Input Arguments
terms — Terms to add to regression model
character vector or string scalar formula in Wilkinson notation | t-by-v terms matrix
Terms to add to the regression model mdl, specified as one of the following:
• Character vector or string scalar formula in Wilkinson Notation representing one or more terms. The variable names in the formula must be valid MATLAB^® identifiers.
• Terms matrix T of size t-by-v, where t is the number of terms and v is equal to mdl.NumVariables. The value of T(i,j) is the exponent of variable j in term i.
For example, suppose mdl has three variables A, B, and C in that order. Each row of T represents one term:
□ [0 0 0] — Constant term or intercept
□ [0 1 0] — B; equivalently, A^0 * B^1 * C^0
□ [1 0 1] — A*C
□ [2 0 0] — A^2
□ [0 1 2] — B*(C^2)
addTerms treats a group of indicator variables for a categorical predictor as a single variable. Therefore, you cannot specify an indicator variable to add to the model. If you specify a categorical
predictor to add to the model, addTerms adds a group of indicator variables for the predictor in one step. See Modify Linear Regression Model Using step for an example that describes how to create
indicator variables manually and treat each one as a separate variable.
Output Arguments
NewMdl — Linear regression model with additional terms
LinearModel object
Linear regression model with additional terms, returned as a LinearModel object. NewMdl is a newly fitted model that uses the input data and settings in mdl with additional terms specified in terms.
To overwrite the input argument mdl, assign the newly fitted model to mdl:
mdl = addTerms(mdl,terms);
• addTerms treats a categorical predictor as follows:
□ A model with a categorical predictor that has L levels (categories) includes L – 1 indicator variables. The model uses the first category as a reference level, so it does not include the
indicator variable for the reference level. If the data type of the categorical predictor is categorical, then you can check the order of categories by using categories and reorder the
categories by using reordercats to customize the reference level. For more details about creating indicator variables, see Automatic Creation of Dummy Variables.
□ addTerms treats the group of L – 1 indicator variables as a single variable. If you want to treat the indicator variables as distinct predictor variables, create indicator variables manually
by using dummyvar. Then use the indicator variables, except the one corresponding to the reference level of the categorical variable, when you fit a model. For the categorical predictor X, if
you specify all columns of dummyvar(X) and an intercept term as predictors, then the design matrix becomes rank deficient.
□ Interaction terms between a continuous predictor and a categorical predictor with L levels consist of the element-wise product of the L – 1 indicator variables with the continuous predictor.
□ Interaction terms between two categorical predictors with L and M levels consist of the (L – 1)*(M – 1) indicator variables to include all possible combinations of the two categorical
predictor levels.
□ You cannot specify higher-order terms for a categorical predictor because the square of an indicator is equal to itself.
Alternative Functionality
• Use stepwiselm to specify terms in a starting model and continue improving the model until no single step of adding or removing a term is beneficial.
• Use removeTerms to remove specific terms from a model.
• Use step to optimally improve a model by adding or removing terms.
Extended Capabilities
GPU Arrays
Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.
This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
Version History
Introduced in R2012a | {"url":"https://nl.mathworks.com/help/stats/linearmodel.addterms.html","timestamp":"2024-11-13T16:03:35Z","content_type":"text/html","content_length":"95896","record_id":"<urn:uuid:3e17f4cc-47df-4c6c-938f-dc503080d92b>","cc-path":"CC-MAIN-2024-46/segments/1730477028369.36/warc/CC-MAIN-20241113135544-20241113165544-00097.warc.gz"} |
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FHSST Physics/Units/How to Change - Wikibooks, open books for an open world
How to Change Units-- the "Multiply by 1" Technique
Also known as fractional dimensional analysis, the technique involves multiplying a labeled quantity by a conversion ratio, or knowledge of conversion factors. First, a relationship between the two
units that you wish to convert between must be found. Here's a simple example: converting millimetres (mm) to metres (m)-- the SI unit of length. We know that there are 1000 mm in 1 m which we can
write as
${\displaystyle {\begin{matrix}1000{\mbox{ mm}}=1{\mbox{ m}}\end{matrix}}}$
Now multiplying both sides by
${\displaystyle {\frac {1}{1000{\mbox{ mm}}}}}$
we get
${\displaystyle {\begin{matrix}{\frac {1}{1000mm}}1000mm={\frac {1}{1000mm}}1m,\end{matrix}}}$
which simply gives us
${\displaystyle {\begin{matrix}1={\frac {1m}{1000mm}}.\end{matrix}}}$
This is the conversion ratio from millimetres to metres. You can derive any conversion ratio in this way from a known relationship between two units. Let's use the conversion ratio we have just
derived in an example:
Question: Express 3800 mm in metres.
${\displaystyle {\begin{matrix}3800mm&=&3800mm\times 1\\&=&3800mm\times {\frac {1m}{1000mm}}\\&=&3.8m\\\end{matrix}}}$
Note that we wrote every unit in each step of the calculation. By writing them in and cancelling them properly, we can check that we have the right units when we are finished. We started with mm and
multiplied by ${\displaystyle {\frac {m}{mm}}}$
This cancelled the mm leaving us with just m, which is the SI unit we wanted! If we wished to do the reverse and convert metres to millimetres, then we would need a conversion ratio with millimetres
on the top and metres on the bottom.
It is helpful to understand that units cancel when one is in the numerator and the other is in the denominator. If the unit you are trying to cancel is on the top, then the conversion factor that you
multiply it with must be on the bottom.
This same technique can be used to not just to convert units, but can also be used as a way to solve for an unknown quantity. For example: If I was driving at 65 miles per hour, then I could find how
far I would go in 5 hours by using ${\displaystyle {\frac {65{\mbox{ miles}}}{1{\mbox{ hour}}}}}$ as a conversion factor.
This would look like ${\displaystyle {\frac {5{\mbox{ hours}}}{1}}{\frac {65{\mbox{ miles}}}{1{\mbox{ hour}}}}}$
This would yield a result of 325 miles because the hours would cancel leaving miles as the only unit.
Problem: Convert 3 millennia into seconds.
Most people don't know how many seconds are in a millennium, but they do know enough to solve this problem. Since we know 1000 years = 1 millennium, 1 year = about 365.2425 days, 1 day = 24 hours,
and 1 hour = 3600 seconds we can solve this problem by multiplying by one many times.
${\displaystyle {\frac {3{\mbox{ millennia}}}{1}}{\frac {1000{\mbox{ years}}}{1{\mbox{ millennium}}}}{\frac {365.2425{\mbox{ days}}}{1{\mbox{ year}}}}{\frac {24{\mbox{ hours}}}{1{\mbox{ day}}}}{\frac
{3600{\mbox{ seconds}}}{1{\mbox{ hour}}}}}$ | {"url":"https://en.m.wikibooks.org/wiki/FHSST_Physics/Units/How_to_Change","timestamp":"2024-11-07T13:17:17Z","content_type":"text/html","content_length":"47428","record_id":"<urn:uuid:88aca333-6643-4d88-ac67-e169edef8cda>","cc-path":"CC-MAIN-2024-46/segments/1730477027999.92/warc/CC-MAIN-20241107114930-20241107144930-00698.warc.gz"} |
Can someone help me solve this problem in C#? - Microsoft Q&A
Can someone help me solve this problem in C#?
There is a number 1 written on the board. We have a sequence a of n natural numbers, and in the i-th step (1 ≤ i ≤ n), we erase the current number on the board and replace it with the product of that
number and the number a[i]. After each step, determine whether the current number on the board is a perfect square.
The first line of standard input contains a natural number n (1 ≤ n ≤ 10,000) representing the length of the sequence a. The next line contains n natural numbers (between 1 and 1 billion) separated
by spaces, which are the elements of the sequence a in the order they are multiplied by the current number on the board.
For each element of the sequence a, in the order of input, print yes if its product with the current number on the board is a perfect square, otherwise print no. Time limit is 0,1 seconde
An object-oriented and type-safe programming language that has its roots in the C family of languages and includes support for component-oriented programming.
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Hi @Uros Zeradjanin, Welcome to Microsoft Q&A,
This forum does not help with homework assignments, etc. It is intended to help people solve coding errors. What specific problem did you encounter? What is your current reproducible code? What
is wrong with the above code? Please provide more explanation.
I dont know to finish. I need a slove code
Have you analyzed this code and found out what is wrong? Or explain your problem?
Time limit is problem. I need a new code
How do you want to limit the time given the input? Is it computation time or runtime?
1 answer
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1. 2024-10-05T21:02:30.02+00:00
To solve this problem, you need to keep track of the number on the board as it is updated by multiplying it with each element from the sequence a, and after each step, we must check if the
resulting number is a perfect square.
1. Read the input.
2. Start with the number 1 on the board.
3. For each element in the sequence, multiply it with the current number on the board and check if the result is a perfect square.
4. Output "yes" or "no" based on whether the current product is a perfect square.
using System;
class PerfectSquareChecker
// Helper function to check if a number is a perfect square
static bool IsPerfectSquare(long x)
if (x < 0) return false;
long sqrtX = (long)Math.Sqrt(x);
return sqrtX * sqrtX == x;
static void Main()
// Read the input
int n = int.Parse(Console.ReadLine()); // Number of elements in the sequence
string[] input = Console.ReadLine().Split(); // Sequence of numbers as strings
long[] a = new long[n];
for (int i = 0; i < n; i++)
a[i] = long.Parse(input[i]); // Parse each element into a long array
// Start with 1 on the board
long currentNumber = 1;
// Process each number in the sequence
for (int i = 0; i < n; i++)
currentNumber *= a[i]; // Multiply current number with a[i]
// Check if the current number is a perfect square
if (IsPerfectSquare(currentNumber))
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Conversion Factors and Problem Solving Lab 2: Report Sheet Answers & Explanations
by Lola Sofia
This guide provides a comprehensive understanding of conversion factors, their application in problem-solving, and specific guidance for your Lab 2 report. We’ll cover everything from basic
definitions to real-world applications, ensuring you’re well-equipped to tackle any unit conversion challenge.
Decoding Conversion Factors
What are Conversion Factors?
Imagine needing 2 cups of flour for a recipe, but your measuring cup only shows milliliters. A conversion factor is your solution. It’s a ratio expressing the equivalent values of two different
units. For example, 1 cup equals approximately 237 milliliters. This relationship, written as 237 mL/1 cup or 1 cup/237 mL, is your conversion factor. It lets you switch between units without
changing the actual amount.
Why are Conversion Factors Important in Lab 2?
In chemistry, precise measurements are crucial. Conversion factors become indispensable when dealing with different units, enabling accurate calculations and meaningful data analysis. They are
particularly useful in stoichiometry (calculating reactant and product quantities) and density calculations (mass/volume relationships). In Lab 2, you’ll likely encounter various scenarios requiring
unit conversions, making mastery of conversion factors essential for accurate results and insightful conclusions.
Mastering Dimensional Analysis
Unit Conversions with Dimensional Analysis
Dimensional analysis is a systematic approach to problem-solving using conversion factors. It’s like planning a route on a map, ensuring you reach your desired unit destination. You multiply your
given value by a series of conversion factors, strategically arranging them so unwanted units cancel out, leaving you with the units you need.
For instance, let’s convert 2500 milliliters (mL) to liters (L):
2500 mL * (1 L / 1000 mL) = 2.5 L
The “mL” units cancel, leaving the answer in liters.
Using Dimensional Analysis in Lab 2
Lab 2 problems will probably involve multiple conversions. Dimensional analysis provides a structured framework for these complex scenarios, guiding you through the steps and minimizing errors. It’s
particularly useful for converting between different systems (e.g., metric to imperial) or calculating derived units like density.
Navigating the Metric System
Understanding Metric Prefixes
The metric system simplifies measurements by using prefixes to denote multiples or fractions of base units (meter, gram, liter). These prefixes represent powers of ten, streamlining conversions.
Prefix Symbol Multiplier
kilo- k 1000 (10³)
hecto- h 100 (10²)
deka- da 10 (10¹)
base unit 1
deci- d 0.1 (10⁻¹)
centi- c 0.01 (10⁻²)
milli- m 0.001 (10⁻³)
micro- µ 0.000001 (10⁻⁶)
Metric Conversions in Lab 2
Lab 2 often involves converting within the metric system. For example, converting milligrams to grams or liters to milliliters. Understanding these prefixes and their corresponding multipliers is key
to accurate and efficient conversions.
Conquering Density Calculations
Density Formula and Units
Density (ρ) measures how much mass (m) is packed into a given volume (V), expressed by the formula: ρ = m/V. Common units include g/mL, g/cm³, kg/m³.
Applying Conversion Factors in Density Problems
Calculating density may require converting mass and volume to consistent units. Conversion factors, along with dimensional analysis, help you align units before applying the density formula. Suppose
you’re given a mass in kilograms and a volume in milliliters; you may need to convert the mass to grams and the volume to liters for consistency.
Check out our density aluminum lb in3 converter for quick density unit conversions.
Solving Lab 2 Report Sheet Questions: Step-by-Step
1. Identify Units: Determine your starting and desired units.
2. Find Conversion Factors: Identify the necessary conversion factors linking the units.
3. Dimensional Analysis Setup: Arrange the problem using dimensional analysis, ensuring unwanted units cancel.
4. Calculate and Round: Perform the calculation and round to the correct number of significant figures.
Example: Convert 15 inches to centimeters (1 inch = 2.54 cm)
15 in * (2.54 cm / 1 in) = 38.1 cm
Real-World Applications of Conversion Factors
Conversion factors extend beyond the lab. They’re used in:
• Cooking: Adjusting recipes, converting between measurement systems.
• Finance: Currency exchange, calculating interest rates.
• Engineering: Designing structures, ensuring precise measurements.
• Healthcare: Determining medication dosages, analyzing patient data.
• Everyday Life: Converting between metric and imperial units (e.g., miles to kilometers).
Beyond the Basics: Uncertainty and Ongoing Research
While we strive for accuracy, some conversion factors are approximations, introducing uncertainty. Ongoing metrology research refines our understanding of units and their relationships, potentially
leading to future revisions. Also, some fields have specific conversion conventions. Researching and using appropriate context-specific conversions demonstrates a thorough understanding.
This guide provides a robust foundation in conversion factors and problem-solving for Lab 2 and beyond. Remember, practice solidifies understanding. Tackle those practice problems, explore further,
and you’ll be converting units with confidence in no time. This knowledge also empowers you to critically evaluate data and calculations, a crucial skill in any scientific endeavor.
Latest posts by Lola Sofia
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What is currying function in JavaScript ? - TechieBundle
Currying function in JavaScript
Currying is a process in functional programming in which we can transform a function with multiple arguments into a sequence of nesting functions. It returns a new function that expects the next
argument inline.
In other words, when a function, instead of taking all arguments at one time, takes the first one and return a new function that takes the second one and returns a new function which takes the third
one, and so forth, until all arguments have been fulfilled.
That is, when we turn a function call sum(1,2,3) into sum(1)(2)(3)
The number of arguments a function takes is also called arity.
function sum(a, b) {
// do something
function _sum(a, b, c) {
// do something
function sum takes two arguments (2-arity function) and _sum takes three arguments (3-arity function).
Curried functions are constructed by chaining closures by defining and immediately returning their inner functions simultaneously.
Why it’s useful ?
• Currying helps we avoid passing the same variable again and again.
• It helps to create a higher order function
Currying transforms a function with multiple arguments into a sequence/series of functions each taking a single argument.
function sum(a, b, c) {
return a + b + c;
sum(1,2,3); // 6
As we see, function with the full arguments. Let’s create a curried version of the function and see how we would call the same function (and get the same result) in a series of calls:
function sum(a) {
return (b) => {
return (c) => {
return a + b + c
console.log(sum(1)(2)(3)) // 6
We could separate this sum(1)(2)(3) to understand it better:
const sum1 = sum(1);
const sum2 = sum1(2);
const result = sum2(3);
console.log(result); // 6
Let’s get to know how it works:
We passed 1 to the sum function:
let sum1 = sum(1);
It returns the function:
return (b) => {
return (c) => {
return a + b + c
Now, sum1 holds the above function definition which takes an argument b.
We called the sum1 function, passing in 2:
let sum2 = sum1(2);
The sum1 will return the third function:
return (c) => {
return a + b + c
The returned function is now stored in sum2 variable.
sum2 will be:
sum2 = (c) => {
return a + b + c
When sum2 is called with 3 as the parameter,
const result = sum2(3);
it does the calculation with the previously passed in parameters:
a = 1, b = 2 and returns 6.
console.log(result); // 6
The last function only accepts c variable but will perform the operation with other variables whose enclosing function scope has long since returned. It works nonetheless because of Closure 🔥
Currying & Partial application 🤔
Some might start to think that the number of nested functions a curried function has depends on the number of arguments it receives. Yes, that makes it a curry.
Let’s take same sum example:
function sum(a) {
return (b, c) => {
return a * b * c
It can be called like this:
let x = sum(10);
// OR
Above function expects 3 arguments and has 2 nested functions, unlike our previous version that expects 3 arguments and has 3nesting functions.
This version isn’t a curry. We just did a partial application of the sum function.
Currying and Partial Application are related (because of closure), but they are of different concepts.
Partial application transforms a function into another function with smaller arity.
function sum1(x, y, z) {
return sum2(x,y,z)
// to
function sum1(x) {
return (y,z) => {
return sum2(x,y,z)
For Currying, it would be like this:
function sum1(x) {
return (y) = > {
return (z) = > {
return sum2(x,y,z)
Currying creates nesting functions according to the number of the arguments of the function. Each function receives an argument. If there is no argument there is no currying.
To develop a function that takes a function and returns a curried function:
function currying(fn, ...args) {
return (..._arg) => {
return fn(...args, ..._arg);
The above function accepts a function (fn) that we want to curry and a variable number of parameters (…args). The rest operator is used to gather the number of parameters after fn into …args.
Next, we return a function that also collects the rest of the parameters as …_args. This function invokes the original function fn passing in …args and …_args through the use of the spread operator
as parameters, then, the value is returned to the user.
Now, we can use the above function to create curry function.
function sum(a,b,c) {
return a + b + c
let add = currying(sum,10);
// 120
// 140
Closure makes currying possible in JavaScript. I hope you have learned something new about currying!
Also Read: What is the JavaScript Execution Context ? | {"url":"https://techiebundle.com/what-is-currying-function-in-javascript/","timestamp":"2024-11-03T19:53:10Z","content_type":"text/html","content_length":"269858","record_id":"<urn:uuid:b5daaa62-374c-4b2c-851f-d423aafd2ff3>","cc-path":"CC-MAIN-2024-46/segments/1730477027782.40/warc/CC-MAIN-20241103181023-20241103211023-00223.warc.gz"} |
NCERT Solutions for Class 9 Maths Chapter 10 - Heron's Formula PDF
NCERT Solutions for Maths Chapter 10 Class 9 Heron’s Formula - FREE PDF Download
NCERT Solutions for heron's formula class 9 maths ch 10 Heron’s Formula curated by our subject experts to facilitate a practical and smooth understanding of the concepts related to Heron's Formula.
These NCERT Solutions can be accessed anytime and anywhere, at your convenience, to understand the concepts in a better way. These solutions to each exercise question in the PDF are explained using a
clear step-by-step method. It acts as an essential tool for you to prepare the chapter quickly and efficiently during exams. You can download and practice these NCERT Solutions for herons formula
Class 9 Maths Chapter 10 to thoroughly understand the concepts covered in the chapter.
1. NCERT Solutions for Maths Chapter 10 Class 9 Heron’s Formula - FREE PDF Download
2. Glance on Maths Chapter 10 Class 9 - Heron's Formula
3. Access Exercise Wise NCERT Solutions for Chapter 10 Maths Class 9
4. Exercises Under NCERT Solutions for Class 9 Maths Chapter 10 Heron's Formula
5. Access NCERT Solutions Maths Chapter 10 - Heron’s formula
6. Overview of Deleted Syllabus for CBSE Class 9 Maths Heron's Formula
7. Other Study Material for CBSE Class 9 Maths Chapter 10
8. Chapter-Specific NCERT Solutions for Class 9 Maths
Glance on Maths Chapter 10 Class 9 - Heron's Formula
• This article deals with Heron's Formula, which is a method to calculate the area of a triangle when the lengths of all three sides are known.
• The questions cover topics such as finding the area of a triangle when the sides are given, finding the missing side when the area and two sides are given, and finding the height of a triangle
when the area and the base are given.
• Chapter 10 Maths Class 9 recalls how to calculate the perimeter of various figures and shapes.
• This article contains chapter notes, exercises, explanation videos, links, and important questions for Chapter 10 - Heron's Formula where you can download a FREE PDF.
• There is one exercise (6 fully solved questions) in class 9th maths chapter 10 Heron's Formula.
Access Exercise Wise NCERT Solutions for Chapter 10 Maths Class 9
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FAQs on NCERT Solutions for Class 9 Maths Chapter 10 Heron's Formula
1. How to find altitude in Heron's Formula class 9?
In class 9 maths herons formula , Heron's formula itself doesn't directly calculate the altitude of a triangle. However, it can be used along with the concept of area to find the altitude in a
scalene triangle (one with all sides unequal). Here's the process:
Heron's formula gives the area (A) of a triangle with sides a, b, and c as:
A = $\sqrt{s(s-a)(s-b)(s-c)}$
where s is the semi-perimeter (s = (a + b + c) / 2)
The area of a triangle can also be calculated as 1/2 * base * height (where base is the side along which the altitude is drawn, and the height is the altitude itself).
By equating these two expressions for area and solving for the height (h) with base (b) known, you can get the formula for altitude using Heron's formula:
h = 2 * $\sqrt{s(s-a)(s-b)(s-c)}$ / b
2. Is Heron's formula applicable for all triangles in chapter 10 class 9 maths solutions pdf?
Yes, In class 9 ch 10 , Heron's formula is applicable to all triangles, irrespective of their type (scalene, isosceles, or equilateral). As long as you know the lengths of all three sides, you can
use the formula to find the area of the triangle.
3. What is the real-life application of Heron's formula?
Suppose you have to calculate the area of a triangular land. What is the probability that the area is of a regular shape? It is impossible that you will come across lands with regular spaces and
sizes, and this is where Heron’s formula comes into use. To calculate the area of real-life objects, the best way to find the exact area of the land is to use Heron’s formula.
4. What is Heron's formula?
Triangle is a three-dimensional closed shape. Heron’s formula calculates the area of a triangle when the length of all three sides is given. Using Heron's formula, we can calculate the area of any
triangle, be it a scalene, isosceles or equilateral triangle. For example, the sides of a triangle are given as a, b, and c. Using Heron’s formula, the area of the triangle can be calculated by Area
= √S (S-a)(S-b)(S-c) where s is the semi-perimeter of the triangle.
5. How do you solve Heron's formula questions?
To solve questions based on Heron’s formula, you need to remember Heron’s formula, Area= Area= √S (S-a)(S-b)(S-c). Here, ‘s’ is the semi-perimeter of the triangle, and a, b, and c are the lengths of
the sides of the triangle. The semi perimeter is denoted by S. It can be calculated by using the formula: S = a+b+c/2. By substituting the values given in these formulas, you can calculate the area
of a triangle.
6. What is the meaning of s in Heron's formula?
In Heron’s formula, Area = √S (S-a)(S-b)(S-c), where ‘s’ stands for the semi-perimeter of the triangle whose area we need to calculate. Semi-perimeter can be calculated by the given formula: S =
a+b+c/2. To learn more about the semi-perimeter and its usage in calculating the area of a triangle, you can download the Vedantu app or check out the official website of Vedantu.
7. What is a Semi-Perimeter?
In Geometry, the Semi-perimeter of any polygon is half of its perimeter. In Class 9, Chapter 12, Heron’s Formula explains the semi-perimeter of a triangle. Semi-perimeter is denoted by ‘s’ in Heron’s
formula, which is Area= √s (s-a)(s-b)(s-c) ‘s’ stands for semi-perimeter, which can be calculated by the given formula: s = a+b+c/2, where a, b, and c are the sides of the triangle of which the area
has to be calculated.
8. What is a Semi-Perimeter?
In Geometry, the Semi-perimeter of any polygon is half of its perimeter. In Class 9, Chapter 12, Heron’s Formula explains the semi-perimeter of a triangle. Semi-perimeter is denoted by ‘s’ in Heron’s
formula, which is Area= √s (s-a)(s-b)(s-c)
‘s’ stands for semi-perimeter, which can be calculated by the given formula:
s = a+b+c/2, where a, b and c are the sides of the triangle of which the area has to be calculated. | {"url":"http://eukaryote.org/ncert-solutions-class-9-maths-chapter-12-herons-formula.html","timestamp":"2024-11-08T15:01:21Z","content_type":"text/html","content_length":"770060","record_id":"<urn:uuid:1f358ebe-0198-42db-a67d-75e1bc91ffe0>","cc-path":"CC-MAIN-2024-46/segments/1730477028067.32/warc/CC-MAIN-20241108133114-20241108163114-00853.warc.gz"} |
confidence interval
Confidence intervals give a lower and upper bound for the true value of an effect. They allow you to be able to answer questions such as "What is the maximum likely value of this?", or "How close to
no difference is there between these two conditions?". They use the same information and mathematics as are used to generate the p-values in a significance test. You can choose to set higher or lower
confidence levels, for example the 95% confidence interval for a value might be [0.3,1.1] and the 99% confidence interval [-0.1,1.5]. In general the higher the level of confidence you ask for, the
wider the range. Like all statistics these bounds are uncertain. However, if you do lots of studies and calculate 95% confidence intervals for them all, then over time the proportion of true values
that lie within the 95% confidence intervals will be at least 95% ... although you can't tell which they are!
Defined on page 65
Used on pages 12, 15, 57, 60, 61, 65, 66, 67, 68, 77, 82, 85, 86, 93, 94, 100, 102 | {"url":"https://alandix.com/glossary/hcistats/confidence%20interval","timestamp":"2024-11-07T19:12:10Z","content_type":"application/xhtml+xml","content_length":"23439","record_id":"<urn:uuid:109d66df-a49b-4337-905a-2f13c04ff587>","cc-path":"CC-MAIN-2024-46/segments/1730477028009.81/warc/CC-MAIN-20241107181317-20241107211317-00304.warc.gz"} |
First, show students diagrams of cylindrical and spherical coordinates. Discuss common notation systems, especially that mathematicians and physicists use opposite notations for the angles \(\theta\)
and \(\phi\). Don't forget to discuss the ranges of each of the coordinates.
It can be very helpful to have a set of coordinate axes, perhaps suspended from the ceiling somewhere in the room, to refer to as needed. | {"url":"https://paradigms.oregonstate.edu/act/2297","timestamp":"2024-11-13T17:50:02Z","content_type":"text/html","content_length":"39143","record_id":"<urn:uuid:4c1e9660-647b-4e04-a184-d49d128e5835>","cc-path":"CC-MAIN-2024-46/segments/1730477028387.69/warc/CC-MAIN-20241113171551-20241113201551-00047.warc.gz"} |
Worksheets and Study Guides Sixth Grade.
Number and Operations (NCTM)
Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Understand and use ratios and proportions to represent quantitative relationships.
Compute fluently and make reasonable estimates.
Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
Grade 6 Curriculum Focal Points (NCTM)
Number and Operations: Connecting ratio and rate to multiplication and division
Students use simple reasoning about multiplication and division to solve ratio and rate problems (e.g., 'If 5 items cost $3.75 and all items are the same price, then I can find the cost of 12 items
by first dividing $3.75 by 5 to find out how much one item costs and then multiplying the cost of a single item by 12'). By viewing equivalent ratios and rates as deriving from, and extending, pairs
of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative sizes of quantities, students extend whole number multiplication and division to ratios
and rates. Thus, they expand the repertoire of problems that they can solve by using multiplication and division, and they build on their understanding of fractions to understand ratios. Students
solve a wide variety of problems involving ratios and rates. | {"url":"https://newpathworksheets.com/math/grade-6/simple-proportions","timestamp":"2024-11-04T11:53:43Z","content_type":"text/html","content_length":"42434","record_id":"<urn:uuid:d9727b9b-a7bc-48f6-baa7-d5d04a4b52d2>","cc-path":"CC-MAIN-2024-46/segments/1730477027821.39/warc/CC-MAIN-20241104100555-20241104130555-00186.warc.gz"} |
Flight of a Projectile on MATHguide
Given: A projectile is launched upward at a velocity of 400 ft/sec on Earth. The projectile starts at an initial vertical distance of 190 feet.
Find: a) Write the function that describes the height of the projectile with respect to time.
h(t) = t^2 + t +
b) At what time does the projectile reach the maximum height?
[Round your answer to the nearest tenth.]
c) Determine the maximum height of the projectile.
[Round the answer to the nearest foot.]
d) At what time does the projectile reach ground level?
[Round your answer to the nearest tenth.] | {"url":"https://www.mathguide.com/cgi-bin/quizmasters2/FP.cgi","timestamp":"2024-11-09T13:48:08Z","content_type":"application/xhtml+xml","content_length":"3353","record_id":"<urn:uuid:f4599c0e-1c67-435a-a2bb-eb6426ef5f91>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00651.warc.gz"} |
How Mortality Tables Work
David J. Kupstas, FSA, EA, MSEA Chief Actuary
When people hear that I am an actuary, the first question they ask (besides, “What is THAT?”) is, “How long am I going to live?” The answer is, I have no idea. Contrary to popular belief, life
expectancy rarely is used in actuarial calculations.
Still, mortality tables are a very important part of actuarial work. This article explains a little bit about what mortality tables are and how we use them.
Mortality Table Basics
In its most basic form, a mortality table consists of two columns of numbers. The first column is a listing of all ages from 0 to 120. The second column contains a q value for that age, which is the
probability of a person that age dying in that year.
The death probabilities at most ages are very low. According to the 2017 IRS applicable mortality table, the probability of someone dying while age 30 is less than three-hundredths of a percent; out
of 10,000 30-year-olds, three are expected die in the next year. On the other hand, the probability of someone dying at age 90 is more than 14 percent (14 out of 100 will die). If you are lucky
enough to reach age 120, the table assumes you have a 100% chance of dying in the next year. Sorry!
People with different characteristics are expected to die at different rates. Separate mortality tables can be constructed for males and females, smokers and non-smokers, people in one state or
country vs. another, different industries, income levels, and more. Data may be obtained from sources such as Social Security, insurance companies, actuarial consulting firms, and public pension
plans. Most pension actuaries do not construct mortality tables themselves, but rather use standard tables that have been prepared by others.
Present Value with a Twist
Present value is what an amount of money in the future is worth in today’s dollars. Assuming 4.00% interest, the present value of $100.00 payable in five years is $100.00 ÷ 1.04^5 = $82.19. Actuarial
present value is present value with a twist. With an actuarial present value, not only are you discounting future payments with interest, you are also reflecting the uncertainty that a given payment
will even be made. If there is only a 50% chance that the $100.00 mentioned above will be paid in five years, the actuarial present value of that payment is 50% x $100.00 ÷ 1.04^5 = $41.10.
Let’s do an actuarial present value example using what we’ve learned about mortality tables. Because mortality means death, we are going to refrain from using an example in which a person might die.
After all, young children may be reading this post. Instead, our example will use a tree – specifically, a money tree.
Suppose someone offers to sell you a money tree which will produce $1,000 at the end of each year it is alive. The tree can live for as long as 10 years. It might live 10 days, it might live four
years, or it might live close to the full 10 years. How much should you pay for the money tree? To determine a fair price, you might want to find the actuarial present value of the money the tree
could produce.
If you buy the money tree, assume you could have invested what you paid for the tree in an account earning 4.00% per year. And let’s say you’ve read up on money trees and know the chances of one
surviving each year from now until year 10. You do an analysis like the following:
Year (t) q[t] p[t] l[t] v^t Harvest Pres. Val. Harvest Act. PV Harvest
1 10% 90% 90.0000% 96.1538% $1,000.00 $961.54 $865.38
2 15% 85% 76.5000% 92.4556% $1,000.00 $924.56 $707.29
3 20% 80% 61.2000% 88.8996% $1,000.00 $889.00 $544.07
4 25% 75% 45.9000% 85.4804% $1,000.00 $854.80 $392.36
5 30% 70% 32.1300% 82.1927% $1,000.00 $821.93 $264.09
6 40% 60% 19.2780% 79.0315% $1,000.00 $790.31 $152.36
7 40% 60% 11.5668% 75.9918% $1,000.00 $759.92 $87.90
8 50% 50% 5.7834% 73.0690% $1,000.00 $730.69 $42.26
9 50% 50% 2.8917% 70.2587% $1,000.00 $702.59 $20.32
10 100% 0% 0.0000% 67.5564% $1,000.00 $675.56 $0.00
Total $3,076.01
That’s a bunch of numbers and symbols. Let’s do a little explaining. Or you can just skip past the bullet points and rejoin us in the last sentence of this section.
• q[t] is the probability that the tree will die in year t. There is a 10% chance the tree will die before it is a year old. There is a 30% chance that a tree surviving four years will die in its
fifth year. There is a 100% chance a tree that makes it to nine years old will die in its 10^th year.
• p[t] is the probability the tree won’t die in year t. It equals 100% minus q[t].
• l[t] is the probability the tree is alive after year t. This is the product of all the p[t]’s prior to and including year t.
• v^t is the fraction of a dollar someone would need to invest today to have a dollar after year t if money grows with interest at 4.00% per year. It takes around 82 cents invested at 4.00% to have
a dollar after year 5. Stated another way, a dollar at year 5 is worth around 82 cents today.
• The Harvest column shows the $1,000 the money tree would pay at the end of each year.
• Pres. Val. Harvest is what each future $1,000 harvest is worth in today’s dollars assuming 4.00% interest. For each year t, it is Harvest × v^t.
• Act. PV Harvest for each year t is $1,000 × v^t × l[t]. Here, not only are the future $1,000 harvests discounted at 4.00% interest, they are discounted further to reflect the probability that the
tree won’t be alive to produce the harvest in a given year.
The sum of the actuarial present values of each year’s $1,000 that could be yielded by the tree is $3,076.01. Upon completing this analysis, you might decide that $3,076 is a fair price to pay for
the money tree.
Examples That Are More Real-Life
Of course, this is not a typical problem an actuary would work on. Haven’t you been told that money doesn’t grow on trees? But there are plenty of real-life examples of when an actuary would use a
mortality table to determine an actuarial present value. Life insurance is a classic example. You know that an insured person is going to die sometime, just not when. One has to figure out the amount
of death benefit payable in each future year, the present value of each year’s death benefit, and the probability using a mortality table that the insured person will die in each of those years. From
there, the premium can be determined.
An example near and dear to our hearts is lump sum payments from a pension plan. When a plan participant retires, he usually may choose from several different monthly pension types, including single
life annuity and joint and survivor annuity. Many plans also offer lump sum settlements. In lieu of the monthly pension, a participant may choose an immediate single large payment and have no further
claims to any benefits from the plan. The lump sum is an actuarial present value of the monthly pension based on interest and mortality assumptions that are prescribed by the government. Like the
person thinking about buying the money tree, the pension plan retiree has to decide which is the better deal between the lump sum payment now or a monthly pension payable for however long he and/or
his beneficiary live.
The IRS has written proposed regulations which, if enacted, would make significant changes starting in 2018 to the mortality tables used in lump sum payouts. We will discuss these proposed
regulations in a future blog post. | {"url":"https://blog.acgworldwide.com/how-mortality-tables-work","timestamp":"2024-11-13T20:51:52Z","content_type":"text/html","content_length":"66162","record_id":"<urn:uuid:26599a27-786f-46bd-81a3-efaf0bf852dd>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00366.warc.gz"} |
Everyday maths 2
6 Fractions, decimals and percentages
You have already worked with decimals in this course and many times throughout your life. Every time you calculate something to do with money, you are using decimal numbers. You have also learned how
to round a number to a given number of decimal places.
Since fractions, decimals and percentages are all just different ways of representing the same thing, we can convert between them in order to compare. Take a look at the video below to see how to
convert fractions, decimals and percentages.
Download this video clip.Video player: bltl_1_6_fractions_decimals_percentages.mp4
Equivalent fractions, decimals, and percentages are all just different ways of representing the same thing, so you can convert between them to compare. First, let's look at turning percentages into
decimals. Take the example of 60%. Remember that a percentage is out of 100. To turn a percentage into its equivalent decimal, you need to divide by 100. 60 divided by 100 = 0.6. So 0.6 is the
equivalent decimal of 60%.
What about 25%? Can you work out the equivalent decimal of 25% before the answer is revealed? Remember that 'cent' means 100.
To turn 25% into its equivalent decimal, you need to divide 25 by 100, which = 0.25.
Now let's find the equivalent fraction for the decimals we've calculated. To do this, you need to first turn it into a percentage and write it as a fraction out of 100. Using our previous examples,
0.6 or 60%, can be written as 60 over 100. 0.25, or 25%, can be written as 25 out of 100.
However, these fractions are not written in their simplest form. Remember that to simplify fractions, you need to divide both parts by the same number and keep going until you can't find a number
that you can divide both parts by. In their simplest form, 0.6 is written as the fraction 3/5. 0.25 is written as the fraction one quarter, or 1 over 4.
Now looking at the table, you can see how percentages, decimals and fractions relate to each other.
Let's try one more example. Can you work out the equivalent decimal and fraction of 72%? 72 divided by 100 gives the decimal 0.72. To find the equivalent fraction, you need to change it into a
percentage and then write it as a fraction out of 100. Then simplify the fraction into its simplest form, 18 out of 25, 18/25.
End transcript
Interactive feature not available in single page view (
see it in standard view
Activity 12: Matching fractions, decimals and percentages
Choose the correct fraction for each percentage and decimal.
Using the following two lists, match each numbered item with the correct letter.
• a.62.5% = 0.625 =
• b.40% = 0.4 =
• c.50% = 0.5 =
• d.35% = 0.35 =
Fractions and percentages deal with splitting numbers into a given number of equal portions, or parts. When dealing with the next topic, ratio, you will still be splitting quantities into a given
number of parts, but when sharing in a ratio, you do not share evenly. This might sound a little complicated but you’ll have been doing it since you were a child.
In this section you have:
• learned about the relationship between fractions, decimals and percentages and are now able to convert between the three. | {"url":"https://www.open.edu/openlearn/mod/oucontent/view.php?id=85562§ion=6","timestamp":"2024-11-06T16:53:05Z","content_type":"text/html","content_length":"150786","record_id":"<urn:uuid:aa02955b-7930-4705-9cf8-5265a692ee32>","cc-path":"CC-MAIN-2024-46/segments/1730477027933.5/warc/CC-MAIN-20241106163535-20241106193535-00026.warc.gz"} |
Correct spelling for kr | Spellchecker.net
KR Meaning and Definition
1. The term "kr" is commonly used to represent kilorotations or kiloradians, a unit of measurement utilized in rotational dynamics and angular displacement. Derived from the metric prefix "kilo"
meaning a thousand, and the rotational unit "radians," "kr" signifies a multiplication factor of 1000 applied to rotational quantities.
In the context of angular displacement, "kr" indicates a rotation of 1000 radians, thus expressing a much larger magnitude of rotation than a single radian. Radians serve as a fundamental unit
for measuring angles in the field of mathematics and physics, facilitating calculations involving circles, circular motions, and trigonometry. The kiloradian, denoted as "kr," extends this unit
by increasing its value considerably.
Similarly, in rotational dynamics, "kr" signifies a rotational speed or angular velocity of 1000 radians per second. This unit enables the measurement of high-speed rotations, such as those
encountered in advanced machinery, engines, or various technological equipment.
The utilization of "kr" allows for concise representations of quantities involving rotational dynamics or angular displacement on a larger scale. By incorporating the kilo prefix, it simplifies
calculations and provides a standard unit of measurement that applies to a range of rotational phenomena.
Chemical symbol of krypton.
A practical medical dictionary. By Stedman, Thomas Lathrop. Published 1920.
Top Common Misspellings for KR *
* The statistics data for these misspellings percentages are collected from over 15,411,110 spell check sessions on www.spellchecker.net from Jan 2010 - Jun 2012.
Other Common Misspellings for KR
Similar spelling words for KR
• km,
• kg,
• kr,
• br,
• krey,
• cr,
• sr.,
• YR.,
• K,
• R,
• RR,
• k's,
• pr,
• kroh,
• Mr.,
• Dr.,
• ky.,
• ks,
• -r,
• ky. | {"url":"https://www.spellchecker.net/kr","timestamp":"2024-11-06T12:16:28Z","content_type":"text/html","content_length":"55545","record_id":"<urn:uuid:163adcc4-e108-4910-af69-b9af7831496d>","cc-path":"CC-MAIN-2024-46/segments/1730477027928.77/warc/CC-MAIN-20241106100950-20241106130950-00775.warc.gz"} |
224 research outputs found
We compute the partition function of the trigonometric SOS model with one reflecting end and domain wall type boundary conditions. We show that in this case, instead of a sum of determinants obtained
by Rosengren for the SOS model on a square lattice without reflection, the partition function can be represented as a single Izergin determinant. This result is crucial for the study of the Bethe
vectors of the spin chains with non-diagonal boundary terms.Comment: 13 pages, improved versio
Using algebraic Bethe ansatz and the solution of the quantum inverse scattering problem, we compute compact representations of the spin-spin correlation functions of the XXZ-1/2 Heisenberg chain in a
magnetic field. At lattice distance m, they are typically given as the sum of m terms. Each term n of this sum, n = 1,...,m is represented in the thermodynamic limit as a multiple integral of order
2n+1; the integrand depends on the distance as the power m of some simple function. The root of these results is the derivation of a compact formula for the multiple action on a general quantum state
of the chain of transfer matrix operators for arbitrary values of their spectral parameters.Comment: 34 page
Using its multiple integral representation, we compute the large distance asymptotic behavior of the emptiness formation probability of the XXZ spin-1/2 Heisenberg chain in the massless
regime.Comment: LPENSL-TH-10, 8 page
We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection
algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general
integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet
interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variable (SOV) representation hence proving that it defines a complete
characterization of the transfer matrix spectrum. The polynomial character of the Q-function allows us then to show that a finite system of equations of generalized Bethe type can be similarly used
to describe the complete transfer matrix spectrum.Comment: 28 page
Using multiple integral representations, we derive exact expressions for the correlation functions of the spin-1/2 Heisenberg chain at the free fermion point.Comment: 24 pages, LaTe
We derive a master equation for the dynamical spin-spin correlation functions of the XXZ spin-1/2 Heisenberg finite chain in an external magnetic field. In the thermodynamic limit, we obtain their
multiple integral representation.Comment: 25 page
We consider the XXX open spin-1/2 chain with the most general non-diagonal boundary terms, that we solve by means of the quantum separation of variables (SoV) approach. We compute the scalar products
of separate states, a class of states which notably contains all the eigenstates of the model. As usual for models solved by SoV, these scalar products can be expressed as some determinants with a
non-trivial dependance in terms of the inhomogeneity parameters that have to be introduced for the method to be applicable. We show that these determinants can be transformed into alternative ones in
which the homogeneous limit can easily be taken. These new representations can be considered as generalizations of the well-known determinant representation for the scalar products of the Bethe
states of the periodic chain. In the particular case where a constraint is applied on the boundary parameters, such that the transfer matrix spectrum and eigenstates can be characterized in terms of
polynomial solutions of a usual T-Q equation, the scalar product that we compute here corresponds to the scalar product between two off-shell Bethe-type states. If in addition one of the states is an
eigenstate, the determinant representation can be simplified, hence leading in this boundary case to direct analogues of algebraic Bethe ansatz determinant representations of the scalar products for
the periodic chain.Comment: 39 page | {"url":"https://core.ac.uk/search/?q=author%3A(Kitanine%2C%20N.)","timestamp":"2024-11-07T09:42:35Z","content_type":"text/html","content_length":"154038","record_id":"<urn:uuid:14b47d05-3ac4-41a5-8bf8-5942ecdb93f8>","cc-path":"CC-MAIN-2024-46/segments/1730477027987.79/warc/CC-MAIN-20241107083707-20241107113707-00707.warc.gz"} |
PVC Pressure Considerations - Vinidex Pty Ltd
PVC Pressure Considerations
Static Stresses
The hydrostatic pressure capacity of PVC pipe is related to the following variables:
1. The ratio between the outer diameter and the wall thickness (dimension ratio).
2. The hydrostatic design stress for the PVC material.
3. The operating temperature.
4. The duration of the stress applied by the internal hydrostatic pressure.
The pressure rating of PVC pipe can be ascertained by dividing the long-term pressure capacity of the pipe by the desired factor of safety. Although PVC pipe can withstand short-term hydrostatic
pressure applications at levels substantially higher than pressure rating or class, the performance of PVC pipe in response to applied internal hydrostatic pressure should be based on the pipe’s
long- term strength.
By international convention, the relationship between the internal pressure in the pipe, the diameter and wall thickness and the circumferential hoop stress developed in the wall, is given by the
Barlow Formula, which can be expressed in the following forms:
and alternatively, for pipe design,
These formulas have been standardised for use in design, routine testing and research work and are thus applicable at all levels of pressure and stress. They form the basis for establishment of
ultimate material limitations for plastic pipes by pressure testing.
For design purposes, P is taken as the maximum allowable working pressure with S being the maximum allowable hoop stress (at 20°C) given below:
│PVC-U pipes up to DN150 │11.0 MPa│
│DN175 PVC-U pipes and larger │12.3 MPa│
│Material Class 400 Oriented PVC pipes (PVC-O) │25 MPa │
│Material Class 450 Oriented PVC pipes (PVC-O) │28 MPa │
│Material Class 500 Oriented PVC pipes (PVC-O) │32 MPa │
│Modified PVC pipes (PVC-M) │17.5 MPa│
Dynamic Stresses
PVC pressure pipes are designed on the basis of a burst regression line for pipes subjected to constant internal pressure. From this long-term testing and analysis, nominal working pressure classes
are allocated to pipes as a first indication of the duty for which they are suitable. However, there are many other factors which must be considered, including the effects of dynamic loading. Whilst
most gravity pressure lines operate substantially under constant pressure, pumped lines frequently do not. Pressure fluctuations in pumped mains result from events such as pump start-up and shutdown
and valves opening and closing. It is essential that the effects of this type of loading be considered in the pipeline design phase to avoid premature failure.
The approach adopted for pipe design and class selection when considering these events depends on the anticipated frequency of the pressure fluctuation. For frequent, repetitive pressure variations,
the designer must consider the potential for fatigue and design accordingly. For random, isolated surge events, for example, those which result from emergency shutdowns, the designer must ensure that
the maximum and minimum pressures experienced by the system are within acceptable limits.
For the purposes of this document, surge is defined as a rapid, very short-term pressure variation caused by an accidental, unplanned event such as an emergency shutdown resulting from a power
failure. Surge events are characterised by high pressure rise rates with no time spent at the peak pressure.
In contrast, fatigue is associated with a large number of repetitive events. Many materials will fail at a lower stress when subjected to cyclic or repetitive loads than when under static loads. This
type of failure is known as (cyclic) fatigue. For thermoplastic pipe materials, fatigue is only relevant where a large number of cycles are anticipated. The important factors to consider are the
magnitude of the stress fluctuation, the loading frequency and the intended service life. Where large pressure fluctuations are predicted, fatigue design might be required if the total number of
cycles over the intended lifetime of the pipeline exceeds 25,000. For smaller pressure cycles, a larger number of cycles can be tolerated.
Pressure Range
Pressure range is defined as the maximum pressure minus the minimum pressure, including all transients, experienced by the system during normal operations
Diurnal pressure changes
Diurnal pressure changes are gradual pressure changes which occur in most distribution pipelines due to demand variation. It is generally accepted that diurnal pressure changes will not cause
fatigue. The only design consideration required for this type of pressure fluctuation is that the maximum pressure should not exceed the pressure rating of the pipe.
Surge design
It has long been recognised that PVC pipes are capable of handling short-term stresses far greater than the long-term loads upon which they are designed. That is, PVC pipes can cope with higher
pressures than they are designed for provided the higher pressures are of only a short duration. However, this characteristic feature is not utilised in design in Australia and design recommendations
advise that the peak pressure should not exceed the nominal working pressure of the pipe. This recommendation is based on the fact the pipes should not be considered in isolation but as part of a
system. Whilst the pipes themselves might be capable of withstanding occasional, short duration exposure to pressures greater than the design pressure, the same assumption may not apply to the
pipeline system.
Where the generation of negative pressures is anticipated, the possibility of transverse buckling should be considered. This topic is addressed elsewhere.
Fatigue design
The fatigue response of thermoplastics pipe materials, particularly PVC, has been extensively investigated (see references). The results of laboratory studies can be used to establish a relationship
between stress range, defined here as the difference between the maximum and minimum stress, and the number of cycles to failure. From these relationships it is possible to derive load factors that
can be applied to the operating pressures, to enable selection of an appropriate class of pipe.
This type of experimental data inevitably has a degree of scatter and it has been Australian practice, after Joseph (3), to adopt the lower bound for design purposes. This approach is retained here
because it ensures the design has a positive safety factor and recognises that pipelines may sustain minor surface damage during installation, which could promote fatigue crack initiation. Note that
for fatigue loading situations, the maximum pressure reached in the repetitive cycle should not exceed the static pressure rating of the pipe.
Recommended fatigue cycle factors for PVC-U, PVC-M and PVC-O are given in Table 1 below:
│ │ │Fatigue Cycle Factors, │
│Total Cycles│Approx No. Cycles/day for 100y life │f │
│ │ ├───────┬───────┬───────┤
│ │ │PVC-U │PVC-M │PVC-O │
│26,400 │1 │1 │1 │1 │
│100,000 │3 │1 │0.67 │0.75 │
│200,000 │5.5 │0.81 │0.54 │0.66 │
│500,000 │14 │0.62 │0.41 │0.56 │
│1,000,000 │27 │0.5 │0.33 │0.49 │
│2,500,000 │82 │0.38 │0.25 │0.41 │
│5,000,000 │137 │0.38 │0.25 │0.41 │
│10,000,000 │274 │0.38 │0.25 │0.41 │
Using Table 1, the Maximum Cyclic Pressure Range for a given class of pipe can be calculated from the following formula:
Charts plotting the MCPR versus the number of cycles for a range of pressure classes of PVC-U, PVC-M and PVC-O pipes are available:
To select the appropriate pipe class for fatigue loading, the following procedure should be adopted:
1. Estimate the likely pressure range, i.e., the maximum pressure minus the minimum pressure.
2. Estimate the frequency or the number of cycles per day which are expected to occur.
3. Determine the required service life and calculate the total number of cycles which will occur in the pipe lifetime
4. Using the appropriate chart draw a vertical line from the x-axis at ΔP and a horizontal line from the y-axis at the total number of cycles in the pipe lifetime
5. Find the intersection point between the horizontal and vertical lines.
6. Select the pipe class that bounds the region of this intersection point as the minimum required for these fatigue conditions.
A sewer rising main has a pump pressure, including static lift and friction losses, of 400 kPa. When the pump starts up, the pressure rises rapidly to 800 kPa before decaying exponentially to the
static pumping pressure. On pump shut down, the minimum pressure experienced by the system is 100 kPa. On average, the pump will start up 8 times per day. A minimum life of 100 years is required.
The maximum pressure experienced indicates that a minimum class of PN 9 will be required. A fatigue analysis is now needed to determine suitability, or otherwise, of PN 9.
In this system, the pressure range is 700 kPa. The pump will start up approximately 292,000 times in a 100-year lifetime. However, the exponential cycle pattern means that this should be doubled for
design purposes. Therefore, the system should be designed to withstand approximately 584,000 cycles in a 100-year lifetime.
Using Table 1 to determine the fatigue load factors for PVC pipes at 5.8 x 10 ^5 cycles gives the following class selection:
│Material│Fatigue Cycle Factor, f (Table 1)│Maximum Cyclic Pressure Range (MPa) │Minimum Pipe Class Selection│
│ │ │PN9 =0.9 x 0.6 =0.54 <ΔP │ │
│PVC-U │0.6 ├────────────────────────────────────┤PN12 │
│ │ │PN12 = 1.2 x 0.6 =0.72 ≥ ΔP │ │
│ │ │PN16 =1.6 x 0.4 =0.64 < ΔP │ │
│PVC-M │0.4 ├────────────────────────────────────┤PN18 │
│ │ │PN18 = 1.8 x 0.4 =0.72 ≥ ΔP │ │
│ │ │PN12.5 =1.25 x 0.54 =0.675 < ΔP │ │
│PVC-O │0.54 ├────────────────────────────────────┤PN16 │
│ │ │PN16 = 1.6 x 0.54 =0.86 ≥ ΔP │ │
The graphical procedure is demonstrated below for PVC-U. Using the charts for PVC-U, draw a vertical line from the pressure range on the x axis and a horizontal line from the number of cycles on the
y axis. Find the intersection of these two lines and read off the pipe pressure class that bounds this region. In this example, the intersection point lies in the region bounded by the PN12 curve so
PN 12 is required for PVC-U pipe for fatigue loading.
The fatigue analysis thus determines that although PN 9 is adequate for the maximum pressure, a minimum PN 12 pipe is needed for PVC-U, PN18 for PVC-M and PN16 for PVC-O, in order to cope with
fatigue effects.
Definition of Pressure Range and effect of surges
For simplicity, the pressure range is defined as the maximum pressure minus the minimum pressure, including all transients, experienced by the system during normal operations. The effect of
accidental conditions such as power failure may be excluded. This is illustrated in the figure below.
This figure also illustrates the definition of a cycle as a repetitive event. In some cases, the cycle pattern will be complex and it may be necessary to also consider the contribution of secondary
Pumping systems are frequently subject to surging following the primary pressure transient on switching. Such pressure surging decays exponentially, and in effect the system is subjected to a number
of minor pressure cycles of reducing magnitude. In order to take this into account, the effect of each minor cycle is related to the primary cycle in terms of the number of cycles which would produce
the same crack growth as one primary cycle.
According to this technique, a typical exponentially decaying surge regime is equivalent to 2 primary cycles. For design purposes, the primary pressure range only is considered, with the frequency
Complex Cycle Patterns
In general, a similar technique may be applied to any situation where smaller cycles exist in addition to the primary cycle. Empirically, crack growth is related to stress cycle amplitude according
to (Δσ)^3.2. Thus n secondary cycles of magnitude Δσ[1], may be deemed equivalent in effect to one primary cycle, Δσ[0]
For example, a secondary cycle of half the magnitude of the primary cycle:
so it would require 9 secondary cycles to produce the same effect as one primary cycle. If they are occurring at the same frequency, the effective frequency of primary cycling is increased by 1.1 for
the purpose of design.
Effect of Temperature
Joseph notes that the available data indicates that there is no evidence of a change in response of PVC fatigue crack growth rates with temperature, at least in the lower temperature region where
results are available. This is logically consistent with known fatigue behaviour, since the propensity to propagate a crack reduces with increasing ductility which results in yielding and blunting of
the crack tip and a reduction in local stress intensity. Thus one would expect that PVC, with increasing ductility and decreasing yield strength, would not be degraded in fatigue performance at
higher temperatures.
It follows that, while normal derating principles must be applied in class selection for static pressures, (ductile burst), no additional temperature derating need be applied for dynamic design.
ie. Select the highest class arrived via:
• Static design including temperature derating; or
• Dynamic design as covered herein.
Safety Factors
The tabulated fatigue cycle factors represent the lower bound of test data generated from a number of different sources over the last few years, on commercially produced PVC pipes. The mean line for
this data is approximately half a log decade higher than this.
It is therefore considered conservative and no additional safety factor need be applied in general. However, where the magnitude or frequency of dynamic stresses cannot be estimated in design with
any reasonable degree of accuracy, appropriate caution should obviously be applied. This judgement is in the hands of the designer.
Whilst it is always possible to predict the steady operating conditions with good accuracy, it will occasionally be the case, in complex systems, that it is impossible to predict the extent of surge
pressures. In such circumstances, relatively low-cost surge mitigation techniques, for example the solid state soft-start motor controllers should be considered. It is of course recommended that
actual operating conditions for all systems should be checked by measurement, as a matter of routine, when the system is commissioned. Should surge pressure amplitudes in the event exceed expected
levels, it is relatively easy matter to retrofit control equipment to ensure that they are kept in check.
Design Hints
To reduce the effect of dynamic fatigue in an installation, the designer can:
1. Limit the number of cycles by:
a. Increasing well capacity for a sewer pumping station;
b. Matching pump performance to tank size to eliminate short demand cycles for an automatic pressure unit; or
c. Using double-acting float valves or limiting starts on the pump by using a time clock when filling a reservoir
2. Reduce the dynamic range by:
a. Eliminating excessive water hammer; or
b. Using a larger bore pipe to reduce friction losses.
PVC fittings present a problem worthy of special consideration. Complex stress patterns in fittings can ‘amplify’ the apparent stress cycle. An apparently harmless pressure cycle can thus produce a
damaging stress cycle leading to a relatively short fatigue life.
This factor is particularly severe in the case of branch fittings such as tees, where amplification factors up four times have been noted. The condition can be aggravated further by the existence of
stress cycling from other sources, for example bending stresses induced flexing under hydraulic thrust in improperly supported systems.
Prudence therefore dictates that a suitable factor of safety be applied to fittings in assessing class requirements. It is recommended that the following factors be applied to the design dynamic
pressure cycle for fittings:
│Tees │equal│D x ¾D│D x ½ D│D x ¼ D│
│Safety Factor │4 │3 │2 │1.5 │
│Bends │90°short│45°short│90°long│45°long│
│Safety Factor │3 │2 │2 │1.5 │
│Reducers │D x ¾D│D x ½ D│D x ¼ D│
│Safety Factor │1.5 │2 │2.5 │
│Adaptors & Couplings │equal size │wyes│
│Safety Factor │1 │6 │
A golf course watering scheme is designed to operate at 0.70 MPa. Balanced loading will ensure no pump cycling during routine watering. However, the system is to be maintained on standby with a
jockey pump for hand watering purposes and this will cut in and out at 0.35 and 0.75 MPa. With normal usage and leakage this may occur every half hour on average for twelve hours a day. A twenty-five
year life is required.
The pressure cycle is 0.4 MPa. Allow 20% for water hammer but no surging is likely in this type of system. Total dynamic cycle 0.48 MPa. The total life cycles predicted is 25 x 365 x 25 = 228,000.
Using the table, the MCPR of a PN 9 pipe is 0.64. Therefore, PN 9 pipe is satisfactory (PN 9 is required to cope with normal operational pressure). For fittings the effective dynamic cycle is:
Equal Tees : 4 x 0.48 = 1.92 MPa
Elbows 90° : 3 x 0.48 = 1.44 MPa
PN 18 fittings are suitable for only 1.8 MPa effective dynamic range. Equal tees may not have an acceptable life in this system.
Solution: Reduce the dynamic range or reduce the frequency or the periods on standby.
1. MARSHALL, G.P., BROGDEN, S. :Final Report of Pipeline Innovation Contract to UKWIR 1997.
2. WATER U.K. Ltd, Water Industry Specification, WIS 4-37-02, March 2000, Issue 1.
3. JOSEPH, S.H., (University of Sheffield) “Fatigue Failure and Service Lifetime in uPVC Pressure Pipes”, Plastics and Rubber Processing and Applications, Vol 4, No. 4, 1984, pp. 325-330, UK.
4. HUCKS, R.T., “Designing PVC pipe for water distribution systems”, J. AWWA, 7 (1972), pp. 443-447.
5. KIRSTEIN, C.E., Untersuchung der Innendruck-Schwellfestigkeit von Rohren aus PVC-hart, “Publication of the Institut fur Kunstsoffprufung and Kunststoffkunde”, Unversitat Stuttgart, 1972.
6. GOTHAM, K.V. AND HITCH, M.J., “Design considerations for fatigue in uPVC pressure pipelines”, Pipes and Pipelines Int., 20 (1975), pp. 10-17.
7. STAPEL, J.U., “Fatigue properties of unplasticised PVC related to actual site conditions in water distribution systems”, Pipes and Pipelines Int., 22 (1977), pp. 11-15 and 33-36.
8. GOTHAM, K.V. AND HITCH, M.J., “Factors affecting fatigue resistance in rigid uPVC pipe compositions”, Brit. Polym. J., 10 (1978), pp. 47-52.
9. JOSEPH, S.H., “The pressure fatigue testing of plastic pipes”, Plastics Pipes 4, PRI, London, 1979, Paper 28.
10. MOORE, D.R., GOTHAM, K.V. AND LITTLEWOOD, M.J., “The long term fracture performance of uPVC pipe as influenced by processing”, Plastics Pipes 4, PRI, London, 1979, Paper 27.
11. MARSHALL, G. P., BROGDEN, S., AND SHEPHERD, M. A., “Evaluation of the surge and fatigue resistance of PVC and PE pipeline materials for use on the U.K. Water Industry”, Plastics Pipes X
Conference, Gothenburg, Sweden, 1998
12. FITZPATRICK, P., MOUNT, P., SMYTH, G. AND STEPHENSON, R. C., “Fracture toughness and dynamic fatigue characteristics of PVC/CPE blends”, Plastics Pipes 9 Conference, Edinburgh, Scotland, 1995.
13. DUKES, B.W., “The dynamic fatigue behaviour of UPVC pressure pipe.”, Plastics Pipes VI, PRI, York, UK, 1985.
14. TRUSS R. W., “Lifetime predictions for uPVC pipes subjected to combined mean and oscillating pressures”, Plastics and Rubber Processing and Applications, Vol 10, No. 1, 1988, pp. 1-9.
15. BURN L. S., “Installation damage: Effect on lifetimes of uPVC pipes subjected to cyclic pressure”, Urban Water Research Association of Australia, Melbourne, Research Report 68 (1993).
16. WHITTLE A. J. AND TEO A., “Resistance of PVC-U and PVC-M to cyclic fatigue”, Plastics, Rubber and Composites, Vol 34, No. 1, 2005, pp. 40-46.
17. WEST D.B., “Fatigue-Life, Fatigue-Limits and Delamination in Oriented Poly(Vinyl Chloride) Pipes”, A thesis submitted for the degree of Doctor of Philosophy at The University of Queensland,
January 2008 | {"url":"https://www.vinidex.com.au/resources/technical-resources/pvc-pressure-pipe/pvc-pressure-considerations/","timestamp":"2024-11-09T13:40:03Z","content_type":"text/html","content_length":"120923","record_id":"<urn:uuid:2fe5fd2d-28b6-4a05-9557-56c3cb4a39d6>","cc-path":"CC-MAIN-2024-46/segments/1730477028118.93/warc/CC-MAIN-20241109120425-20241109150425-00151.warc.gz"} |
LaTeX notation polls
A LaTeX notation poll is where you ask and test complex mathematical formulas or expressions. It is typically used in higher education like science, statistics, computer science, physics,
engineering, math classes etc. It is also used in multilingual materials for Sanskrit or Greek.
Example of LaTeX being used for a poll in Present view.
For detailed steps on how to set up LaTeX notation polling read our article.
Example of a poll using LaTeX on a participant device screen. | {"url":"https://help.vevox.com/hc/en-us/articles/4402092515217-LaTeX-notation-polls","timestamp":"2024-11-11T23:53:04Z","content_type":"text/html","content_length":"21046","record_id":"<urn:uuid:b120795b-1779-43dc-89fe-b537175b09b7>","cc-path":"CC-MAIN-2024-46/segments/1730477028240.82/warc/CC-MAIN-20241111222353-20241112012353-00721.warc.gz"} |
100 Years of Markov Chains
On January 23, 1913, the Russian mathematician Andrei Andreyevich Markov addressed the Imperial Academy of Sciences in St. Petersburg, reading a paper titled “An example of statistical investigation
of the text Eugene Onegin concerning the connection of samples in chains.” The idea he introduced that day is the mathematical and computational device we now know as a Markov chain.
On January 23, 2013, the Institute for Applied Computational Science of the Harvard School of Engineering and Applied Sciences will celebrate the centenary of this event. If you are in the Boston
area and would like to attend, consider this your invitation. See the announcement for details of when and where. There will be three talks:
First Links in the Markov Chain: Poetry and Probability.
Brian Hayes, American Scientist magazine
From Markov to Pearl: Conditional Independence as a Driving Principle
for Probabilistic Modeling.
Ryan P. Adams, SEAS Computer Science
Applications of Markov Chains in Science.
Pavlos Protopapas, Harvard-Smithsonian Center for Astrophysics and SEAS
Markov’s 1913 paper was not his first publication on “samples in chains”; he had written on the same theme as early as 1906. So why celebrate now? Well, for one thing, it’s too late to do it in 2006.
But there is another reason: It was the 1913 paper that was widely noticed, both in Russia and abroad, and that inspired further work in the decades to come. The earlier discussions were abstract and
technical, giving no hint of what the new probabilistic method might be good for; in 1913 Markov demonstrated his technique with a novel and intriguing application—analyzing the lexical structure of
Alexander Pushkin’s poem Eugene Onegin. Direct extensions of that technique now help to identify genes in DNA and generate gobbledygook text for spammers.
Sticklers for calendrical accuracy might raise another question about the timing of this event. On January 23 it will not yet be 100 years since Markov spoke in St. Petersburg. In 1913 Russia had not
yet adopted the Gregorian calendar; when the nation did so in 1918, it skipped ahead by 13 days. If you are troubled by this calendrical lacuna, you may want to organize your own symposium on
February 5.
Update: Coverage of Markov Day in the Harvard Gazette.
Update 2013-02-24: Recordings of the talk are now available. Audio: MP3. Video: FLV, MP4. The slides are also on the web: HTML. Finally, my latest American Scientist column covers much of the same
material: HTML, PDF.
3 Responses to 100 Years of Markov Chains
1. George Washington was in the same position, and changed the date of his birthday from February 11 (Julian) to February 22nd (Gregorian) in order to maintain the same actual day. So Washington was
not born on Washington’s Birthday (much less on President’s Day, of course, which can be any time from February 15th to February 21st).
2. Is there a webcast available, or in the works?
□ Any day now. I’ll post the link when it’s up.
This entry was posted in computing, mathematics. | {"url":"http://bit-player.org/2013/100-years-of-markov-chains","timestamp":"2024-11-10T04:59:35Z","content_type":"text/html","content_length":"36159","record_id":"<urn:uuid:0e98fcbe-f112-4e36-a42f-d96566d6a2fc>","cc-path":"CC-MAIN-2024-46/segments/1730477028166.65/warc/CC-MAIN-20241110040813-20241110070813-00268.warc.gz"} |
Multiplying Polynomial Expressions by Monomials
What are Polynomial Expressions?
Polynomials are regarded as the sums of specific terms of a given form k⋅xⁿ. Here, the k stands for any number and n stands for a positive integer.
How do you Multiply x – y – z by -8x²?
One has to first Multiply each term of the polynomial x – y – z by making use of the monomial -8x².
To multiply x – y – z by -8x², one needs to follow these steps:
1. Multiply each term of the polynomial by -8x²:
□ -8x² * (x – y – z) = (-8x² * x) – (-8x² * y) – (-8x² * z)
2. Add the like terms together:
Polynomials are algebraic expressions consisting of variables and coefficients, combined using arithmetic operations such as addition, subtraction, multiplication, and division. The general form of a
polynomial is represented as k⋅xⁿ, where k is a numerical coefficient and n is a non-negative integer representing the degree of the term.
When multiplying a polynomial by a monomial, each term in the polynomial is multiplied by the monomial separately, and then the resulting terms are combined based on the rules of arithmetic. In the
given example of multiplying x – y – z by -8x², the process involves distributing the monomial -8x² to each term of the polynomial and then simplifying the result by adding any like terms.
It is important to pay attention to the signs and coefficients when multiplying polynomials by monomials to ensure the correct computation of the final expression. By following the steps outlined
above, one can successfully multiply polynomial expressions by monomials and obtain the desired result. | {"url":"https://www.mantrapaloalto.com/sat/multiplying-polynomial-expressions-by-monomials.html","timestamp":"2024-11-07T16:32:25Z","content_type":"text/html","content_length":"22861","record_id":"<urn:uuid:2a523c64-2d1d-4601-8cd2-e5922c0a0b8a>","cc-path":"CC-MAIN-2024-46/segments/1730477028000.52/warc/CC-MAIN-20241107150153-20241107180153-00617.warc.gz"} |
Standard Deviation (2 of 4)
Learning Objectives
• Use mean and standard deviation to describe a distribution.
A More Common Measure of Spread about the Mean: The Standard Deviation
The standard deviation (SD) is a measurement of spread about the mean that is similar to the average deviation. We think of standard deviation as roughly the average distance of data from the mean.
In other words, the standard deviation is approximately equal to the average deviation. We develop the formula for standard deviation in the following example.
Calculating the Standard Deviation
Let’s consider the same data set we used on the previous page: 2, 2, 4, 5, 6, 7, 9. We already know that the mean is 5. We compute the standard deviation similarly to the way we compute the average
deviation. We begin by computing the deviation of each point from the mean, but instead of taking the absolute value of the differences, we square them. Here are the steps:
1. We start by finding the differences between each value and the mean (just like before):[latex]\begin{array}{l}2-5=-3\\ 2-5=-3\\ 4-5=-1\\ 5-5=0\\ 6-5=1\text{}\\ 7-5=2\text{}\\ 9-5=4\end{array}[/
2. We square each of the differences:[latex]\begin{array}{l}{(2-5)}^{2}={(-3)}^{2}=9\\ {(2-5)}^{2}={(-3)}^{2}=9\\ {(4-5)}^{2}={(-1)}^{2}=1\\ {(5-5)}^{2}={0}^{2}=0\\ {(6-5)}^{2}={1}^{2}=1\\ {(7-5)}^
{2}={2}^{2}=4\\ {(9-5)}^{2}={4}^{2}=16\end{array}[/latex]
3. As before, we find the average of these squared differences. We add the squared differences and divide by n − 1 (the count minus 1). Note that we divide by n − 1 instead of n. (The reason is
subtle. We do not discuss it in this course.)[latex]\frac{9+9+1+0+1+4+16}{6}=\frac{40}{6}\approx 6.67[/latex]
4. To scale back the value to account for the squaring we did in step 2, we take the square root of the value we found in step 3:[latex]\sqrt{6.67}\approx 2.58[/latex]
Notice that the standard deviation is a little bit larger than the average deviation (which was 2). We can get a good approximation of the standard deviation by estimating the average distance from
the mean. The shaded box on the following dotplot indicates 1 SD to the right and left of the mean.
The formula for the standard deviation of a data set can be described by the following expression. However, we will always use technology to perform the actual computation of the standard deviation.
The symbols in the expression are defined as follows:
• n is the number of values in the data set (the count).
• Recall that ∑ means to add up (compute the sum).
• [latex]\stackrel{¯}{x}[/latex] is the mean of the data set.
• The individual values are denoted by x.
Note: In the formula you can see
• the deviations from the mean [latex](x-\stackrel{¯}{x})[/latex].
• the squaring of these deviations.
• the averaging of the squared deviations: add them up (∑) and divide by (n − 1).
Before we learn to use technology to compute the standard deviation, we practice estimating it. We can estimate standard deviation in the same ways we estimated ADM. Think of standard deviation as
roughly equal to ADM, so standard deviation is roughly the average distance of data from the mean.
Learn By Doing
Let’s consider the same collection of cereals we worked with previously, except this time we’ll look at the calorie content. | {"url":"https://courses.lumenlearning.com/atd-herkimer-statisticssocsci/chapter/standard-deviation-2-of-4/","timestamp":"2024-11-03T13:34:35Z","content_type":"text/html","content_length":"51175","record_id":"<urn:uuid:a86e0cee-bba2-482f-8597-50d73b6d9b19>","cc-path":"CC-MAIN-2024-46/segments/1730477027776.9/warc/CC-MAIN-20241103114942-20241103144942-00643.warc.gz"} |
Name CVE-2024-42131
In the Linux kernel, the following vulnerability has been resolved: mm: avoid overflows in dirty throttling logic The dirty throttling logic is interspersed with assumptions that dirty
limits in PAGE_SIZE units fit into 32-bit (so that various multiplications fit into 64-bits). If limits end up being larger, we will hit overflows, possible divisions by 0 etc. Fix these
Description problems by never allowing so large dirty limits as they have dubious practical value anyway. For dirty_bytes / dirty_background_bytes interfaces we can just refuse to set so large
limits. For dirty_ratio / dirty_background_ratio it isn't so simple as the dirty limit is computed from the amount of available memory which can change due to memory hotplug etc. So when
converting dirty limits from ratios to numbers of pages, we just don't allow the result to exceed UINT_MAX. This is root-only triggerable problem which occurs when the operator sets dirty
limits to >16 TB.
Source CVE (at NVD; CERT, LWN, oss-sec, fulldisc, Red Hat, Ubuntu, Gentoo, SUSE bugzilla/CVE, GitHub advisories/code/issues, web search, more)
References DSA-5747-1
Vulnerable and fixed packages
The table below lists information on source packages.
Source Package Release Version Status
linux (PTS) bullseye 5.10.223-1 fixed
bullseye (security) 5.10.226-1 fixed
bookworm 6.1.115-1 fixed
bookworm (security) 6.1.112-1 fixed
trixie 6.11.5-1 fixed
sid 6.11.7-1 fixed
The information below is based on the following data on fixed versions.
Package Type Release Fixed Version Urgency Origin Debian Bugs
linux source bullseye 5.10.223-1 DSA-5747-1
linux source bookworm 6.1.98-1
linux source (unstable) 6.9.9-1
https://git.kernel.org/linus/385d838df280eba6c8680f9777bfa0d0bfe7e8b2 (6.10-rc7) | {"url":"https://security-tracker.debian.org/tracker/CVE-2024-42131","timestamp":"2024-11-14T07:56:02Z","content_type":"text/html","content_length":"5659","record_id":"<urn:uuid:32e5b36f-a472-4db2-b7f6-e6557ffe0e96>","cc-path":"CC-MAIN-2024-46/segments/1730477028545.2/warc/CC-MAIN-20241114062951-20241114092951-00599.warc.gz"} |
Differing Neighbors
Algorithm A
Given an array of N integers, sort the array, and find the 2 consecutive numbers in the sorted array with the maximum difference. Example - on input [1,7,3,2] output 4 (the sorted array is [1,2,3,7],
and the maximum difference is 7-3=4).
Algorithm A runs in O(NlogN) time.
Implement an algorithm identical in function to algorithm A, that runs in O(N) time.
Thanks to Yossi Richter for this riddle!
Spoiler Alert - Solution Ahead
We first find the minimum and maximum of the array. We then allocate another array, call it B, of size N + 2. We let each cell in B contain two integers. Let \(s = \frac{max(A) - min(A)}{N + 2}\). We
pseudo-sort the elements of the original array into array B, such that the first bucket of B represents the integers [min(A), s), the second bucket represents [s, 2s), …, the bucket before last
represents [max(A) - s, max(A)), and the last bucket represents [max(A), max(A) + s). We only keep the minimum and the maximum in each bucket (hence the 2 integers in each cell of B). Now, we are
splitting N numbers between N + 2 cells, so there must be at least one empty cell. We know that the first and last cells are not empty (as they respectively contain the minimum and maximum elements
of the original array), so that means that there are 2 consecutive elements with a distance of at least s. That means that the 2 consecutive elements with maximum distance are the maximum and minimum
elements in two cells of B such that all cells between them are empty. \(\blacksquare\)
In python:
from dataclasses import dataclass
class Bucket:
min : int = None
max : int = None
def add(self, x : int):
if not self.min or x < self.min:
self.min = x
if not self.max or x > self.max:
self.max = x
def empty(self):
return self.min == self.max == None
def max_diff_neighbors(A):
N = len(A)
mn, mx = min(A), max(A)
tot = mx - mn
B = [Bucket() for i in range(N + 2)]
for x in A:
bucket = (x - mn) * (N + 1) // tot
max_diff = 0
cur = B[0].max
for i in range(1, N+2):
if B[i].empty(): continue
diff = B[i].min - cur
if diff > max_diff: max_diff = diff
cur = B[i].max
return max_diff | {"url":"https://yanivle.github.io/puzzles/2011/01/12/differing-neighbors.html","timestamp":"2024-11-11T14:34:15Z","content_type":"text/html","content_length":"40268","record_id":"<urn:uuid:9812ba04-8c96-4eb0-a330-f4efa6778322>","cc-path":"CC-MAIN-2024-46/segments/1730477028230.68/warc/CC-MAIN-20241111123424-20241111153424-00624.warc.gz"} |
How to Create 2D Geometry from Cross Sections of 3D Designs
COMSOL Multiphysics^® contains many specialized features for creating specific types of geometry, such as those that enable you to create 2D geometry from 3D objects by cutting the solid using a
plane. In this article, we will discuss the use cases for doing this, the advantages of implementing this approach, and demonstrate how to do it.
Use Cases and Functionality
Taking a cross section from a 3D geometry and then using that cross-sectional slice in a 2D model is particularly useful when building models that involve any of the following elements:
• Symmetric or axisymmetric 3D geometries, in which you can create a 2D model based on your 3D model and obtain significantly faster solutions for the same geometry
• Thin-walled parts, in which you can use the plane stress assumption for your 2D model
• Long parts, in which you can use the plane strain modeling assumption for your 2D model
The 3D geometry (left) and 2D cross-sectional geometry (right) for a model of an h-bend waveguide. The physics of the model make it possible to set up and compute in 2D.
The 3D geometry (left) and 2D cross-sectional geometry (right) for a model of an optical ring resonator notch filter. The 2D model is used to investigate different designs for the optical ring
resonator before any 3D model simulations are computed.
A model of a rectangular plate with a hole in the center. The plate is modeled in 2D using a plane stress assumption (left) before turning to model the full, 3D model (right).
Using this approach is also helpful for developing large or complex 3D models. By taking advantage of 2D model approximations when possible, you can more quickly and easily investigate specific
aspects of your simulation, all before applying them to the 3D model. This could include analyzing any small or complex geometric features, material property values, physics settings, multiphysics
couplings settings, mesh settings, or solver settings.
In COMSOL Multiphysics, cutting through a solid using a plane is done in two steps. First, add a work plane to the geometry for your 3D model component. The work plane needs to be placed so that the
appropriate cross section can be obtained from where the work plane cuts through the 3D solid object.
The COMSOL Multiphysics UI showing the Model Builder with the Work Plane feature selected, the corresponding Settings window, and the Graphics window with the 3D geometry of a pipe fitting model.
The 3D model geometry for a pipe fitting, with transparency enabled to see the inside. A work plane is added in the xz-plane along the center of the design, from which the cross section is obtained.
Next, add a second model component, either 2D or 2D axisymmetric, and subsequently add a Cross Section geometry operation. With this, you obtain the cross section from the work plane defined
previously in the 3D model component. (You can also do this with the Projection operation, but note this functionality requires the Design Module.)
The COMSOL Multiphysics UI showing the Cross Section feature selected, the corresponding Settings window, and the Graphics window with the 2D axisymmetric geometry of the pipe fitting model.
The cross section used to create a 2D axisymmetric model of the pipe fitting.
Watch the video below and follow along step-by-step in the software using the exercise files to learn how to convert 3D geometries for use in 2D model components. We also discuss how it may be
necessary or optimal for you to prepare your 3D geometry so that you can obtain an appropriate, representative 2D cross section. This process largely depends on the design and the features contained
Tutorial Video: Create 2D Geometry from Cross Sections of 3D Designs
Modeling Exercises
Demonstrate the knowledge you have gained from this article by putting it into practice with the follow-up modeling exercises listed below. The directions provided are intentionally generalized to
encourage self-guided problem solving. You can manually check your implementation with the solution model files provided here or use the comparison tool to identify any differences.
Recommended exercises:
1. For the shell-and-tube heat exchanger model geometry (MPH-file), determine how to convert the 3D model into 2D to obtain the geometry pictured below
2. For the corrugated circular horn antenna model geometry (MPHBIN-file), create a geometry that can be used in a 2D axisymmetric model of the design
3. For the ball check valve model geometry (MPHBIN-file), create a geometry that can be used in a 2D axisymmetric model of the design
For the shell-and-tube heat exchanger model geometry, you can download the MPH-file from the linked text. For the MPHBIN files, download the files linked above and then import them into the software.
The 3D model geometry for a shell-and-tube heat exchanger (left) and the geometry used for a 2D model of the design (right).
Further Learning
To learn more about reducing your model geometry from 3D to 2D, see our Learning Center article "Using Symmetry to Reduce Model Size". In the Application Libraries, there are also several tutorial
models that simulate a device or system in both 2D and 3D, which you can inspect further to see how the 2D approximation for the model was implemented. Available models (and their corresponding
application area) include: | {"url":"https://www.comsol.fr/support/learning-center/article/72361","timestamp":"2024-11-08T17:00:12Z","content_type":"text/html","content_length":"53037","record_id":"<urn:uuid:ad0c0c85-327a-4054-9fd2-86d0dc1d2b79>","cc-path":"CC-MAIN-2024-46/segments/1730477028070.17/warc/CC-MAIN-20241108164844-20241108194844-00520.warc.gz"} |
Problem A
You’re playing hopscotch! You start at the origin and your goal is to hop to the lattice point $(N, N)$. A hop consists of going from lattice point $(x_1, y_1)$ to $(x_2, y_2)$, where $x_1 < x_2$ and
$y_1 < y_2$.
You dislike making small hops though. You’ve decided that for every hop you make between two lattice points, the x-coordinate must increase by at least $X$ and the y-coordinate must increase by at
least $Y$.
Compute the number of distinct paths you can take between $(0, 0)$ and $(N, N)$ that respect the above constraints. Two paths are distinct if there is some lattice point that you visit in one path
which you don’t visit in the other.
Hint: The output involves arithmetic mod $10^9+7$. Note that with $p$ a prime like $10^9+7$, and $x$ an integer not equal to $0$ mod $p$, then $x(x^{p-2})$ mod $p$ equals $1$ mod $p$.
The input consists of a line of three integers, $N$ $X$ $Y$. You may assume $1 \le X, Y \le N \le 10^{6}$.
The number of distinct paths you can take between the two lattice points can be very large. Hence output this number modulo 1000000007 ($10^9 + 7$).
Sample Input 1 Sample Output 1
Sample Input 2 Sample Output 2 | {"url":"https://purdue.kattis.com/courses/CS311-CP2/2024-Spring/assignments/rig735/problems/hopscotch","timestamp":"2024-11-06T02:29:47Z","content_type":"text/html","content_length":"28796","record_id":"<urn:uuid:15ee30bd-e6dc-4a32-b6a7-314f4b9958c7>","cc-path":"CC-MAIN-2024-46/segments/1730477027906.34/warc/CC-MAIN-20241106003436-20241106033436-00044.warc.gz"} |
For each of the following statements indicate if - Genius Papers
For each of the following statements indicate if
Chem 452 – HW 6Winter 2012CHEM 452 – Homework 6•••••Due by 11:00 PM on Friday February 10 2012 in the CHEM 452 catalystmailbox. Please scan you HW into a single PDF file and then submit it online.The
link to the dropbox is: https://catalyst.uw.edu/collectit/dropbox/mkhalil/19205No late homework will be accepted on this or future homework sets.This homework is worth a total of 10 points.Please box
your answers.Please give all answers in SI units.Refer to the article below and answer the following questions (same article as homework 2)Article: Lewis Nathan S. and Daniel G. Nocera. “Powering the
planet: Chemicalchallenges in solar energy utilization.” PNAS 103: (Oct. 2006) 15729-35.1.For each of the following statements indicate if they are True or False.a. The challenge with solar energy is
to dramatically decrease the cost per Watt forelectricity production.b. Three carbon-neutral fuel options according to this article are nuclear fission carbon capture and storage and clean coal.c.
Multi-layer quantum dots could be used to overcome the Shockley-Queisser limit(the maximum efficiency of most solar cells).d. Based on the statistics in the article the resource base of fossil energy
resourcescould support the world’s energy needs for several centuries to come.e. Without cost-effective storage solar energy could be the primary source ofelectrical power.2.Answer the following
questions. Your answers do not need to be in full sentences.a. Why will higher levels of carbon-neutral power be required by 2050 if theirintroduction does not start immediately with a constant
ramp-up?b. Why is it important to have low material costs in order to make a solar-basedprocess economical?c. What would the result be of having multiple-bandgap absorbers in a cascadedjunction
configuration?d. In the water splitting process what needs to happen in order to close the catalyticcycle?e. What are the only established molecular electro-catalysts for generating O2 fromH2O?
3.Assume that we can harness the energy stored in the bonds of H2 and O2. Using theenergy consumption rates tabulated in table 1 answer the following questions for2001 2050 and 2100. Looking back at
the calculation from homework 2 will behelpful.a. How many moles of water do you need to store all the energy consumed in oneyear in the form of hydrogen and oxygen bonds? Assume standard conditions.
(Answers: 1.49×1015 3.044×1015 and 4.76×1015 molH2O/year)1 Chem 452 – HW 6Winter 2012b. For comparison how many gallons is that? (Remember density of water 1g =1mL and 1gal = 3.785 L). (Answers:
7.09×1012 1.45×1013 and 2.26x1013gal/year)The following questions do not pertain to the article.4. Find ?S for the following processes:a. The isothermal reversible expansion at room temperature of
one mole of an ideal gasfrom 2L to 3L. (Answer: 3.4 J/K)b. One mole of ice at 265K is melted under usual lab conditions to form water at 329K.(Answer: about 37 J/K)5. Two blocks of the same metal and
mass are at different initial temperatures T1 and T2. Theblocks are brought into contact and come to a final temperature Tf. Assume the system istotally isolated from the surroundings and that the
surroundings are at equilibrium.a. Your intuition probably tells you that for 2 identical blocks Tf is the average of theinitial temperatures so that Tf = 1/2 (T1 T2). Show that for a system of two
blockstotally isolated from the surroundings that this is true.b. Show that the change in entropy is? S = CP ln(T1 T2 )24 T1 T2c. How does this expression show that this process is spontaneous?Hints:
(1) Since the blocks are of the same material they will have the same Cp. (2)Remember that the system is isolated. (3) For part c you can show this in various waysincluding assuming numerical values
for T1 and T2 in different cases.6. Steam is condensed at 100 °C and the water is cooled to 0 °C and frozen to ice. Whatis the molar entropy change of the water? Consider that the average specific
heat ofliquid water at constant pressure to be 4.2 J K-1 g-1 ?Hvap is 2258.1 J g-1 and ?Hmelt is333.5 J g-1. (Ans: -154.6 J K-1 mole-1)7. In class we said that the entropy of mixing for two ideal
gases is:?Smixing = – R ( nA ln( x A ) nB ln( xB ) )Say we have nA moles of gas A initially at 1 atm pressure mix with nB moles of gas Balso at 1 atm pressure to form 1 mole of a mixture of A and B
at a final pressure of 1 atm.The entire process occurs at a constant temperature T.a. Show that the entropy change is given by :?S mixing = – Rx A ln( x A ) – RxB ln( xB )b. What can you say about
the sign of ?S? Is it in keeping with the second law?c. Gibbs Free Energy can be written as dG = dH – d(TS). Given that temperature isconstant in this problem what is the sign of dG? Based upon this
is the mixingprocess spontaneous?2 Chem 452 – HW 6Winter 2012d. Given that xA xB = 1 what values of xA and xB give the largest entropy ofmixing? (Prove it). Hint: Recall how to prove that a function
is at its maximum.8. Text 5.26A 25.0-g mass of ice at 273 K is added to 150.0 g of H2O(l) at 360. K at constantpressure. Is the final state of the system ice or liquid water? Calculate ?S for the
process.Is the process spontaneous?9.a.b.c.d.e.f.Describe in one short phrase whether ?S will generally increase or decrease inthe following:Neutralization of charges in an aqueous solutionA polar
molecule is placed in a nonpolar solvent (or the opposite)DNA mixed with cationic lipids in an aqueous solution (drawing a picture ishelpful)A phase change from a gas to a liquidIncreasing the number
of molecules in the system (i.e. doubling the system)Increasing the temperature of the systemStretching a rubberband.3 | {"url":"https://geniuspapers.net/for-each-of-the-following-statements-indicate-if/","timestamp":"2024-11-04T05:34:00Z","content_type":"text/html","content_length":"54697","record_id":"<urn:uuid:2e35e168-cb5a-4c24-9d47-a4efeb8e5243>","cc-path":"CC-MAIN-2024-46/segments/1730477027812.67/warc/CC-MAIN-20241104034319-20241104064319-00143.warc.gz"} |
Programmes in Mathematics Department
1. Programme: B.Sc. Mathematics
Programme Code: 5208
A BRIEF HISTORY OF THE Department
Mathematics Department is one of the pioneer units in the Faculty of Sciences that was established as a servicing unit at the inception of the University in 2004. The unit continued in its servicing
capacity until the 2008 academic session when the full B.Sc. (Mathematics) programme took off.
The BSc. Mathematics programme is designed to equip the undergraduate students with the basic requirement for serving in a professional capacity in most areas of computation Mathematics as well as
develop knowledge in theory of applied Mathematics. The degree programme would explore all the basic rudimentary or foundation knowledge of computing technology not known to most of students with the
tools for computational techniques and thinking, as they would be exposed to the fundamentals of computing processes.
The BSc. Mathematics programme is aimed at taking you through the fundamental of the sciences of computation and the latest technologies that make the application of Mathematical Science an all-round
catalyst in the design of any new market driven technological designs and devices without the constraints of face to face teaching.
• To produce competent graduates of Mathematics with sound knowledge and skills to contribute to the rapid technological growth of the Nigerian society and the word at large.
• To produce competent graduates who will seek to advance and exploit entrepreneurial opportunities in the field of Mathematics.
• To produce graduates who will utilize their Mathematics knowledge, skills and abilities to enhance safety, health and welfare of the public through the simulation, construction and maintenance of
industrial equipment.
• To produce graduates that will satisfy the manpower needs of our society in sectors of energy, industry, communication, science, engineering and research.
To be admitted into the B.Sc. Mathematics programme, a candidate is expected to possess at least one of the following:
1. Five (5) credit passes in Senior School Certificate Examination (SSCE) or at the School Certificate (SC), General Certificate of Education (GCE) Ordinary Level, National Examinations
Council (NECO) or 6 merit passes in National Board for Technical Education (NABTEB) or Teachers Grade Two Certificate (TC II) examinations. The credit passes must include Mathematics and
Physics. Credit pass in English language is required.
2. General Certificate of Education (GCE) Advanced level in Mathematics and Physics for entry into 200 level of the programme.
3. National Certificate in Education (NCE) with merit passes in Mathematics and Physics or Physics and Chemistry for entry into 200 level of the programme
4. National Diploma (N.D.) in the Mathematical sciences or equivalent qualification from an institution recognized by Senate for entry into 200 level of the programme.
5. Degree or Higher National Diploma (HND) or equivalent qualification in any physical science from an institution recognized by Senate for entry into 200 level of the programme.
Note: All direct entry candidates must satisfy the ordinary level requirement.
Evaluation: There are two aspects to the assessment of this programme. First, there are tutor marked assignments (TMA) which is 30% of the total course mark. At the end of the course, 100 – 200 level
students would sit for Computer Based Test CBT (e-examination) and 300-400 level students would sit for written examination called Pen on Paper (POP) which has a value of 70% of the total course
Structure of the Programme:
Subjects taught in the Unit are based on the ‘course system’ in which the subject areas are broken down into courses which are examinable. The courses are organized into levels (100-400 levels) in an
order according to the academic progress.
• Classification of Courses
The courses in the Unit are classified as follows:
1. Compulsory courses: These are the core courses that must be offered and passed by students at a grade not below E
2. Elective Courses: These are optional courses which may be offered based on the interest of the student or for the purpose of fulfilling the minimum requirement for the award of the degree.
3. General Studies Courses: These consist of the university general studies courses coded GST. They are compulsory courses for all students of the university and are being offered by the University
in compliance with the National University Commission (NUC) minimum Bench Mark.
• Criteria for the award of B.Sc (Mathematics) degree
The student is required to pass all compulsory courses and complete a minimum of 140 credits units of core courses and at least 12 units of electives for 8 semesters to qualify to be admitted into
the B.Sc. Mathematics degree. Direct entry must pass minimum of 110 credit units of core courses and at least 10 units of elective courses for a 6 semesters to qualify to be admitted into the B.Sc.
Mathematics degree. The compulsory courses are made up of those courses specifically labeled as compulsory (C) and the required elective courses labeled as elective (E).
1. OUTLINE PROGRAMME PROPOSAL (OPP)
Outline of Course Structure Mathematics Programme
100 Level 1^st Semester
Course Code Course Title Unit(s) Status
BIO101 General Biology I 2 C
BIO191 General Biology Practical I 1 C
CHM101 Introductory Inorganic Chemistry 2 C
CHM103 Introductory Physical Chemistry 2 C
CHM191 Introductory Practical Chemistry I 1 C
CIT104 Introduction to Computer Science 2 C
MTH101 Elementary Mathematics I 3 C
MTH103 Elementary Mathematics II 3 C
PHY101 Elementary Mechanics, Heat and Properties of Matter 2 C
PHY191 Introductory Practical Physics I 1 C
GST101 Use of English and Communication Skills 2 C
GST107 The Good Study Guide 2 C
Total Credit Units 23
100 Level 2^nd Semester
Course Code Course Title Unit(s) Status
BIO102 General Biology II 2 C
BIO192 General Biology Practical II 1 C
CIT102 Software Application Skills 2 C
CHM102 Introductory Organic Chemistry 2 C
CHM192 Introductory Practical Chemistry II 1 C
MTH102 Elementary Mathematical II 3 C
STT102 Introductory Statistics 2 C
PHY102 Electricity, Magnetism and Modern Physics 3 C
PHY192 Introductory Physics Laboratory II 1 C
GST102 Use of English and Communication Skills II 2 C
200 Level – 1^st Semester
Course Code Course Title Unit(s) Status
CIT215 Introduction to Programming Languages 3 C
MTH211 Abstract Algebra 3 C
MTH213 Numerical Analysis I 3 C
MTH241 Introduction to Real Analysis 3 C
MTH281 Mathematical Methods I 3 C
STT211 Probability Distribution I 3 C
GST201 Nigerian Peoples and Culture 2 C
GST203 Introduction to Philosophy and Logic 2 C
Elective 2 E
Total Credit Units 24
Elective Courses
PHY207 Thermodynamics 2 E
PHY201 Classical Dynamics 3 E
200 Level – 2^nd Semester
Course Code Course Title Unit(s) Status
MTH212 Linear Algebra II 3 C
MTH232 Elementary Differential Equation 3 C
MTH210 Introduction to complex analysis 3 C
MTH251 Mechanics 3 C
MTH282 Mathematical Methods II 3 C
GST202 Fundamentals of Peace Studies and Conflict Resolutions 2 C
Elective 2 E
Total Credit Units 19
Elective Courses
PHY204 Electrodynamics 2 E
PHY206 Optics I 2 E
300 Level – 1^st Semester
Course Code Course Title Unit(s) Status
MTH301 Functional Analysis I 3 C
MTH304 Complex Analysis I 3 C
MTH311 Calculus of Several Variables 3 C
MTH341 Real Analysis 3 C
MTH381 Mathematical Methods III 3 C
MTH303 Vector and Tensor Analysis 3 C
GST301 Entrepreneurial Studies 2 C
Elective 3 E
Total Credit Units 23
Elective Courses
MTH307 Numerical Analysis II 3 E
STT311 Probability Distribution II 3 E
300 Level 2^nd Semester
Course Code Course Title Unit(s) Status
MTH302 Elementary Differential Equation II 3 C
MTH305 Complex Analysis II 3 C
MTH308 Introduction to Mathematical Modeling 3 C
MTH312 Abstract Algebra II 3 C
MTH382 Mathematical Methods IV 3 C
Elective Course 3 E
Elective Courses
MTH309 Optimization Theory 3 E
MTH315 Analytical Dynamics I 3 E
400 Level – 1^st Semester
Course Code Course Title Unit(s) Status
MTH401 General Topology I 3 C
MTH411 Measure Theory and Integration 3 C
MTH421 Ordinary Differential Equation 3 C
MTH423 Integral Equation 3 C
Elective Course 3 E
Total Credit Units 15
Electives Courses
MTH417 Electromagnetic Theory 3 E
CIT425 Operation Research 3 E
400 Level – 2^nd Semester
Course Code Course Title Unit(s) Status
MTH402 General Topology II 3 C
MTH412 Functional Analysis II 3 C
MTH422 Partial Differential Equation 3 C
MTH499 Project 6 C
Total Credit Units 15
1. SYNOPSES OF COURSES AND DETAILED PROGRAMME PROPOSAL (DPP)
BIO101: GENERAL BIOLOGY I (2 UNITS)
Characteristics of living things; cell as the basic unit of living things, cell structure, organization, cellular organelles, tissues, organs and systems.
Classification of living things, general reproduction and concept of inter-relationships of organism. Heredity and evolution. Elements of ecology (introduction) and habitats.
BIO102 GENERAL BIOLOGY II (2 UNITS)
Systematic studies of diversity of life including monera, protista, plants (Algae, Fungi, Bryophytes, Pteridophytes, Gymnosperms and angiosperms) and animals (Protozoa, Platyhelminthes, Annelids,
Arthropods, Fishes, Amphibians, Reptiles, Birds and Mammals) based on similarities and differences in external morphology. Taxonomic divisions of plant and animal kingdoms. Ecological adaptations of
these forms.
BIO191 GENERAL BIOLOGY PRACTICAL I (1 UNIT) What practical work in biology involves. Laboratory organization. Handling common laboratory equipment. Microscopic handling and maintenance. Making
microscopic measurements. Procuring animal materials for practicals. Killing, preserving and maintaining animal materials. Procuring plant materials. External features of plants (differences and
similarities). Preparation of temporary slides. Preparation of stains and reagents. Techniques for microbial culture and grain staining. Setting up demonstration for physiological processes in
plants. Setting up apparatus for demonstrating physiological processes in animals. Preparation required for dissection.
BIO192 GENERAL BIOLOGY LABORATORY II (1 UNIT) Observation and description of the morphological and diagnostic features as well as the differences among the different phyla of the plant, animal,
archebacteria, eubacteria, fungi and protista kingdoms. Identification of the taxonomic hierarchy of the members of the above groups. Study of the structure and functions of their parts and habitats
CHM101: Introductory Inorganic Chemistry (2 units)
Hypothesis, theory and law with appropriate illustrations, Nature of matter – 3 states of matter, Atomic structure, electronic energy levels and orbital. Periodic classification of elements and its
relationship to their electronic configurations, Chemical bonding, Survey of properties and trends in groups I, II, IV, VI and transition metal,
CHM102: introductory organic chemistry (2 units)
Simple reactions of hydrocarbons, alcohols, and acids. Petroleum chemistry, Oils and fats, hydrogenation of oils, polymer and biologically important molecule.
CHM103: Introductory Physical Chemistry (2 units)
Mole concepts and calculations based on it, methods of expressing concentrations, Chemical Kinetics and equilibrium, and related calculations, Important application of equilibrium – pH, solubility
products and solubility of ionic solids, Thermo chemistry and simple calculations based on Hess’s law, Electrochemistry and working of various cells, Brief mentions of corrosion; chemical
thermodynamics; DG = DH – TDS
CHM191: Introductory practical chemistry I (1 unit)
Practical based of CHM 101 and CHM 103: Cations and anions – identification, Acid- base titrations, Redox reactions and determinations
CHM192: Introductory practical chemistry II (1 unit)
Practical based on general chemistry CHM 101 and introductory organic chemistry I CHM 102- Determination of melting and boiling points and reaction of functional groups.
GST101: USE OF ENGLISH AND COMMUNICATION SKILLS I (2 UNITS)
Listening enabling skills, listening and comprehending comprehension, note taking and information retrieval. Including data, figures, diagrams and charts. Listening for main idea, interpretation and
critical evaluation. Effective reading. skimming and scanning. Reading and comprehension at various speed levels. Vocabulary development in various academic contexts. Reading diverse texts in
narratives and expository. Reading and comprehension passages with tables, scientific texts. Reading for interpretation and critical evaluation.
GST102: USE OF ENGLISH AND COMMUNICATION SKILLS II (2 UNITS)
Writing paragraphs: Topic sentence and coherence. Development of paragraphs: illustration, Description, cause and effect including definitions. Formal letters; essential parts and stylistic forms,
complaints and requests; jobs, ordering goods, letters to government and other organizations. Writing reports; reporting event, experiments. Writing summaries: techniques of summarizing letters and
sounds in English, vowels and consonants. Interviews, seminar presentation, public speech making, articles, concord and sentences including tenses. Gerund, participles, active, passive and the
infinitive. Modal auxiliaries.
GST105 HISTORY AND PHILOSOPHY OF SCIENCE (2 UNITS)
Nature of science, scientific methods and theories; Law of nature,; History of science. Lost sciences of Africa, science, technology and inventions. Nature and scope of philosophy in science. Man,
nature and his origin. Man, environment and resources. Great Nigerian Scientists.
GST107: THE GOOD STUDY GUIDE. (2 UNITS)
Getting started: How to use the book, why read about skills, getting yourself organised ; what is studying all about, reading and note taking; Introduction, reactions to reading, your reading
strategy, memory, taking notes, conclusion. Other ways of studying: Introduction, learning in groups, talks and lectures, learning from TV and radio broadcasts, other study media. Working with
numbers; Getting to know numbers, describing the world, describing with the tables, describing with diagrams and graphs; What is good writing? The Importance of writing, what does an essay look
like, what is a good essay? Conclusion. How to write essays: Introduction, the craft of writing, the advantages of treating essay writing as a craft, making your essay flow, making a convincing case,
the experience of writing. Preparing for examination.
MTH101 ELEMENTARY MATHEMATIC I: (3 Units)
Elementary set theory, subsets, union, intersection, complements, venn diagrams. Real numbers; integers, rational and irrational numbers, mathematical induction, real sequences and series, theory of
quadratic equations, binomial theorem. Complex numbers; algebra of complex numbers; the Argand Diagram. Re Moivre’s theorem, nth roots of unity. Circular measure, trigonometric functions of angles of
any magnitude, addition and factor formalae.
MTH102 ELEMENTARY MATHEMATICS III: (3 UNITS) CALCULUS:
Function of a real variable, graphs, limits and idea of continuity. The derivative as limit of rate of change, Techniques of differentiation, Extreme curve sketching. Integration as an inverse of
differentiation, Methods of integration, Definite integrals; Application to areas and volumes
MTH103 ELEMENTARY MATHEMATICS III: (3 Units) PRE-REQUISITE – MTH 101
Geometric representation of vectors in 1-3 dimensions, components, direction cosines. Addition and Scalar multiplication of vectors and linear independence. The Scalar and vector products of two
vectors. Differentiation and integration of vectors with respect to a scalar variable. Two-dimensional co-ordinate geometry. Straight lines, circles, parabola, ellipse, hyperbola. Tangents, normals.
STT102 INTRODUCTORY STATISTICS (2UNITS)
Measures of Central Tendency and dispersion, (grouped and ungrouped); mean: – arithmetic and geometric, harmonic, median, mode quartiles, deciles, modes, relative and absolute dispersion, sample
space and events as sets. Finite probability space properties of probability. Statistical independenceand conditional probability. Tree diagram. Bayes theorem. Discrete and continuous random
variables. Expectation, independent Bernoulli trials. Binomial Poisson and Normaldistributions. Normal approximation to binomial and Poisson distribution, Hyper geometric.
PHY101: Elementary Mechanics, Heat and Properties of Matter (3 UNITS)
Space and Time: Physical quantities: Units and dimensions of physical quantities; Kinematics: Uniform velocity motion, uniformly accelerated motion; Dynamics: Newton’s laws of motion; Impulse and
Linear Momentum, Linear Collision, Newton’s universal law of gravitation; Work, energy and power; Conservation laws; Concept of mechanical equilibrium; Centre of mass and centre of gravity; Moment of
a force; Rotational kinematics and dynamics: Torque; Moment of Inertia; angular momentum; Total mechanical energy. Simple harmonic motion
Heat and temperature, work and heat, Quantity of heat: heat capacities, latent heat; Thermal expansion of solids, liquids and gases; Gas laws, heat transfer; Laws of thermodynamics: Isothermal and
Adiabatic changes, Carnot cycle; Application kinetic theory of gases; van der Waals gas.
Classification of matter into (solids, liquids and gases, forces between atoms and molecules, molecular theory of matter, Elasticity, plasticity, Hook’s Law, Young’s Shear and bulk Moduli)
Crystalline and non-crystalline materials, Hydrostatics: pressure, buoyancy, Archimedes’ principle; Hydro-dynamics-streamlines, Bernouli and Continuity equations, turbulence, Reynold’s number,
Viscosity, laminar flow, Poiseuille’s equation; Surface tension, adhesion, cohesion, capillary, drops and bubbles.
PHY102: ELECTRICITY, MAGNETISM AND MODERN PHYSICS (3 UNITS)
Electrostatics: Coulomb’s law, Gauss’s law, potential and capacitance, dielectrics, production and measurement of static electricity. Current: Ohm’s law, resistance and resistivity, heating.
Galvanometers, Voltmeters and Ammeters; D.C. circuits: sources of emf and currents, Kirchhoff’s laws; Electrochemistry; The Earth’s magnetic field; Magnetic fields and induction, Faraday’s and Lenz’s
laws; Force on a current-carrying conductor. Biot-Savart law. Flemming’s right and left-hand rules, motors and generators. A.C. Theory. Atomic structure; Production and properties of X-rays;
Radioactivity; Photoelectric emission.
PHY191: Introductory Practical Physics I (1 unit)
Graphs, Measurement, Error Analysis, Determination of Acceleration due to Gravity by Means of Simple Pendulum, Determination of force constant of a spiral spring, Determination of effective mass of a
spiral spring and the constant, Determination of surface tension of water, Determination of specific latent heat of fusion of ice, Determination of the co-efficient of limiting static friction
between two surfaces, Determination of the co-efficient of static friction on two surfaces using an inclined plane, Determination of Relative Density of kerosene using the specific Gravity Bottle,
Determination of the Relative Density of a Granular substance not soluble in water using the specific gravity bottle.
PHY192: Introductory Practical Physics II (1 unit)
Refraction through the glass block; Image formed by a concave mirror; Determination of the focal length of the convex mirror; Refraction through the triangular prism; Determination of the focal
length of a converging lens and the refractive index of groundnut; Determination of resistance of resistors in series and in parallel in simple circuits; Determination of internal resistance of a dry
cell using a potentiometer; To compare the E.M.F. of cells using potentiometer; Determine the unknown resistance of a resistor using Wheatstone Bridge; To determine the relationship between current
through a Tungsten and a potential applied across it.
CIT215: INTRODUCTION TO PROGRAMMING LANGUAGES (3UNITS)
FORTRAN programming language; Comparison of various versions of the language. Programming exercises using FORTRAN with emphasis on scientific application problems. Elements of Pascal language.
Exercises in Pascal Program structures and programming concepts; Structured design principles; abstraction, modularity, stepwise refinement, structured design techniques teaching of a structured
programming language, e.g. PASCA/JAVA, C^++.
GST203: INTRODUCTION TO PHILOSOPHY AND LOGIC (2 UNITS)
General introduction to logic; clarity of thought; expression and arguments as basis for conclusion. Fundamentals of logic and critical thinking, types of discourse, nature of arguments; validity and
soundness ; distinction between inductive and deductive inferences etc; illustrations from familiar texts, including literature materials, novels, law reports and newspaper publications.
MTH210: INTRODUCTION TO COMPLEX ANALYSIS (3UNITS)
Complex number, the topology of complex plane. Limits and continuity of function of complex variables, properties and example of analytic functions, branch-points, Cauchy-Riemann equations. Harmonic
MTH211: ABSTRACT ALGEBRA I (3UNITS)
Set: Binary operations, mapping, equivalence relations integers: Fundamental theorem of arithmetic, congruence equations, Euler’s function (n) Group Theory: Definition and examples of groups.
Subgroups, coset decomposition, Lagrange’s theorem. Cyclic groups. Homeomorphisms, isomorphism. Odd and even permutations. Cayley’s theorem. Rings: Definition and examples of rings. Commutative
rings. Integral domain. Order, well-ordering principles. Mathematical induction.
MTH212: LINEAR ALGEBRA II (3UNITS)
Vector spaces. Liner independence. Basis, change of basis and dimension. Linear equations and matrices. Linear maps. The diagonal, permutation, triangular matrices. Elementary matrix. The
inverse of a matrix. Rank and nullity. Determinants. Adjoint, cofactors, inverse matric. Determinant rank. Crammer’s rule. Canonical forms, similar matrices, Eigen values and vectors, quadratic
MTH213: NUMERICAL ANALYSIS I (3UNITS) PRE-REQUISITE – MTH 102
Interpolation: Lagrange’s and Hermite interpolation formulae, divided differences and difference schemes. Interpolation formulas by use of divided differences. Approximation: Least-square
polynomial approximation, Chebychev polynomials continued fraction and rational fraction orthogonal polynomials.
Numerical Integration: Newton’s-cotes formulae, Gaussian Quadrature. Solution of Equations: Graffe’s method (iterative method) Matrices and Related Topics: Definitions, Eigenvalue and Eigenvectors,
Algebraic Eigenvalue problems-power method, Jacobi method.
Systems of linear Equations: Gauss elimination, Gauss-Jordan method. Jacobi iterative method, Gauss-field iterative method.
MTH232: ELEMENTARY DIFFERENTIAL EQUATION (3UNITS)
PRE-REQUISITE – MTH 103
Introduction, equation of first order and first degree, separable equations, homogeneous equations, exact equations, linear equations, Bernoulli’s and Riccati equations. Applications to mechanics
and electricity. Orthogonal and oblique trajectories. Second order equations with constant coefficients.
MTH241: INTRODUCTION TO REAL ANALYSIS (3UNITS)
Sets: Cartesian products, functions and mappings direct and inverse images. Countable sets. Limits: Elementary properties of limits. Upper and lower bounds, supremum, infimum, convergence of
sequences. Limit of monotone functions and sequences. Cauchy convergence principles. Continuity: Real-Valued functions of a real variable Monotone functions, periodic functions, bounded
functions. Continuity of functions using neighborhood. Elementary properties of continuous functions. Uniform continuity. Series: convergence of series, tests for convergence, absolute
convergence, power series, uniform convergence.
MTH251: MECHANICS
Static: System of live vectors. Coyoles and wrenches. Principles of virtual work. Stability of equilibrium. Dynamics of systems of particles: Elastic strings. Hooks law. Motion in resisting
media. Changing mass. Motion along a curve. Frenets formulae.
Coplanar Motion: Energy equation. Motion in a vertical circle. Simple pendulum. The cycloid and cycloidal motion. Orbital motion-disturbed orbits and stability.
MTH281: MATHEMATICAL METHOD I (3UNITS) PRE-REQUISITE – MTH 103
Sequences and Series: Limits, continuity, Differentiability, implicit functions, sequences. Series, test for convergence sequences and series of functions. Calculus: partial differentiation, total
derivatives, implicitly functions, change of variables. Taylor’s theorem and maxima and minima functions, of two variables. Lagrangian multiplier. Numerical Methods: Introduction to iterative
methods, Newton’s method applied to finding roots. Trapezium and Simpson’s rules of integration.
MTH282: MATHEMATICAL METHODS II (3UNITS) PRE-REQUISITE – MTH 281
Vector Theory: Vector and scalar field functions. Grad, divi, curl, directional derivatives. Orthogonal curvilinear coordinates.
Complex Numbers: The algebra and geometry of complex numbers; de’moivre’s theorem. Elementary transcendental functions. The n^th root of unity and of a general complex number.
PHY202: MODERN PHYSICS I (3 UNITS) PREREQUISITES: PHY102
Atomic structure: Experimental basis of quantum theory: Black body radiation; electrons and quanta; Charge quantization, Mass spectra, the plum pudding model, Rutherford model and Bohr models of the
atom, Hydrogen spectra, Magnetic moment and Angular momentum of an atom, Electron spin, Pauli Exclusion Principle and electronic configuration, X-ray spectra, De Broglie hypothesis, the uncertainty
principle; Wave-particle duality, Schrodinger’s equation and simple applications; Nuclear Structure: nomenclature, binding energy and stability, Radioactivity, The radioactive series, Accelerators,
Detectors. Bohr’s theory of atomic structure.
PHY204: ELECTROMAGNETISM (2 UNITS) PREREQUISITES:PHY102, Macroscopic properties of dielectrics: polarisation, Gauss’s law in a dielectric, the displacement
vector, boundary conditions on D and E, dielectric strength and breakdown; Capacitor: capacitance, the parallel plate capacitor, effect of a dielectric, energy stored in a dielectric medium,
capacitors in series and parallel, practical capacitors; Microscopic properties of dielectrics: microscopic picture of a dielectric in a uniform electric field, determination of local field,
Clausius-Mossotti equation, behaviour of dielectric in alternating fields; Magnetism of materials: response of various substances to a magnetic field, magnetic moment and angular momentum of an atom,
diamagnetism and paramagnetism, Lamor precession, magnetization of paramagnets, ferromagnetism, magnetic field due to a magnetized material, magnetic intensity, relationship between E and H for
magnetic material, magnetic circuits.
PHY206: OPTICS I (2 UNITS)
Nature of light: the corpuscular model, the wave model, light as an electromagnetic wave; Reflection and refraction of light: electromagnetic waves at the interface separating two media, idealization
of waves as light rays, Fermat’s principle; Perception of light: human vision, colour vision; Polarization of light: simple states of polarized light, principles of producing linearly polarized
light, wave plates.
STT211: PROBABILITY DISTRIBUTION I (3UNITS) PRE-REQUISITE – STT 102
Discrete sample spaces: Algebra and probability of events, combinatorial analysis. Sampling with and without replacement. Conditional probability, Bayes theorem and stochastic independence.
Discrete distributions: Binomial, Poisson, negative binomial-hyper geometric and multinomial. Normal approximation to binomial and Poisson, Poisson approximation to binomial. Random variables and
expectations: mean, variance, covariance. Probability generating function and moment generating function. Cheycher’s inequality. Continuous joint distributions: marjind as conditional density.
Expectations: movement, movement generating functions. Uniform normal, beta Cauchy and hop-normal distributions.
MTH301: FUNCTIONAL ANALYSIS I (3UNITS) PRE-REQUISITE – MTH 241
Metric Spaces – Definitions and examples. Open Sphere of (balls) closed sets, interior, exterior, frontier, limit points and closure of a set. Dense subsets and separable space. Convergence in
metric space, homeomorphism, continuity and compactness.
MTH302: ELEMENTARY DIFFERENTIAL EQUATION II (3UNITS)
PRE-REQUISITE – MTH 282
Series, solution of second order linear equations. Bessel, legendry and hyper geometric equations and functions. Gamma and Beta functions. Storm Lionvelle problems. Orthogonal polynomial and
functions, Fourier, Fourier, Bessel and Fourier – legendry series. Expansion in series of orthogonal functions. Fourier transformation. Laplace transforms solution of wave and heat equations by
Fourier method.
MTH303: VECTOR AND TENSOR ANALYSIS (3UNITS) PRE-REQUISITE – MTH 103
Vector algebra, Vector dot and cross products. Equation of curves and surfaces. Vector differentiation and application. Gradient, divergence and curl. Vector integration, line, surface and volume
integrals, Green stoke’s and divergence theorems. Tensor products and vector spaces tensor algebra, symmetry, Cartesian tensors.
MTH304: COMPLEX ANALYSIS I (3UNITS) PRE-REQUISITE – MTH 101
Functions of a complex variable. Limits and continuity of functions of a complex variables. Deriving the Cauchy-Riemann equations. Analytic functions. Bilinear transformations, conformal mapping.
Contour Integrals, Cauchy’s theorems and its main consequences. Convergence of sequences and series of functions of complex variables.Power Series, Taylor Series.
MTH 305: COMPLEX ANALYSIS II (3UNITS) PRE-REQUISITE – MTH 304
Laurent expansions, isolated singularities and residues, residue theorem, calculus of residue and application to evaluation of integrals and to summation of series. Maximum modulus principle.
Argument principle. Rouche’s theorem. The fundamental theorem of algebra. Principle of analytic continuation, multiple valued functions and Riemann surfaces.
MTH307: NUMERICAL ANALYSIS II (3UNITS) PRE-REQUISITE – MTH 213
Polynomial and Splines approximations: Orthogonal polynomials and chebychev approximations, least squares, cubes spline, Hermits approximations, Numerical Integration. Boundary value problem.
Introduction to numerical solution of partial differential equations.
MTH308: INTRODUCTION TO MATHEMATICAL MODELING (3UNITS)
Methodology of the Model building. Identification, formulation and solution of problems, cause – effect diagrams, equation types, algebraic, ordinary differential, partial differential, difference,
integral and functional equations. Application of Mathematical model to physical, biological, social and behavioural sciences.
MTH309: OPTIMIZATION THEORY (3UNITS)
Linear programming models. The simplex method, formulation and theory. Duality, integer programming. Transportation problem, two-person zero-sum games. Non – linear programming, quadratic
programming Kuhn tucker methods, optimality criteria simple variable optimization. Multivariable techniques, Gradient methods.
MTH311: CALCULUS OF SEVERAL VARIABLES (3UNITS) PRE-REQUISITE – MTH 282
Value, Limit and Continuity of functions of several variables. Partial derivatives of function of several variables. Total derivative of a function. Partial Differentials and Total Differentials
of f(x,………x [n )) . ]Composite differentiation. Fuller’s Theorem. Implicit Differentiation. Taylor’s Series for function of two variables. Maxima and Minima of functions
of several variables. Lagrange’s Multipliers. Differentials under integral sign, The Jacobians
MTH312: ABSTRACT ALGEBRAII (3UNITS) PRE-REQUISITE – MTH 241
Normal subgroups and quotient groups. The isomorphism theorem. Symmetric groups, automorphism, conjugate classes, Normalisers. The sylow theorems. Normal and composition series. The
Jordan-Holder theorem. Direct product. Solvable group. Isomorphism theorems for rings. Ideals and quotient rings. Commutative ring, maximal ideals. Euclidean rings, principal ideal domain and
unique factorization domain.
MTH315: ANALYTICAL DYNAMICS I (3UNITS) PRE-REQUISITE – MTH 251
Degrees of freedom, Holonomic and non-holonomic constraint. Generalized coordinates. LaGrange’s equation for holonomic systems, force dependent on coordinates only, force obtainable from a
potential, Impulsive force, variational principles, calculus of variation, Hamilton principles. Canonical transformation, normal modern of variation, Hamilton Jacobi equation.
The notion of displacement, speed, velocity and acceleration of a particles. Newton’s law of notions and applications to simple problems. Work, power and energy. Application of the principle of
conservation of energy to notion of particles and those involving elastic string and springs. Simple Harmonic motion. Resultant of any number of forces acting on a particle. Reduction of coplanar
forces acting on a rigid body to a force and a couple. Equilibrium of coplanar forces, parallel forces, couples Laws of friction. Application of the principle of moments. Moments of Inertia of
simple bodies.
MTH318: FLUID MECHANICS I (3UNITS) PRE-REQUISITE – MTH 251
Real and Ideal fluid. Differentiation following the motion of fluids particles, Equation of continuity. Equation of motion for incompressible in viscid fluids. Velocity potential and stoke’s stream
function. Bernoulli’s equation with applications. Kinetic Energy. Sources, sinks, doublets in 2 and 3 dimensions stream lines. Images. Use of conformal transformation.
MTH341: REAL ANALYSES (3UNITS) PRE-REQUISITE – MTH 312
Integration: The integral as the area of the ordinate set of a function. Definition of the Riemann integral of bounded functions. Conditions for integrality. Properties of the integral.
Relations between integrals and derivatives. Approximation to integrals by sum.
The Riemann Integral: Riemann-Sieltejes integral. Properties, functions of bounded variation and extension to the notion of integration. Sequences and Series of Functions: Convergence of
sequences and series of functions. Uniform convergence. Continuity of sum of a uniform convergent series of continuous functions. Terms by term integration and differentiation of a series of
continuous functions. Applications to power spaces metric spaces.
MTH381: MATHEMATICAL METHODS III (3UNITS) PRE-REQUISITE – MTH 303
Functions of several variables: Jacobian, functional dependence and independence. Multiple integrals, line integrals. Improper integrals. Vector Field theory: Relations between vector field
functions. Integral theorems. Gauss’s. Stoke’s and Green’s theorems. Elementary tensor calculus. Functions of a complex variable: The Cauchy-Riemann equations. Integration of complex plane.
Cauchy’s theorem Cauchy’s inequality. The residue theorem and the evaluation of integrals. Integral Transforms: Fourier and Laplace transforms. Convolution properties and their applications.
MTH382: MATHEMATICAL METHODS IV (3UNITS) PRE-REQUISITE – MTH 281
Ordinary Differential Equations: The concept of existence and uniqueness of solutions. Operational methods of solution of linear equations. Sturm-Lionvelle theory, Green’s functions, series
solution. Special functions and some of their elementary properties; Gamma and Beta functions. Partial Differential Equations: Solutions of boundary and eigenvalue problems of partial differential
equations by various methods which include: Separation of variables, transform techniques. Sturn-Liouville theory; Green’s functions; method of characteristics.
STT311: Probability Distribution II (3units) PRE-REQUISITE – STT 211
Probability spaces measures and distribution. Distribution of random variable spaces. Product probabilities. Independence and expectation of random variables. Convergence of random variables.
Week convergence almost everywhere, laws of large numbers. Characteristic function and inversion formula.
STT316: MULTIVARIATE ANALYSIS AND APPLICATION (3UNITS)
PRE-REQUISITE – STT 311
Vector random variables. Expectations of random vectors and matrices. Multivariate normal distribution and distribution of quadratic forms. Application to linear models: Tests of general linear
hypothesis and estimation. Least square theory: Guass-Markoff and general linear hypothesis with applications to regression and experimental design models. Estimation: partial and multiple
correction coefficients, mean vector and co-variance matrix. Hatelting’s T^2 and Wishart distribution: multivariate ANOVA.
MTH401: GENERAL TOPOLOGY I (3UNITS) PRE-REQUISITE – MTH 301
Point Set Topology: The space R^n Euclidean metric. Metrics, open spheres, metric topologies, metric spaces, properties of metric topologies. Equivalent metric. Heine-Borel theorem.
Bolzano-wierstress theorem. Properties of separable, complete, compact, locally-compact and connected spaces. Cantor’s set. Continuity and uniform continuity of mappings on metric space.
Topological spaces: Definitions, examples, accumulation points, closes sets, closure, interior, exterior and boundary of a set. Neighborhoods and neighborhood systems. Coarser and finer topologies,
subspaces and relatives topologies. Base for a topology sub bases.
MTH402: GENERAL TOPOLOGY II (3UNITS) PRE-REQUISITE – MTH 401
Separation axioms: T-spaces, Hausdorff spaces, Regular spaces. Normal spaces, Urgsohn’s lemma. Category and seperability: Dense sets, nowhere dense sets. Sets of the first and second categories.
Perfectly separable spaces. Separable spaces. The axiom of count ability. Compactness: Covers, compact sets, subsets of compact spaces. Sequentially, count ably and locally sets.
Compactification. Product spaces: product topology. Base for a finite product topology. Tychonoff product theorem. Connectedness: separated sets, connected sets, connected spaces. Connectedness of
the real line. Components. locally-connected spaces. Homotopic paths. Homotopy relations. Simple connected spaces.
MTH411: MEASURE THEORY AND INTEGRATION (3UNITS)
PRE-REQUISITE – MTH 301
Measure Theory: Measure of open, closed sets. Outer and inner measure. Measurable sets. Properties of measure. Non-measurable sets. Measurable in the scene of Borel. Measurable space.
Measurable functions. Simple function Algebra. The Lebesgue integral: Lebesgue measure. Integral of non-negative function. Integral as measure of ordinate set, as a limit of approximate sums.
Integral of an unbounded function. Integral over an infinite range. Simple properties of the integral. Sequences of integral (Positive functions; functions with positive and negative values).
Lesbesgue monotone convergence theorem. Fatou’s Lemma, Dominated convergence. Bepo’s Lemma-Bounded Convergence. Sets of measure zero. Integration by parts. Fubini’s theorem and applications to
multiple integrals.
MTH412: FUNCTIONAL ANALYSIS II (3UNITS) PRE-REQUISITE – MTH 411
Normal Linear Space: Definition and examples. Convex sets. Norms. Holder’s minkowski’s inequalities. Riese-Fisher theorem. Linear operations on finite dimensional spaces. Linear functionals
spaces. Banach spaces, examples. Quotient spaces. Inner product spaces. Topological linear spaces. Hilbert space, examples. Linear operators in Hilbert spaces. Adjoint operators. Hermitian
operators. Orthogonality; orthogonal complement and projections in Hilbert spaces.
MTH414: ANALYTICAL DYNAMICS II (3UNITS) PRE-REQUISITE – MTH 315
Lagrange’s equations for non-holonomic systems. Lagrangran multipliers. Variational principles. Calculus of variation, Hamilton’s principle, Lagrange’s equation from Hamilton’s principles.
Canonical transformation Normal modes of vibrations. Hamilton-Jacobian equations.
MTH415: SYSTEM THEORY (3UNITS) PRE-REQUISITE – MTH 341
Lyapunov theorems. Solution of Lyapunov stability equation
A^TP + PA= -Q. Controllability and observability. Theorems on existence of solution of linear systems of differential operations with constant coefficient.
MTH416: ALGEBRAIC NUMBER THEORY (3UNITS)
Algebraic numbers; quadratic and cyclotomic fields. Factorization into irreducible, ideals, Murkowski’s theorem, class-group and class number, Fermat’s last theorem Dirichilet’s unit theorem.
MTH417: ELECTROMAGNETIC THEORY (3UNITS) PRE-REQUISITE –PHY 204
Maxwell’s field equations. Electromagnetic waves and electromagnetic theory of lights. Plane detromagnetic waves in non-conducting media, reflection and refraction of plane boundary. Wave guide
and resonant cavities. Simple radiating systems. The Lorentz-Einstein transformation. Energy and momentum. Electromagnetic 4-Vectors. Transformation of (E.H) fields. The Lorentz force.
MTH421: ORDINARY DIFFERENTIAL EQUATIONS (3UNITS)
Existence and uniqueness theorems, dependence of solution on initial data and parameters. Properties of solutions. General theory for linear differential equation with constant coefficients, the
two-point Sturm-Liouville boundary value problem, self-adjointness, linear and non-linear equations, Theorems and solution of Lyapunov equation. Controllability and observability.
MTH422: Partial Differential Equation (3units) PRE-REQUISITE – MTH 421
Theory and solutions of first order equations. Second order linear equations. Classification, characteristics canonical forms, Cauchy problem. Elliptic equations. Laplace’s and Poisson’s
formulae, properties of harmonic functions. Hyperbolic equations, retarded potential transmission line equation, Riemann methods, parabolic equation, diffusion equation, singularity function
boundary value and initial value problems.
MTH423: INTEGRAL EQUATION (3UNITS) PRE-REQUISITE – MTH 103
Integral Equation: Classification – Volterra and Fredholm types. Transformation of Differential Equations. Neumann series. Fredholm alternative for degenerate Hilbert – Schmidt Kernels. Reduction
of ordinary differential equation to Integral equations. Symmetric Kernels, eigen function expansion with applications.
MTH424: ABSTRACT ALGEBRA III (3UNITS) PRE-REQUISITE – MTH 341
Minimal polynomial of an algebraic number. Eisenstein’s irreducibility criterion. Splitting fields and normal extension. Primitive element theorem. Galois group of a polynomial. Field degrees and
group orders. The Galois group of a polynomial. Field degrees and group orders. The Galois correspondence. The fundamental theorem
MTH418: FLUID MECHANICS II (3UNITS) PRE-REQUISITE – MTH 318
Governing equations of viscous flow, exact solutions, Low Reynolds’s number solutions, Boundary layers, compressible flows.
MTH499: PROJECT
Individual or Group projects of approved topics related to the current research interests in the department. | {"url":"https://fos.nou.edu.ng/b-sc-mathematics/","timestamp":"2024-11-08T02:54:53Z","content_type":"text/html","content_length":"187945","record_id":"<urn:uuid:82147db0-572b-460b-ba2e-478722daceab>","cc-path":"CC-MAIN-2024-46/segments/1730477028019.71/warc/CC-MAIN-20241108003811-20241108033811-00747.warc.gz"} |
B. Sidney Smith
December 17, 1997
The idea of the continuum seems simple to us. We have somehow lost sight of the difficulties it implies… We are told such a number as the square root of 2 worried Pythagoras and his school almost
to exhaustion. Being used to such queer numbers from early childhood, we must be careful not to form a low idea of the mathematical intuition of these ancient sages; their worry was highly
— Erwin Schroedinger
It was a sometime paradox, but now time has given it proof.
— Shakespeare
Born to a culture of Greek and Cartesian geometrical intuitions, it is perhaps inevitable that we conceptualize time as a linear progression from earlier events to later. The contingencies of our
subjective experience are as tiny beads on a temporal string, ordered—like the points of the real number line—and likewise dense perhaps, but nonetheless slavishly constrained to the trichotomy of
past, present, and future, just as in the mathematician’s richly packed continuum a given number is either less than, equal to, or greater than another. From ancient times to our own century, careful
thinkers have detected and pondered a lurking absurdity in this conceptualization of time, and it is the thesis of this paper that only by reconceptualizing the texture of time’s fabric can the
paradoxes be resolved.
These paradoxes may be analyzed into two distinct but related difficulties, exemplified by Zeno’s ancient Paradox of the Arrow and McTaggart’s contemporary paradox of the unreality of time,
[McTaggart, 1927] respectively. The first demonstrates the difficulty inherent in our analysis of the time-line into a succession of dimensionless points, and the second reveals an inconsistency in
our natural ordering of these points into past, present, and future events. I submit that we are under no obligation to view time and its passage in such a way as to make it subject to these
objections; that there are alternative conceptualizations which, while not in themselves meeting every objection, still clearly demonstrate the possibility that time is endowed with such a texture as
to render these traditional paradoxes obsolete. I will argue that the paradoxes rest ultimately upon a notion of the continuum that is rooted in the formalizations of modern mathematics,
formalizations which are in deep trouble philosophically, and to which there are philosophically desirable (even perhaps mathematically desirable) alternatives.
In the first and most notorious of the time paradoxes, the Paradox of the Arrow, Zeno of Elea (circa 450 BCE) begins with the tacit assumption that one may speak of a state of affairs in a
dimensionless instant of time, and he founds the ensuing argument on the intuition that no change can occur in the confines of such an instant:
Zeno asked his contemporaries to imagine an arrow in flight. If time consists of discrete instants, then at any particular instant of time, the arrow must be at a particular location, a certain point
in space. At that instant, the arrow is indistinguishable from a similar arrow at rest. But this will be true of any instant of time, Zeno argued, so how can the arrow move? Surely, if the arrow is
at rest at every instant, then it is always at rest [Devlin, 1997].
Although Zeno, a Parmenidean monist, was arguing against an atomistic view of time (and space), the paradox is not weakened by assuming he thought of time as moderns are inclined to do, as consisting
of a continuum of ‘instants’ such as is modelled by the points of the standard real number line.
This can be made vivid by casting the paradox in more contemporary terms. Consider the ‘transporter machine’ on the popular television program Star Trek. This device was originally conceived by the
writers of the program as a means of cutting production costs (characters could be moved from point of action to point of action instantaneously, eliminating the need for shuttle footage, etc.) but
it quickly became a defining element of the show, and intricate story lines were generated by the implications of the existence of such a device. It is imagined to function in the fictional world of
that program by somehow deconstructing objects—and people in particular—and ‘beaming’ them to a remote location, where they are reconstructed. Its exact mode of operation is left intentionally vague,
but it is generally presumed to work by transmitting the actual substance of the transportee via a particle/energy beam, together with the encoded information specifying how his/her substance is to
be reassembled. Hosts of reasons why such a device probably could not exist, whatever its mode of operation, are ready to hand, but for our purposes it is enough to consider the necessity for such a
device to render the precise make-up at a given instant of the person to be transported. There are two difficulties. The first is a class of problems raised by quantum indeterminacy which I wish to
put aside as involving difficult theoretical questions beyond the scope of this discussion, and in any event not essential to the main issue. The second difficulty is precisely the Paradox of the
Arrow, in more compelling guise.
Suppose the transporter succeeds in reassembling my physical self precisely at the point of destination, down to the last follicle, corpuscle, and neuron. What is my first act upon
re-materialization? It is to fall down dead, and this for many reasons. First, several pints of blood which had been in motion in my body are stopped in an instant, bringing about immediate cardiac
arrest. The heart, indeed, swings very like a pendulum in its complex rhythms, and would act precisely as a pendulum does when an intervening obstacle is placed in its path. At best it would jolt and
fibrillate, unable to recover its rhythm on its own. This effect, which we might term ‘the removal of momentum,’ would be mirrored on smaller scales all through the body, which is a vessel of
countless cycles of matter in motion, from the large scale motion of blood and other fluids to the tiny scale of cellular processes. (Conceivably the most devasting effect would be on the nervous
system, which particularly in the autonomous functions maintains a delicate rhythm of actions and reactions. Hiccoughing is one innocuous symptom of a disturbance in this cycle; an epileptic seizure
is a more dramatic one.)
This undesirable fate of the transportee is an inevitable consequence of fixing his physical state at an instant of time, which necessarily leaves out any information about where an object (or the
state of a system) has been and where it is going—for a human being is a very dynamical system indeed.
One objection to this analysis of the transporter is that the machine may record not just the position but the momentum of each particle, so that the information transmitted includes a complete
phase-space portrait of the subject. However, this objection confuses properties of a mathematical model with the actual properties of an object. In mathematics one may speak of an ‘instantaneous
rate of change,’ but mathematicians themselves are careful to note that to speak in this way is to indulge a mental fiction; the actual mechanisms of mathematical analysis appeal to a limit process
or a notion of infinitessimal precisely so as to avoid the logical absurdity of change occuring in a dimensionless point.
Zeno’s question remains as relevant as ever: if there is no change in a dimensionless point of time, and time itself is ‘made up’ of such points, then where, precisely, does change reside? Where is
the motion? It seems to me that only two conclusions are possible: either motion (through space as well as time) is illusory, or there is something incoherent in the notion of picking out an
‘instant’ of time. To accept the first conclusion, as many philosophers have done (and as Zeno might have intended), is to do considerable violence to our naive conceptions about the world, for that
things certainly seem to move and change through time is a universal experience across cultures and epochs. But what of the second conclusion, the one Zeno may after all have intended his listeners
to infer? Here it seems that a deep reconsideration of our conceptual models of space and time is in order. But, as I will show below, this need not entail the abandonment of our most basic
intuitions about the world. Indeed, our current conceptual models of space and time, which invite Zeno’s old argument anew, are not only historically recent and culturally influenced, but involve
mathematical abstractions and empirical assumptions which have no special claim, over and above other equally natural abstractions and assumptions, to ontological verity.
McTaggart presents us with a subtler difficulty than did Zeno. He begins in the same place, by defining an event as ‘the contents of any position in time.’ [McTaggart, 1927, p.24] (I believe this use
of the term ‘event’ is unfortunate, as the term itself tends to connote change and may thereby mislead the reader as to McTaggart’s meaning. ‘State of affairs’ might have been better, but I will keep
McTaggart’s terminology for simplicity’s sake, hoping to forestall confusion by means of this parenthetical.) He then notes two kinds of relations among such events, the relation of being either
‘earlier than’ or ‘later than,’ and that of being either ‘past, present, or future.’ The first relation gives what he calls the B-series of events, and he notes that this series is forever fixed,
i.e., that the relations of ‘earlier than’ and ‘later than’ which events enjoy with respect to one another are unchanging. The beheading of Anne Stuart has always been ‘earlier than’ World War II,
and always will be. In short, the B-series is a strict linear ordering not subject to flux or flow. The relations arising from an event’s being ‘past, present, or future,’ by contrast, are changing
relations, and give rise to what he terms the A-series. Although, with respect to a given event, other events have the fixed relations of being more or less past, present, or less or more future,
with respect to the ‘present’ itself these relations constantly change. On this account, it is in (or with respect to) the A-series, and there only, that change occurs.
Armed with these definitions, McTaggart argues to the unreality of time in two steps. First, he appeals to the ‘universally admitted’ notion that without change there can be no time. He then
abolishes the A-series by a sophisticated reductio ad absurdum, and with it all actual change, and ipso facto the reality of time. The B-series necessarily follows the A-series into oblivion, for it
is an essentially temporal series and hence, absent actual time, cannot exist. That McTaggart’s paradox is still prominent in the literature, indeed a touchstone for current debate, some ninety years
after publication, is a testament to the robustness of an argument which initially strikes one as implausible. The reductio argument abolishing the A-series, in particular, looks weak at first blush:
that ‘past, present, and future’ are relations or properties characterizing events, that they are mutually exclusive relations or properties, but that any given event must have all three.
Contradiction. The obvious riposte, that events don’t have these relations or properties concurrently but in temporal succession, turns out to involve a new set of similar relations or properties,
e.g., ‘will be past,’ ‘was present,’ et cetera, in which arise the same contradiction. And so on ad infinitum.
I find the cage of this argument unpalatably linguistic, but nonetheless difficult to escape on that account, so it is well that I’ve no need to address it here. The flaw in McTaggart’s argument is
at its roots, in his definitions and assumptions, and when exposed will obviate the reductio argument entirely. It is his ‘universally admitted’ assertion that there can be no time without change,
and his insistence that only the A-series and not the B-series permits change, that together provide the fulcrum, the point of leverage, for the strength of his argument. If these points be conceded,
he has won the field. But if one of these be denied, he falls. If there can be time without change, then the nature or consistency of neither the B-series nor the A-series is relevant to its
existence. Similarly, if change is necessary to time, but the A-series is not necessary to change, then we may deny that the reality of time is dependent on the consistency or otherwise of the
I have no idea whether change is necessary to time (I am inclined to doubt it), but I am prepared to deny that McTaggart’s A-series is necessary to change. This will of course place the burden on the
B-series, and leave us with the difficulty of showing how a static set of temporal relations can give rise to change, or indeed how B-series relations can even be spoken of as ‘temporal’ except in
some formal sense, e.g., as the anisotropic fourth dimension of the cosmologist’s ‘block-universe.’ But at least this difficulty has the virtue of familiarity—for it is precisely the Paradox of the
Arrow! Once again we look for change in the ordered, dimensionless points of a continuum whose relations are eternally fixed.
To meet this challenge, it will be necessary to analyze the ‘Greek and Cartesian geometrical intuitions’ with which this essay began. That the modern real number line of mathematics has become the
universal model for time and space wants some investigating. What is this real number line? Whence did it arise, and how is its use as the preferred model of the continuum justified? Do alternatives
exist, and if so what implications do they have for the paradoxes under consideration?
In fact the ‘set of real numbers,’ which I will denote hereafter by R, has only been with us in its modern conception for about 100 years. Prior to its formalization in set-theoretic terms during the
19th century by Dedekind and others, R was very problematic. Its dual identity, as a model of the continuum and as the numerical field—the very stuff—of quantitative mathematics, created a rich
interplay of often conflicting ideas about the nature of both space and number. It was this philosophical breathing room that made possible such ideas as Newton’s fluxions and Leibniz’s
infinitessimals, which were conceived of as quantities so small as to be less than any given magnitude but still greater than nothing; ideas long discredited and only recently (partially)
rehabilitated. Even so, by the beginning of the 20th century the tension between the spatial and numerical aspects of R was firmly decided in favor of the numerical aspect, and this has had a
profound consequence for the paradigms of modern science. It is this consequence that is the crux of this essay.
It will be necessary to briefly review the standard formalization of R. We begin with the integers, {…, -3, -2, -1, 0, 1, 2, 3, …}, which I will take as given. The set of rational numbers, i.e., of
ratios, is then defined in the obvious way as the collection of all ratios of integers. It was learned by the ancient Greeks that the rational numbers fail to exhaust all possible relative
magnitudes; e.g., the ratio of the diagonal of a square to its side. To wit: the square root of 2 cannot be represented exactly as a ratio of integers. As late as 1870, a sound formalization of the
so called irrational numbers was still elusive. It had long been known that one could ‘sneak up’ on irrational quantities by means of rational ones, and this is reflected in the fact that one may
approximate the ‘actual value’ of the square root of 2, for example, by longer and longer terminating decimals (1.4, 1.41, 1.415, et cetera), each of which is expressible as a ratio of (larger and
larger) integers. However, this is an infinite process, and classical mathematicians retained a strong aversion (rooted in Aristotle: see Physics, Book III) to admitting actual infinities into their
methods. Then, in 1872, a crucial conceptual shift occured, voiced first by Karl Weierstrass and subsequently made precise by Richard Dedekind. [Boyer, p. 563.] Rather than taking the approximating
series of rational quantities as a means of approaching the irrational quantity, take the described collection of rational quantities actually to be the irrational quantity. Dedekind took this idea
further, defining the square root of 2, for example, to be the set of all rational numbers whose square is less than 2. This is now called a Dedekind cut, for the definition neatly cuts the real
numbers into two halves, with the square root of 2 being defined as the ‘point’ of the cut. Boyer sums up Dedekind’s intuition in this way:
Upon pondering this matter, Dedekind came to the conclusion that the essence of the continuity of a line segment is not due to a vague hang-togetherness, but to an exactly opposite property: the
nature of the division of the segment into two parts by a point on the segment.
Gone are Newton’s fluxions and Leibniz’s infinitessimals, to be replaced by a continuum each of whose points is merely a division into two parts, separating that which has come before from that which
comes after.
The analogy to instants of time, conceived of as merely the dividing points between past and future, is striking. Indeed the matter goes beyond analogy, for in all modern science time is parametrized
by a real variable t, real in precisely Dedekind’s sense of being a mere cut, a slice through the substance of the actual. Hence of necessity the Now has for us no structure, no texture; it is a
dimensionless nothing between that which was and that which will be.
I concede that this model of the continuum has served both mathematics and the physical sciences very well. Like all good formalizations, it has provided the structures needed to push forward
mathematical discovery, and by extension physical insight, and is thereby justified. The immense edifice of 20th century mathematics and theoretical physics is founded upon Dedekind’s
conceptualization, and there is no compelling empirical reason (that we can properly identify) for researchers in these sciences to abandon it. Yet, for the philosopher, Zeno’s ghost lingers to haunt
us, for the empirical fact that things actually happen stands in direct contradiction to his conclusion that instants of time, thus conceived, without structure or parts, cannot provide an arena for
change. On this basis, the pragmatic justification given above for the use of our contemporary model of the continuum begins to look alarmingly weak.
But what alternative exists? Let us first note that, since the 19th century, certain results in the foundations of mathematics assert that this formalization of the real numbers is, in some important
sense, incomplete. For instance, the Continuum Hypothesis of Georg Cantor, which asserts certain basic facts about the nature of the continuum in its modern conception, is undecidable from the
established axioms of set theory. (This was proved jointly by Kurt Gödel and Paul Cohen; see Rucker, 1995, p. 252.) Similar considerations led the great 20th century logician Kurt Gödel to conjecture
that the points of the (Dedekind/Cantor) real number line are insufficient to exhaust an absolutely continuous line, but rather form “some kind of scaffold on the line.” In other words, Dedekind’s
conceptualization somehow fails to capture the richness of structure of the continuum, providing instead a mere skeleton outline, so to speak, much the way that the map of a city, however brilliantly
drawn, can never capture all of the relationships among its features. These difficulties remind us that a given formalization can never reach, conceptually, beyond what it assumes, but can only serve
to clarify and make precise those very assumptions, and to reveal their logical consequences. Dedekind began with the assumption that the points on a geometrical line can be put into a one-to-one
correspondence with the real numbers, and implicit within that assumption was Dedekind’s own (and his peers’) conception of the structure of a geometrical line.
The question of an alternative remains. Although at present there does not exist an alternative formalization of the continuum which meets every philosophical objection, it is clear that even within
the strictures of mathematical formalisms meaningful alternatives are possible. In the 1960’s, Abraham Robinson developed the foundations of non-standard analysis, at the heart of which was an
alternative formalization of the real-numbers in which points themselves have a deep structure. In particular, it makes possible the resurrection of a version of infinitessimals that is logically
precise. Unlike the standard real line, this non-standard real line is not unique, but may be constructed in a variety of ways. However, it always contains the standard real line embedded within it,
and to each of the ‘standard’ points is associated a cloud of new points having a complex structure of its own. One way of imagining this is to think of the points on the continuum as consisting, not
of discrete ordinates, but of infinitessimal regions, typically called monads, each fully as rich with structure as the entire standard set of real numbers. (Monads should not be thought of, however,
as tiny copies of the real line, for typically their structure is far more complex.) Robinson’s construction relies on advanced concepts in set theory (see for example Lengyel, 1996), and it may be
objected on that account that the construction is ad hoc or artificial. But such an objection misses the point that our common conceptualization (Dedekind’s) is equally dependent on set theoretical
constructions and is similarly ad hoc.
On the basis of such possible conceptualizations, it is clear that we are free to consider the modern real numbers to be an imperfect map of time’s continuum, but no more. Specifically, I suggest
that what we call ‘moments of time’ are not dimensionless points at all; they are frameworks fully capable of embodying the dynamics that give rise to time and change. In short, time is textured.
This is admittedly hard to picture, but then there is no a priori reason to expect the nature of time to be so simple as the traditional stick-figure conception would have it. We are called upon to
set aside our Euclidean notion of ‘point’ as ’that which has no parts,’ and replace it instead with the very intuitions that led Newton and Leibniz to develop the calculus, which, as so many freshman
learn to their wonderment each autumn, is the very science of change, and arguably the most successful, effective, and far-reaching intellectual achievement in human history. No longer is change
constrained to abide within the dimensionless; infinitessimal change can and must occur in infinitessimal time, and the large scale flow of happenings which we seem to witness is the aggregate of
these processes.
This idea of textured time should be carefully distinguished from certain other suggestions that have been put forth in recent years. In particular, the idea of merely compounding the points of the
real line by adding new dimensions or by replacing them with ‘tiny copies’ of real spaces is to be avoided. This proposal should also be distinguished from 2-dimensional ‘subjective’ time, such as
that suggested by Elliott Jaques [1982], for the structure I suggest for time is neither ‘2-dimensional’ nor merely subjective. Neither does it amount to meta-time, as proposed for instance by George
Schlesinger [1980]. Instead, by textured time I intend that ‘moments’ be conceived neither as space-like entities of the sort typically modelled by a standard real variable, nor as psychological
artifacts, but rather as a new category of object in the world, one which demands and deserves metaphysical investigation.
How we are to form a positive conception, one capable of metaphysical investigation, of what I have here only adumbrated is by no means clear. It is not even clear whether an examination of
mathematical structures (as possible models of time’s texture) is helpful or harmful to this effort. In particular, I wish to emphasize that I do not construe Robinson’s non-standard model of the
continuum to be the correct model for time; its present value is to force the recognition that the notion of textured time may be susceptible to formalization and is not mathematically
inconsistent—briefly, that such a conception is intellectually possible. The real impetus for our investigation is provided by the paradoxes: Zeno and McTaggart, between them, have demonstrated that
without some such conception we must abandon our given, universal intuitions about time and change altogether, leaving us in an unintelligible world where our most basic perceptions are the blankest
Aristotole: 1941, The Basic Works of Aristotle, Ed. Richard McKeon. New York: Random House.
Boyer, Carl: 1968, A History of Mathematics. New York: Wiley.
Devlin, Keith: 1997, Goodbye, Descartes. New York: Wiley.
Frege, Gottlob: 1884, The Foundations of Arithmetic. Breslau: Verlag von William Koebner. (Reprinted 1953 with Engl. trans. by J.L. Austin, Oxford: Blackwell.)
Gödel, Kurt: 1959, “A Remark about the Relationship between Relativity Theory and Idealistic Philosophy.” Albert Einstein: Philosopher-Scientist. Ed. Paul Arthur Schilpp. New York: Harper and Row.
Jaques, Elliot: 1982, The Form of Time. New York: Crane.
Lengyel, Eric: 1996, Hyperreal Structures Arising from an Infinite Base Logarithm. Preprint: Virginia Polytechnic Institute and State University.
McTaggart, J.M.E.: 1927, “The Unreality of Time,” The Philosophy of Time. Ed. Robin Le Poidevin and Murray MacBeath. Oxford: Oxford UP.
Rucker, Rudolf Von Brucker: 1995, Infinity and the Mind. New York: Dover.
Schlesinger, George: 1980, “Temporal Becoming,” The New Theory of Time, Eds. L. Nathan Oaklander and Quentin Smith. New Haven: Yale UP. | {"url":"https://bsidneysmith.com/index.php/writings/exposition/textured-time","timestamp":"2024-11-13T21:11:03Z","content_type":"text/html","content_length":"46237","record_id":"<urn:uuid:db2177c6-7bd8-4879-85a7-512a61db09a8>","cc-path":"CC-MAIN-2024-46/segments/1730477028402.57/warc/CC-MAIN-20241113203454-20241113233454-00140.warc.gz"} |